Reactive flow modeling using the lattice Boltzmann method Interpretation of lab experiments by
Janne Pedersen Thesis submitted in fulfillment of the requirements for degree of PHILOSOPHIAE DOCTOR (PhD)
Faculty of Science and Technology Department of Petroleum Engineering 2015
University of Stavanger N-4036 Stavanger NORWAY www.uis.no ©2015 Janne Pedersen ISBN: 978-82-7644-619-7 ISSN: 1890-1387
Acknowledgements I would like to express my gratitude and appreciation to those who have contributed to this work. My two supervisors, Prof. Aksel Hiorth and Dr. Espen Jettestuen, have guided me throughout the work, been valuable discussion partners to me and contributed signi cantly to the publications. I appreciate all their help and support. My gratitude goes also to all the co-authors of the publications included in this thesis; Dr. Jan Ludvig Vinningland, Dr. Tania HildebrandHabel, Dr. Reidar I. Korsnes, Prof. Merete V. Madland, Prof. Rune Time, Dr. Herimonja A. Rabenja manantsoa, Ørjan Tveteraas, and Prof. Lawrence M. Cathles, in addition to Dr. Espen Jettestuen and Prof. Aksel Hiorth. All of you have contributed to this work, and for that, I am grateful. Finally, I want to acknowledge those who have nanced this work; the Norwegian Research Council, the companies in the Eko sk and the Valhall lisence, and the National IOR Centre of Norway and its industry partners; ConocoPhillips Scandinavia AS, BP Norge AS, Det Norske Oljeselskap AS, Eni Norge AS, Maersk Oil Norway AS, DONG Energy A/S, Denmark, Statoil Petroleum AS, GDF SUEZ E&P NORGE AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS.
Abstract The lattice Boltzmann method is used to interpret lab experiments. A pore scale simulator consisting of a 3D lattice Boltzmann advection and diusion model coupled to a full geochemical solver has been used to simulate the chemical alterations in chalk during ooding of dierent brines. The model has been further developed to account for moving boundaries, and to facilitate use as a core scale simulator, mapping the surface of a core onto a cylinder to match the kinetic rates in a core scale experiment quantitatively. The work has contributed to improved interpretation of lab experiments, and better understanding of processes that take place on the pore scale and are important to nd an appropriate upscaling of rate equations to the eld scale. A study of the localization of mineral alterations due to dissolution and precipitation in the pore space showed that more alterations are seen in pore bodies than in pore throats for transport limited ow. To study the dynamics of structural alterations in the pore space, a moving boundary model that is independent of the underlying mathematical grid is proposed. A chemical ux boundary condition is coupled to a volume of uid interface reconstruction routine to keep track of the solid- uid interface position and normal. For diusion limited growth, a surface tension term is included in the boundary condition in order to obtain a model that is independent of the initial orientation of a growing seed on the underlying grid. A method is proposed that maps the mass and surface area of a core onto a cylinder, to obtain a model that gives results that are quantitatively comparable to experimental data. A rescaling of the dissolution and precipitation rate equations, to account for dierences in actual surface area and pore volume of the core and of the cylinder, as measured in lattice units, is used. The model is then used to interpret e uent data from a Liège chalk core ooded with MgCl2 for 1072 days. An overgrowth model, i.e. a model that reduces the reactive surface area of calcite due to surface coverage of precipitating magnesite, is proposed to match the experimental e uent pro le. In addition to core ooding experiments, the lattice Boltzmann model has been used to simulate a pressure decay experiment where CO2 gas diuses into water, undergoes chemical reactions with the water, which leads to convective mixing. Simulations are performed in two dimensions and compared to experimental data. A good match is obtained when a
lower Rayleigh number is used in the simulations. The relative dierence in the Rayleigh number agrees well with the dierence in mass transfer rate between the CO2 gas phase and the water phase in two and three dimensions.
List of papers Paper 1: Pore Scale Modeling of Brine Dependent Pore Space Evolution. J. Pedersen, J. L. Vinningland, E. Jettestuen, M.V. Madland and A. Hiorth. Paper SPE-164849-MS prepared for presentation at the EAGE Annual Conference & Exhibition incorporating SPE Europec held in London, United Kingdom, 10-13 June, 2013.
Paper 2: Pore Scale Modeling of Brine Dependent Permeability. J. Pedersen, E. Jettestuen, J. L. Vinningland, M.V. Madland and A. Hiorth. Paper SCA2013-064 prepared for presentation at the International Symposium of the Society of Core Analysts held in Napa Valley, California, USA, 16-19 September, 2013.
Paper 3: Improved Lattice Boltzmann Models for Precipitation and Dissolution. J. Pedersen, E. Jettestuen, J. L. Vinningland and A. Hiorth. Transp. Porous Med. 104, 593-605 (20014) Paper 4: A dissolution model that accounts for coverage of mineral surfaces by precipitation in core oods. J. Pedersen, E.
Jettestuen, M. V. Madland, T. Hildebrand-Habel, R. I. Korsnes, J. L. Vinningland and A. Hiorth. Submitted to Advances in Water Resources.
Accepted after major revision
Paper 5: Lattice Boltzmann simulations of advection driven ow by CO2 diusion into water. J. Pedersen, E. Jettestuen, H. A. Raben-
ja manantsoa, Rune Time, Ørjan Tveteraas and A. Hiorth. In prepara-
tion.
Abstracts Abstract 1: A study on the eect of pore geometry on mineral changes. J. Pedersen, E. Jettestuen, J. L. Vinningland, M. Madland, L.
M. Cathles III and A. Hiorth. Presented at the Goldschmidt conference, Prague, Czech Republic, August 14-19, 2011.
Abstract 2: Models for evolution of reactive surface area during dissolution and precipitation. J. Pedersen, E. Jettestuen, T.
Hildebrand-Habel, J. L. Vinningland, M. V. Madland, R. I. Korsnes and
A. Hiorth. Presented at European Geosciences Union General Assembly, Vienna, Austria, 27 April - 02 May, 2014.
Additional abstracts A multi scale approach to interpret chalk core experiments. J. Pedersen, A. Hiorth, R. I. Korsnes, A. Nermoen, J. L. Vinningland, E. Jettestuen, T. Hildebrand-Habel, U. Zimmermann and M. V. Madland. Presented at IEA collaborative project on enhanced oil recovery, EOR Workshop and Symposium, Beijing, China, 15-17 October, 2014.
Nomenclature Roman letters a
ab ac amag A A0 Ac Acov Adef Alb Amag Aplug,j Arot Aslow b B1 B2 c cs ceq c∗ dˇ dadef D
eα E
fα fαeq
Activity of chemical species Vector of activities for basis species Vector of activities for complexes Area occupied by one mole of magnesite on the calcite surface Surface area Area of shape rotated 0o on the underlying grid Area of composite shape made up from two single shapes Surface area of calcite covered by magnesite Area of defects on the calcite surface (used in overgrowth model) Surface area of a geometry calculated by the lattice Boltzmann method Magnesite surface area Surface area of mineral j in a physical core plug Area of shape rotated 19o or 45o on the underlying grid Calcite surface area without defects Estimated intercept from volume of uid method Temperature dependent constant in the Debye-Hückel theory Temperature dependent constant in the Debye-Hückel theory Concentration of chemical species Lattice sound speed Equilibrium concentration Dimensionless concentration Eective diameter of hydrated ion in Debye-Hückel theory Part of the calcite surface consisting of defects Diusion coe cient Lattice velocity vector Error between estimated and correct solid fraction for the volume of uid method Single particle distribution function for uid eld Equilibrium distribution function for uid eld
fmc fn g0 gα gα¯ gαeq (0) (1) (2) gα ,gα ,gα g˜α G HF I JM JR Jcal Jmag k kα k1 ,k2 kcov
kr kslow K
Kb l0 LLB Lphys m ˜ x,y b,c,f mlb,j wn M Mjmol
Geometrical factor describing 3D growth (overgrowth model) Fluid node Gravitational acceleration Single particle distribution function for chemical species Distribution function for direction pointing away from the surface Equilibrium distribution function for chemical species Expansion of distribution function used in Chapman-Enskog expansion Distribution function for direction pointing towards a surface Gibbs free energy Height function (for curvature calculations) Ionic strength Mineral ux Reaction ux on the surface Flux of calcite Flux of magnesite Chemical reaction rate constant Discrete lattice reaction rate Reaction rate constants Calcite dissolution rate constant at areas covered by magnesite relative to that at defect ares kcal (overgrowth model) Rate constant for magnesite precipitation onto calcite relative to that onto magnesite Calcite dissolution rate constant in areas without defects relative to that at defect areas Chemical equilibrium constant Vector of equilibrium constants for basis species Length scale LB length Physical length Estimated slopes from volume of uid method Mole number of mineral j in wall node wn Mineral (solid) concentration Molecular weight of mineral j
n ng Nt Nx p∗ P q qα rs,j R S Sj t t0 t1 ,t2 tb tD tLB tphys tν t∗ T T
u u∗
ub uν V Vlb −1 VM
Vplug wn xW
x z
Surface normal Unit vector in the direction of the gravitational force Number of iterations Number of cells in x-direction Dimensionless pressure Virtual node used in interpolation scheme Distance from uid node to wall Distance from uid node to wall along direction α Rescaling factor for rate equation of mineral j Universal gas constant Entropy Speci c surface area of mineral j Time Dimensionless time scale Time scales used in Chapman-Enskog expansion Buoyancy time scale Time scale for diusion LB time Physical time Viscous time scale Dimensionless time Temperature Time scale Fluid velocity Dimensionless velocity Buoyancy velocity Viscous velocity Volume Pore volume of a geometry calculated with the lattice Boltzmann method (dimensionless) Density dierence between uid and solid for use in solid fraction Pore volume of a physical core plug Wall node Position of wall Position vector Valence
Greek letters α α ¯ β γ δc δt δtLB δtphys δx δxLB δxphys δij ζ η κ µ ν ξ ρ σ σ est τ τD ωα Ω Ωα
Discrete lattice direction number Lattice direction pointing away from the surface Constant (used in the de nition of the Rayleigh number) Activity coe cient Physical concentration interval (unity in all simulations) Time interval LB time interval (unity) Physical time interval Physical cell length (= δxphys ) LB cell length (unity) Physical cell length Kronecker delta Order of reaction in dissolution/precipitation rate equation Order of reaction in dissolution/precipitation rate equation Surface curvature Stoichiometric matrix Kinematic viscosity Curvature constant Fluid density Solid fraction (actual) Estimated solid fraction Dimensionless relaxation time for uid ow Dimensionless relaxation time for diusion Lattice weight coe cients Saturation index Collision operator
Abbreviations AFM BC BGK CEC CFD
Atomic force microscopy Boundary Condition Bhatnagar-Gross-Krook Cation Exchange Capacity Computational Fluid Dynamics
ELVIRA EOR EOS FDM FEM FIB-SEM FVM IAP LB LBE LBGK LBM LBS LGCA LS NCS Pe PeDa PLIC Pr PV Ra SCAL SEM SSW VOF VSI
E cient Least squares Volume of uid Interface Reconstruction Algorithm Enhanced Oil Recovery Equation Of State Finite Dierence Method Finite Element Method Focused ion beam - scanning electron microscopy Finite Volume Method Ion activity product Lattice Boltzmann Lattice Boltzmann Equation Lattice Boltzmann Bhatnagar-Gross-Krook Lattice Boltzmann Method Lattice Boltzmann Simulator Lattice Gas Cellular Automata Level set Norwegian continental shelf Peclét number Peclét-Damköhler number Piecewise Linear Interface Calculation Prandtl number Pore volume Rayleigh number Special Core Analysis Scanning Electron Microscopy Synthetic Seawater Volume Of Fluid Vertical scanning interferometry
Table of Contents Acknowledgements Abstract List of papers Nomenclature
PART I 1 Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7
Reservoir simulation Rock- uid interactions - water weakening of chalk Core ooding experiments on Chalk - water weakening and enhanced oil recovery A multiscale problem Geochemical pore scale modeling Objectives Structure of thesis
2 The Lattice Boltzmann Method 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
A CFD method based on the Boltzmann equation Brief history The lattice Boltzmann equation Lattice Boltzmann equation for diusion DnQm discretization of phase space Boundary conditions Conversion between LB and physical units Chapman-Enskog expansion for the advection-diusion equation Derivation of ux boundary condition from the ChapmanEnskog expansion
21
32
3 Petroleum geochemistry
48
4 Lattice Boltzmann pore scale simulator
53
5 Location of mineral alterations in a pore geometry
58
6 Moving boundaries within the LBM
64
7 Reactive surface area models
80
8 Use of LBM to model convection caused by CO2 diusion into water
89
3.1 3.2 3.3 3.4 4.1 4.2 4.3 5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4 6.5 7.1 7.2 7.3 7.4 7.5
8.1 8.2 8.3 8.4 8.5
Introduction Chemical equilibrium Ion exchange Adsorption Introduction Fluid solver Geochemical solver
Introduction Results Discussion Summary & conclusions
Introduction Method Some details Results Summary & Conclusions
Introduction Arti cial microporosity to increase reactive surface area Mapping of actual reactive surface area onto a cylinder Overgrowth model Conclusions
Introduction Experimental setup Simulation setup Results Summary & conclusions
9 Introduction to papers
101
10 Summary, conclusions, impact and future work
105
Bibliography
120
10.1 Summary & conclusions 10.2 Impact 10.3 Future work
PART II
Paper 1: Pore Scale Modeling of Brine Dependent Pore Space Evolution in Carbonates Paper 2: Pore Scale Modeling of Brine Dependent Permeability Paper 3: Improved Lattice Boltzmann Models for Precipitation and Dissolution Paper 4: A dissolution model that accounts for coverage of mineral surfaces by precipitation in core oods. Paper 5: Lattice Boltzmann simulations of advection driven ow by CO2 diusion into water Abstract 1: A study on the eect of pore geometry on mineral changes Abstract 2: Models for evolution of reactive surface area during dissolution and precipitation
PART I
Chapter 1
Introduction 1.1 Reservoir simulation Reservoir simulation [Fanchi, 2006, Thomas, 1982] is part of the reservoir management function, in which the primary goal is to maximize economic recovery, i.e. optimize the recovery while minimizing the cost. The typical reservoir simulation activity consist of two parts; 1. history matching to produce a baseline for comparison with other reservoir management strategies and 2. Simulating dierent recovery processes for use in the planning of operational changes in order to achieve the primary goal of maximum economic recovery. In order to simulate reservoir behavior, some initial data must be provided. Geological and petrophysical properties of the reservoir are used in the construction of a geological model in which the motion of the reservoir uids will be simulated. Such properties include e.g. texture, mineralogy, sedimentary structure (geological data), porosity, permeability, uid saturations (petrophysical data) etc. In addition, reservoir uid properties are needed to simulate the evolution in the reservoir. These initial data are gathered through laboratory (core sample analysis) and eld tests (well log analysis, seismic analysis). Core samples are typically at the centimeter scale which might be something like a few parts in a million of the size of the reservoir. Hence the core sample will only give data from a very limited area and might not be a good representation for dierent locations or the whole reservoir. The range of a well log is at the scale of one meter, and the range of seismic analysis is at the dekameter (10 m) scale. A seismic analysis extends the sample area from a core sample with a factor of approximately 1000, but the interpretation 21
Chapter 1. Introduction of seismic data is less reliable. Some properties can be measured with more than one method, which can improve the reliability of the measured data. However dierences are seen in values measured at dierent scales due to reservoir heterogeneity. For example a property measured with well log is the average value over an area substantially larger than the area covered by a laboratory measurement of a core sample, and hence might give a dierent value. In order to give a good representation of the reservoir, the information gathered at dierent scales must be collected and integrated into a single consistent representation. The process of integrating data collected at dierent scales is called upscaling, and it is a di cult task. History matching is performed to improve the quality of the initial data. A simulation is run with the best parameter set available, and the simulated production pro le is compared to the actual historical production pro le. If the simulated pro le is not satisfactory, a set of simulations with varying initial data is performed in order to obtain a set of initial data that gives a better match to the historical production pro le. This set of initial data can now be used in simulations with dierent production conditions in order to determine the conditions that will maximize the economic recovery. History matching is an inverse problem, that might have several solutions, i.e. dierent combinations of the initial data parameters might give the same production pro le. Hence, one can not be sure that the initial data is a good representation of the reservoir. Bad initial data will give bad or wrong predictions for future production. History matching for a longer production time will increase the reliability of the resulting initial data set. Hence, history matching should be a continuous process in order to have the best initial data available at all times. The need for good initial data for the eld scale reservoir simulators has resulted in simulation activities down to the pore scale. Traditional special core analysis (SCAL) programs on core samples can be extremely time consuming and challenging to perform under realistic reservoir conditions. The time frame for a SCAL program is typically between 1-2 years. An alternative to the time consuming SCAL is to obtain reservoir properties through pore scale ow modeling in a 3D microstructure representative for the reservoir rock. Several methods for the production of 3D microstructure images have been proposed in the last decades; examples being X-ray microtomography [Dunsmoir et al., Spanne et al., 1994, Arns 22
1.2. Rock- uid interactions - water weakening of chalk et al., 2004], process or geologically based reconstructions [Bryant et al., 1993, Bakke and Øren] and statistical reconstructions [Adler et al., 1990, Yeoung and Torquato, 1998, Wu et al., 2006]. Flow simulations in 3D microstructures have been used in the last decades to predict petrophysical properties of porous rocks, such as NMR, permeability, relative permeability, capillary pressure, resistivity index, elastic properties etc. [Arns et al., 2002, 2004, Knackstedt et al., Jin et al., Øren and Bakke, 2002, Øren et al., 2007]. Today commercial companies oer pore scale simulations as an alternative to SCAL. Examples of such companies are Lithicon (www.lithicon.com) and InGrain (www.ingrainrocks.com).
1.2 Rock- uid interactions - water weakening of chalk When water in ltrates into the Earth's crust, e.g. in a hydrocarbon reservoir, chemical reactions will take place if the water is out of equilibrium with the surrounding rocks. Reactions such as dissolution and precipitation, adsorption and ion exchange will change both the composition of the water and of the rock. Formation water present in the hydrocarbon reservoirs are in equilibrium with the reservoir rock, but seawater injected for pressure maintanence and increased oil recovery is not in equilibrium with the reservoir. Hence, rock- uid interactions will take place in the reservoir. One example of seawater injection on the Norwegian continental shelf (NCS) is the Eko sk eld. The Eko sk reservoir is the oldest reservoir on the NCS, starting oil production in 1971. In 1984 a seabed subsidence of approximately 10m was observed due to compaction of the reservoir caused by pressure depletion during production. Figure 1.1 shows the Eko sk tank in 1973 and in 1984, where we easily see that the level of the tank compared to the sea level has changed. The observed subsidence corresponds to a compaction rate of 35 cm/year [Doornhof et al., 2006] until 1984. Seawater injection was initiated at the Eko sk eld in 1987, both to increase the pressure to withstand compaction, and to increase the oil recovery. Even though the pressure increased (after approximately 4 years of water injection), the subsidence was only reduced to approximately 1/3, i.e. 10 cm/year [Jensen et al., 2000, Awan et al., 2008]. 23
Chapter 1. Introduction Several mechanisms have been proposed as responsible for weakening of chalk and the resulting compaction of chalk reservoirs. In particular, mechanical properties such as pore collapse [Johnson and Rhett, 1986, Schroeder and Shao, 1996], capillary and wetting eects [Andersen and Foged, 1992, Piau and Maury, 1994, Brignoli et al., 1994, Delage et al., 1996, Papamichos et al., 1997], and temperature eects [DaSilva et al., 1985, Brignoli et al., 1994, Risnes, 1990, Charlez et al., 1992, Addis] among others have been studied and proposed as mechanisms for chalk compaction. Pressure solution, i.e. dissolution at grain-to-grain contacts caused by localized stress maxima, has been suggested as a physiochemical mechanism of compaction by several researchers [Mimran, 1975, 1977, Hancock and Scholle, 1975, Newman, 1983, Heugas and Charlez, 1990, Piau and Maury, 1994, Gutierrez et al., 2000, Sylte et al., 1999, Hellmann et al., 1996, 2002a,b, De Gennaro et al., 2003]. The continued subsidence at the Eko sk eld after water injection has given birth to the term "`water weakening of chalk"'. Rock- uid interactions, as described in the beginning of this section, has been proposed as a mechanism behind this water weakening and the resulting compaction [Newman, 1983, Heggheim et al., 2005, Madland et al., 2011].
(a)
(b)
Figure 1.1: The Eko sk tank in (a) 1973 and (b) 1984. The pictures show that the seawater level has raised from 1973 to 1984. In addition to the water weakening eect causing unwanted subsidence, experiments have shown that water ooding and spontaneous imbibition of chalk cores with dierent brines can give dierent degree of oil recovery [Shehata et al., 2014, Fathi et al., 2011, Strand et al., 2008, Zhang et al., 2007]. If the injection water could be designed in a way that increases the recovery in a viable manner, this would hold an enormous 24
1.3. Core ooding experiments on Chalk - water weakening and enhanced oil recovery economic upside. According to the Norwegian Minister of Petroleum and Energy, Tord Lien, an increase in oil recovery of only 1% will give a revenue of approximately 300 million NOK with an oil price as per early 2014 (regjeringen.no). Hence, rock- uid interactions can have major consequences, and it is thus important to understand these mechanisms in order to nd injection solutions that are viable.
1.3 Core ooding experiments on Chalk - water weakening and enhanced oil recovery Numerous core ooding experiments have been performed with dierent brines under dierent experimental conditions to study both physical and chemical processes taking place in the core during water ooding [Heggheim et al., 2005, Korsnes et al., 2006a,b, Madland, 2005, Madland et al., 2008, 2011, Zangiabadi et al., 2009]. It is crucial to understand the important mechanisms on laboratory scale in order to nd an appropriate upscaling to the eld scale. The eect of brine chemistry is interpreted in terms of measurements like e uent pro les and strain rates, and visual and elemental analyses with SEM. The e uent pro le is the ion concentrations measured at the outlet end of the core. Together with knowledge of the concentrations in the injection water this holds information about chemical changes inside the core. A typical e uent pro le plot is presented in gure 1.2. The data is taken from an experiment where MgCl2 was ooded through a chalk core at 130o C. The e uent water contains calcium (Ca2+ ), magnesium (Mg2+ ) and chlorine (Cl− ) ions. Mg2+ and Cl are both in the injection water, while the presence of Ca2+ is interpreted as calcite dissolution. Cl is used as a tracer species, i.e. it does not undergo any chemical reactions inside the core. The Mg2+ curve is retained compared to the tracer curve, which is interpreted as a loss of Mg2+ in the core due to precipitation of magnesite (MgCO3 ).
25
Chapter 1. Introduction
concentration [mol/L]
0.5
Ca2+ Mg2+
0.4
Cl−
0.3 0.2 0.1 0
0
5
10
15
days
Figure 1.2: E uent pro les from a Chalk core ooded with MgCl2 . Horizontal dotted lines give the injected concentrations of the species, and solid lines with crosses give the e uent concentrations (concentrations at the outlet of the core). Interpretations from e uent pro les can be supported by observations with scanning electron microscopy (SEM). Figure 1.3 show typical SEM images from (a) before, and (b) after ooding with MgCl2 . Before ooding the core consist mostly of rounded calcite grains and some clay minerals, while after ooding the core holds newly formed magnesite crystals and clay minerals. These observations are veri ed by elemental analysis [see e.g. Zimmermann et al., 2015].
26
1.4. A multiscale problem
(a) Before ooding
(b) After ooding with MgCl2
Figure 1.3: SEM image of chalk core before and after ooding with MgCl2 . Even though dissolution and precipitation are indicated from both e uent pro les and SEM analysis, the exact e uent pro les can not be matched from a simple core ow simulation using e.g. PHREEQC [Parkhurst and Appelo, 1999]. This can be because several mechanisms are taking place at once, making it di cult to interpret the data, or that microscopic processes taking place at the pore scale are important, etc. Regardless, this tells us that it is important to have suitable models at the pore scale in the search for the important mechanisms taking place in the core.
1.4 A multiscale problem Deviations between eld and lab measurements of geochemical reaction rates have been observed by several authors [White and Brantley, 1995, Murphy et al., 1998, Malmstrom et al., 2000, White and Brantley, 2003, Malmstrom et al., 2004]. Sometimes the rates measured in the eld are found to dier by several orders of magnitude from the rates observed in laboratory measurements. The reason for these deviations are probably due to dierences between lab systems and natural systems. Lab measurements of geochemical reaction rates have typically been performed with crushed mineral to obtain a homogeneous system, and with well-mixed solution to avoid transport eects on the measured rates; i.e. to avoid that transport to and from the reaction sites, but rather the reaction itself is the limiting process [Lasaga, 1998, Misra, 2012]. Atomic force microscopy (AFM) and vertical scanning interferometry (VSI) measurements 27
Chapter 1. Introduction on cleaved surfaces have become an alternative way to estimate reaction rates. These measurements give estimates of calcite dissolution rates up to two orders of magnitude lower than that measured with crushed minerals [see e.g. Arvidson et al., 2003]. This discrepancy has been ascribed to dierences in step densities in powders and on cleaved surfaces [Dove and Platt, 1996]. Natural geochemical systems are inherently heterogeneous, and measurements are performed on scales smaller than the scale of heterogeneity. Scaling eect of geochemical reaction kinetics have been observed in several studies [Szecsody et al., 1998, Lichtner and Tartakovsky, 2003, Li et al., 2006]. E.g. Lichtner and Tartakovsky [2003] suggest that the discrepancy between laboratory and eld measurements may be partly caused by heterogeneous grain size distributions in natural systems. Bryant and Thompson [2001] give a literature review of developments in reactive porous media transport until 2001, and discuss challenges with predictive simulations. Fist of all, many of the existing models are empirical because of the complexity of natural porous media. Observations have often been seen that can not be explained by existing models or theories. The reason for this has often turned out to be important aspects of small-scale behavior that gives a behavior at larger scale that was not foreseen. This indicates the importance of integrating information from multiple scales, i.e. upscaling. Studies of applications that involve multiple mechanisms has revealed that the rate limiting mechanism can vary with time. Hence, as already argued, it is important to have a separate description of all the mechanisms at the pore scale.
1.5 Geochemical pore scale modeling As mentioned in previous sections, rock- uid interactions are important for the understanding of petroleum reservoir properties during water ooding. It has also been argued that the mechanisms taking place should be understood at the pore scale in order to nd appropriate upscaling laws to describe the reservoir at the eld scale. Commercial pore scale simulators (ref. section 1.1) do not include rock- uid interactions. Since these mechanisms aect the predicted petrophysical properties from the simulators, they should be part of future versions of the simulators, but before these mechanisms are implemented into the commercial simulators, there should be a solid understanding of the important mechanisms. Historically, most experiments have been performed at the core scale, 28
1.6. Objectives but due to advances in technology, experiments are now performed down to sub-micron scale. Imaging techniques such as focused ion beam SEM (FIB-SEM) and X-ray microtomography can be used to study pore space geometries and the distribution of uids. However, while core scale experiments are performed at realistic reservoir pressure and temperature [see e.g. Heggheim et al., 2005, Omdal et al., 2010], these techniques are still restricted to relatively low pressure and temperature conditions. Nevertheless, in order to interpret experimental results, good interpretation tools are necessary, and as already argued, the interpretation should be done down to the pore scale. When the technology is ready for pore scale experiments at realistic reservoir conditions, it is also a huge advantage to have the interpretation tools ready. In this thesis we present a geochemical pore scale simulator and application of it to interpret core ooding experiments. Additional functionality of the simulator is also presented in this thesis. The simulator is based on the lattice Boltzmann method (LBM). The LBM is a good choice for complex geometries and boundary conditions, and it has a major advantage in that it is applicable for parallel computing, which decreases computational times dramatically. Although many advantages, the LBM, as any other model, also has some limitations. E.g. there are limitations on the maximum velocity in the system that can make it di cult to achieve the wanted values for the physical parameters. Relation between simulation parameters and physical parameters can be challenging, and many authors ignore this task because they nevertheless don't have experiments to compare with. However, in this thesis we present physical data from the simulations, in order to compare the model predictions to experimental data.
1.6 Objectives The aim of this thesis has been: 1. to implement important physical and chemical processes necessary to describe pore space evolution during chemical ooding into a lattice Boltzmann simulator 2. to apply this simulator to interpret lab experiments in order to extract parameters that can be used to make better models and suggest new injection strategies 29
Chapter 1. Introduction
1.7 Structure of thesis Part I Part I gives an overview of the lattice Boltzmann method and relevant geochemistry, and to the pore scale simulator used in the simulation studies. Further, it gives a summary of results, and a short introduction to the papers. Conclusions and thoughts about future work are included in the end. An overview of the chapters in Part I is given below.
Chapter 2
The lattice Boltzmann method is presented. Its origin and its place in uid dynamics are discussed rst, followed by a description of the method applied to uid ow and advection-diusion of chemical species, and its implementation in a computer program. At the end, the recovery of the advection-diusion equation, and the derivation of the most general boundary condition used at the uid-solid interfaces, are shown by the so-called Chapman-Enskog expansion.
Chapter 3
Chemical processes related to the ow of water in petroleum reservoirs are discussed.
Chapter 4
A lattice Boltzmann based pore scale simulator for the ow of water in a natural porous media is described. This simulator was developed before the work presented in this thesis, and it has been used for simulations in the work presented in this thesis.
Chapter 5
A method for moving boundaries that couples the LBM to an interface tracking routine is proposed. Dierent models with varying complexity is studied, and a method that is independent of the initial orientation of a solid seed on the underlying mathematical grid is found. A simpler method is implemented into the pore scale simulator described in Chapter 4 as a rst implementation, and simulation results are discussed.
Chapter 6
The importance of the mineral surface areas on the dissolution rate is discussed. A model that reduces the reactive surface area of primary calcite minerals due to precipitation of magnesite minerals is proposed, together with a mapping of the surface of a core onto a cylinder, that enables direct, quantitative comparison between LB simulations and core ooding experiments. 30
1.7. Structure of thesis
Chapter 7
The LBM is used to model convective currents caused by CO2 diusing into a water column from the top. 2D simulations are compared to a pressure decay experiment.
Chapter 8
given.
A short introduction to the 5 papers included in Part II is
Chapter 9
In this last chapter in Part I, summary and conclusions, and some ideas about future work are presented.
Part II Part II includes 5 papers and 2 abstracts that treat the work presented in this thesis. Paper 1 and Paper 2 are proceedings, of which Paper 2 has been under peer review. Paper 3 - Paper 5 are journal papers, where Paper 3 has been published, Paper 4 has been reviewed by four reviewers and will be accepted after major revision, and Paper 5 is in preparation. The three journal papers cover dierent topics, while the conference proceedings cover similar topics as in Paper 3. Abstract 1 deals with work that has not been covered by any of the papers, while Abstract 2 deals with the same topic as Paper 5.
31
Chapter 2
The Lattice Boltzmann Method 2.1 A CFD method based on the Boltzmann equation Classical uid dynamics can be described either at a macroscopic level or at a microscopic level. At the macroscopic level, uid dynamics is described by the Navier-Stokes equation (1822), found from applying Newton's second law to a uid substance. Together with conservation laws for other macroscopic quantities such as mass and energy, the Navier-Stokes equation forms the continuum formulation of uid dynamics. The counterpart to the continuum formulation is the statistical formulation. In this formulation the Boltzmann equation (1872) is used to describe the motion of molecules that constitute the uid. The Boltzmann equation was rst introduced by Ludwig Boltzmann [see e.g. Boltzmann, 1872, 1995, Brush and Hall, 2003] to describe the dynamics of an ideal gas in terms of statistical mechanics. However, the Boltzmann equation can also be used for uid dynamics [Cercignani, 1975]. In the statistical formulation the uid is considered as a collection of molecules and not as as a continuous substance. The motion of the molecules is described by a probability distribution function, representing the probability that a number of molecules occupy an (in nitely) small region in space. In contrast to the continuum formulation, the statistical formulation can relate equations of state and transport coe cients such as heat conductivity and viscosity to the molecular motion. Even though the Navier32
2.1. A CFD method based on the Boltzmann equation Stokes equation was rst derived from Newtons second law several decades before the Boltzmann equation was introduced, it has been showed later that the Navier-Stokes equation can be derived directly from the Boltzmann equation [McLennan, 1989, Bardos and Ukai, 1991]. Both the continuum formulation and the statistical formulation of uid dynamics are useful, however one formulation might be more suitable for a given problem.
Figure 2.1: Overview of the eld of uid dynamics, including continuum and statistical formulations, and classical and numerical solution methods. Several methods has been proposed to derive the Navier-Stokes equation from the Boltzmann equation. Since the Chapman-Enskog method was introduced rst, this is used in the gure. While traditional computational uid dynamics (CFD) methods numerically solve the Navier-Stokes equation together with conservation equations typically for mass and energy, using discretization methods such as the nite volume method (FVM), nite element method (FEM) or nite dierence method (FDM), the lattice Boltzmann method numerically solves the lattice Boltzmann equation (LBE). The LBE is a discrete veloc33
Chapter 2. The Lattice Boltzmann Method ity version of the Boltzmann equation [He and Luo, 1997]. An advantage using the Boltzmann equation, contrary to the Navier-Stokes equation, is its ability to describe uids in non-hydrodynamic regimes and easily represent complex physical phenomena such as multiphase ow [Shan and Chen, 1993] and chemical interactions between the uid and its surroundings [Kingdon and Scho eld, 1992, Cali et al., 1992, Dawson et al., 1993, Chen et al., 1995]. This makes the LBM a natural choice when modeling physical and chemical phenomena in porous media. Another bene t with the LBM compared to classical CFD methods is the gain in computational time due to its suitability for parallelization.
2.2 Brief history Even though the LBM can be directly derived from the Boltzmann equation [He and Luo, 1997], it was rst introduced as an alternative to the lattice gas cellular automata (LGCA) [McNamara and Zanetti, 1988, Higuera and Jiménez, 1989]; a class of cellular automata. Cellular automata, introduced by Stanislaw Ulam and John von Neumann (1940s) consist of a lattice of cells where each site can have a nite number of states that evolve in discrete time steps according to well de ned (random or deterministic) rules. The main dierence between LBM and LGCA is the replacement of the particle occupation number by the single-particle distribution function, being the local ensemble average of the occupation number. LGCA was subject to some important shortcomings; non-Galilean invariance, and unphysical velocity-dependent pressure and an inherent statistical noise. The statistical noise was completely eliminated with the LBM proposed by McNamara and Zanetti [McNamara and Zanetti, 1988], while nonGalilean invariance and velocity-dependent pressure remained. The lattice Boltzmann model proposed by Chen et al. [Chen et al., 1991, 1992] and Qian et al. [Qian, 1990, Qian et al., 1992a] apply a single relaxation time approximation for the collision operator, rst introduced by Bhatnagar, Gross and Krook [Bhatnagar et al., 1954], and eliminates the remaining problems with non-Galilean invariance and velocitydependent pressure. This model is called the lattice Boltzmann BhatnagarGross-Krook model (LBGK), and is the model that has been used in this thesis.
34
2.3. The lattice Boltzmann equation
2.3 The lattice Boltzmann equation The lattice Boltzmann equation (LBE) (2.1) describes the evolution of single particle distribution functions, fα , for a set of discrete directions, α:
fα (x + eα δt, t + δt) − fα (x, t) = Ωα .
(2.1)
eα is the velocity in the α-direction and Ωα is a collision operator.
A common choice for this term is the Bhatnagar-Gross-Krook (BGK) collision operator [Bhatnagar et al., 1954];
1 [fα (x, t) − fαeq (x, t)] , (2.2) τ where the dimensionless relaxation time, τ , is the average time between collisions related to the uid viscosity, ν (τ = ν/c2s + 1/2), and fαeq is an equilibrium distribution function. Using the BGK collision operator in the LBE we form the lattice Boltzmann-BGK equation [Qian et al., 1992b]: Ωα =
fα (x + eα δt, t + δt) − fα (x, t) =
1 [fα (x, t) − fαeq (x, t)] , τ
(2.3)
The single particle distribution function, fα (x, t), is the probability of nding a particle with velocity eα at (x, t) and the macroscopic variables are computed by
ρ =
X
fα =
α
ρu =
X
X α
eα fα =
α
fαeq ,
X
eα fαeq .
(2.4) (2.5)
α
The equilibrium distribution function, fαeq , is chosen as a function of the macroscopic quantities in such a way that it assures recovery of the NavierStokes equation in the continuum limit [Qian et al., 1992b]: " # 2 e · u ( e · u ) u · u α α fαeq = ωα ρ 1 + + − , (2.6) c2s 2c4s 2c2s
35
Chapter 2. The Lattice Boltzmann Method where ωα are weight coe cients for the dierent velocities (directions) and cs is the lattice sound speed. The discretization of phase space must be done in a way that the conservation constraints (2.4-2.5) are preserved and the necessary symmetries required by the Navier-Stokes equation [Gusyatnikova and Yumaguzhin, 1989] are retained:
X
ωα = 1
(2.7)
ωα eα,i = 0
(2.8)
ωα eα,i eα,j = c2s δi,j
(2.9)
ωα eα,i eα,j eα,k = 0
(2.10)
ωα eα,i eα,j eα,k eα,l = c4s (δij δkl + δik δjl + δil δjk )
(2.11)
α
X α
X α
X α
X α
Programming
The LBE can be be viewed as two principal operations on virtual particles; collision (2.12) and streaming (2.13):
fα (x, t + δt) = fα (x, t) + Ωα ,
fα (x + eα δt, t + δt) = fα (x, t + δt) .
(2.12) (2.13)
One can imagine several particles coming in to the same point in space (node), colliding and shifting directions of propagation. This will be the collision step. In the next step - the streaming step - the new distribution of particles (after collision) is propagated along the new directions. New particles will then meet at the dierent points in space and make new collisions. The collision and streaming steps are repeated. This is how the LBE is programmed into a computer algorithm.
2.4 Lattice Boltzmann equation for diusion The diusion equation can be described by the lattice Boltzmann equation [Wolf-Gladrow, 1995]. With a BGK collision operator and an equilibrium 36
2.5. DnQm discretization of phase space distribution of the form of eq. (2.6) the LBGK equation can recover the advection-diusion equation in the continuum limit. This will be shown in section 2.8 where we go through the Chapman-Enskog expansion for the advection-diusion equation. We will use the following notation for the LBGK advection-diusion equation;
gα (x + eα δt, t + δt) − gα (x, t) =
1 [gα (x, t) − gαeq (x, t)] , τD
(2.14)
where τD is the relaxation time associated with the diusion process, given in terms of the diusion coe cient, D (τD = D/c2s + 1/2). The macroscopic concentration, c, is computed by
c=
X
gα ,
(2.15)
α
and the equilibrium distribution function is " # 2 e · u ( e · u ) u · u α α gαeq = ωα c 1 + + − . c2s 2c4s 2c2s
(2.16)
For the advection-diusion equation only constraints (2.7-2.10) are necessary.
2.5 DnQm discretization of phase space One group of discretization of phase space is the DnQm group for square lattices, where n denotes number of spatial dimensions and m the number of discrete velocities. Two models have been used for the work presented in this thesis; the D2Q9 model ( gure 2.2a) for two dimensional simulations and the D3Q15 model ( gure 2.2b) for three dimensional simulations. In the D2Q9 model there are eight non-zero velocities; four of which is along direct links between nodes, i.e. horizontal and vertical links (e1 −e4 ), and four along diagonal links (e5 − e8 ). In the D3Q15 model there are 14 non-zero discrete velocities; six along direct (horizontal and vertical) links and 8 along diagonal links through the cell corners.
37
Chapter 2. The Lattice Boltzmann Method
(a) D2Q9
(b) D3Q15
Figure 2.2: Phase space discretizations used in the papers. 9 velocities are used in 2 dimensions, and 15 velocities in 3 dimensions.
2.6 Boundary conditions In the uid nodes next to the solid wall the distribution functions leaving the surface are not found by the regular colliding and streaming steps. A boundary condition (BC) is needed that relate the distribution functions leaving the wall to the macroscopic quantities given at the wall. Boundary conditions for the LBM is generally di cult because there is no one-toone correspondence between the particle distribution functions and the macroscopic quantities given at the wall. When formulating a boundary condition within LBM the problem usually consist of nding an expression for the distributions leaving the surface (unknown) as function of the distributions entering the surface (known) and the macroscopic quantities. For a stationary boundary/wall we have used the mid-grid approach where the boundary is located halfway between the grid points. Figure 2.3 shows the position of the wall between a uid node and a solid node, and the distribution functions after collision at an initial time (2.3a) and the distributions after streaming, i.e. at the next time step (2.3b). Known distributions are marked with solid arrows, while unknown distributions are marked with dotted arrows. Initially we know all the distributions (gα¯ and gα ) in the uid node. In the next time step we know only the gα -distributions pointing towards the wall in the uid node. However, we also know the gα -distributions in the solid node inside the wall. These are simply the distributions after 38
2.6. Boundary conditions collision streamed from the uid nodes towards the wall. We denote these distributions g˜α , hence g˜α (x + eα δt, t + δt) = gα (x, t).
Figure 2.3: Illustration of distributions used in the boundary condition. (a) At the initial time t all distributions in the uid (gα¯ and gα ) are known . The wall is positioned midway between the nodes. (b) At time t + δt the distributions gα¯ are unknown. We want to express these as functions of g˜α ≡ gα (x + eα δt, t + δt). Three dierent lattice boundary conditions have been used in this thesis; bounce-back (no-slip/zero ux) [Succi, 2001], specular re ection (freeslip) [Succi, 2001] and a ux boundary condition. Only an introduction to the boundary conditions will be given in this section. A detailed derivation of the ux boundary condition from the Chapman-Enskog expansion will be given in a later section. The bounce-back rule is used on a rigid wall where the roughness (friction) is large enough to prevent uid motion. The unknown distributions leaving the surface are set equal to the known incoming distributions in the opposite direction:
gα¯ = g˜α .
(2.17)
The specular re ection rule apply to rigid walls with negligible friction. The tangential uid motion is free, and there is no momentum transfer 39
Chapter 2. The Lattice Boltzmann Method with the wall in this direction. The bounce-back rule is used for the normal direction. For the D2Q9 lattice, the specular re ection rule for a horizontal wall with uid below is (see gure 2.2a for directions):
g4,7,8 = g˜2,6,5 .
(2.18)
The ux boundary condition is a modi ed bounce-back rule that accounts for chemical reactions (such as dissolution and precipitation) between the uid and the solid wall. The boundary condition was proposed by Bouzidi et al. [2001], Verhaeghe et al. [2006], and as for the bounceback rule the unknown outgoing distributions are given as functions of the known incoming distributions.
gα¯ − g˜α = 6ωα eα¯ · JR n,
(2.19)
gα¯ + g˜α = 2ωα c.
(2.20)
where JR is the macroscopic chemical ux on a surface that lies halfway between the grid points, and eα¯ is the velocity vector pointing away from the surface. The concentrations are related through a similar expression [Verhaeghe et al., 2006]
This expression will be derived in section 2.8. These two expressions together with a rate law for JR make up the ux boundary condition. In Paper 3 we use a linear rate law with a surface tension term;
JR = −k(c − ceq ) − ξκ,
(2.21)
where k is the rate constant, ceq is the equilibrium concentration, ξ is a constant, and κ is the surface curvature. Solving Eq. (2.20) for c, inserting that into Eq. (2.21) and then inserting the ux expression (JR ) into Eq. (2.19) yields the following moving boundary condition
gα¯ =
kα 1 − kα 6ωα ξκ 2ωα ceq + g˜α − eα¯ · n, 1 + kα 1 + kα 1 + kα
(2.22)
where kα = 3k eα¯ · n. Eq. (2.22) equals the bounce-back condition (zero ux BC) in the limit k, ξ → 0 and is valid for a boundary that is located half-way between the computational nodes. For low reaction rates the system is approximated to diusive equilibrium, and the concentration in the pore space is approximately constant. Hence the system reduces 40
2.7. Conversion between LB and physical units to a constant ux boundary condition. For high reaction rates (when ξ = 0) the boundary condition reduces to a Dirichlet boundary condition, gα¯ + g˜α = 2ωα ceq , and is independent of the surface normal. Dissolution and precipitation constantly change the position and shape of the solid- uid interface and it is no longer constrained to lie half-way between the computational nodes. To allow sub-grid resolution of the interface the boundary condition in Eq. (2.22) must be modi ed. This is the topic of Paper 3.
2.7 Conversion between LB and physical units In order to simulate real experiments we must have a method to convert dimensionless lattice units to physical units. Usually, a lot of the work within the LBM is theoretical and hence the conversion of units has got little attention. Since the goal of the LB modeling is to compare to experiments a short description is included here. When a physical domain is transformed into a lattice for simulation purposes the physical length (Lphys ) of the system is divided into Nx cells of physical length δxphys = Lphys /Nx . Similarly the physical time (tphys ) is divided into Nt intervals of length δtphys = tphys /Nt . In the LB system the equations are: δxLB = LLB /Nx and δtLB = tLB /Nt . For programming simplicity the LB cell length and LB time interval are both set to unity: δxLB = δtLB = 1. It follows that the number of cells and number of time intervals equals the LB length and the LB time respectively, so the physical length and time can be expressed by the LB length and time and the physical cell length and time interval length:
δxphys = Lphys /LLB ,
(2.23)
δtphys = tphys /tLB .
(2.24)
This means that the physical value of every parameter with dimension Lp Tq can be found by multiplying the LB value with δxpphys · δtqphys , e.g. the physical velocity with dimension L/T is found by multiplying the LB velocity with δxphys /δtphys . For a complete conversion scheme we need similar rules for all other dimensions such as mass (M) and moles (N) etc. For every dimension one has the freedom to choose the physical interval length, which is a scaling factor between the physical and the LB system. 41
Chapter 2. The Lattice Boltzmann Method The length scale δxphys is set by choosing the grid size (resolution) in the simulation. Then the time scale, δtphys , can be determined from any known parameter with dimension Lp Tq . However, in some cases there might be more than one parameter that can be used to determine δtphys . In case of advection and diusion, both the viscosity [m2 /s] and the diusion coe cient [m2 /s] could be used to determine the time interval. However, they don't necessarily end up with the same value, because these processes take place on dierent time scales (advection is faster than diusion). This is the case for the problem studied in Paper 5. By choosing one of the parameters (viscosity or diusion coe cient) to determine the time interval, the other parameter will not obtain its correct value. In Paper 5 we instead studied the problem in terms of dimensionless numbers. The equations that describe the system (NavierStokes and advection-diusion equation) can be written on dimensionless form to nd some dimensionless parameters that describe the system. In the case of the Navier-Stokes equation and the advection-diusion equation, two dimensionless numbers, the Rayleigh number and the Prandtl number, describe the system. By matching the dimensionless numbers to the value in the experiment, one can perform simulations on systems that are dynamically similar to the experimental system.
2.8 Chapman-Enskog expansion for the advection-diusion equation The Chapman-Enskog expansion was introduced independently by Sydney Chapman and David Enskog between 1910 and 1920, as a derivation of the Navier-Stokes equation and its transport coe cients from the Boltzmann equation. The expansion is a type of multiscale method used to solve problems involving description of processes at multiple time scales. It is an expansion of both dependent and independent variables in a smallness parameter in order to split contributions at dierent scales. Since the lattice Boltzmann equation is a discretization of the Boltzmann equation, we can go directly from LBE to the continuum equation. In the following we will demonstrate a Chapman-Enskog like expansion for the advectiondiusion equation, following the approach used by Hou and Zou [1995]. Starting from the LBE;
42
2.8. Chapman-Enskog expansion for the advection-diusion equation
gα (x + eα δt, t + δt) = gα (x, t) −
1 [gα (x, t) − gαeq (x, t)] , τ
(2.25)
and Taylor expanding the left side around (x, t);
gα (x + eα δt, t + δt) (2.26) δt2 = gα (x, t) 1 + δt (eα ∇ + ∂t ) + (eα ∇ + ∂t )2 + O δt3 , 2
gives the lattice Boltzmann equation (2.25) as:
δt2 2 gα (x, t) δt (eα ∇ + ∂t ) + (eα ∇ + ∂t ) + O δt3 2
(2.27)
1 = − [gα (x, t) − gαeq (x, t)] τ The next step is the Chapman-Enskog like expansion of the distribution function. As the continuum equations (Navier Stokes equation and advec-tion-diusion equation) are only valid for low Knudsen numbers (Kn ), i.e. the ratio between mean free path and characteristic ow length, the distribution function is expanded using Kn . By making some assumptions, as done by Hou and Zou [1995], we will show that the time increment δt is of the same order as the Knudsen number, and hence can be used in the expansion. First, since the relaxation time represents the characteristic collision time, τ is of the same order as the mean free path. The value of τ in the LB formulation is limited downwards to 1/2 in order to give positive viscosity (see section 2.3). Also, since τ is of the same order as the mean free path, which is of the same order as the Knudsen number, τ should not be too large in order to have low Kn (validity of continuum equations). Hence, it is okay to assume that τ is of order one. The physical value of τ is τ δt, and since this has the same order as the Knudsen number, we nd that δt ∼ Kn . Therefore we will use δt as the expansion parameter in the following Chapman-Enskog expansion of the distribution function. The expansion is gα (x, t) = gα(0) (x, t) + δtgα(1) (x, t) + δt2 gα(2) (x, t) + O δt3 . 43
(2.28)
Chapter 2. The Lattice Boltzmann Method As advection and diusion happen on dierent time scales (advection being the fastest), the following expansion of time is introduced
t = t1 + δt · t2 ,
∂t = ∂t1 + δt · ∂t2 .
(2.29) (2.30)
As only the zeroth order expansion of space is necessary to obtain the continuum equation, this expansion is omitted. The expansions of time and distribution function (2.28-2.30) are then inserted into the Taylor expanded lattice Boltzmann equation (2.27), and by requiring that the coe cient of each power of δt vanish, we nd the following equations up to second order in δt:
∂t2 gα(0) (
gα(0) (x, t) = gαeq (x, t) ,
(2.31)
1 (eα ∇ + ∂t1 ) gα(0) (x, t) = − gα(1) (x, t) , τ
(2.32)
1 1 (eα ∇ + ∂t1 ) gα(1) (x, t) = − gα(2) (x, t) . (2.33) x, t) + 1 − 2τ τ
Summing over α in eq. (2.32) and eq. (2.33) respectively, and using eq. (2.31) together with the expression for the equilibrium distribution (2.16) and the relations (2.7-2.9) gives;
∂t1 c + ∇(cu) = 0,
(2.34)
X 1 ∇ eα gα(1) (x, t) = 0. ∂t2 c + 1 − 2τ α
(2.35)
Multiplying eq. (2.32) with relations as above gives;
eα
and summing over α, using the same
c2s ∇c + ∂t1 (cu) = −
1X eα gα(1) (x, t) . τ α
(2.36)
Multiplying eq. (2.35) with δt, adding eq. (2.34), and inserting eq. (2.36) gives 44
2.9. Derivation of ux boundary condition from the Chapman-Enskog expansion 1 τ− δt∇2 c 2
(∂t1 + δt∂t2 ) c + ∇(cu) − 1 − τ− δt∇ · ∂t0 (cu) = 0. 2 c2s
(2.37)
The term in the rst parenthesis is recognized as ∂t . The last term on the left hand side is of higher order in the Knudsen number (δt) as we consider only uids with small velocities, i.e. u ∼ O (δt). Finally, by recognizing the diusion constant as D = c2s τ − 12 δt, we end up with the advection-diusion equation:
∂t c + ∇(cu) − D∇2 c = 0.
(2.38)
2.9 Derivation of ux boundary condition from the Chapman-Enskog expansion In order to express distribution functions in terms of macroscopic variables on the wall, the gα¯ -distributions in the uid node and the g˜α -distributions in the solid node (ref. section 2.6) must be expressed in terms of the distributions on the wall. This is done by a Taylor expansion of the two:
g˜α
eα δt
(eα δt)2 2 ∇ gα¯ (xW ) + O δt3 , 2 4 eα δt (eα δt)2 2 = g˜α (xW ) + ∇˜ gα (xW ) + ∇ g˜α (xW ) + O δt3 , 2 4
gα¯ = gα¯ (xW ) −
∇gα¯ (xW ) +
where xW = x + 21 eα δt is the position of the wall midway between two nodes. To derive the boundary condition we start by adding and subtracting the expanded distribution functions for the two opposite directions gα¯ and g˜α ;
(0) (1) (2) gα¯ + g˜α = gα¯ + g˜α(0) + δt gα¯ + g˜α(1) + δt2 gα¯ + g˜α(2) , (0) (1) (2) gα¯ − g˜α = gα¯ − g˜α(0) + δt gα¯ − g˜α(1) + δt2 gα¯ − g˜α(2) . 45
(2.39) (2.40)
Chapter 2. The Lattice Boltzmann Method The zeroth, rst and second order distribution functions are given by equations (2.31-2.33). They can all be expressed in terms of the equilibrium distribution (2.16). For the distributions on the wall we have:
gα(0) (xW ) = gαeq (xW ),
gα(1) (xW ) = −τ (∂t1 + eα ∇) gαeq (xW ), 1 2 (2) (∂t1 + eα ∇) gαeq (xW ). gα (xW ) = −τ ∂t2 − τ − 2 By inserting the expression for the equilibrium distribution (2.16) for all the terms in (2.39) and using ωα¯ = ωα and eα¯ = −eα , we nd:
x
(0) gα¯ ( W )
=
g˜α(0) (xW ) = gα¯ (xW ) = (1)
g˜α(1) (xW ) =
"
# (eα u)2 u · u ωα c 1 − 2 + + , cs 2c4s 2c2s " # eα u (eα u)2 u · u ωα c 1 + 2 + + , cs 2c4s 2c2s # " eα u (eα u)2 u · u + , −τ (∂t1 − eα ∇) ωα c 1 − 2 + cs 2c4s 2c2s " # eα u (eα u)2 u · u −τ (∂t1 + eα ∇) ωα c 1 + 2 + + , cs 2c4s 2c2s
eα u
gα¯ (xW ) (2)
" # 2 e u ( e u ) u · u 1 α α = −τ ∂t2 − τ − (∂t1 − eα ∇)2 ωα c 1 − 2 + + , 2 cs 2c4s 2c2s
g˜α(2) (xW )
" # 1 eα u (eα u)2 u · u 2 = −τ ∂t2 − τ − (∂t1 + eα ∇) ωα c 1 + 2 + + , 2 cs 2c4s 2c2s
where the distributions are given on the wall and c and tration and uid velocity on the wall respectively. 46
u are the concen-
2.9. Derivation of ux boundary condition from the Chapman-Enskog expansion Now we have all the expressions we need to calculate the boundary condition (2.39-2.40). We have the distributions gα¯ and g˜α as functions of the distributions on the wall, and we have the expressions for the wall distributions as function of the macroscopic quantities. To second order in δt we get, by inserting everything into the boundary condition (2.39-2.40);
(eα · u)2 u · u gα¯ + g˜α = 2ωα c 1 + + 2c4s 2c2s ( (eα · u)2 u · u + δt −2ωα τ ∂t1 c 1 + + 2c4s 2c2s ) ωα + (1 − 2τ ) 2 (eα · ∇)eα · (cu) cs + O δt2 gα¯ − g˜α =
2ωα eα · (cu) c2s ( 2ωα τ + δt ∂t1 eα · (cu) c2s ) (eα · u)2 u · u + − (1 − 2τ )ωα (eα · ∇)c 1 + 2c4s 2c2s + O δt2
(2.41)
−
(2.42)
The term −δt(1 − 2τ )ωα (eα · ∇)c can be rewritten as
−
2ω 2ωα eα · c2s δt(τ − 1/2)∇c = 2α eα · (D∇c) , 2 cs cs
(2.43)
where D is the diusion coe cient. We nd that the ux boundary condition (2.19) is valid to rst order in δt, recognizing the term cu − D∇c as a macroscopic ux JR :
gα¯ − g˜α = 6ωα eα¯ · JR n
If the uid velocity at the surface is zero, the boundary condition is second order accurate. To obtain the wanted relation (2.20) from the sum of the distribution functions we must be con ned to the low velocity limit. Then the relation is rst order accurate in δt. 47
Chapter 3
Petroleum geochemistry 3.1 Introduction Petroleum geochemistry is a branch within the vast subject of geochemistry. While geochemistry deals with the chemistry of the composition of the Earth, its oceans and the atmosphere, and how it changes with time [Walther, 2009], petroleum geochemistry has its focus on the origin, generation, migration, accumulation and alteration of oil and gas in petroleum reservoirs [Hunt, 1979, Brooks and Welte, 1984]. In the beginning, drilling sites for petroleum was based solely on observations of surface seepages, dating back to ancient times. As the technology matured, using geochemistry to understand the origin, composition and migration of petroleum became important in the search for it [Hunt, 1863, 1979]. Today quantitative geochemistry is a necessity for the understanding of migration and alteration of ow from petroleum reservoirs during production, and in the search for good enhanced recovery methods. A common way of introducing quantitative geochemistry is the "`concepts of chemical equilibrium"' approach [Walther, 2009]. Searching through textbooks on geochemistry, some principles seem to reoccur; chemical equilibrium, dissolution and precipitation kinetics, ion exchange and adsorption of ions on the mineral surface. These are also the concepts used in the lattice Boltzmann modeling in the work presented in this thesis. Hence, a brief introduction to these concepts is appropriate and will ease the subsequent description of the geochemical solver used in the LBM simulator (section 4.3).
48
3.2. Chemical equilibrium
3.2 Chemical equilibrium In addition to hydrocarbon oil and gas, a petroleum reservoir contains formation water; water that is in equilibrium with the surrounding rocks. When fresh water comes into contact with minerals that constitute the rocks, minerals will dissolve until a chemical equilibrium is reached between the water and the rock. The equilibrium state is the state that minimizes the Gibbs free energy of the system. The excess free energy of a system will be used to break and/or create new chemical bonds until there is no more excess energy and the system is in equilibrium. If the water that contacts a rock already contain aqueous species due to dissolved minerals, new minerals can form in solution and precipitate on the rock until the system reaches the equilibrium state between the rock and the water solution. The mineral composition might vary throughout a reservoir, hence formation water initially in equilibrium with its surroundings might lead to dissolution and/or precipitation when migrating through the reservoir. In many reservoirs, seawater injection is used to maintain pressure and increase the recovery. The seawater will mix with the formation water and change the equilibrium state, and hence dissolution and/or precipitation can take place. Consequently dissolution/precipitation reactions are frequent chemical reactions in the reservoir. Dissolution/precipitation reactions are governed by equilibrium kinetics described by chemical reactions of the sort;
Am Bn * ) mA + nB,
(3.1)
where Am Bn is a mineral phase that can dissolve into A and B. For each equilibrium reaction there is an associated equilibrium constant K that is related to the activities of the reactants and the products through the law of mass action (Peter Waage and Cato M. Guldberg 1864-1879):
K=
anA am B . aAm Bn
(3.2)
ai is the activity of species i. The activity is a measure of the eective concentration (in mol/kg H2 O) of a species, which can be dierent from the absolute concentration because of electrostatic shielding and/or presence of aqueous complexes [Appelo and Postma, 2005]. If these eects are negligible the activity equals the value of the species concentration. The activity is proportional to the concentration and inversely proportional to 49
Chapter 3. Petroleum geochemistry a standard state of 1 mol/kg H2 O, hence the value (dimensionless) of the activity is;
a = γ · c,
(3.3)
where c is the concentration and γ is the activity coe cient. Activity coe cients can be measured experimentally or estimated theoretically using for example the Debye-Hückel theory. In this theory the activity coe cient is given by; √ B1 z 2 I √ . logγ = − (3.4) 1 + B2 dˇ I
B1 and B2 are temperature dependent constants, dˇ is an empirical ion-size parameter, a measure of the eective diameter of the hydrated ion, z is the charge number, and I is the ionic strength. The value (dimensionless) of the ionic strength is [Atkins and de Paula, 2002]; I=
1X 2 ci zi , 2
(3.5)
i
where the sum runs over all charged species in solution. The activity of a pure solid phase is usually unity and omitted in the mass action expression. The equilibrium constant (K ) for a reaction is related to the change in Gibbs free energy for the reaction, which is given by the dierence in Gibbs free energy of formation (∆Gr ) for products and reactants of the reaction;
∆G0r =
X
∆G0r,products −
X
∆G0r,reactants = −RT lnK,
(3.6)
where the superscript denotes a standard state, typically 25o C and 1 atm, R is the universal gas constant and T is temperature. ∆G0r for numerous reactions are tabulated in thermodynamical databases. To nd the equilibrium constant of a reaction at a speci c pressure and temperature, one needs an additional equation of state (EOS). Chemical reactions might be fast, but they are not instantaneous. Even on the microscopic scale, the time scale is nite. The equilibrium constant will tell the direction of the reaction, but says nothing about the time frame to reach the equilibrium state. In addition to the equilibrium 50
3.3. Ion exchange constant the reactions are associated with a reaction rate, which states how much material will undergo reaction during a given time interval. Rate data are measured in laboratories and models are made, that relate the rate data to fundamental mechanisms. A standard rate law for dissolution is on the form
A k(1 − Ω)η (3.7) V where A is the surface area, V is the pore volume, k is a rate constant, Ω is the saturation state, and η is a constant describing the order of reaction. The saturation state is a measure of how far the solution is from equilibrium, and it is given by the ion activity product (IAP) divided by the equilibrium constant: rate =
an am IAP = A B. (3.8) K K Hence, the rate of dissolution increases as the distance from equilibrium increases. Ω=
3.3 Ion exchange Ion exchange is a process where an ion on the solid surface is exchanged by an ion from solution [Appelo and Postma, 2005]. The process might be confused with dissolution and precipitation, however a dierence between ion exchange and precipitation is that the ion exchange process is dependent on a preexisting solid surface. The ion exchange process is described by the law of mass actions (3.2), as is the equilibrium reaction. An example of an ion exchange reaction is the exchange of Na+ by Ca2+ on the solid surface:
Ca2+ + 2 NaX * ) CaX2 + 2 Na+ ,
(3.9)
where X is a cation exchange site. The mass action expression becomes;
K=
CaX2 2Na+ Ca + 2NaX
.
(3.10)
2
The activity of an exchangeable site () is expressed as a fraction of the total cation exchange capacity (CEC) of the exchanger, expressed in mol/l [Gaines and Thomas, 1953]. The exchange capacity is a function 51
Chapter 3. Petroleum geochemistry of for example surface area/grain size [Appelo and Postma, 2005]. Anion exchange is much more rare than cation exchange, because there are fewer anions than cations.
3.4 Adsorption Adsorption is the process of ions adhering to a solid surface. It should not be confused with absorption, where ions penetrate through the structure of the solid surface. Atoms within the solid structure have all their bonds connected to other atoms. On the surface, however, atoms can have free bonds and attract ions from solution and form surface complexes. The surface complexes can be charged or uncharged. Charged complexes will attract ions with opposite charge from solution and hence some ions will be adsorbed in a diuse layer close to the surface instead of following the bulk ow. Surface complexation is part of the geochemical solver of the LBM simulator, and is described by mass action equations as ion exchange and chemical equilibrium. The diuse layer is, at the time of writing, not part of the simulator, but is a possible implementation for the future.
52
Chapter 4
Lattice Boltzmann pore scale simulator 4.1 Introduction Motivated from core ooding experiments and the di cult task to interpret the results, a lattice Boltzmann based pore scale simulator was developed prior to the work described in this thesis [Hiorth et al., 2013]. The simulator has numerous purposes; in addition to simulating the ow through the pore space the simulator is used to nd out which physical and chemical mechanisms that are most important when water with dierent chemical composition is injected into a hydrocarbon chalk reservoir. This will in the future be used to nd an appropriate way of upscaling chemical reaction rates to the core scale and eld scale. Despite the present interest in chalk reservoirs, the simulator is general in that any mineral can be added to the database, and can be used for any sort of reservoir rock. The version of the simulator published by Hiorth et al. [Hiorth et al., 2013] includes non-linear dissolution-precipitation kinetics, surface complexation and ion exchange. This version does not include moving boundaries, i.e. the geometry of the pore space does not change due to chemical reactions at the uid-solid boundaries. A simple ' rst approach' model for moving boundaries was implemented during the course of this work, and has been used for the work presented in Paper 1. The simulator consist of two main components; a uid solver and a geochemical solver. These two will be described in the following sections.
53
Chapter 4. Lattice Boltzmann pore scale simulator
4.2 Fluid solver The uid solver is based on the lattice Boltzmann method described in section 2. One set of distribution functions (fα ) is used to evolve the ow eld, and a new set of distribution functions (gαi ) is used for each chemical species in solution. The ow eld is solved prior to the advection of species, since the velocity is input to the advection calculations (equilibrium distributions). Flow in the bulk simply consists of streaming and collision steps for the dierent distribution functions as described in section 2.3. In addition to the relatively simple bulk calculations, the distributions near the surface are calculated through boundary conditions. The boundary condition for the ow eld near a solid node is the bounce-back/no slip condition (2.17). For the chemical species the boundary condition is the chemical ux boundary condition (2.22), which simpli es to the bounce-back rule if no chemical reactions take place. If dissolution occurs, the distributions of the dissolved species increase in solution, and if precipitation occurs, the distributions of the precipitated species decrease. The simulator is usually run with a constant dierential pressure over the geometry in order to induce ow through the geometry. This is implemented using a force term. A force can be added to the lattice Boltzmann ow eld in several ways, and in the simulator the altered velocity method is used [Buick and Greated, 2000]. A moving boundary routine was implemented into the simulator as an improved functionality. Only the simplest model has been implemented at the time of writing, as a rst approach. The ux boundary condition (2.22) is used as is, meaning that the position of the wall between the uid node and the solid node is not taken into account; it is approximated to 0.5 at all times. Also, the surface normal is not being estimated, and the product of the microscopic velocities and the surface normal is set to unity (eα ·n = 1). Even though formally better approaches has been tested and identi ed (Paper 3), this rst approach has already given insight to the importance of having dynamically changing boundaries in the model (Paper 1).
4.3 Geochemical solver The geochemical solver handles chemical reactions between solution species, and surface reactions between solution species and solid minerals. A 54
4.3. Geochemical solver set of basis species is de ned in the geochemical model, and after the basis species concentrations have been altered by diusion and advection in the uid solver, the geochemical solver performs equilibrium calculations necessary to determine the concentrations of basis species and secondary species (complexes) that are present in solution [Johnson et al., 1992]. The calculations performed by the geochemical solver can be divided into three: 1. Calculate uid-solid equilibrium 2. Update basis species concentrations (rate equation) 3. Speciation calculation in solution These three calculations will now be described in separate paragraphs.
Fluid-solid equilibrium The equilibrium between solution basis species and the surrounding solid mineral phases includes dissolution and precipitation reactions, ion exchange and surface complexation. Dissolution and precipitation equilibrium is determined from mass action expressions for each mineral phase. The logarithmic form of the mass action equation is used to set up a system of linear equations; log10 Kb = µ · log10 ab ,
(4.1)
Ca2+ + 2 NaX * ) CaX2 + 2 Na+ ,
(4.2)
where we have set the solid (mineral) phase activities to unity, and where Kb is the equilibrium constants for the dissolution and precipitation reactions, µ is the stoichiometric matrix for the rock buers, and ab is the vector containing the activities for the basis species. Ion exchange is described by a cation exchange basis species, X− , as described in section 3.3. The unit of concentration for this species is moles of cationic charge per liter of pore uid. Anion exchange is not included due to its infrequency. In order to treat the exchange reactions similar to the equilibrium reactions, they are divided into half-reactions as described by Appelo and Postma (2005). The exchange of Na+ by Ca2+ ;
is described by the two half reactions;
55
Chapter 4. Lattice Boltzmann pore scale simulator
Na+ + X− * ) NaX, + − * Ca2 + 2 X ) CaX2 .
(4.3) (4.4)
The logK for exchange reactions are assumed to be small compared to dissolution/precipitation reactions, such that the activity of X is negligible. Note that exchange complexes are electrically neutral. Surface complexes are added following the approach of Van Cappelen et al. [1993]. The surface complexes are related to surface basis species and solution basis species through surface complexation equations and mass action equations, similar to the equilibrium equations for dissolution/precipitation. The equations for logK for the surface complexation reactions contain an extra term compared to the usual equilibrium logK . This term contains the surface potential, which is related to the surface charge through the Grahame equation [see e.g. Israelachivili, 1985]. Contrary to the ion exchange complexes, the surface complexes are charged.
Rate equation
Basis species concentrations are updated by shifting the concentrations towards equilibrium according to a speci ed rate. The rate is calculated by a rate equation that are speci c for each dissolution/precipitation reaction. There are several proposed rate equations to be found in the literature. The rate equation used in the simulator is an equation suggested by Morse and Berner [1972], Steefel and Van Cappelen [1990] and Lasaga [1998]:
dM i Ai i A = JM = sgn (1 − Ωi ) k1i + k2i aH |1 − Ωζi |η , (4.5) dt V V where Mi is the concentration of mineral i (mol/L), Ai is the surface area of mineral i (m2 ), V is the total uid volume (L), Ωi is the saturation state for mineral i (dimensionless), k1,2 are rate constants (mol/m2 /s), and ζ and η are exponents indicating the order of reaction. The change in solute concentration for basis species i, ci , is given by; dci A = JRi , dt V where JRi is the chemical reaction ux on the surface;
56
(4.6)
4.3. Geochemical solver
JRi =
X j
νij sgn (1 − Ωi ) k1i + k2i aH |1 − Ωζi |η .
(4.7)
j runs over all minerals, and νij is the stoichiometric matrix. JRi is the macroscopic ux used in the ux boundary condition (2.19).
Speciation calculation
After equilibrium between solution and rock has been established, the solution equilibrium between basis species and complexes is determined. Basis species activities are determined from charge balance (H+ ), mass balance and setting the activity of water to unity (aH2 O =1). Calculation of the activity of the secondary species is performed using the logarithmic form of the mass action equation for all the reactions, as for the mineral phases; log10 ac = µ · log10 ab − log10 K,
(4.8)
where ab is a vector containing the activities of all basis species (b), ac is a vector containing the activity of the complexes (c), the K-vector is the collection of equilibrium constants for all the reactions, and µ is the stoichiometric matrix. For each chemical reaction, either in solution or with the solid phase, an equilibrium constant (K) is needed. The equilibrium constants are calculated with the HKF equation of state [Helgeson and Kirkham, 1947a,b, Helgeson et al., 1981], for which a recent review is given by Oelkers et al. (2009) [Oelkers et al., 2009]. Thermodynamic data are taken from the SUPCRT geochemical database [Johnson et al., 1992] and the program EqAlt [Cathles, 2006]. It has been shown that the LB chemical simulations are nearly identical to that from the PHREEQC program by Parkhurst and Appelo [1999] [Hiorth et al., 2013].
57
Chapter 5
Location of mineral alterations in a pore geometry 5.1 Introduction Where in the pore space minerals dissolve and precipitate are important for the evolution of porosity and permeability of the rock. E.g. if dissolution reactions take place in the pore throats, rather than in the pore bodies, there can be a signi cant increase in the permeability even for relatively small changes in porosity. However, if precipitation reactions take place preferentially in the pore throats, one might see a situation where precipitates ll the pore throats and hence the permeability decreases to zero. Abstract 1 deals with the location of mineral alterations due to dissolution and precipitation as function of the dimensionless Peclét number, i.e. the ratio between advective and diusive transport.
5.2 Results Three sine tube geometries, as shown in Figure 5.1 were used for this study. To better understand the results from the dierent geometries, the wide pore throat geometry (Figure 5.1a) was used as a reference geometry, being very close to a straight pipe. In all the simulations presented in this chapter, the initial composition is assumed to be 100% calcite, and the ooding brine is 0.291 M MgCl2 . When ooding with MgCl2 , calcite dissolves and magnesite precipitates, and when ooding with seawater, anhydrate precipitates as well. The simulator described in chapter 4 was 58
5.2. Results used, only with a simpli ed linear rate law
dci = −k(ci − ceq (5.1) i ), dt where ci is the concentration of species i, k is a rate constant and ceq i is the equilibrium concentration of species i.
(a)
(b)
(c)
Figure 5.1: Three sine tube geometries used to study the eect of pore geometry on localization of mineralogical changes due to dissolution and precipitation. The three geometries are referred to as (a) wide, (b) medium and (c) narrow pore throats.
5.2.1 Mineral alterations as function of dimensionless distance from the inlet Before comparing simulation results in the three geometries above, the wide pore throat geometry was used for a reference simulation to establish how mineral alterations distribute in a more or less straight pipe. 0.219 M MgCl2 was used as the ooding brine, the ooding rate was 1.2 PV/day and temperature and pressure were 130o C and 8 bar, respectively. Figure 5.2 shows the normalized concentration of calcite and magnesite. For calcite we have plotted the loss rather than the remaining amount. From the plot, we nd two things to notice; i) more mineral alterations, both dissolution and precipitation, are found near the inlet of the tube, and ii) the amount of precipitated magnesite equals the amount of dissolved calcite. The reason for higher mineral alterations near the inlet lies in the rate law. As the ooding brine enters the calcite geometry, the solution is subsaturated with respect to calcite, which leads to dissolution of calcite into Ca2+ and CO32 ions. As carbonate ions are available, the solution becomes supersaturated with magnesite. As calcite dissolves and magnesite 59
Chapter 5. Location of mineral alterations in a pore geometry precipitates, the magnesium concentration in solution will decrease and calcium concentration will increase, hence the solution will go towards less subsaturation with respect to calcite as it ows further into the geometry. We see from the rate law that as the solution becomes less supersaturated/subsaturated, the rate of precipitation and dissolution decreases, and hence less alterations are seen in the core as the distance from inlet increases. The equality of alteration in calcite and magnesite concentrations indicate a one-to-one exchange between calcite and magnesite.
Figure 5.2: Normalized concentration pro le of calcite (loss) and magnesite from MgCl2 ooding through the wide pore throat geometry.
5.2.2 Mineral alterations as function of Peclét number The solution concentrations near the pore walls depend on the speed of the transport mechanism. Both advection and diusion contribute to transport of chemical species in the pores. In the wide pore throat geometry where the walls are more or less straight, advection is only in the direction parallel to the pore walls, while diusion dominates the transport in the normal direction. The ratio between advective and diusive transport, i.e. the Peclét number, will aect the distribution of mineral alterations in the geometry. Figure 5.3 shows simulated normalized magnesite concentrations for three dierent Pe after ooding 1.2 pore volumes 60
5.2. Results with MgCl2 . We see that the lowest Pe number gives more alterations. This can be explained by a lower ooding rate, which gives a higher residence time within the geometry, and hence more alterations. For higher Pe numbers we see that the alterations pro le attens out, i.e. there are less discrepancies between inlet and outlet. Also this is believed to be explained by the ooding rate. When the ooding rate is higher, the residence time is lower and hence less alterations can take place near the inlet than for a lower Pe number. Hence, the solution is further away from equilibrium as it ows through the geometry than for a lower Pe number. As the distance from equilibrium is maintained at a higher level, so is also the rate of dissolution and precipitation relative to the inlet.
Figure 5.3: Normalized magnesite concentration pro le for dierent Pe numbers from MgCl2 ooding simulations through the wide pore throat geometry after 1.2 pore volumes.
5.2.3 Mineral alterations as function of pore geometry Figure 5.4 shows normalized magnesite concentration pro les after 1 pore volume of MgCl2 ooding in the three sine tubes shown in Figure 5.1. The gray eld represent a sine geometry for easy comparison between the form of the curves and the location of pore bodies and pore throats. We 61
Chapter 5. Location of mineral alterations in a pore geometry see from the gure that the medium and narrow pore throat geometries dier from that of the wide pore throat geometry (green curve) in that they are aected by the shape of the pores. The decrease in magnesite concentration is reduced in the large pore body compared to the wide pore throat geometry without pores. This means that more chemical alterations take place in the large pores. If we calculate the Pe number in the pore body and in the pore throat, keeping in mind that the velocity is higher in the pore throat, we nd that the Pe number is ve times higher in the pore throat than in the pore body in the narrow geometry (blue curve). From the discussion in the previous section, we would then expect higher alteration in the large pore, as observed.
Figure 5.4: Normalized magnesite concentration pro les for simulations in the three sine tubes shown in Figure 5.1. The gray area represents a sine tube for easy comparison between the curves and the location of pore bodies and pore throats in the geometry.
62
5.3. Discussion
5.3 Discussion The location of mineralogical alterations have been studied as function of Peclét number, which is the ratio between advective and diusive transport. However, there is another parameter that is important for this system, namely the rate of chemical reactions. The ratio between reaction rate and transport rate is given by the dimensionless Damköhler number. It is believed that for lower Damköhler numbers the distribution of alterations might look dierent as the ow is limited by reaction rather than advection (residence time). The eect of the Damköhler number was not included in the above described study, because it was believed that in order to get a realistic picture of the dynamics, it would be necessary to include a moving boundary routine that changes the geometry as mass dissolved and precipitates. The work on implementing a moving boundary routine is described in the next chapter.
5.4 Summary & conclusions The localization of mineral alterations was studied as function of i) distance from inlet, ii) Peclét number and iii) pore geometry. It was found that more alterations take place near the inlet because of a higher supersaturation and subsaturation is found in the solution as it enters the geometry than further into the pores. When varying the Pe number, we in addition nd that a higher Pe number gives lower absolute alterations and a lower gradient through the geometry. This is explained by a lower residence time for the uid which will maintain a higher super/subsaturation further into the geometry. Simulations with three dierent sine tubes varying in size of pore bodies and pore throats showed that more alterations take place inside the large pore bodies compared to the narrow pore throats for the cases studied. It is discussed that an extended study should have included the eect of the Damköhler number, which is the ratio between chemical reaction rate and transport rate. However, it was decided to rather proceed with an implementation of moving boundaries, which will be important for the dynamic behavior of the system.
63
Chapter 6
Moving boundaries within the LBM 6.1 Introduction When dissolution/precipitation processes take place inside the pore space, the geometry of the pore space, and hence porosity and/or permeability of the core, might change. As the rate and location of chemical reactions depend on the pore space geometry, a model that captures the evolution of the pore space is necessary. To be able to predict the evolution of the pore space, a model in which the solid/ uid contacts can move as minerals dissolve and/or precipitate is needed. LB models with moving boundaries are found in the literature. Lallemand and Luo [2003] propose a model that describe solid objects moving in a uid phase. The model is based on a model by Bouzidi et al. [2001], where the bounce-back rule is combined with an interpolation scheme for boundaries that do not lie half-way between lattice nodes. The model proposed by Bouzidi et al. [2001] was applied to curved surfaces. Verhaeghe et al. [2005] use a linear kinetic boundary condition and interpolations/extrapolations to model evolution of a surface due to dissolution. This method is later updated with a ux boundary condition [Verhaeghe et al., 2006]. Arnout et al. [2008] build on the model by Verhaeghe et al. [2005, 2006], and approximate the true position of the interface with two simpli ed formulas. Kang et al. have proposed a LB reactive transport model that incorporates moving boundaries due to dissolution/precipitation [Kang et al., 2004, 2010].
64
6.1. Introduction We have experienced that the above-mentioned models are aected by the underlying mathematical grid. The reason is either that they don't include a sub-grid resolution of the surface (Kang et al.), that they don't use information about the position and/or direction of the surface (Verhaeghe et al.), and that they don't take into account the eect of surface curvature, which is important especially for growth phenomena, such as precipitation. It is important to have a model that is not aected by the underlying mathematical grid in order to give robust results. For mineral growth, it is a challenge to avoid unstable growth, which can lead to errors in surface area and pore volume, and hence give errors in porosity and permeability estimates. There are several ways to estimate an interface position on a xed grid. Examples of some commonly used methods are volume of uid (VOF) methods and level set (LS) methods. The VOF method has been widely used to simulate two-phase and free surface ows since it was introduced by Hirt and Nichols in 1981 [Hirt and Nichols, 1981]. VOF is very good in its mass conserving properties, but it lacks accuracy of surface normal and curvature calculations due to discontinuous spatial derivatives of the volume fraction eld near the interface. In the LS method [Osher and Sethian, 1988], the surface normal and curvature can be easily and accurately calculated, and is easy to implement in both two and three dimensions. However, the LS method don't have a good mass conservation property, due to numerical dissipation. In Paper 1, Paper 2 and Paper 3 we study three dierent moving boundary routines within the LBM. All methods are based on the boundary condition proposed by Bouzidi et al. [2001], Verhaeghe et al. [2006], but dier in whether i) a surface normal is estimated and ii) a sub-grid location of the uid-solid boundary is calculated. We believe that a good moving boundary method should conserve mass, converge for increasing grid resolution, and be rotational invariant. In the most complex moving boundary routine, we couple the LBM to a VOF interface tracking method. In Paper 1, we use the simplest of the three methods with the LB simulator described in chapter 4 to simulate experimental data from a core ooding tests when the uid-solid interface is allowed to move. In Paper 2, we study how the porosity, surface area and permeability evolves around a seed growing from supersaturated solution for dierent reaction limiting processes. In Paper 3, we study how the initial orientation of a solid seed on the underlying mathematical grid aects the growth pattern 65
Chapter 6. Moving boundaries within the LBM due to precipitation from a supersaturated solution, in order to test the rotational invariance of the three methods.
6.2 Method 6.2.1 The moving boundary methods The three moving boundary methods are all based on the boundary condition given by eq. 2.22:
gα¯ =
1 − kα 6ωα ξκ kα 2ωα ceq + g˜α − eα¯ · n. 1 + kα 1 + kα 1 + kα
This boundary condition is valid for an interface that is half-way between a uid node and a solid node. However, when the uid-solid boundary moves on a static grid, the interface distance to the nodes will vary with time. The three methods are described below.
Method 1
The uid-solid interface is assumed to always lie half-way between the uid and solid nodes, i.e. the interface follows the grid lines. Hence, in a quadratic grid the interface must be described by horizontal and vertical line segments, giving a staircase geometry. When the boundary moves, it moves with a step equal to the grid spacing, δx. We assume that eα¯ · n = 1 for all α.
Method 2
The surface normal is estimated using a volume of uid interface tracking routine, while the position if it, as in Method 1, is assumed to be half-way between the uid node and the solid node.
Method 3
Both the surface normal and the position of the interface is estimated using the VOF method. An interpolation scheme, as described by Bouzidi et al. [2001], is used to obtain the unknown distribution functions in uid nodes with neighboring solid nodes.
6.2.2 Volume of uid approach A VOF approach was chosen for the estimation of interface position and surface normal for use in the ux boundary condition (2.22). VOF is one kind of interface tracking procedure, where the surface is reconstructed from a scalar volume fraction eld. In our usage the scalar eld is the solid fraction (σ ), giving the volume fraction of solid in each computational cell. 66
6.2. Method In every iteration step the solid fraction is updated in the boundary cells according to the total microscopic ux into the solid phase; ! X ∆σ = Vm−1 (6.1) gα¯ − g˜α , links
where Vm−1 accounts for the dierence in densities between the uid and the solid phase. The VOF procedure takes σ as input and estimates interface normal and position. These properties are then used in interpolation routines to estimate distribution functions close to the interface (o-grid), and in the boundary condition (n). A 2D version of the E cient Least Squares VOF Interface Reconstruction Algorithm (ELVIRA)[Pilliod Jr. and Puckett, 2004] has been implemented to simulate mineral growth. The ELVIRA method is a piecewise linear interface calculation (PLIC) procedure; i.e. the interface is reconstructed by linear segments in each boundary cell. What separates the dierent VOF PLIC algorithms is how the interface normal is calculated from the volume fraction eld. In the ELVIRA algorithm this is done by rst estimating six candidate slopes from backward, central and forward dierences of the column sums in the x- and y-direction for each boundary cell;
m ˜ xb =
1 X
l=−1
m ˜ xc =
σi,j+l − σi−1,j+l ,
1 1 X σi+1,j+l − σi−1,j+l , 2
(6.2) (6.3)
l=−1
m ˜ xf =
1 X
l=−1
m ˜ yb
=
1 X
l=−1
m ˜ yc
=
σi+1,j+l − σi,j+l ,
(6.4)
σi+l,j − σi+l,j−1 ,
(6.5)
1 1 X σi+l,j+1 − σi+l,j−1 , 2 l=−1
67
(6.6)
Chapter 6. Moving boundaries within the LBM m ˜ yf
=
1 X
l=−1
σi+l,j+1 − σi+l,j ,
(6.7)
where the m ˜ y slopes (6.5-6.7) are with respect to the coordinate system rotated 90o from the original coordinate system. A linear function y = mx ˜ + b is then estimated to each of the six candidate slopes from the correct solid fraction in the cell. From the six candidate linear functions we choose the one that yields least error in estimated solid fraction in the eight neighboring cells (6.8); s X E= (σ − σ est )2 . (6.8) neighbors
The estimated solid fractions (σ est ) are found by extrapolating the linear function in cell (i, j ) into the nearest neighbor cells (8 cells in the D2Q9 discretization) and calculating the solid fractions from the volume fractions on the solid side of the interface.
6.2.3 Interpolation scheme The boundary condition (2.22) can be updated to hold for any position of the uid-solid interface by the use of interpolation. The interpolation routine is somewhat dierent for the cases q < 1/2 and q > 1/2. No interpolation is needed for q = 1/2. The position of the surface is given by the distance from the surface to the uid node closest to the surface (fn) along each discrete direction, qα . We follow the procedure described by Bouzidi et al. [2001] and introduce a point P such that for qα < 1/2 the distance from P to the surface and back to fn is unity (see Fig. 6.1), and for qα > 1/2 the distance from fn to the surface and back to P is unity (see Fig. 6.1). For qα < 1/2 we rst interpolate to nd the outgoing distribution at point P (6.9), then this distribution undergoes the BC and travels to fn (6.10);
gα (P ) = gα (fn, t) + (1 − 2qα ) [gα (fn + eα¯ , t) − gα (fn, t)] , (6.9) kα 1 − kα 6ωα ξκ gα¯ (fn) = 2ωα ceq + gα (P ) − eα¯ · n (6.10) 1 + kα 1 + kα 1 + kα
68
6.3. Some details For qα > 1/2 the distribution at fn undergoes the BC and travels to point P (6.11), then we interpolate to nd the new distribution at fn (6.12);
kα 1 − kα 6ωα ξκ 2ωα ceq + gα (fn) − eα¯ · n, (6.11) 1 + kα 1 + kα 1 + kα 2qα − 1 gα¯ (fn) = gα¯ (P ) + [gα¯ (fn + eα¯ ) − gα¯ (P )] . (6.12) 2qα gα¯ (P ) =
Figure 6.1: Sketch over symbols and distances used in the interpolation routine. Cells are marked with black lines and nodes by crosses. The length of the two arrows is 1.
6.3 Some details Distance function
When surface normal and position has been calculated by the VOF routine, the lattice boundary condition can be performed. Some details are not included in Paper 3. For example, since there is no constraint that the reconstructed VOF surface (the collection of linear segments) is continuous, there will sometimes be linear segments in two neighboring cells along one lattice link (see gure 6.2). In order to calculate the distance from node to interface (q ), a test is implemented to check which line segment is met rst along the link from the node to the interface. This is a straightforward procedure, and the three dierent options are shown in gure 6.2 (a)-(c).
69
Chapter 6. Moving boundaries within the LBM
Figure 6.2: Examples of calculation of the distance from node to wall when two neighboring cells along a link contain a linear segment. (a) Wall is met in the rst cell (q < 1/2). (b) Wall is met in the second cell (q > 1/2). (c) Wall is not de ned by any of the line segments (q = 1/2).
Full or empty cells Boundary cells usually have 0 < σ < 1. However sometimes the interface is exactly midways between two nodes (q = 1/2) and the boundary cell has σ = 1. The update in solid fraction, σ → σ + ∆σ , is performed in the solid cell (σ ≥ 1/2) or in the uid cell (σ < 1/2) depending on the sign of ∆σ . If ∆σ ≥ 0 we have precipitation, and the update is performed in the solid cell if σ < 1 in this cell, else it is updated in the uid cell. If ∆σ < 0 we have dissolution, and the update is performed in the solid cell if σ = 0 in the uid cell, else it is updated in the uid cell. Despite this splitting of updates, one can experience that the solid fraction exceeds 1 or falls below 0 during one iteration. Therefore a redistribution routine has been implemented. If σ > 1 the excess solid (σ − 1) is redistributed among the neighboring uid cells, weighted according to the orientation of the link compared to the surface normal (eα · n). This way cells with links to the wall that are normal to the surface will be given more of the excess solid than cells with links inclined compared to the surface, motivated from the ux concept. For cells with σ < 0 solid is taken from the neighboring solid cells and redistributed into the uid cell 70
6.4. Results according to the same weighting as for σ > 1.
6.3.1 Curvature calculation The curvature of the interface is needed in the surface tension term in the boundary condition (2.22), and it is estimated using the HF method [Cummins et al., 2005]. In the HF method a height function is given by the sum of solid fractions along rows or columns. The sum is taken along the direction in which the component of the estimated surface normal is largest, i.e. if nx < ny the height function, HF, is de ned as
HF (i, j) =
j+3 X
σ(i, j).
(6.13)
j−3
The curvature, κ, is de ned in cell (i, j) if 3 < HF < 4. The curvature is then given by
κ= h
∂ 2 HF ∂x2
1+
i3/2 ∂HF 2 ∂x
.
(6.14)
where the derivatives are estimated using a central dierences scheme. In boundary cells with HF ≤ 3 or HF ≥ 4 we assign the curvature from the neighbor cell along a direct link with the same condition for the normal components as cell (i, j), i.e. if nx < ny in cell (i, j) the curvature in cell (i, j −1) or (i, j +1) is assigned to cell (i, j). If none of these two cells have a de ned curvature, the curvature from a cell along the other direction, in this case the x-direction, is assigned to cell (i, j).
6.4 Results 6.4.1 Evolution of solid fraction during synthetic seawater ooding In Paper 1 we have implemented Method 1 to the lattice Boltzmann simulator described in chapter 4. We present simulation results from a synthetic seawater ood through a sinusoidal geometry, initially consisting of 100% calcite. Calcite dissolves, while magnesite and anhydrite precipitate. The solid fractions of calcite, magnesite and anhydrite at two
71
Chapter 6. Moving boundaries within the LBM dierent times (t1 and t2) are shown in Figure 6.3. The time t1 represent the beginning of the simulation, while t2 is later, when anhydrite has almost clogged the tube entirely.
Figure 6.3: Solid fractions of calcite, magnesite and anhydrite from a simulation where synthetic seawater was ooded through a sinusoidal geometry initially consisting of 100% calcite. We see that there is not much change in the calcite distribution (Figure 6.3a-b). Changes in calcite solid fraction are only found in the rst layer of solid nodes. Since magnesite precipitation is dependent on available carbonate from calcite dissolution, the solid fraction of magnesite (Figure 6.3c-d) changes to the same degree as calcite. As magnesite precipitates, it will screen the calcite surfaces from the ow and hence the dissolution will stop. Precipitated anhydrite (Figure 6.3e-f) gives the most signi cant change to the geometry. At time t2, anhydrite has almost clogged the rst pore throat completely (Figure 6.3f). The total solid fraction is shown in Figure 6.3g-h. In the experiment, a total clogging of the core due to anhydrite precipitation was seen after 2 weeks [Madland et al., 72
6.4. Results 2011]. Although the time scale for the clogging is dierent because of dierent geometries and spatial scales, the qualitative behavior matches the behavior of the experiment well.
6.4.2 Dependency on the underlying mathematical grid comparison between Method 1, 2 and 3 As mentioned earlier, it is important to have a model that is independent on the underlying mathematical grid. We have studied how the three moving boundaries methods behave when a solid seed is rotated on the grid. The growth depends on whether the system is limited by surface reactions or transport of species to and from the surface. We use the same approach as [Kang et al., 2003] and study the growth as function of the dimensionless Peclét-Damköhler number, PeDa, de ned as the rate of reaction over rate of diusion;
kL , (6.15) D where k is the chemical reaction rate, L is the system length scale, and D is the diusion coe cient. PeDa =
(a) Method 1
(b) Method 2 and 3
Figure 6.4: Contours from constant ux growth. Initial circle in black. Black dotted curves represent perfect circles. In the limit of PeDa → 0, the ux onto the surface is constant. Therefore, we have tested how the three moving boundary methods behave in a constant ux system. In Paper 1, we have started the growth from a square that is either coincident with the grid lines or rotated 45o compared to the grid lines, and compared Method 1 and 2. In Paper 3, the study 73
Chapter 6. Moving boundaries within the LBM was extended to cover also Method 3, and the initial seed was changed to a circle so that the surface don't coincide with lattice directions or grid lines. This should be a better test than a square on a square lattice. Figure 6.4 shows countors for the position of the surface (solid fraction of 1/2) for the three methods. Method 2 and 3 give the same results since the ux is constant and no interpolation is needed. These methods give an almost perfect growing circle, while Method 1 gives some deviations that increase as the circle grows larger. In Paper 2 and 3 we studied the growth of the initial seeds presented in Figure 6.5 for varying PeDa. The interfaces of the seeds are given by [López et al., 2010],
x(s) = [0.1 + 0.02cos(8πs)] cos(2πs),
(6.16)
y(s) = [0.1 + 0.02cos(8πs)] sin(2πs),
(6.17)
and dier only in the orientation on the underlying mathematical grid. The three seed are rotated respectively 0o , 19o and 45o on a quadratic grid.
(a) 0o
(b) 19o
(c) 45o
Figure 6.5: Initial seeds used in the surface growth simulations. The seed in a) is given by eq. 6.16-6.17, while the seeds in b) and c) are rotated with respect to a). Figure 6.6 shows simulation results for PeDa = 10 and PeDa = 100 obtained with the three models. Results from the three initial seeds are plotted together to compare the growth patterns for dierent orientations on the grid. Blue contours are from the initial seed rotated 0o on the grid, black contours from that rotated 19o and blue from the seed rotated 45o on the grid. It is clear that the discrepancies between the simulations 74
6.4. Results with dierent orientations of the initial seed is smaller for low PeDa. All three methods are more or less independent of the orientation of the initial seed for PeDa = 10. Method 3 shows less discrepancies between the initial seeds for PeDa = 100, but all the methods give unstable growth at scales larger than the grid resolution. To improve the behavior of the growth at high PeDa, we included a surface tension term. Figure 6.7 shows countors obtained with the three models when surface tension is included in the boundary condition. All method gives less discrepancies between the seeds, but Method 3 stands out as the best method.
(a) Method 1, PeDa=100
(b) Method 2, PeDa=100
(c) Method 3, PeDa=100
(d) Method 1, PeDa=10
(e) Method 2, PeDa=10
(f) Method 3, PeDa=10
Figure 6.6: Simulated interface positions for PeDa = (10, 100) using method 1-3. Blue, red and black contours respectively represent seeds oriented 0o , 45o , and 19o with respect to the grid, see Fig. 6.5. The contours are rotated until the initial seeds overlap in order to show the deviation between contours of dierent methods, which indicate that the underlying mathematical grid aects the results.
75
Chapter 6. Moving boundaries within the LBM
(a) Method 1, PeDa=100
(b) Method 2, PeDa=100
(c) Method 3, PeDa=100
Figure 6.7: Results from simulations with surface tension (ξ = 0.01) for method 1-3 with PeDa=100. Blue: 0o , Red: 45o , Black: 19o (only in c). For each method, three simulations have been performed with varying orientation of the initial seed (see Fig. 6.5) for PeDa=100 and surface tension strength ξ = 0.01. The simulated contours have then been rotated until the initial seeds overlap, in order to show the deviation between the contours at later times.
0.5
0.07
method 3, γ=0, 0 vs 45 method 3, γ=0, 0vs 19 method 2, γ=0, 0 vs 45 method 1, γ=0, 0 vs 45
0.4
e
0.05
e
0.3
0.06
method 3, γ=0.01, 0 vs 45 method 3, γ=0.005, 0vs 45 method 2, γ=0.01, 0 vs 45 method 1, γ=0.01, 0 vs 45
0.04
0.2 0.03 0.1
0.02
0 0
10
20 3
30
40
0.01 0
50
10
20
30
40
50
60
103 timesteps
10 timesteps
(a) No surface tension ξ = 0
(b) With surface tension ξ = 0.01
Figure 6.8: Relative error (e) between contours. Figure 6.8 shows the relative errors between contours from the dierent shapes. For example, the error between the 0o shape and the 45o shape is calculated as the area between the two shapes divided by the area of the 0o shape. The area between the two shapes is given by 2Ac − (A0 − Arot ), where Ac is the area of the composite shape from the two shapes, A0 is the are of the 0o shape and Arot is the are of the 0o shape. Figure 6.8a shows relative errors for PeDa = 100 when surface tension is not included. 76
6.4. Results The relative error starts at approximately 5% for all the methods and increases as the shape grows, to approximately 40% for Method 1 and 2 and 30% for Method 3. Figure 6.8b shows the relative error for PeDa = 100 when surface tension is included (γ = 0.01). The error is reduced from approximately 40% to 6% for Method 1 and from 30% to 2% for Method 3. This veri es that Method 3 behaves better than Method 1 and 2 for PeDa = 100. For low PeDa one of the simpler methods could be considered to save computational cost, as the errors are less for low PeDa.
6.4.3 Porosity and permeability evolution around a growing seed In Paper 2, we calculated the evolution of porosity, permeability and speci c surface area of the growing shapes at dierent PeDa. Permeability was related to porosity through the Carman-Kozeny relation. Figure 6.9a shows the porosity evolution. The porosity is calculated as the area of the surrounding circle divided by the area of the solid shape. The results show that the porosity drops faster for increasing PeDa. The surface area is calculated as the length of the solid- uid interface. Clearly this increases with increasing PeDa, as the growth gets more unstable for higher PeDa. The evolution of surface area is shown in Figure 6.9b. The permeability decreases at the shapes grow and take up more space inside the circular pore. The permeability evolution is shown in Figure 6.9c. It shows that the permeability drops faster for higher PeDa. The surface area increases and the permeability decreases with an increasing rate as more details grow on the solid surface. This is clear from the changing slopes seen for PeDa = 104 .
77
Chapter 6. Moving boundaries within the LBM 1
0.14
0.9
0.12
Specific surface area (SSA)
PeDa = 104
Porosity
0.8 0.7 0.6 0.5
PeDa = 104
0.4
PeDa = 102 PeDa = 10
0.3 0
1
2
3 Timestep
4
2
PeDa = 10 PeDa = 10
0.1 0.08 0.06 0.04 0.02 0 0
5
1
2
4
x 10
(a)
3 Timestep
4
5 4
x 10
(b) 1 4
PeDa = 10 PeDa = 102 PeDa = 10
0.8
k/k
0
0.6
0.4
0.2
0 0
5
10 Timestep
15 4
x 10
(c)
Figure 6.9: (a) Porosity, (b) speci c surface area, and (c) permeability development around a growing seed for varying PeDa.
6.5 Summary & Conclusions We have presented three methods that can be used to move uid-solid interfaces within the LBM. In the most complex method, the LBM is coupled to a VOF interface tracking routine, which estimates the position of the surface and its normal. This method is the least dependent on the underlying mathematical grid, but a surface tension term is needed to decrease the relative errors between shapes grown from seeds that were rotated relative to each other, from 30% to 2%, for high PeDa systems. However, for small PeDa, the growth is more stable and one of the simpler methods can be used in order to save computational time. When the simplest method is used to simulate synthetic seawater ooding in a sinusoidal geometry, the tube clogges at the rst pore throat due to anhydrite precipitation. Because of the dierent time scales and geometries in the simulations and in the experiment, the results don't 78
6.5. Summary & Conclusions compare quantitatively, but a clogging due to anhydrite was seen in the experiment after two weeks. The dissolution of calcite in the simulation stops as magnesite has lled the node layer in contact with the water, and screens the ow from the calcite surface. A model that allows dissolution further into the solid matrix is necessary to explain full transformation from calcite to magnesite as has been seen near the inlet of a Liége chalk core ooded with MgCl2 for 516 days [Zimmermann et al., 2015]. Porosity, speci c surface area and permeability development around the growing seeds for varying PeDa show that the porosity decrease and the speci c surface area increase, accelerates as more details start to grow on the solid surface for high PeDa. The curve is more smooth for low PeDa, when the growth is more stable.
79
Chapter 7
Reactive surface area models 7.1 Introduction In standard dissolution rate equations, the dissolution rate is proportional to the reactive surface area, i.e.;
A k(1 − Ω)η , (7.1) V where A is the reactive surface area, V is the pore volume, k is the rate constant, Ω is the saturation state, and η is a constant describing the order of the reaction [Morse and Arvidson, 2002]. In measurements of the calcite dissolution rate, the surface area are sometimes assumed to be constant. The assumption can be veri ed by a constant dissolution rate throughout the experiment [see e.g. Plummer et al., 1978]. However, for experiments run over a longer time period, the evolution of the surface area can be signi cant for the overall dissolution rate. In a recent experiment where a Liége chalk core was ooded with MgCl2 for 516 days, the composition of the core near the inlet changes almost completely from calcite to magnesite, due to calcite dissolution and magnesite precipitation [Zimmermann et al., 2015]. In this case it is clear that the calcite surface area should be important to the overall dissolution rate as it approaches zero. rate =
80
7.2. Arti cial microporosity to increase reactive surface area
7.2 Arti cial microporosity to increase reactive surface area In Paper 1 we introduce a microporosity by slicing through the geometry, making narrow channels (one pixel wide) in the solid which increases the reactive surface area of the geometry. Simulations are compared to experimental e uent data from two experiments; one ooded with MgCl2 and one ooded with synthetic seawater (SSW) [Madland et al., 2011]. These data are shown in Figure 7.1. The data shows a peak in the calcium concentration followed by a transient period where the concentration decreases slowly. The initial peak can be captured by using a relatively high ion exchange capacity, but the slow transient behavior has been hard to explain [Andersen et al., 2012, Hiorth et al., 2013]. LB simulations were performed in a sinusoidal tube used to represent a simple pore geometry. In addition, simulations were performed in the same geometry, only with narrow channels perpendicular to the ow direction to increase the reactive surface area. The two geometries are shown in Figure 7.2a and 7.2b. Figure 7.2c and 7.2d show simulated data from simulations in both the geometries. Data from the geometry without micropores are represented by solid circles, while data from the geometry with micropores are represented by open squares. The injected concentrations are given by the diamonds on the right axis. Simulations with both geometries produce an initial peak in the calcium concentration. Further, for the MgCl2 ooding, the micropore geometry gives a peak that is two times higher than without micropores. This is expected from the proportionality to the reactive surface area in the dissolution rate equation. However, when ooding with SSW, the hight of the peak dier to a much smaller extent. Instead, the peak is extended in the temporal direction. One possible explanation of this, as presented in the paper, is that micropores are blocked by anhydrite in the SSW ooding. Paper 1 also describes a moving boundary routine, and simulation data are presented that show that anhydrite will block pores (see chapter 6 for more details on the moving boundary routine). Since we perform pore scale simulations, the simulated data cannot be quantitatively compared to the e uent data from the core oods. However, the results show good qualitative agreement between simulations and experiments. We conclude that the initial peak in the calcium e uent data can hold information about microporosity, while the following slow transient give information about the kinetics of precipitating miner81
Chapter 7. Reactive surface area models als in the core.
(a)
(b)
Figure 7.1: E uent pro les from a Liége chalk core ooded with (a) 0.219 M MgCl2 , and (b) synthetic seawater with 0.445 M Mg2+ , 0.125 M Cl , 0.130 M Ca2+ , 0.024 M SO42 , 0.01 M K+ and 0.050 M Na+ . The Cl concentration is scaled to t into the plot for the seawater case.
(a)
(b)
(c)
(d)
Figure 7.2: Sinusoidal geometry (a) without micropores, and (b) with micropores, and simulated e uent data using both geometries for (c) MgCl2 ooding and (d) synthetic seawater ooding.
82
7.3. Mapping of actual reactive surface area onto a cylinder
7.3 Mapping of actual reactive surface area onto a cylinder Motivated from the results presented in paper 1, in Paper 4 we present a model that maps the reactive surface area of a core onto a cylinder which can be used in LB simulations. This way, the LB simulations, performed over a relatively short time frame, can be quantitatively compared to data from core ooding experiments. We use a cylinder where we distribute the mass of the physical core in one voxel layer, as seen in Figure 7.3. Since the LB area is given by the geometry, we use a rescaling factor, rs , in the rate equation for each mineral, j , to obtain the same rate as in the experiment;
rs,j =
Aplug,j , Alb,j δx2
(7.2)
where Aplug,j is the reactive surface area of mineral j in the physical core plug, Alb,j is the LB area of mineral j and δx is the step length (length of one voxel).
Figure 7.3: The geometry used to map the physical core data. The entire core mass is distributed in one voxel layer. The dimensions of the cylinder is 40x9x9. The volume within the cylinder also diers from the pore volume in the physical plug. In the proposed model, we keep the geometry, and hence the volume constant. Thus we only need to account for the dierence in pore volume in the initialization. The mole number of mineral j in each wall node is initialized using the following formula:
mlb,j wn,init =
Aplug Alb wn Vlb δx , lb Atot Vplug Sj Mjmol δcδx3 83
(7.3)
Chapter 7. Reactive surface area models lb where Alb wn is the LB area of wall node wn, Atot is the total LB area of the cylinder, Vlb is the LB volume of the cylinder, Vplug is the physical pore volume of the core, Sj is the speci c surface area of mineral j [m2 /g], Mjmol is the molar weight of mineral j [g/mol] and δc is the concentration step length in the simulations, which is set to unity.
7.4 Overgrowth model 7.4.1 Mathematical model The mapping of reactive surface area described above is used in Paper 4 together with a proposed overgrowth model to describe the evolution of the reactive surface area during dissolution and precipitation. The overgrowth is a covering of primary calcite surfaces by precipitating magnesite crystals, that will reduce the reactive surface area of calcite as magnesite precipitates, and hence reduce the dissolution rate. The overgrowth model is motivated from previous SEM investigations of MgCl2 ooded cores, and can describe scenarios from no overgrowth, where the calcite surface area is unaected by the precipitating magnesite, to full overgrowth, where magnesite forms as a monomolecular layer on the calcite surface. The model is used to explain the long term behavior of a Liége core ooding test lasting for 1072 days [Nermoen et al., 2014]. We have included three terms in the calcite dissolution rate and two terms in the magnesite precipitation rate:
∆mcal wn ∆t ∆mmag wn ∆t
= (Adef + kcov Acov + kslow Aslow ) J cal ,
(7.4)
= (kr Adef + Amag ) J mag ,
(7.5)
mag where mcal wn and mwn are the mole numbers of calcite and magnesite in wall node wn, Adef is the defect area on the calcite surface which is associated with the standard calcite dissolution rate constant kcal (part of J cal ) and Acov and kcov are the calcite area covered by magnesite and the dissolution rate constant associated with the covered area. We allow for kcov > 0 to include transformations where the primary mineral is exchanged by a denser mineral and lead to microporosity, which is the case for calcite to magnesite transformation. Further, Aslow and kslow are
84
7.4. Overgrowth model the remaining calcite area (no defects and not covered by magnesite) and the corresponding dissolution rate constant. Note that kcov and kslow are the values of the rate constants relative to the standard dissolution rate constant, kcal . Both kcov and kslow are assumed to be less than one. In the magnesite precipitation rate equation, kr is the rate constant for magnesite precipitation onto calcite relative to than onto magnesite (kmag which is part of J mag ). Magnesite is only allowed to precipitate on the defect parts of the calcite surface. Amag is the surface area of magnesite associated with kmag . Finally, J cal and J mag are the dissolution and precipitation uxes. We have used a rate law suggested by Morse and Berner [1972], Steefel and Van Cappelen [1990] and Lasaga [1998], where the ux of mineral j is given by: η J j = sgn(1 − Ωj )(k1j + k2j aH ) 1 − Ωζj . (7.6)
Ωj is the saturation state of mineral j , k1j and k2j are rate constants. k2 is only important in acidic solutions, while in the experiment studied in Paper 4 the pH is always above 6.5, hence we have neglected k2j in the simulations. kcal and kmag described above are used for k1 for calcite and magnesite, respectively. ζ and η are constant exponents indicating the order of reaction. The dierent surface areas are initialized as follows: mol cal Ainit = S cal Mcal minit dadef , def
Ainit cov init Aslow Ainit mag
(7.8)
= 0, = S
cal
(7.7)
mol cal Mcal minit (1
= 0,
− dadef ),
(7.9) (7.10)
where dadef is the defect fraction of the calcite surface. The changes in surface areas due to dissolution and precipitation are given as:
∆Adef
= −∆Acov + dadef δAcal ,
∆Acov = fmc amag ∆mcov , ∆Aslow = ∆Amag =
cal
mol S Mcal ∆mslow , mol ∆Acov + S mag Mmag
85
(7.11) (7.12)
(7.13) mag ∆mmag + (1 − fmc )∆mcov ,(7.14)
Chapter 7. Reactive surface area models where fmc is a geometrical factor describing 3D growth (zero if all mass accumulates i one point, and 1 if the mass distributes as a monolayer), amag is the area occupied by one mole of magnesite on the calcite surface, ∆mcov = kcov Acov J cal ∆t, and ∆mmag mag is the change in magnesite area due mag ∆t). to magnesite precipitation onto magnesite (∆mmag mag = Amag J
7.4.2 Results In Paper 4, the above overgrowth model is used to explain data from a core ooding test where a Liége chalk core was ooded with MgCl2 for 1072 days [Nermoen et al., 2014]. The experimental e uent data are shown in Figure 7.4. The ooding rate was 1 PV/day in interval I and III, and 3 PV/day in interval II and IV. The injected concentrations are indicated by the horizontal dotted lines for each species. 1 Cl− Mg2+ Ca2+
0.8
0.2 0.6
0.4 0.1
I 0 0
II 200
III 400
600 days
IV 800
1000
Cl− concentration [mol/L]
Ca2+, Mg2+ concentrations [mol/L]
0.3
0.2
0 1200
Figure 7.4: Experimental e uent pro les from a Liége core ooded with 0.219 M MgCl2 for 1072 days. The ooding rate was 1 PV/day in interval I and III, and 3 PV/day in interval II and IV. The dotted lines indicate the injected concentrations. We show that with literature rate constants with values kcal = 3.88 · mol/m2 /s [Plummer et al., 1978] and kmag = 2.74 · 10−11 mol/m2 /s [Saldi et al., 2009] and no overgrowth, we only obtain a good match to the rst week of the experiment. Even with reduced rate constants to improve the t, the no-overgrowth limit of the model in inadequate to
10−6
86
7.4. Overgrowth model describe the experimental data. In the opposite limit of full overgrowth, the entire calcite surface is covered by magnesite after only 2.5 days, and the dissolution/precipitation process stops. As described in Paper 4, by optimizing the model parameters, we obtain a better description of the experiment, but not satisfactory. However, by simulating only interval III and IV we obtain a very good match to the experimental data when the evolution of the rate is dominated by overgrowth, i.e. reduction in the calcite surface area due to magnesite coverage; see Figure 7.5. We performed a sensitivity analysis towards the initial distribution of calcite and magnesite in the core, and found that the results depend only to a little extent on the initial distribution. However, the nal distribution of magnesite in the core seems to be homogeneous independent of the initial distribution. A homogeneous distribution of magnesium was also found by SEM-EDS measurements of the ooded core after 1072 days, which indicates a homogeneous distribution of magnesite. 0.08 exp sim
concentration [mol/L]
0.07 0.06 0.05 0.04 0.03 0.02
III
0.01 0 0
200
400
IV 600 days
800
1000
1200
Figure 7.5: Simulated e uent pro le from the start of interval III, when overgrowth is the dominating mechanism in the dissolution rate equation. The values of the model parameters are: kr = 1, dadef = 1 (⇒ Aslow = 0), fmc = 1, kcov = 10−3 , amag = 500, kmag = 10−9 and kcal = 1.8 · 10−10 . In the paper, we discuss several limitations to the model that can possibly explain the poor match to the experimental data in interval II; i) The core compacted 10% in the axial direction during the experiment. 80% of the compaction took place in interval I and II. Compaction is not part of the LB model. ii) We assume initially 100% calcite in the 87
Chapter 7. Reactive surface area models simulations. Liége chalk typically consist of 91-95% CaCO3 [Hjuler and Fabricius, 2009, Megawati et al., 2013, Zimmermann et al., 2015], which means that other components might be important to the overall behavior. iii) We use bulk saturation indices as an approximation to the value at the mineral surfaces. This is believed to be a good approximation when the dissfusion is fast, as in this experiment. iv) The pore volume is kept constant in the simulations. However, the porosity of the plug only changes from 41.32% to 40.02% from the beginning to the end of the experiment, and hence this is believed to be a good approximation. To match the experimental data at the start of interval III, we used a calcite dissolution rate constant that is four orders of magnitude lower than the literature value (from bulk powder experiment) of 3.88 · 10−6 mol/m2 /s. This might be explained by all easily dissolving sites on the calcite surface already being dissolved. Measurements of the dissolution rate performed with atomic force microscopy (AFM) and vertical scanning interferometry (VSI) on cleaved calcite surfaces give rates that are up to two orders of magnitude lower that that from bulk powder experiments. It has been suggested that this discrepancy is due to a higher step density in powders than on cleaved surfaces [Dove and Platt, 1996]. In addition, it has been shown that Mg2+ ions can have an inhibitory eect on calcite dissolution by adsorbing on the surface and acting as barriers to step retreat [see e.g. Xu and Higgins, 2011, and references therein].
7.5 Conclusions From the studies on reactive surface area, we conclude that this might be an important factor in the dissolution rate evolution in chalk core ooding experiments at both short term and long term. On short term, the existence of micropores that increase the total surface area of calcite substantially, can be important to describe the peak in the calcium e uent pro le after only a few days of ooding. On a longer term, the surface area evolution is important to obtain a more or less complete transformation from calcite to magnesite in MgCl2 oods. Overgrowth, i.e. the reduction in reactive calcite surface area due to coverage of precipitated magnesite, seems to be important to describe the trends in the e uent data.
88
Chapter 8
Use of LBM to model convection caused by CO2 diusion into water 8.1 Introduction In Paper 5, we use the LBM to simulate an experiment where CO2 diffuses into water and causes convective mixing. The process of CO2 -water mixing is important to e.g. CO2 injection into hydrocarbon reservoirs to increase oil recovery, or to the storage of CO2 into deep saline aquifers to reduce emission to the atmosphere. Simulations of CO2 -water mixing have been performed both in a PVT cell [Yang and Gu, 2006, Farajzadeh et al., 2009] and in porous media [see e.g. Chen and Zhang, 2010]. We study a system similar to that of Farajzadeh et al. [2009]. Farajzadeh et al. [2009] used the Finite Element Software package Comsol Multiphysics, while we use the lattice Boltzmann method, which has not been used before for this particular system. We study the mixing process in a transparent PVT cell, where not only the pressure drop in the gas phase is measured, but the convective ow pattern is also observed by the use of a pH indicator in the water phase. The pH indicator gives us an additional observation that can be compared to simulation results in order to verify the model. In addition to simulating the experiment in two dimensions, we study the information that can be obtained by using dierent pH indicators.
89
Chapter 8. Use of LBM to model convection caused by CO2 diusion into water
8.2 Experimental setup The experimental setup is shown in gure 8.1. A 20 cm long cylindric transparent acrylic PVT cell was placed in a water bath for temperature control. The water bath was a rectangular box with four lateral sides made of glass. In addition to temperature control it also gave good optical conditions and corrected the refraction of light caused by the PVT cell curvature. The water bath is represented by the dashed line (black) in Figure 8.1. The temperature in the water bath was held constant at 21o C by use of a refrigerated/heating circulator model F34-HL (Julabo Laborthecnik GMBH). Temperature was measured with a PT-104 thermometer connected via a PT-100 monitor from Picotech to the monitoring computer using a RS232 interface.
Figure 8.1: Sketch of the experimental setup used in the experiment. De-ionized water was pumped into the cell through the bottom valve and at a low rate to avoid air bubbles. Water was lled to the top of the cylinder before the injection of CO2 was initiated. CO2 (99.99% pure CO2 ) was injected from the top valve at a low pressure in order to avoid reaction with water, while water was drained o to reach the targeted water level (hw ). CO2 pressure was then increased to 5 bar. Pressure was measured with a Rosemount 3051C absolute pressure transmitter through pressure taps on top of the PVT cell. The range of the pressure transmitter is 0-20.684 bar, the accuracy is 0.065% of full range and the response time is 100 msec. 90
8.2. Experimental setup Brom Thymol blue pH indicator, was added to the water to serve as a contrast agent for local determination of CO2 concentration in the water. The measured pressure decay is shown in Figure 8.2, while pictures showing pH pro les taken during the experiment is shown in Figure 8.3.
Figure 8.2: Measured pressure decay.
Figure 8.3: pH pro les captured during the experiment.
91
Chapter 8. Use of LBM to model convection caused by CO2 diusion into water
8.3 Simulation setup We have performed LB simulations of the water phase in two dimensions, and compared the pressure decay curve and pH pro les to the experimental data. The height of the box equals the height of the water column from the experiment, and the width of the box is set equal to the diameter of the PVT cell used in the experiment. A quadratic grid is employed, and hence the ratio between grid points in the height (y) and width (x) direction equals the height to diameter ratio of the water column, i.e. 2.8. An instability was initiated in the middle of the box by an sinusoidal perturbation. The imposed instability is described in more detail in appendix B in Paper 5. From the dimensionless form of the Navier Stokes equation and the advection-diusion equation;
r ∂ u∗ Pr ∗ ∗ ∗ ∗ ∗ + (u · ∇ ) u = −∇ p + (∇∗ )2 u∗ + c∗ ng , ∂t∗ Ra r ∂c∗ 1 ∗ ∗ ∗ + u · (∇ c ) = (∇∗ )2 c∗ , ∗ ∂t P rRa
(8.1) (8.2)
we nd the dimensionless numbers that describe the system; the Prandtl number (Pr) and the Rayleigh number (Ra). They are given by:
Pr = Ra =
ν , D βg0 l03 , νD
(8.3) (8.4)
where ν is the kinematic viscosity, D is the diusion coe cient, g0 is the gravitational acceleration, l0 is the system length scale and the constant β is given by mol β = c0 MCO /ρ. 2
(8.5)
mol is the Here c0 is the equilibrium concentration of CO2 at 5 bar, MCO 2 molecular weight of CO2 and ρ is the density of water. The starred quantities in equations (8.1-8.2) are the dimensionless variables rescaled by
92
8.4. Results p the system length scale (l0 ), the time scale t0 = l0 /βg0 , water density (ρ) and the equilibrium concentration of CO2 at 5 bar (c0 ). The experimental value of the Pr and Ra numbers are Pr = 500 and Ra = 5 · 1010 . Simulations were performed for dierent Pr and Ra.Pr is varied in the simulations by varying the diusion coe cient, while Ra is varied by varying the gravitation acceleration.
8.4 Results 8.4.1 Comparison with experimental data Figure 8.4 shows pressure decay curves and their derivatives for simulations at dierent Pr and Ra, together with the experimental curve. Pr in the simulations is limited by computational e ciency to a value of 200. We found that a lower Ra than in the experiment (the blue curve in Figure 8.4) ts the data best. The value of Ra is a factor 240 lower than in the experiment. A higher Ra gives a pressure decay that declines too fast. In Figure 8.5 we see the concentration patterns of CO2 in the water phase for dierent times. We see that we get a mushroom instability that has almost reached the bottom of the container after 5.9 min. Several secondary instabilities form as time passes. As described in Paper 5, we explain the dierence in Ra by dierences in the rate of mass transfer between the gas and the water phase for two and three dimensions. In three dimensions the area where most of the transport between the gas and the water phase takes place, is in the stem of a mushroom instability. In two dimensions, we simulate only a cross section of the mushroom, which would correspond to a channel transfer area between the gas and the water phase in three dimensions. The ratio between the disk area and the channel area scales as ∼ L/r, where L is the length of the container and r is the radius of the mushroom stem. If this ratio equals the ratio in Ra between simulation and experiment, it predicts a mushroom stem diameter of 2.5 nodes. This ts well with the simulated data, where the mushroom stem span in the order of 1-10 nodes. The pH pattern from the experiment is shown in Figure 8.6. It shows that the rst mushroom instability causes large parts of the container to have reduces pH. The low pH water has almost reached the bottom of the container after 5.9 min. After 11.8 min, the low pH water is distributed more or less in the entire container, and before 17.7 min has passed, the whole container consist of low pH water. The simulated pH pro les don't 93
Chapter 8. Use of LBM to model convection caused by CO2 diusion into water match the pro les seen in the experiment exactly. This is believed to be caused by the sharp transition from high pH color to low pH color at pH 6 in the simulation results. Brom Thymol blue, used in the experiment, has a transition pH range that is 6-7.6.
94
8.4. Results 0 -0.1 -0.2
log(P/Pi)
-0.3 -0.4 -0.5 -0.6 -0.7 -0.8 0
10
20
30
40
50
60
70
80
60
70
80
t1/2 [min1/2] Experimental: Ra=5.0e10, Pr=500 Simulated: Ra=2.1e07, Pr=200 Simulated: Ra=2.1e08, Pr=200 Simulated: Ra=2.1e09, Pr=200
(a) Pressure decay curves 0.02 0
dlog(P/Pi)/d(t1/2)
-0.02 -0.04 -0.06 -0.08 -0.1 -0.12 0
10
20
30
40
50
t1/2 [min1/2] Experimental: Ra=5.0e10, Pr=500 Simulated: Ra=2.1e07, Pr=200 Simulated: Ra=2.1e08, Pr=200 Simulated: Ra=2.1e09, Pr=200
(b) Derivative of pressure decay curves
Figure 8.4: Pressure decay curves for (Pr, Ra) = (200, 2.1·107 ), (Pr, Ra) = (200, 2.1 · 108 ), (Pr, Ra) = (200, 2.1 · 109 ) and for the experimental results. Grid sizes were 150x420 for Ra = 2.1 · 107 , 300x840 for Ra = 2.1 · 108 and 900x2520 for Ra = 2.1 · 109 .
95
Chapter 8. Use of LBM to model convection caused by CO2 diusion into water
(a) t = (b) t = (c) t = (d) t = (e) t = (f) t = (g) t = (h) t = (i) t = 5.9 min 11.8 min 17.7 min 23.6 min 29.5 min 35.4 min 41.3 min 47.2 min 53.1 min
(j) t = (k) t = (l) t = (m) t = (n) t = (o) t = (p) t = (q) t = (r) t = 59.0 min 64.9 min 70.8 min 76.7 min 82.6 min 88.5 min 94.4 min 100.3 min106.2 min
(s) t = (t) t = 112.1 min118.0 min
Figure 8.5: Concentration patterns for Pr = 200 and Ra = 2.1 · 108 at dierent times.
(a) t = 5.9 min
(b) t = 11.8 min
(c) t = 17.7 min
Figure 8.6: pH pattern for Pr = 200 and Ra = 2.1 · 109 at dierent times. 96
8.4. Results
8.4.2 Information obtainable with dierent pH indicators The choice of pH indicator will determine what information that can be obtained from the visual observations during the experiment. Figure 8.7 shows pH pro les at two dierent times for two dierent transition pH of 4 and 6. The gures show that with a lower transition pH, more of the structure of the convective currents can be seen, especially at later times, when a higher transition pH will have resulted in a color change in the entire volume. A higher transition pH will however give better information on the rate of the chemical reactions taking place at the CO2 water interface, and the onset of instability.
(a) Transition pH 6. t=0.7 min.
(b) Transition pH 4. t=0.7 min.
(c) Transition pH 6. t=5.9 min.
(d) Transition pH 4. t=5.9 min.
Figure 8.7: pH pro les at times t=0.7 min and t=5.9 min for Pr = 200 and Ra = 2.1 · 109 , with transition pH 6 and 4. For better resolution (a) and (b) shows only the top water layer, while (c) and (d) show the full water column. Figure 8.8 shows the time at which the water will have fully changed 97
Chapter 8. Use of LBM to model convection caused by CO2 diusion into water color as function of transition pH. When the entire water volume has changed color, no more information can be obtained from visual observations. The y-axis has been scaled with the time scales for the dierent processes Costa [2002];
t0 = t/tν tν
= l02 /ν
tD = l02 /D tb = ub /(βg0 ) where tν is a viscous time scale, tD a diusive time scale and tb a buoyancy time scale. τb is the time needed for the acceleration βg0 to act in such a way as to induce a vertical velocity change ∆u = ub . In order for the Rayleigh number to take the form of equation (8.4), ub must equal the viscous velocity uν = ν/l0 . From the simulations it seem like the dimensionless time t0 = t/tν √ scales as t0 ∼ tD tb , which we interpret as the geometric mean of the diusive and buoyancy time scales. In Figure 8.8 we have also indicated the range of transition for some pH indicators. It can be used as a pointer to which indicator one would use in order to get the desired information from an experiment.
98
8.5. Summary & conclusions
14
t0/(τDτb)1/2
12
Screened methyl orange (second transition)
Pr200RaE09 Pr200RaE08 Pr200RaE06 Pr100RaE06 Pr10RaE06
10
8
6 Bromocreosol green 4
4.5
5 5.5 transition pH
Bromthymol blue 6
Figure 8.8: Time at which the entire water volume has changed color as function of transition pH.
8.5 Summary & conclusions We have simulated the mixing of CO2 into water using the lattice Boltzmann method in two dimensions, and compared to experimental data. The pressure decay is explained quite well with a Rayleigh number a factor 240 lower than its physical value. We explain this dierence with a dierence in the area through which CO2 is transported after onset of convection, in two and three dimensions. The rst instability is prominent after only a few minutes, and several secondary instabilities follow as time passes. Even though the simulations show that the entire volume of water has changed to low pH color, the concentration pro les show that convective currents are still ongoing after approximately 2 hours. The use of pH indicator in the experiment gives an additional observation to verify the numerical model. We have studied the information obtainable with the use of dierent pH indicators. With a pH indicator with a high transition pH, the best information about the rate of reaction at the CO2 -water interface is obtained, as well as about the onset 99
Chapter 8. Use of LBM to model convection caused by CO2 diusion into water of instability. However, with a pH indicator with a lower transition pH, more of the structure in the water phase can be seen at later times of the experiment. Our simulations suggest that the time at which the entire water volume has changed to low pH color scales as the geometric mean of the diusive and buoyancy time scales.
100
Chapter 9
Introduction to papers Paper 1: Pore Scale Modeling of Brine Dependent Pore Space Evolution in Carbonates
The work presented in this paper is based on simulations with the LB simulator with moving boundaries (section 4) on two dierent simpli ed pore geometries. The two geometries are sine tubes, only diering in that one of them in addition have perforated planes perpendicular to the ow direction, in order to increase the surface area in contact with the ow. The simulations were performed to get information on the eect of chemical reactions, reactive surface area and moving boundaries on geometry evolution and e uent pro les. E uent data from core ooding with MgCl2 and synthetic seawater (SSW) showed Ca-pro les with an initial peak followed by a transient period of decreasing Ca concentration. The initial peak can be captured by an ion exchange process with a relatively high ion exchange capacity [Hiorth et al., 2013, Andersen et al., 2012], but the transient behavior was not matched with this type of process. The simulation results presented in this paper compares qualitatively well to the experimental data, in that both the initial peak and the following transient period are captured. The conclusions from this paper is that the reactive surface area, altered by the perforated planes in this study, in uences the Ca-peak and that this peak potentially hold information about the initial reactive surface area of the ooded core material. Also it is proposed that the transient behavior can be partly or fully due to the dynamic change in reactive surface area caused by overgrowth of secondary minerals on the solid sur101
Chapter 9. Introduction to papers faces.
Paper 2: Pore Scale Modeling of Brine Dependent Permeability
In this paper we give a brief overview of methods for modeling dynamical boundaries (moving boundaries) within the lattice Boltzmann framework based on the chemical ux boundary condition [Bouzidi et al., 2001, Verhaeghe et al., 2006]. We have coupled the LB model to a volume of uid model that estimates the surface normal from a solid fraction scalar eld, and introduced an extra term to the boundary condition that accounts for surface tension. We show that the surface tension term is necessary in order to achieve a model that is independent of the orientation of the initial solid structure on the underlying mathematical grid for high Peclét-Damk'ohler numbers (PeDa). In the limit of low PeDa the system behaves as constant ux growth and the surface tension term might be excluded. In this limit it is also possible to make use of a simpler calculation routine that does not use the surface normal in order to enhance computational e ciency. From simple mineral growth simulations we present porosity and speci c surface area calculations, and we observe an increasing drop in porosity and increasing growth of the surface area as more and more lower level details appear on the surface as the mineral grows. Permeability is calculated with the Carman-Kozeny relationship, and it is observed that permeability can decrease extensively without signi cant porosity loss.
Paper 3: Improved Lattice Boltzmann Models for Precipitation and Dissolution
This paper presents a moving boundary method for dissolution and precipitation within the LB framework that is independent of the orientation of the underlying mathematical grid. A VOF method was used to estimate surface normal and position used in the boundary condition. Surface tension was implemented to get rid of numerical instabilities for diusion limited growth, reducing errors from approximately 20% to 2%. Reaction limited growth is more stable, and a routine without the surface tension can be used in trade for a more computational eective algorithm. The boundary condition is based on an existing model, but diers in that it is 102
coupled to a VOF model that estimates the surface position and normal. Also the existing model has only been used for diusion, which is a stable process in contrast to the precipitation process.
Paper 4: A dissolution model that accounts for coverage of mineral surfaces by precipitation in core oods
In this paper, we propose an overgrowth model where the dissolution rate of calcite is reduced due to surface area coverage of precipitating magnesite. In the model we allow for separate dissolution rate constants at easily dissolving sites, such as grain contacts and crystal dislocations, and at normal sites. Dissolution at sites covered by precipitated magnesite is also facilitated by a low rate constant. This can capture e.g. dissolution through a layer of microporous substance. Magnesite has a higher density than calcite, and can potentially create microporosity when calcite is transformed to magnesite through either ion exchange or a coupled dissolution/precipitation process. The model is used to interpret e uent data from a Liége chalk core ooded with MgCl2 for 1072 days. It is capable of capturing the trends of the e uent data after approximately one year, when the rate determining process in calcite dissolution. To match the simulations to the experimental e uent curve, magnesite must precipitate as larger crystals or aggregates of smaller size crystals, and not as a thin layer on the calcite surface. The model is also capable of predicting a homogeneous distribution of magnesite in the core after 1072 days, as seen in the experiment. A calcite dissolution rate constant four orders of magnitude lower than measured in bulk powder experiments was used to match the experimental e uent data after one year. We argue that as easily dissolving sites are dissolved, the rate constant decreases. It might also decrease due to adsorption of Mg2+ ions on the surface, acting as barriers to step retreat (dissolution).
Paper 5: Lattice Boltzmann simulations of advection driven ow by CO2 diusion into water
The lattice Boltzmann method is used to model CO2 dissolution into water from a pressurized CO2 gas phase overlying a distilled water column, and the subsequent convection that occur due to the higher density of 103
Chapter 9. Introduction to papers the carbon rich water compared to the distilled water. Simulations were performed for varying Rayleigh (Ra) and Prandtl (Pr) numbers, and the concentration patterns and pressure decay curves for the CO2 gas phase are presented for varying Pr and Ra. The results are compared qualitatively to an experiment where pH indicator was used in the water phase to visually monitor concentration changes in the water column. The pH pro les are calculated from concentration pro les from the simulations and a discussion around the choice of pH indicator is given. One might want to choose a dierent pH indicator dependent on what information is preferred from the experiment. E.g. a pH indicator with a lower transition pH will give information for a longer experiment time.
104
Chapter 10
Summary, conclusions, impact and future work 10.1 Summary & conclusions The lattice Boltzmann method has been used to interpret lab experiments, and a method for moving boundaries within the LBM has been proposed. Two types of experiments has been studied; i) chalk core ooding tests with dierent injection brines, and ii) a pressure decay experiment where CO2 mix with water through convection. Interpretation of core ooding experiments has been done both qualitatively, from inherent pore scale simulations, and quantitatively by the use of a proposed method for mapping the core surface onto the surface of a cylinder. The importance of the surface area in the dissolution rate equation has been studied qualitatively by the introduction of microporosity in a simpli ed sinusoidal pore geometry, and quantitatively by a proposed overgrowth model that reduces the reactive surface area of calcite due to surface coverage by precipitating magnesite. The studies clearly show that a dynamical evolution of the surface area is necessary to explain the long term behavior of core ooding tests where the mineral composition of the core changes almost entirely. The pressure decay experiment where CO2 mixes into water is simulated in two dimensions. The simulated pressure decay compare well to the experimental pressure decay when a lower Rayleigh number is used in two dimensions. The relative dierence in Rayleigh number in two and three dimensions agree well with the dierence in interface area where
105
Chapter 10. Summary, conclusions, impact and future work CO2 is transported into the water phase by advection, i.e. the interface area of the instability. Hence, the dierence in Rayleigh number is thought to be correlated to the enhanced mass transfer rate between the gas and the water phase in two dimensions. The simulated concentration pro les show the formation of a primary mushroom instability, followed by several secondary instabilities. Advection is still important after two hours of mixing. A moving boundary routine within the LBM, that is independent of the underlying mathematical grid, has been proposed. An existing LB boundary condition was coupled to a volume of uid interface tracking routine to keep track of the surface position and normal. The LB distribution functions for surfaces not positioned half-way between nodes are calculated with an interpolation scheme. Surface tension is needed in order to obtain independence of the underlying mathematical grid. The inclusion of surface tension reduces the relative errors between seeds initially rotated relative to each other from 30% to 2% for diusion limited growth. Reaction limited growth is more stable, and a simpler model can be used in order to reduce computational time.
10.2 Impact The work presented in this thesis has given contribution to better interpretation of core ooding experiments, both on the core scale and on the pore scale. A method that easily can test dierent mechanisms on the core scale, as proposed in Paper 4, is of great value, since relevant experiments are mainly on the core scale. Further, pore scale investigations are inevitable when studying mechanisms that take place on the pore scale and even on sub-micron scale. The work on nding a good moving boundary model for use within the LB framework has enhanced the capability of pore scale investigations. The study of overgrowth (surface coverage) models and its impact on the rate of dissolution is de nitely important for long term core ooding tests, and might also be of importance in the eld where at least parts of the reservoir will be subject to long term contact with the injection brine.
106
10.3. Future work
10.3 Future work LBM to simulate pore scale experiments
So far, the pore scale lattice Boltzmann simulator (LBS) described in chapter 4 has only been tested towards core scale experiments. Since it is unrealistic in terms of computational capacity to perform pore scale simulations on a full core geometry, it has been di cult to test the LBS quantitatively towards experiments. This has resulted in some qualitative analysis of the simulation results towards experimental results, and to a proposed method to map the surface of a core onto a cylinder used for simulations. In a newly started project, micro uidic experiments on chalk will be designed and performed, that can be simulated directly with the LBS. With these experiments the potential of the LBS can be utilized to the fullest. Flow paths and the location of chemical reactions such as dissolution and precipitation will be monitored directly as a function of time through various microscopic techniques; e.g. uorescent molecules will be added to measure ow paths and velocities. The same measures can be found with the LBS and compared to the experimental values. The pore scale experiments will serve as a good test for the LBS, and the results will more easily be interpreted to give information on mechanisms that should be implemented into the LBS. Once the LBS manage to match the micro uidic experimental data well, it will be used to predict enhanced oil recovery behavior in core scale experiments. Experimental and simulation results from both pore scale and core scale will be used to establish upscaling routines for the kinetics.
Pore scale overgrowth model The study of overgrowth, i.e. reduction in surface area due to coverage by precipitating minerals, is naturally extended by the implementation of a pore scale overgrowth model. The results from the core scale model should be used to propose a model at the pore scale, and the pore scale model should be able to predict the values of parameters in the core scale model. A pore scale overgrowth model could also be coupled to a moving boundaries routine, giving a model that can predict both long term evolution of mineral distributions and e uent pro les, and porosity and permeability evolutions.
107
Bibliography M. A. Addis. The behavior and modeling of weak rocks. RM International Symposium, August 30 - September 2, 1989 , Pau, France. P. M. Adler, C. G. Jacquin, and J. A. Quiblier. Flow in simulated porous media. Int. J. Multiphase Flow, 16:691712, 1990. M. A. Andersen and N. Foged. The link between water ood-induced compaction and rate-sensitive behavior in a weak north sea chalk. Fourth North Sea Chalk Symposium, Deauville, France., 1992. P. Ø. Andersen, S. Evje, M. V. Madland, and A. Hiorth. A geochemical model for interpretation of chalk core ooding experiments. Chemical engineering science, 84:218241, 2012. C. A. J. Appelo and D. Postma. Geochemistry, groundwater and pollution. CRC Press, second edition, 2005. S. Arnout, F. Verhaeghe, B. Blanpain, and P. Wollants. Lattice boltzmann model for diusion-controlled indirect dissolution. Computers and Mathematics with Applications, 55:13771391, 2008. C. H. Arns, M.A. Knackstedt, V. Pinczewski, and E. J. Garboczi. Computation of linear elastic properties from microtomographic images: Methodology and agreement between theory and experiments. Geophysics, 7:13961405, 2002. C. H. Arns, M. A. Knackstedt, V. Pinczewski, and N. S. Martys. Virtul permeametry on microtomographic images. J. Petroleum Science and Engineering, 45:4146, 2004. S. Arvidson, I. E. Ertan, J. E. Amonette, and A. Luttge. Variation in calcite dissolution rates: A fundamental problem? Geochim. Cosmochim. Acta, 67:16231634, 2003. 108
BIBLIOGRAPHY P. W. Atkins and J. de Paula. Atkins' Physical chemistry. Oxford University Press, seventh edition, 2002. A. R. Awan, R. Teigland, and J. Kleppe. A survey of north sea enhancedoil-recovery projects initiated during the years 1975 to 2005. SPE Reservoir Evaluation & Engineering, 11:497512, 2008. S. Bakke and P. E. Øren. 3-d pore-scale modeling of sandstones and ow simulations in the pore networks. C. Bardos and S. Ukai. The classical incompressible navierstokes limit of the boltzmann equation. Math. Models Methods Appl. Sci., 1:235257, 1991. P. L. Bhatnagar, E. P. Gross, and M. Krook. A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems. Physical Review, 94(3):511525, 1954. L. Boltzmann. Weitere studien über das wärmegleichgewicht ubter gasmolecülen. Sitzungsberichte der Akademie der Wissenschaften, Mathematische-Naturwissenschaftliche Klasse, pages 275370, 1872. L. Boltzmann. Lectures on Gas Theory. Dover, 1995. M. Bouzidi, M. Firdaouss, and P. Lallemand. Momentum transfer of a boltzmann-lattice uid with boundaries. Physics of uids, 13(11), 2001. M. Brignoli, F. J. Santarelli, and C. Righetti. Capillary phenomena in an impure chalk. Eurock 94, Delft, Netherland, 1994. J. Brooks and D. Welte, editors. Advances in Petroleum Geochemistry, volume 1. Academic Press, London, UK, 1984. S. G. Brush and N. S. Hall. The Kinetic theory of Gases. Imperial College Press, 2003. S. L. Bryant and K. E. Thompson. Theory, modeling and experiment in reactive transport in porous media. Current Opinion in Colloid & Interface Science, 6:217222, 2001. S. L. Bryant, C. A. Cade, and D. W. Melor. Permeability prediction from geological models. AAPG Bulletin, 77:13381350, 1993.
109
BIBLIOGRAPHY J. M. Buick and C. A. Greated. Gravity in a lattice boltzmann model. Phys. Rev. E, 61(5):53075320, 2000. A. Cali, S. Succi, A. Cancelliere, R. Benzi, and M. Gramignani. Diusion and hydrodynamic dispersion with the lattice boltzmann method. Phys. Rev. A, 45(8):57715774, 1992. L. M. Cathles. Eqalt-equilibrium chemical alteration. Technical report, Combined Physical and Chemical Geo uids Modeling, University of Windsor, Windsor, Ontario, 2006. C. Cercignani. Theory and Application of the Boltzmann Equation. Elsevier, New York, 1975. P. A. Charlez, O. Heugas, and J. F. Shao. Eect of temperature on mechanical properties of chalk. Fourth North Sea Chalk Symposium, Deauville, France, 1992. C. Chen and D. Zhang. Pore-scale simulation of density-driven convection in fractured porous media during geological CO2 sequestration. Water Resour. Res., page DOI: 10.1029/2010WR009453, 2010. H. Chen, S. Chen, and W. H. Matthaeus. Recovery of the navier-stokes equations using a lattice-gas boltzmann method. Physical Review A, 45:53395342, 1992. S. Chen, H. Chen, D. Martinez, and W. H. Matthaeus. Lattice boltzmann model for simulation of magnetohydrodynamics. Phys. Rev. Lett., 67: 37763779, 1991. S. Chen, S. Dawson, G. Doolen, D. Janecky, and A. Lawniczak. Lattice methods and their applications to reacting systems. Comput. Chem. Eng., 19(6-7):617646, 1995. V. A. F. Costa. A time scale-based analysis of the laminar convective phenomena. International Journal of Thermal Sciences, 41:11311140, 2002. S. J. Cummins, M. Francois, and D. B. Kothe. Estimating curvature from volume fractions. Comput. Struct., 83:425434, 2005. F. DaSilva, J. P. Sarda, and C. Shroeder. Mechanical behavior of chalks. Second North Sea Chalk Symposium, Book II, Stavanger, Norway, 1985. 110
BIBLIOGRAPHY S. Dawson, S. Chen, and G. Doolen. Lattice boltzmann computations for reaction-diusion equations. J. Chem. Phys., 98(2):15141523, 1993. V. De Gennaro, P. Delage, Y. J. Cui, C. Shroeder, and F. Collin. Timedependent behaviour of oil reservoir chalk: A multiphase approach. Soils and Foundations, 43(4), 2003. P. Delage, C. Schroeder, and Y. J. Cui. Subsidence and capillary eects in chalks. Eurock 96, Torino, Italy, 1996. D. Doornhof, T. G. Kristiansen, N. B. Nagel, P. D. Pattillo, and C. Sayers. Compaction and subsidence. Oil eld Review, 18:5068, 2006. P. M. Dove and F. M. Platt. Compatible real-time reaction rates for in situ imaging of mineral-water interactions using scanning force microscopy. Chem. Geol., 127:331338, 1996. J. H. Dunsmoir, S. R. Ferguson, K. L. D'Amico, and J. P. Stokes. Xray microtomography: a new tool for the characterization of porous media. Proceedings of the 1991 SPE Annual Technical Conference and Exhibition, Dallas, Oct. 6-9 (1991). J. R. Fanchi. Principles of Applied Reservoir Simulation. Gulf Professional Publishing, USA, third edition, 2006. R. Farajzadeh, P. L. J. Zitha, and H. Bruining. Enhanced mass transfer of CO2 into water: Experiments and modeling. Ind. Eng. Chem. Res., pages 64236431, 2009. S. J. Fathi, T. Austad, and S. Strand. Water-based enhanced oil recovery by "`smart water"': Optimal ionic composition for eor in carbonates. Energy & Fuels, 25(11):51735179, 2011. G. L. Gaines and H. C. Thomas. Adsorption studies on clay minerals, ii, a formulation of the thermodynamics of exchangeadsorption. J. Chem. Phys., 21:714718, 1953. V. N. Gusyatnikova and V. A. Yumaguzhin. Symmetries and conservation laws of navier-stokes equations. Acta Applicandae Mathematica, 15(12):6581, 1989. M. Gutierrez, L. E. Øino, and K. Høeg. The eect of uid content on the mechanical behaviour of fractures in chalk. Rock Mechanics and Rock Engineering, 33(2):93117, 2000. 111
BIBLIOGRAPHY J. M. Hancock and P. A. Scholle. Chalk of the north sea. Petroleum and
the Continental Shelf of Northwest Europe, Vol. 1 - Geology, pp.413425, John Wiley and Sons, New York City, 1975.
X. He and L.-S. Luo. Theory of the lattice boltzmann method: From the boltzmann equation to the lattice boltzmann equation. Physical Review E, 56(6):68116817, 1997. T. Heggheim, M. V. Madland, R. Risnes, and T. Austad. A chemical induced enhanced weakening of chalk by seawater. J. Pet. Sci. Eng., 46:171184, 2005. H. Helgeson and D. Kirkham. Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures; i, summary of the thermodynamic/electrostatic properties of the solvent. Am. J. Sci., 274:10891198, 1947a. H. Helgeson and D. Kirkham. Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures; ii, debye-hückel parameters for activity coe cients and relative partial molal properties. Am. J. Sci., 274:10891198, 1947b. H. Helgeson, D. Kirkham, and G. Flowers. Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures; iv, calculation of activity coe cients, osmotic coe cients, and apparent molaland standard and relative partial molal properties to 600 degrees c and 5 kb. Am. J. Sci., 281:12491516, 1981. R. Hellmann, J. P. Gratier, and P. Renders. deformation of chalk by pressure solution. V. M. Goldschmidt conference, Heidelberg, 1996. R. Hellmann, P. J. N. Renders, J. P. Gratier, and R. Guiguet. Experimental pressure solution compaction of chalk in aqueous solutions. part 1. deformation behaviour and chemistry. Water-rock interactions, ore de-
posits, and environmental geochemistry: A tribute to David A. Crerar. The geochemical Society, Special Publication No. 7, 2002a.
R. Hellmann, P. J. N. Renders, J. P. Gratier, and R. Guiguet. Experimental pressure solution compaction of chalk in aqueous solutions. part 2. deformation examined by sem, porosimetry, synthetic permeability, and x-ray computerized tomography. Water-rock interactions, ore de-
posits, and environmental geochemistry: A tribute to David A. Crerar. The geochemical Society, Special Publication No. 7, 2002b. 112
BIBLIOGRAPHY O. Heugas and P. Charlez. Mechanical eect of water injection on eko sk chalk. Third North Sea Chalk Symposium, Copenhagen, Denmark, 1990. F. J. Higuera and J. Jiménez. Boltzmann approach to lattice gas simulations. Europhysics Letters, 9(7):663668, 1989. A. Hiorth, E. Jettestuen, L. M. Cathles, and M. V. Madland. Precipitation, dissolution, and ion exchange processes coupled with a lattice boltzmann advection diusion solver. Geochimica Cosmochimica Acta, 104:99110, 2013. C. W. Hirt and B. D. Nichols. Volume of uid (vof) method for the dynamics of free boundaries. J. Comput. Phys., 39:201225, 1981. M. Hjuler and I. Fabricius. Engineering properties of chalk related to diagenic variations in the upper cretaceous onshore and oshore chalk in the north sea area. Journal of Petroleum Science and Engineering, 68:151170, 2009. S. Hou and Q. Zou. Simulation of cavity ow by the lattice boltzmann method. Journal of Computational Physics, 118:329347, 1995. J. M. Hunt. Petroleum Geochemistry and Geology. W. H. Freeman and Company, San Francisco, USA, 1979. T. S. Hunt. Canadian geological survey report: Progress to 1863. Canadian Geological Survey. 983 p., 1863. J. Israelachivili. Intermolecular and Surface Forces. Academic Press, New York City, 1985. T. B. Jensen, K. J. Harpole, and A. Østhus. Eor screening for eko sk. SPE European Petroleum Conference, 24-25 october, Paris, France, 2000. G. Jin, C. Torres-Verdin, F. Radaelli, and E. Rossi. Experimental validation of pore-level calculations of static and dynamic petrophysical properties of clastic rocks. SPE paper 109547, Proceedings of the 2007
SPE Annual Technical Conference and Exhibition, Anaheim, CA, Nov. 11-14 (2007).
J. P. Johnson and D. W. Rhett. Compaction behavior of eko sk chalk as function of applied stress. Paper SPE 15872 presented at the 1986 SPE European Petroleum Conference, London, October 20-22, 1986. 113
BIBLIOGRAPHY J.W. Johnson, E. H. Oelkers, and H. Helgeson. Supcrt92: A software package for calculating the standard molal thermodynamic properties of minerals, gases, aqueous species, and reactions from 1 to 5000 bar and 0 to 1000o c. Comp. Geol.Sci., 18(7):899947, 1992. Q. Kang, D. Zhang, and S. Chen. Simulation of dissolution and precipitation in porous media. Journal of Geophysical research. Solid Earth., 108(B10):doi:10.1029/2003JB002504, 2003. Q. Kang, D. Zhang, P. C. Lichtner, and I. N. Tsimpanogiannis. Lattice boltzmann model for crystal growth from supersaturated solution. Geophysical Research letters, 31:doi:10.1029/2004GL021107, 2004. Q. Kang, P. C. Lichtner, H. S. Viswanathan, and A. I. Abdel-Fattah. Pore scale modeling of reactive transport involved in geologic co2 sequestration. Transp. Porous. Med., 82:197213, 2010. R. Kingdon and P. Scho eld. A reaction- ow lattice boltzmann model. J. Phys. A: Math. General, 25(14):L907L910, 1992. M.A. Knackstedt, C. H. Arns, A. Limaye, A. Sakellariou, T. J. Senden, A. P. Sheppard, R. M. Sok, and W. V. Pinczewski. Digital core laboratory: Properties of reservoir core derived from 3d images. SPE paper 87009, Kuala Lumpur, Malaysia (2004). R. I. Korsnes, M. V. Madland, and T. Austad. Eurock 2006: Multiphysics Coupling and Long Term behaviour in Rock Mechanics, chapter 18. Impact of brine composition on the mechanical strength of chalk at high temperature, pages 133140. Taylor & Francis Group, London, 2006a. R. I. Korsnes, S. Strand, Ø. Ho, T. Pedersen, M. V. Madland, and T. Austad. Eurock 2006: Multiphysics Coupling and Long Term behaviour in Rock Mechanics, chapter 61. Does the chemical interaction between seawater and chalk aect the mechanical properties of chalk?, pages 427434. Taylor & Francis Group, London, 2006b. P. Lallemand and L. Luo. Lattice boltzmann method for moving boundaries. J. Comp. Phys., 184:406421, 2003. A. C. Lasaga. Kinetic theory in the earth sciences. Princeton University Press, New Jersey, 1998.
114
BIBLIOGRAPHY L. Li, C. A. Peters, and M. A. Celia. Upscaling geochemical reaction rates using pore-scale network modeling. Advances in Water Resources, 29: 13511370, 2006. P. C. Lichtner and D. M. Tartakovsky. Stochastic analysis of eective rate constants for heterogeneous reactions. Stochastic Environ. Res. Risk Assess., 17(6):419429, 2003. J. López, P. Gómez, and J. Hernández. A volume of uid approach for crystal growth simulation. J. Comp. Phys., 229:66636672, 2010. M. V. Madland. Water weakening of chalk. A mechanistic study. PhD thesis, University of Stavanger, 2005. M. V. Madland, K. Midtgarden, R. Manafov, R. I. Korsnes, T. G. Kristiansen, and A. Hiorth. The eect of temperature and brine composition on the mechanical strength of kansas chalk. International Symposium of the Society of Core Analysts, Abu Dhabi, UAE, SCA2008-55, 2008. M. V. Madland, A. Hiorth, E. Omdal, M. Megawati, T. Hildebrand-Habel, R. I. Korsnes, S. Evje, and L. M. Cathles. Chemical alterations induced by rock- uid interactions when injecting brines in high porosity chalks. Transp. Porous Med., 87:679702, 2011. M. E. Malmstrom, G. Destouni, S. A. Banwart, and B. H. E. Stromberg. Resolving the scale-dependence of mineral weathering rates. Environ. Sci. Technol., 34(7):13751378, 2000. M. E. Malmstrom, G. Destouni, and P. Martinet. Modeling expected solute concentrations in randomly heterogeneous ow systems with multicomponent reactions. Environ. Sci. Technol., 38(9):26732679, 2004. J. A. McLennan. Introduction to non-equilibrium statistical mechanics. Prentice-Hall, Inc., New Jersey, 1989. G. R. McNamara and G. Zanetti. Use of the boltzmann equation to simulate lattice-gas automata. Physical Review Letters, 61(20):2332 2335, 1988. M. Megawati, A. Hiorth, and M. V. Madland. The impact of surface charge on the mechanical behavior of high-porosity chalk. Rock Mech. Rock Eng., 46:10731090, 2013. 115
BIBLIOGRAPHY Y. Mimran. Fabric deformation induced in cretaceous chalks by tectonic stresses. Tectonophysics, 26:309316, 1975. Y. Mimran. Chalk deformation and large-scale migration of calcium carbonate. Sedimentology, 24:333360, 1977. K. C. Misra. Geochemistry. Principles and Applications. Wiley-Blackwell. A John Wiley & Sons, Ltd., Publication., 2012. J. W. Morse and R. S. Arvidson. The dissolution kinetics of major sedimentary carbonate minerals. Eatrh-Science Reviews, 58:5184, 2002. J. W. Morse and R. A. Berner. Dissolution kinetics of calsium carbonate in seawater: Ii. a kinetic origin for lysocline. Am. J. Sci., 272:840851, 1972. S. F. Murphy, S. L. Brantley, A. E. Blum, A. F. White, and H. L. Dong. Chemical weathering in a tropical watershed, luquillo mountains, puerto rico: Ii. rate and mechanism of biotite weathering. Geochim. Cosmochim. Acta, 62(2):227243, 1998. A. Nermoen, R. I. Korsnes, A. Hiorth, and M. V. Madland. Porosity and permeability development in compacting chalks during ooding of non-equilibrium brines - insights from long term experiment. Journal of Geophysical Research - Solid Earth (Resubmitted after major revisions), 2014. G. H. Newman. The eect of water chemistry on the laboratory compression and permeability characteristics of north sea chalks. J. Petr. Tech., 35:976980, 1983. E. H. Oelkers, P. Benezeth, and G. S. Pokrovski. Thermodynamic database for water-rock interaction. Thermodynamics and Kinetics of Water-Rock Interactions, 70, 2009. E. Omdal, M. V. Madland, T. G. Kristiansen, N. B. Nagel, R. I Korsnes, and A. Hiorth. Deformation behavior of chalk studied close to in situ reservoir conditions. Rock Mechanics and Rock Engineering, 43(5):557 580, 2010. P. E. Øren and S. Bakke. Process based reconstruction of sandstones and prediction of transport properties. Transport in Porous Media, 46: 311314, 2002. 116
BIBLIOGRAPHY P. E. Øren, S. Bakke, and R. Held. Direct pore-scale computation of material and transport properties for north sea reservoir rocks. Water Resources Research, 43:W12S04, 2007. S. Osher and J. Sethian. Fronts propagating with curvature-dependent speed: algorithms based on hamiltonjacobi formulations. J. Comput. Phys., 79:1249, 1988. E. Papamichos, M. Brignoli, and F. J. Santerelli. An experimental and theoretical study of partially saturated collapsible rocks. Mech. Cohesive-Frictional Mater., 2:251278, 1997. D. L. Parkhurst and C. A. J. Appelo. User's guide to phreeqc (version 2) - a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations. US Geol. Surv. Water res. Inv. Report 99-4259, 1999. J. M. Piau and V. Maury. Mechanical eects of water injection in chalk reservoirs. Eurock 94 SPE. ISRM Int. Conf. 819-828), 1994. J. E. Pilliod Jr. and E. G. Puckett. Second-order accurate volume-of- uid algorithms for tracking material interfaces. Journal of Computational Physics, 199:465502, 2004. L. N. Plummer, T. M. L. Wigley, and D. L. Parkhurst. The kinetics of calcite dissolution in co2-water systems at 5 to 60 degc and 0.0 to 1.0 atm co2. Am. J. Sci., 278:179216, 1978. Y. Qian, D. d'Humières, and P. Lallemand. Lattice bgk models for navierstokes equation. Europhysics Letters, 17:479484, 1992a. Y. H. Qian. Lattice gas and lattice kinetic theory applied to navier-stokes equation. PhD Thesis, Université Pierre et Marie Curie, Paris, 1990. Y. H. Qian, D. D'Humières, and P. Lallemand. Lattice bgk models for navier-stokes equation. Europhysics Letters, 17(6):479484, 1992b. R. Risnes. A laboratory study of chalk extrusion. Third North Sea Chalk Symposium, Copenhagen, Denmark, 1990. G. D. Saldi, G. Jordan, J. Schott, and E. H. Oelkers. Magnesite growth rates as a function of temperature and saturation state. Geochim. Cosmochim. Acta, 73:56465657, 2009. 117
BIBLIOGRAPHY C. Schroeder and J. Shao. Plastic deformation and capillary eects in chalk. Fifth North Sea Chalk Symposium, Reims, France., 1996. X. Shan and H. Chen. Lattice boltzmann model for simulating ows with multiple phases and components. Physical Review E, 47(3):18151819, 1993. A. M. Shehata, M. B. Alotaibi, and H. A. Nasr-El-Din. Water ooding in carbonate reservoirs: Does the salinity matter? SPE Reservoir Evaluation & Engineering, 17(3):304313, 2014. P. Spanne, J. F. Thovert, C. J. Jacquin, W. B. Lindquist, K. W. Jones, and P. M. Adler. Synchrotron computed microtomography of porous media: toplogy and transports. Phys. Rev. Lett., 73:20012004, 1994. C. J. Steefel and P. Van Cappelen. A new kinetic approach to modeling ater-rock interaction: the role of nucleation, precursors, and ostwald ripening. geochim. Cosmochim. Acta, 54:26572677, 1990. S. Strand, T. Puntervold, and T. Austad. Eect of temperature on enganced oil recovery from mixed-wet chalk cores by spontaneous imbibition and forced displacement using seawater. Energy & Fuels, 22(5): 32223225, 2008.
The Lattice Boltzmann Equation for Fluid Dynamics and Beyaond. Oxford University Press, New York, 2001.
S. Succi.
J. E. Sylte, L. K. Thomas, D. W. Rhett, D. D. Bruning, and N. B. Nahel. Water induced compaction at the eko sk feild. SPE Annual Coference and Exhibition, Houston, Texas, 1999. J. E. Szecsody, J. M. Zachara, A. Chilakapati, P. M. Jardine, and A. S. Ferrency. Importance of ow and particle-scale heterogenity on coii/iiiedta reactive transport. J. Hydrol., 209(1-4):112136, 1998. G. W. Thomas. Principles of Hydrocarbon Reservoir Simulation. International Human Resources Development Corporation, Boston, second edition, 1982. P. Van Cappelen, L. Charlet, W. Stumm, and P. Wersin. A surface complexation model for the carbonate mineral-aqueous solution interface. Geochim. Cosmochim. Acta, 57:35053518, 1993. 118
BIBLIOGRAPHY F. Verhaeghe, S. Arnout, B. Blanpain, and P. Wollants. Lattice boltzmann model for diusion-controlled dissolution of solid structures in multicomponent liquids. Phys. Rev. E, 72:036308, 2005. F. Verhaeghe, S. Arnout, B. Blanpain, and P. Wollants. Lattice-boltzmann modeling of dissolution phenomena. Phys. Rev. E, 73:036316, 2006. J. V. Walther. Essentials of geochemistry. Jones and Bartlett Publishers, USA, second edition, 2009. A. F. White and S. L. Brantley. Chemical weathering rates of silicate minerals. Reviews in mineralogy, vol. 31., chapter Chemical weathering rates of silicate minerals: an overview, pages 122. Mineralogical Society of America, Washington DC, 1995. A. F. White and S. L. Brantley. The eect of time on the weathering of silicate minerals: why do weathering rates dier in the laboratory and eld? Chem. Geol., 202(3-4):479506, 2003. D. Wolf-Gladrow. A lattice boltzmann equation for diusion. Journal of Statistical Physics, 79(5/6):10231032, 1995. K. Wu, M. I. J. van Dijke, G. D. Couples, Z. Jiang, J. Ma, K. S. Sorbie, J. Crawford, I. Young, and X. Zhang. 3d stochastic modeling of heterogeneous porous media - application to reservoir rocks. Transport in Porous Media, 65:443467, 2006. M. Xu and S. R. Higgins. Eects of magnesium ions on near-equilibrium calcite dissolution: Step kinetics and morphology. Geochim. et Cosmochim. Acta, 75:719733, 2011. C. Yang and Y. Gu. Accelerated mass transfer of CO2 in reservoir brine due to density driven natural convection at high pressures and elevated temperatures. Ind. Eng. Chem. Res., pages 24302436, 2006. C. L. Y. Yeoung and S. Torquato. Reconstructing random media. Phys. Rev. E, 57:495506, 1998. B. Zangiabadi, R. I. Korsnes, T. Hildebrand-Habel, A. Hiorth, I. K. Surtarjana, A. Lian, and M. V. Madland. Chemical water weakening of various outcrop chalks at elevated temperature, 2009.
119
BIBLIOGRAPHY P. M. Zhang, M. T. Tweheyo, and T. Austad. Wettability alteration and improved oil recovery by spontaneous imbibition of seawater into chalk: impact of the potential determining ions Ca2+ , Mg2+ , and SO42 . Colloid. Surf. A: Physicochem. Eng.Aspects, 301(1-3):199208, 2007. U. Zimmermann, M. V. Madland, A. Nermoen, T. Hildebrand-Habel, S. R. A. Bertolino, A. Hiorth, R. I. Korsnes, J-N. Audinot, and P. Grysan. Challenging the chalk: SEM, nano-SIMS and geochemical data from ooded chalk and its consequenses for enhanced oil recovery applications. In press in AAPG Bulletin, DOI 10.1306/12221412196, 2015.
120
PART II
Paper 1: Pore Scale Modeling of Brine Dependent Pore Space Evolution in Carbonates. J. Pedersen, J. L. Vinningland, E. Jettestuen, M.V.
Madland and A. Hiorth. Paper SPE-164849-MS prepared for presentation at the EAGE Annual Conference & Exhibition incorporating SPE Europec held in London, United Kingdom, 10-13 June, 2013.
SPE-164849-MS Pore Scale Modeling of Brine Dependent Pore Space Evolution in Carbonates Janne Pedersen, IRIS, Jan Ludvig Vinningland, IRIS, Espen Jettestuen, IRIS, Merete Vadla Madland, UiS, and Aksel Hiorth IRIS/UiS.
Copyright 2013, Society of Petroleum Engineers This paper was prepared for presentation at the EAGE Annual Conference & Exhibition incorporating SPE Europec held in London, United Kingdom, 10–13 June 2013. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Abstract Rock fluid interactions play a crucial role in chemical EOR methods. Possible EOR methods are usually screened at core scale before being considered for implementation on field scale. It is therefore essential to have reliable methods for interpreting core scale experiments and to extract relevant parameters to be used in field scale investigations. In general, core scale experiments targeting rock-fluid interactions are not easy to interpret by standard Darcy scale models. In this paper we present a pore scale lattice Boltzmann model that is capable of simulating fluid flow in the pore space while interacting with individual rock minerals to induce mineral precipitation and/or dissolution causing dynamical changes in the pore space geometry. We discuss different mineral growth models, and compare chemical flooding lattice Boltzmann simulations with core scale experiments. The pore scale model we present in this paper can only be qualitatively compared with experiments, but we demonstrate that by incorporating mineral growth we are able to capture important trends in the experimental data which is not possible by using standard rate equations in combination with Darcy scale models.
Introduction Several core flooding studies (Webb et al. 2008; Madland et al., 2009, 2011; Zhang et al., 2007; Yildiz and Morrow, 1996; Tang and Morrow, 1999; Jerauld et al., 2006) have demonstrated that geochemical (rock-fluid) reactions in the pore space is important for the flow paths of oil and water. The results from field studies are not always clear, see (Thyne, 2011 for a review). A particular interesting case on the Norwegian continential shelf (NCS) is the Ekofisk and Valhall field. At the Ekofisk field, more than 10 m of subsidence has occurred over the last 30 years. Injection of brine to maintain fluid pressure and increase oil recovery has reduced the compaction rate by one third but not eliminated it. Injections of different chemistry brines at elevated temperatures cause different amounts of compaction and different degrees of oil recovery in the lab (Heggheim et al. 2005; Madland et al. 2008, Korsnes et al. 2008, Madland et al. 2011). Investigations of the core material by SEM studies before and after core floods have demonstrated that seawater injection causes dissolution of the chalk framework and secondary mineral formation (Madland et al. 2011). Geochemical modelling predicts that in order for the rock to equilibrate with the core, magnesite and anhydrite will precipitate and calcite will dissolve. However, although the overall interpretation of the rock fluid interactions seems to be clear, there are still some open questions: 1) What is the role (e.g. amount and distribution) of non-carbonate minerals? Chalk cores with lower content of silicates shows longer response times before a chemical induced water weakening effect is observed (Madland et al. 2011, Megawati et al. 2011). 2) What causes transient effects observed in the effluent concentrations? 3) How does the exact mineral location and alteration relate to permeability, porosity, and wettability changes? In order to upscale core data to larger scale and evaluate the field potential, it is extremely important to search for answers to these questions. Thus, when interpreting lab experiments, and also aiming to improve the design of new experiments, it is important to have models that can simulate flow through the pore space, in which the fluid can contact individual minerals, interact with them, and predict how the pore space characteristics will change. To achieve this we have developed a full 3D geochemical pore scale simulator using the lattice Boltzmann (LB) model, with moving boundaries. In the next sections we will briefly describe both the geochemical model and the LB method. The main challenge on the development side is the choice of the moving boundary method based on ease of implementation and accuracy; both current and possibly future methods are discussed. At the end we compare our results to core scale experiments, and
2
SPE-164849-MS
demonstrate how the pore geometry and evolution impact the effluent concentrations. The Lattice Boltzmann Method for Advection and Diffusion The BGK lattice Boltzmann method (LBM) (Succi, 2001; Sukop and Thorne, 2006) is used to simulate fluid flow and diffusion of chemical species at the pore scale. The fluid-rock interactions are described by a bounce back boundary condition (Hiorth et al. (2012)) and will only be reviewed briefly here. The evolution of the solid is tracked by a volume of fluid like approach where each mineral phase is represented by a scalar field. The LBM describes the evolution of velocity distribution functions ( ) for a discrete set of microscopic velocities ( ⃗ ). The model is applied to regular quadratic/cubic grids with grid constant and a constant timestep . The step lengths associated with a given velocity ⃗ will always connect nodes on the grid. Two nodes connected by ⃗ is termed a link. The macroscopic density ( and velocity ( ⃗⃗) at a given grid point ( ⃗) at time is calculated by summing the contributions from each distribution function: (⃗
∑
∑
⃗⃗( ⃗
(1)
,
⃗ .
(2)
The evolution of the distributions is given by (⃗
(⃗
⃗
[ (⃗
( ⃗⃗ ],
(3)
where the right hand side is the BGK collision operator. is the relaxation time that describes how fast the liquid equilibrates ⁄ . and relates to the physical fluid kinematic viscosity ( ) through is the equilibrium distribution function for particles moving along direction , and it is a function of the macroscopic variables fluid density ( ) and velocity ( ⃗⃗), in such a way that one can recover the macroscopic Navier Stokes equation in the continuum limit, and is given by: [
⃗ ⃗⃗
( ⃗ ⃗⃗
⃗⃗
].
(4)
are weights that distinguish between direct and diagonal links between grid points. For simulations in two dimensions we use the d2q9 discretization with weights for direct links, for diagonal links and for the stationary state. For simulations in three dimensions we use the d3q15 discretization with weights for direct links, for diagonal links and for the stationary state. is the lattice sound speed and the value is ⁄√ for both the d2q9 model and the d3q15 model. The chemical species are described in a similar manner as the fluid flow. A BGK-LB equation can be used to describe evolution of distribution functions for the chemical species, , such that ∑
(5)
where is the distribution functions for chemical basis species . In this case the relaxation time relates to the diffusion coefficient rather than the fluid kinematic viscosity. An equilibrium distribution function similar to equation (4) is introduced for the total concentration of each chemical basis species. The equilibrium distribution is a function of total concentrations, ’s, and the velocity, given by the fluid simulations (2), in such a way that the BGK-LB equation can be shown to recover the macroscopic diffusion equation. The pore walls consist of minerals (calcite minerals in the cases studied in this article) that can react with the pore fluid, and form secondary minerals on the pore walls due to precipitation (as fluid is super saturated). The complexes (e.g. CaSO40, CaHCO3+…) is treated as part of the boundary condition. Mineral water boundary condition All chemical reactions are implemented in the boundary conditions. The difference between the incoming and outgoing flux on the wall is set equal to the chemical flux from dissolution/precipitation processes and the outgoing distributions are given in terms of the incoming distributions (Verhaeghe et al., 2006):
̅
̃
(
̅
⃗̅ ⃗⃗
(6)
SPE-164849-MS
3
where ̃ denotes the incoming distribution function and ̅ is the distribution reflected from the surface. Opposite directions are assigned equal weights so that pointing away from the surface, ⃗⃗ is the surface ̅ . ⃗̅ is the velocity along link normal pointing away from the surface and ⃗⃗ is the chemical reaction flux for species . In general is a function of the activities of all the chemical species at the pore wall. The equation relating the velocity distributions along a link to the concentration is (Verhaeghe et al., 2006): ̅
̃
(7)
The chemical flux is determined from the rate equation (8). In the study by Hiorth et al. 2012 we used non-linear rate equations, in this work we include moving boundaries at the mineral surface, but simplify to a linear rate equation: (
)
(8)
where is the reaction rate constant and is the equilibrium concentration for species . The equilibrium concentration is a function of the aqueous concentrations and the mineral phases. Inserting the expression for the concentration (7) into the expression for the flux (8) gives the as a function of the velocity distribution for each species. Hence, for each surface link we need to solve equation (6) for . Note that equation (6) is in general a nonlinear equation in ̅ . The chemical solver is based on log K values of dissociation reactions for solution complexes, which are calculated using the HKF equation of state (Helgeson and Kirkham 1974a,b; Helgeson et al., 1981) with thermodynamic data from the SUPCRT database (Johnsonet al., 1992) and the program EQAlt (Cathles, 2006). The Lattice Boltzmann Growth Model When dissolution or precipitation takes place the geometry of the pore space can change, e.g. when minerals dissolve the volume of the pore space and the porosity changes, and this has in return shown to affect the physical and chemical processes in the system (Madland et al. 2011; Megawati et al. 2011). Two approaches for implementing moving boundaries in LBM are presented in studies of crystal growth from supersaturated solution by Kang et al. (2004; 2009) and Verhaeghe et al. (2005; 2006). Kang et al. 2004, use a node based scheme, where all the ’s entering a node are used in the reaction calculations. In the moving boundary routine they monitor the solid mass in every boundary wall node and when the mass increases above a given threshold, one of the fluid neighbor nodes is transformed to a solid node with the probability 1:4 for direct versus diagonal links. Kang et al. (2009) improve this routine by updating volume fractions during precipitation in the fluid nodes neighbouring the solid wall nodes, instead of doing the update in the solid nodes and then make a choice to which fluid node the structure grows. Verhaeghe at al. (2005; 2006) uses a linked based approached, contrary to the node based approach used by Kang et al. (2004), so that only one incoming is known for each link. This allows for an easy implementation as all links intersected by the boundary is treated in an equivalent manner. To calculate the outgoing distributions they use a combination of bounce-back and interpolations/extrapolations to model evolution of a boundary due to dissolution. This method is based on a bounce-back scheme for curved and moving boundaries in fluid flows presented by Bouzidi et al. (2001) and Lallemand et al. (2003). Both Kang et al. (2004; 2009) and Verhaeghe et al. (2005; 2006) base their calculations on a scalar solid fraction, giving for each node the fraction of solid phase. This approach is motivated by Volume of Fluid (VOF) methods for interface reconstruction. However, only monitoring the solid fraction in individual nodes does not give detailed information about the orientation of the surface. In the boundary condition (6) the normal vector is needed for a general orientation of the surface element. This can be estimated using a VOF method as suggested by Verhaeghe et al. (2006), where the solid fractions in a small neighborhood of nodes are used to estimate the surface normal. We use scalar fields to keep track of the volume fraction of each mineral phase. When only one mineral phase is present only one scalar field, the solid fraction ( ), is needed. is defined in every computational cell/node, and updated in the boundary condition. takes values between 0 and 1, being less than ⁄ in fluid cells, and larger or equal to ⁄ in solid cells, equal to Verhaeghe et al. node classification. The change in sf from interactions between one fluid cell and one solid cell equals the difference between incoming and outgoing distributions on the wall, which is calculated from equation (6). The total change in in one solid cell is given, in a one component case, by the sum of the contributions from each neighbouring fluid cell (Verhaeghe et al. (2005; 2006)): (∑
̅
̃ )
(9)
where is the molar volume to the mineral. In more complex systems the change in solid fraction is found by using the stoichiometric matrix and the molar volumes of the minerals together with the sum over distributions, as described in Hiorth et al. (2012). For positive changes in (precipitation), is updated in the solid cell, and for negative changes (dissolution) is
4
SPE-164849-MS
updated in the fluid cell. Hence can at some point exceed 1 in solid cells or become negative in fluid cells. For these cases a routine is implemented to redistribute solid from the solid cell to the neighboring fluid cells. For the case sf is set to one in the solid cell and the excess amount ( ) is distributed between the neighbouring fluid cells weighted by the change in originating from the different cells. Hence the following solid fractions are redistributed into the fluid cells: ̅
(
̃
(10)
For the case sf is set to zero and the amount ( ) is removed from the solid cells that have received solid from the fluid cell at the current time step. As for the case the redistribution is weighted by the change in in the different solid cells originating from the fluid cell under consideration: ̅
̃ (11)
When more than one mineral phase is present we repeat this for each mineral phase and use the sum of all mineral phases to classify nodes and to find the pore boundary. Results The experimental data that motivated the numerical study are obtained from series of outcrop chalk cores exposed to continuous flooding of 0.219M MgCl2 and synthetic seawater with low NaCl (SSW) content at an injection rate equal to either 1 or 2 pore volumes per day (PV/D) at 130°C (Madland et al. 2011). In the study by Madland et al. 2011 the experimental data was discussed in terms of ion substitution, calcite dissolution, and precipitation of magnesite and anhydrite (in the case of seawter) and these chemical processes including ion exchange have also been further discussed by Hiorth et al. (2012) and Andersen et al. (2012). The effluent profiles are shown in Figure 1. The core flooded with SSW was blocked after only two weeks of flooding probably due to anhydrite blocking the pores, which seems to affect the flow through pore channels and thus the permeability. As pointed out in Hiorth et al. (2012) and Andersen et al. (2012) the initial peak in the calcium concentration may be captured by using a relatively high ion exchange capacity, however it is not easy to capture the slow decrease of calcium, and increase in magnesium concentration by the use of traditional rate equations. In the following we will present simulations using two simplified pore geometries to show that these effects can be captured in a natural way by using linear kinetics on the pore scale.
Figure 1: Experimental effluent profiles obtained from Liege outcrop chalk cores flooded with a) MgCl, [Mg2+]=0.219, [Cl-]=0.438M and b) synthetic seawater (SSW) with low NaCl concentration, [Mg2+]=0.0445, [Cl-]=0.125, [Ca2+]=0.130, [SO42-]=0.024, [K+]=0.01, [Na+]=0.050M. Note that the Cl- is scaled to fit the plot.
The pore geometries are 1) a sinusoidal tube representing an idealized pore geometry (macro porosity), and 2) the same tube as in 1 but perforated with narrow channels perpendicular to the flow direction to introduce porosity at the smallest spatial scale (micro porosity). These channels also increase the surface area of the geometry which is very high for Liège chalk samples; the specific surface area (SSA) is typically ≥4 m2/g (Megawati et al. 2012). The two different geometries are shown in Figure 3 as 2D slices. Micropores are intracrystal pores that are substantially smaller than the pores between crystals. Micro porosity can
SPE-164849-MS
5
contribute considerably to the chalk porosity but will not contribute to permeability (accessible through diffusion, not through advection). It will thus increase the surface area accessible for chemical reaction, and can possibly explain the high surface area to volume ratio found in chalk. This is an effect that can be important for the chemical alteration of the chalk rock and the composition of the effluent from experiments. Figure 2 shows the solid fractions for calcite, magnesite and anhydrite at two different times during the SSW simulation of geometry 1). The first time (t1) is early in the simulation, but late enough for magnesite and anhydrite to already be present on the solid surface. The second time (t2) is much later in the simulation and anhydrite has now almost clogged the pipe. For visualization purposes the snapshots are 2D cuts from the 3D simulation. Figure 2 a) and b) shows the calcite solid fraction at early and late times and it is clear that the calcite reduction only takes place in the narrow band of nodes in direct contact with the pore fluid. Consequently, the change in magnesite in Figure 2 c) and d) is equally limited to a narrow band of nodes since magnesite replaces the dissolved calcite. Calcite is screened by the precipitated magnesite and leads to a cease of both calcite dissolution and magnesite precipitation. Precipitated SO4 in the form of anhydrite represents the most significant change of the solid fraction, as is clearly seen in Figure 2 e) and f). Finally, the total solid fraction is shown in Figure 2 g) and h). Figure 3 summarizes the effluent results obtained for the two geometries (with and without micropores) using both MgCl and SSW brines. Figure 3 a) shows plots of effluent concentration profiles for MgCl2 flooding, and the initial increase in Ca and decrease in Mg concentrations (inlet concentrations are shown on the right axis) agree well with calcite dissolution and magnesite precipitation. Both geometries qualitatively reproduce the peak in the Ca concentration observed in experiments, and in addition the micro pore geometry leads to a significant increase of the peak. This is indeed expected since the micro pore geometry has a higher surface area and consequently more calcite is dissolved. A similar transient behavior of the Ca concentration is found in Figure 3 b) showing effluent profiles for SSW for both geometries. However, the shift of the Ca peak caused by the micro pores is somewhat different for the two brines. For MgCl2 the amplitude of the Ca peak more than doubles for the micro pore geometry, while for SSW the Ca peak is almost unchanged but extended in the temporal direction. A possible interpretation of this difference in behavior is the sealing of micro pores by anhydrite in the SSW case that will quench the release of Ca from the micro pores to the pore fluid. Finally, when all channels are sealed off, the pore space in contact with the brine is identical to the pore space in geometry 1) and the effluent profiles coincide. The simulated effluent profiles cannot be compared directly to the experimental effluents because of different dimensions and timescales. In addition, the reaction rate and the advective velocity are increased compared with molecular diffusion to effectively speedup the boundary velocity. However, we are still in the reaction-limited domain (in contrast to the transport-limited domain) which is the relevant domain for core plug experiments, and the qualitative comparison shows that the simulations can capture the experimentally observed evolution. Also, we now have a method for matching surface area to volume ratio to a better degree than before.
Figure 2: Solid fractions for calcite, magnesite and anhydrite for flushing with SSW through pore geometry 1). a) calcite solid fraction at t=t1, b) calcite solid fraction at t=t2, c) magnesite solid fraction at t=t1, d) magnesite solid fraction at t=t2, e) anhydrite solid fraction at t=t1, f) anhydrite solid fraction at t=t2, g) total solid fraction at t=t1, h) total solid fraction at t=t2
6
SPE-164849-MS
Figure 3: Numerical effluent concentrations measured in the two different 3D pore geometries (shown here as 2D slices with the solid part in black) for MgCl2 in a) and synthetic seawater (SSW) in b). The open squares (filled circles) give the effluent chemistry for the geometry with (without) micro structures. The initial transient behavior in the numerical data is significantly affected by the change in geometry; the concentrations reach steady-state later when the surface area is increased.
Discussion of the Growth model The growth model used for the work presented in this article is a first approach to the moving boundaries problem. Ultimately we want to have a model that satisfies the three demands 1) conserve mass, 2) converge for increasing resolution and 3) invariant under rotation, i.e. independent of alignment of geometry compared to the underlying grid. In this section we look at the rotational symmetry of the current growth model and compare in two dimensions to a partial VOF method. In the basic model we do not have the surface normal and therefore we set ⃗ ⃗⃗ in equation (6). This assumption gave good results for the simulation of NMR signals in square and triangular pores (Hiorth et al. 2009). By using the VOF method we obtain an estimate for the surface normal and use this in ⃗ ⃗⃗ in the boundary condition. We have implemented the ELVIRA method in two dimensions (Pilliod Jr. and Puckett, 2004) to calculate the surface normal. Figure 4 shows how the surface evolves for the two models, starting initially from a 20x20 square. Figure 4 a) uses the current growth model while the Figure 4 b) uses the surface normal estimated from the ELVIRA VOF method. The dominant bulk transport mechanism on the pore scale is diffusion, at least in relation to core scale experiments. The surface reactions in chalk are also usually rate limited so the important limit in these simulations is the constant flux case. Hence in the test presented here we will confine our self to this case, that is is treated as a constant in equation (6) so that is independent of the velocity distribution. The curves are the contour lines for , which mimics the position of the interface. Simulations have been performed for initial square aligned with the grid (blue curves) and initial square rotated 45° with respect to the grid lines (red curves). The red curves have been rotated back after simulation for comparison between the two cases. Clearly there is a gain in rotational symmetry by estimating the surface normal; however, the current model is not too bad for constant flux growth.
SPE-164849-MS
7
(a)
(b)
Figure 4: Simulations with constant flux into the surface using the two models a) ⃗ ⃗⃗
and b) ⃗ ⃗⃗ with ⃗⃗ from VOF routine. The curves show the contour where . The blue curves are simulations starting from a square aligned with the grid (as in the figure) and the red curves are simulations starting from square rotated 45 deg. Figure 5 shows the contours from the two simulations with initially a square aligned with the grid lines. The blue curves are from simulation using the current model without the surface normal, and the red curves are from simulation using the estimated surface normal from the VOF method. For early times we see that the contours for the two models overlap on the straight edges where ⃗ ⃗⃗ , but differs only in the corners where ⃗ ⃗⃗ . Also the method using the surface normal grows slightly slower than the ‘ ⃗ ⃗⃗ ’-method, which is expected since ⃗ ⃗⃗ is assumed to be lower than one with ⃗ and ⃗⃗ both being unit vectors.
Figure 5: Contours for sf=0.5 for the two models for simulations with square initially aligned with the grid lines. Blue: ⃗ ⃗⃗ with ⃗⃗ from VOF routine
In the 3D simulations following in this article we use the simple model in which ⃗ ⃗⃗ computational heavy, and the simplified model gives fair results for the cases studied in this article.
, red: ⃗ ⃗⃗
. The VOF method is
Discussion and Conclusions We have presented a numerical model for simulations of chemical induced surface reaction on the pore scale with moving boundaries. Two approaches for the moving boundary problem were presented; a simple model that treat all link direction equally, and one using the volume of fluid (VOF) method to calculate surface the normal. These models were tested for the rate limiting surface reactions. These test shows that the model without surface normal developed an artificial surface roughness which increased the surface area and lead to an over-estimation of the amount of reactions as seen from Figure 4 where the results for the two approaches are compared. We observed similar types of errors in the tests for rotational invariance without surface area, whereas the VOF model only showed sub-pixel discrepancies. That being said, we would like to stress that the VOF model is computational expensive. Probably the best approach is to test the growth model with the typical Peclet and Damkohler number used in the experiment, and choose the model without the surface normal if it gives similar results as the VOF model. Simulations using artificial 3D pores flooded with synthetic sea water and MgCl2 water were conducted for two different geometries, one with and and one without micro pores. The simulations show clearly that non monotonic behavior in
8
SPE-164849-MS
the observed effluent profiles can be generated although the chemical rate equations are linear at the mineral water interface. Furthermore the simulated effluent profiles compares qualitatively well with the experimental profiles; there is an initial calcium peak and slow transient increase in magnesium and sulphate, and a decrease in calcium. More numerical work needs to be done to obtain rigorous quantitative results. However, we might state a tentative qualitative hypotheses: the shape of the Ca-peaks observed in the initial effluent profiles can be related to the existence of micro pores, but the long term behavior is similar for both types of pore systems. Simulations on both pore geometries show that the reactive surface area of calcite decreases as a consequence of secondary mineral formation. Relating this to the effluent of the core scale experiments shown in Figure 1, we could argue that the peak in concentration during the first 25 hours holds information on the micro porosity, while the slow varying trends after about 25 hours give information on the kinetics of secondary mineral precipitation along the flow pathways. The simulations are computational expensive: The geochemical solver needs to resolve a nonlinear problem for each wall link, and the VOF method for finding surface normal is both cumbersome to implement and computationally time consuming, especially in 3D. Another challenge is the separation of time scales, between the advective velocity, diffusion and reaction rates. As a consequence we have chosen high reaction rates in our simulations so that the growth velocity of the pore walls are faster than what would be reasonable for a core scale experiment, but we make sure that the reactions are still rate limited. Another choice is to use the simpler of the two presented surface growth models. To improve the efficiency of our model in these regards are the main challenges for future developments. Acknowledgements The authors would like to thank the Research Council of Norway, BP Norge AS andthe Valhall co-venturers, including Hess Norge AS, ConocoPhillipsand the Ekofisk co-venturers, including Total E& P Norge AS, ENINorge AS, Statoil Petroleum AS and Petoro AS for financial support.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11.
12.
13. 14. 15. 16.
17. 18. 19. 20. 21.
Andersen, P.O., Evje, S, Madland, M. V., Hiorth, A, A geochemical model for interpretation of chalk core flooding experiments, Chemical engineering science, 2012. 84: p. 218-241 Bekri, S., Thovert, J. F. and Adler, P. M. (1995) Dissolution of porous media, Chem. Eng. Sci. 50, 2765-2791. Bouzidi M., Firdaouss M. and Lallemand P. (2001) Momentum transfer of a Boltzmann-lattice fluid with boumdaries. Phys. Fluids 13, 3452-3459. Buick, J. M. and Greated, C. A. (2000) Gravity in a lattice Boltzmann model. Phys. Rev. E 61(5), 5307-5320. Cathles L. M. (2006) Eqalt-Equilibriom chemical alteration. Combined Physical and Chemical Geofluids Modeling. University of Windsor, Windsor, Ontario. Childs, E. C. (1969) The physical basis of soil water phenomena. Wiley and Sons. New York. Geoffrey Thyne (2011), Evaluation Of The Effect Of Low Salinity Waterflooding For 26 Fields In Wyoming SPE 147410 Heggheim, T., M.V. Madland, R. Risnes, and T. Austad, A chemical induced enhanced weakening of chalk by seawater. Journal of Petroleum Science and Engineering, 2005. 46: p. 171-184. Helgeson H. and Kirkham D. (1974a) Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: I. Summary of the thermodynamic/electrostatic properties of the solvent. Am. J. Sci. 274, 1089-1198. Helgeson H. and Kirkham D. (1974b) Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: II. Debye-hückel parameters for activity coefficients and relative partial molal properties. Am. J. Sci. 274, 1199-1261. Helgeson, H., Kirkham, D. and Flowers, G. (1981) Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coefficients, osmotic coefficients, and apparent molal and standard and relative partial molal properties to 600 °C and 5 kb. Am. J. Sci. 281, 1249-1516. Hiorth, A., Lad, U. H. A., Evje, S. and Skjaeveland, S. M. (2009) A lattice Boltzmann-BGK algorithm for a diffusion equation with Robin boundary condition – application to NMR relaxation, International Journal for Numerical Methods in Fluids, 59(4), 405-421 Hiorth A., Cathles L. M. and Madland M. V. (2010) The impact of pore water chemistry on carbonate surface charge and oil wettability. Trans. Porous Media, 85. Hiorth A., E. Jettestuen, L.M. Cathles, M.V. Madland, Precipitation, dissolution, and ion exchange processes coupled with a lattice Boltzmann advection diffusion solver. Geochim. Cosmochim. Acta (2012), http://dx.doi.org/10.1016/j.gca.2012.11.019 Jerauld G., Lin C., Webb K. and Seccombe J. (2006) Modeling low-salinity waterflooding. In: The 2006 SPE Annual Technical Conf. & Exhib., San Antonio, Texas, USA, 24–27 September, SPE102239. Johnson J. W., Oelkers E. H. and Helgeson H. (1992) Supcrt92: A software package for calculating the standard molal thermodynamic properties of minerals, gases, aqueous species, and reactions from 1 to 5000 bar and 0 to 1000 °C. Comp. Geol. Sci. 18(7), 899-947. Kang, Q., Zhang, D., Lichtner, P. C. and Tsimpanogiannis, I N. (2004) Lattice Boltzmann model for crystal growth from supersaturated solution, Geophysical Research letters, 31, doi:10.1029/2004GL021107. Kang, Q. and Lichtner, P. C. (2009) Pore Scale Modeling of Reactive Transport Involved in Geologic CO2 Sequestration, Transp. Porous Med, 82 197-213 Kevin Webb, Arnaud Lager, Cliff Black (2008) Comparison of High/low salinity water oil relative permeability, SCA2008-39 Korsnes, R.I., M.V. Madland, T. Austad, S. Haver, and G. Røsland, The effects of temperature on the water weakening of chalk by seawater. Journal of Petroleum Science and Engineering, 2008. 60: p. 183-193. Lallemand, P. and Luo, L. (2003) Lattice Boltzmann method for moving boundaries. J. Comp. Phys. 184, 406-421.
SPE-164849-MS
9
22. Madland, M.V., K. Midtgarden, R. Manafov, R.I. Korsnes, T.G. Kristiansen, and A. Hiorth. The effect of temperature and brine composition on the mechanical strength of Kansas chalk. in International Symposium of the Society of Core Analysts. 2008. Abu Dhabi, UAE. 23. Madland M. V., Hiorth A., Korsnes R. I., Evje S. and Cathles L. (2009) Rock fluid interactions in chalk exposed to injection of seawater, MgCl2, and NaCl2 brines with equal ionic strength. EAGE-2009, A22. 24. Madland M. V., Hiorth A., Omdal E., Megawati M., Hildebrand-Habel T., Korsnes R. I., Evje S. and Cathles L. M. (2011) Chemical alterations induced by rock–fluid interactions when injecting brines in high porosity chalks. Trans. Porous Media 87(3), 679–702 25. M. Megawati, P. Ø. Andersen, R. I. Korsnes, S. Evje, A. Hiorth, and M. V. Madland. The effect of aqueous chemistry pH on the time dependent deformational behaviour of chalk-experimental and modelling studies. Flows and mechanics in natural porous media from pore to field scale. Pore2Field. 16-18 November 2011, IFP Energies nouvelles (France). 26. M. Megawati, P. Ø. Andersen, R. I. Korsnes, S. Evje, A. Hiorth, and M. V. Madland. The effect of aqueous chemistry pH on the time dependent deformational behaviour of chalk-experimental and modelling studies. Flows and mechanics in natural porous media from pore to field scale. Pore2Field. 16-18 November 2011, IFP Energies nouvelles (France). 27. M. Megawati, A. Hiorth, M.V. Madland. The Impact of Surface Charge on the Mechanical Behavior of High-Porosity Chalk. Rock Mechanics and Rock Engineering 2012. DOI 10.1007/s00603-012-0317-z 28. Pilliod Jr, J. E. and Puckett, E. G. (2004) Second-order accurate volume-of-fluid algorithms for tracking interfaces, J. Comp. Phys. 199, 464-502. 29. Succi S. (2001) The Lattice Boltzmann Equation, for Fluid Dynamics and Beyond. Oxford University Press. 30. Sukop M. C. and Thorne D. T. (2006) Lattice Boltzmann Modelling: An Introduction for Geoscientists and Engineers. Springer. 31. Tang G. and Morrow N. R. (1999) Oil recovery by waterflooding and imbibition – invading brine cation valency and salinity. SCA, 11. 32. Verhaeghe, F., Arnout, S. Blanpain, B. and Wollants P. (2005) Lattice Boltzmann model for diffusion-controlled dissolution of solid structures in multicomponent liquids, Phys. Rev. E, 72(3), 036308 33. Verghaeghe, F., Arnout, S., Blanpain, B. and Wollants, P. (2006) Lattice-Boltzmann modeling of dissolution phenomena. Phys. Rev. E 73, 1-10. 34. Yildiz H. O. and Morrow N. R. (1996) Effect of brine composition on recovery of moutray crude oil by waterflooding. J. Pet. Sci. Eng. 14, 159–168. 35. Zhang P. M., Tweheyo M. T. and Austad T. (2007) Wettability alteration and improved oil recovery by spontaneous imbibition of seawater into chalk: impact of the potential determining ions Ca2+, Mg2+, and SO42- Colloid. Surf. A: Physicochem. Eng.Aspects 301(1-3), 199–208.
Appendix A: Description of the ELVIRA VOF method The ELVIRA method is a Piecewise Linear Interface Construction (PLIC) method using VOF. Hence in every boundary cell ( ) the model gives a linear segment that represents the interface. The ELVIRA method estimates six candidate slopes from backward, central and forward differences of the column sums in the x-direction and y-direction for each boundary cell: ̃ ̃
∑ ∑
̃
∑
̃
∑
(12)
̃ ̃
∑ ∑
Note that the ̃ slopes are with respect to the coordinate system rotated 90° from the original coordinate system. A linear function ̃ is estimated from each of the six candidate slopes by matching the correct solid fraction in the cell. From the six candidate linear functions we choose the one resulting in least error in estimated solid fraction in the eight neighbouring
10
SPE-164849-MS
cells: (
∑
√
(13)
The linear function that minimizes the error ( ) is chosen as the best representation of the interface in the cell under consideration. We use this method to estimate the unit surface normal which can be calculated directly from the slope and the demand that | ⃗⃗| :
̃ ̃
√ √
̃
(14)
Paper 2: Pore Scale Modeling of Brine Dependent Permeability.
J. Pedersen, E. Jettestuen, J. L. Vinningland, M.V. Madland and A. Hiorth. Paper SCA2013-064 prepared for presentation at the International Symposium of the Society of Core Analysts held in Napa Valley, California, USA, 16-19 September, 2013.
PORE SCALE MODELING OF BRINE DEPENDENT PERMEABILITY Janne Pedersen1, Espen Jettestuen1, Jan Ludvig Vinningland1, Merete V. Madland2 and Aksel Hiorth1,2 1
International Research Institute of Stavanger (IRIS), 2University of Stavanger (UiS)
This paper was prepared for presentation at the International Symposium of the Society of Core Analysts held in Napa Valley, California, USA, 16-19 September, 2013
ABSTRACT The numerical study of pore surface evolution due to precipitation and dissolution of minerals requires a reliable representation of the time dependent evolution of the pore surface, as the evolution of the pore volume and surface area impacts both permeability and effective rate equations used in the “macroscopic” modeling of porous media. In this paper we present three different implementations of a moving boundary routine and discuss them in terms of 1) conservation of mass, 2) convergence for increasing grid resolution and 3) rotational invariance. We also study growth as a function of PeclétDamköhler number, we use the Carman-Kozeny equation to relate the predicted porosity and surface area evolution to permeability evolution. We observe that for high PeclétDamköhler numbers the permeability reduction is more affected by a large increase in surface area than porosity loss.
INTRODUCTION In several rock mechanical compression tests it has been observed that the permeability of the core decreases during the experiment. Part of this permeability loss is simply due to porosity loss caused by pure mechanical deformation, and part is due to chemical precipitation and dissolution processes in the pore space and/or intergranular contacts. Dissolution and precipitation will change the porosity and/or the specific surface area, which in turn affect the permeability of the rock. Flooding with seawater at 130°C through chalk cores makes the permeability drop to zero over a period of two weeks, while flooding with NaCl brine has a much lower effect on the permeability [1]. In order to predict the permeability-porosity evolution during core floods from pore scale simulations, it is necessary to have numerical methods that can describe the dynamic
evolution of the pore space as a consequence of rock-fluid interactions. The BGK lattice Boltzmann method (LBM) [2,3] is well suited to describe multiphase, reactive flow on the pore scale, and has been used in numerous studies. Lattice Boltzmann methods with a dynamic evolution of the pore space caused by reactive flows have been published [4,5], but these methods do not respect rotational invariance. By rotational invariance we mean that the growth of minerals should be independent of the orientation of the grid. In this paper we develop rotationally invariant numerical methods that allow reliable studies of changes in surface topology due to chemical reactions on the pore scale. On the pore scale diffusion is usually much faster than advection, thus it is a reasonable approximation to only consider diffusion. In this paper we study to which degree three different moving boundary routines (MBRs) fulfill conservation of mass, convergence for increased grid resolution, and rotational invariance for different Peclét-Damköhler (PeDa) numbers for the growth of an initial mineral seed in a diffusion field. We also study how porosity and surface area evolves with time, and the permeability evolution through a Carman-Kozeny relation.
METHOD The MBRs discussed in this paper are based on the Volume of Fluid (VOF) approach where each computational cell is assigned a solid fraction (sf) ranging from 0 (empty) to 1 (completely filled). In this work we consider only one solid mineral phase, which is coupled to one aqueous chemical species. However, the MBRs presented are easily extended to more complex cases. The update in sf is given by the difference in incoming and outgoing microscopic flux at the mineral surface. The microscopic fluxes are related to the macroscopic surface flux and species concentration, , by [4]:
̅
̃
(
̅)
⃗̅ ⃗⃗
̅
̃
(
̅)
(1)
where ̃ denotes the incoming distribution function and ̅ is the distribution reflected from the surface for direction . and ̅ are the LBM weights that distinguish between direct and diagonal links ( for direct links, for diagonal links and for the stationary state in the d2q9 model used here, where 9 discrete directions are given on a 2 dimensional square lattice [2]). ⃗̅ is the discrete LBM velocity pointing away from the surface, ⃗⃗ is the surface normal pointing away from the surface, ⃗⃗ is the chemical reaction flux for species , and is the lattice sound speed ( ⁄√ for d2q9). For the surface flux we have assumed linear kinetics complemented ( ) with a simplified surface tension term giving , where is the reaction rate, is the equilibrium concentration at the surface, is the strength of the surface tension, and is the cell curvature. Inserting this expression into Eq. (1) yields:
̅
̃
⃗
⃗⃗
(2)
where ⃗ ⃗⃗. PeDa is related to the reaction rate through ( ), where L is the system length. Below we describe three MBRs that use this boundary condition formulation. In method 1 the surface normal is not calculated from the solid fraction, but simply assumed to behave as ⃗ ⃗⃗ for all link directions, and we put . The change in solid fraction is given by (∑ where
̅
̃ )
(3)
is the molar volume of the mineral.
Method 2 differs from method 1 only by the calculation of ⃗⃗ used in equation (1). The surface normal is estimated with the Elvira VOF method [6], although other methods could also be used. In this VOF method the true surface is approximated by linear segments in boundary cells, i.e. cells with 0 < sf < 1. Method 3 uses the subgrid VOF estimation of the surface position to approximate the distance between nearby fluid nodes and the surface. This distance is then used to interpolate the distribution functions near the wall [7]. We have used the VOF height function (HF) technique described in [8] to estimate the cell curvature used in the surface tension term.
RESULTS The mineral growth simulations start from an initial solid seed rendered at different resolutions, similar to what is used in López et.al [9]. Figure 1 shows the growth of the initial seed when rotated 0° and 45° for PeDa = 10 and 100 for each of the three MBR methods. The initial concentration field used in all these simulations is a stationary solution obtained with and rate on the seed surface, a fixed boundary condition and rate on the circular outer boundary, and an initial concentration . All three methods conserve mass since the amount of solid mass precipitated on the surface equals the amount of mass extracted from the fluid phase. Rotational invariance is examined in Figure 1. The results show that the deviation from rotational invariance increases with increasing PeDa. However, method 3 shows a much smaller deviation for higher PeDa than the two simpler methods. In order to obtain better rotational invariance for higher PeDa numbers a surface tension term is introduced which also sets a limit on
the smallest spatial scales of the growing mineral. Hence, for increasing grid resolution the mineral shapes converge. Without this term more and more details will appear as the grid resolution increases, given that PeDa is high enough. Figure 2 shows the deviation from rotational invariance (E) as a function of grid resolution (L) for PeDa 100 with surface tension strength , for time step 12.000 at resolution 120x120 (outer circle radius of 50). We observe that as the resolution increases with a factor two, the gain in rotational invariance is a factor four for the deviation.
PeDa=10
PeDa=10
PeDa=10
PeDa=102
PeDa=102
PeDa=102
Figure 1: Rotational symmetry for PeDa 102 and 10 for methods 1, 2 and 3, from left to right respectively. Initial seed shown as the innermost contour.
Figure 3 shows the evolution of permeability, , porosity, , and specific surface area (SSA), S, for different PeDa numbers for method 3 without the surface tension term (same results as shown in Figure 1.). Permeability is related to porosity and SSA through ⁄ ) . The drop in porosity and increase the Carman-Kozeny (CK) relation ( in surface area are faster with increasing PeDa. The SSA will increase and the permeability will decrease at an increasing rate as more details grow on the solid surface. This is shown by the changing slopes seen for PeDa = 104. Edge-effects caused by approaching the outer boundary can also contribute to varying slopes at late times. Using the surface tension term would smooth the SSA evolution curves for high PeDa. For high PeDa numbers the permeability can approach zero in relatively short time.
1
E
10
slope = -2
0
10
2
10
L
Figure 2: Deviation from rotational invariance (E) as a function of grid resolution (L). Dotted line shows a straight line with slope -2 for comparison of slopes.
S
0.1
0.05
-2 0 0
-4
10
4
1
PeDa = 102 PeDa = 10
0.8
PeDa = 10
-6
10
2 timestep
4
2 timestep
4
4
x 10
/ 0
10
0.6
0
1
2 3 timestep
4
5 4 x 10
0.4 0
4
x 10
Figure 3: The evolution of the permeability based on the Carman-Kozeny relation (on the left), based on the evolution of the surface area (top right hand figure) and porosity (bottom right hand figure).
CONCLUSION It is important to have physical consistent models that can describe how the injected water moves through the pore space, interacts with minerals, and dynamically changes the properties of the pore surface, such as changes in geometry and wettability in order to interpret lab experiments. Therefore, in this paper, we have investigated the behavior of three moving boundary routines for conservation of mass, rotational invariance, and convergence for increasing grid resolutions. All methods are mass conserving, while the subgrid VOF method (method 3) displays better rotational symmetry compared to the two simpler methods. The deviation from rotational invariance increases with increasing PeDa for all three methods. To obtain rotational invariance and convergence for high
PeDa numbers we introduce a surface tension term which sets a minimum unstable wave length that is larger than the grid resolution. For low PeDa numbers we note that methods 1 and 2 give reasonable results and might be preferred over method 3 due to computational effectivity. Method 3 shows an increasing drop in porosity (increase in SSA) for increasing PeDa numbers. The rate at which SSA increases seems to increase at the time when higher resolution details start to grow on the mineral surface for high PeDa numbers. Permeability profiles also show that the permeability can approach zero for high PeDa numbers, without a significant porosity loss.
ACKNOWLEDGEMENTS The authors would like to thank the Research Council of Norway, BP Norge AS and the Valhall co-venturers, including Hess Norge AS, ConocoPhillips and the Ekofisk coventurers, including Total E&P Norge AS, ENI Norge AS, Statoil Petroleum AS and Petoro AS for financial support.
REFERENCES 1. Madland, M. V., Hiorth, A., Omdal, E., Megawati, M. and Hildebrand-Habel, T. et al. Chemical Alterations Induced by Rock-Fluid Interactions When Injecting Brines in High Porosity Chalks, Transp. Porous Med. (2011) 87, 679-702 2. Succi S., The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford University Press, Oxford (2001). 3. Sukop, M. C. and Thorne D. T. Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers, Springer, Netherlands (2006) 4. Verghaeghe, F., Arnout, S., Blanpain, B. and Wollants, P., Lattice-Boltzmann modeling of dissolution phenomena. Phys. Rev. E, (2006) 73, 1-10. 5. Kang, Q. and Lichtner, P. C., Pore Scale Modeling of Reactive Transport Involved in Geologic CO2 Sequestration, Transp. Porous Med, (2009) 82, 197-213 6. Pilliod Jr, J. E. and Puckett, E. G. Second-order accurate volume-of-fluid algorithms for tracking interfaces, J. Comp. Phys. (2004) 199, 464-502. 7. Bouzidi, M., Firdaouss, M. and Lallemand, P., Momentum transfer of a Boltzmannlattice fluid with boundaries, Physics of Fluids, (2001) 13, 11, 3452-3459. 8. Cummins, S. J., Francois, M. and Kothe, D. B., Estimating curvature from volume fractions, Computers and structures, (2005) 83, 425-434 9. López, J., Gómez, P and Hernández, J., A volume of fluid approach for crystal growth simulation, J. Comp. Phys. (2010) 229, 6663-6672.
Paper 3: Improved Lattice Boltzmann Models for Precipitation and Dissolution. J. Pedersen, E. Jettestuen, J. L. Vinningland and A. Hiorth. Transp. Porous Med. 104, 593-605 (20014)
Transp Porous Med (2014) 104:593–605 DOI 10.1007/s11242-014-0353-0
Improved Lattice Boltzmann Models for Precipitation and Dissolution J. Pedersen · E. Jettestuen · J. L. Vinningland · A. Hiorth
Received: 1 July 2013 / Accepted: 24 June 2014 / Published online: 12 July 2014 © Springer Science+Business Media Dordrecht 2014
Abstract A challenge when modeling mineral growth inside the pore space of a porous media is to minimize the effect of the computational grid on the shape of the minerals being formed. Pore surface area and volume are important quantities in estimating upscaled permeability and effective rate equations, which emphasize the importance of models that minimize or completely eliminate grid effects. In this paper, we study how the initial orientation of the solid structure on the numerical grid affects the growth pattern due to precipitation in a lattice Boltzmann model. We have implemented a volume of fluid method to represent the solid interface, and we introduce a surface tension term that extensively reduces the dependency on the underlying numerical grid. We study both diffusion-limited and reaction-limited precipitation. In the diffusion-limited case, instabilities will develop on small scales. The surface tension term effectively introduces a short wavelength cut off which limits the unstable precipitation and reduces grid effects. We argue that the surface tension term is needed to obtain a growth pattern independent of the initial orientation on the underlying grid in the diffusion-limited case, and that simpler models can be used in the reaction-limited case. Keywords
Lattice Boltzmann · Mineral growth · Moving boundary · Pore space
1 Introduction In the last decades pore-scale modeling has been used to predict petrophysical properties of porous rocks, such as permeability, relative permeability, capillary pressure, resistivity index, elastic properties, etc. (Arns et al. 2002, 2004; Knackstedt et al. 2004; Jin et al. 2007; Øren and Bakke 2002; Øren et al. 2007). Part of the reason for the growing interest in porescale modeling is the fact that special core analysis (SCAL) programs to determine these petrophysical properties could be extremely time consuming and challenging to perform under realistic reservoir conditions. As an example, to generate a capillary pressure curve
J. Pedersen (B) · E. Jettestuen · J. L. Vinningland · A. Hiorth International Research Institute of Stavanger (IRIS), Prof. Olav Hanssensvei 15, 4021 Stavanger, Norway e-mail:
[email protected]
123
594
J. Pedersen et al.
(one cycle) for a chalk core using the porous plate technique takes approximately 1 year (Ashan et al. 2012). Within the oil and gas industry, the need for alternatives to traditional SCAL analyses has resulted in commercial companies that offer pore-scale simulations as a service. Fluid–rock interactions such as dissolution/precipitation or wetting properties of the surface are usually not considered, or are taken as static input to the model. However, fluid–rock interactions are essential in order to explain the effect of brine chemistry on oil recovery. Enhanced oil recovery has been observed when injecting low salinity water in sandstone (Tang and Morrow 1997; Webb et al. 2004; Morrow and Buckley 2011), and by changing the divalent ion concentration of the injected water in carbonates (Zhang and Austad 2005; Al-Shalabi et al. 2013). Still there is no consensus in the research community as to why the additional oil is released. Most likely this is due to the fact that there is not one single explanation for the release of additional oil but rather an interplay of several mechanisms taking place at the same time. Pore-scale models that can describe how water flows through the pore space and interacts with individual minerals that lead to changes in ion concentrations, and in the oil components that are adsorbed on the surface would be extremely valuable, not only to interpret lab-scale experiments, but also to optimize the brine chemistry to maximize oil recovery. In Hiorth et al. (2013), we developed a lattice Boltzmann (LB) model that was capable of simulating ion exchange, adsorption/desorption, and non-linear precipitation/dissolution interactions. That model did not include textural changes in the pore space, which is the topic of this paper. In this paper, we do not consider flow and geochemical reactions in general, because it is not essential for developing a consistent moving boundary routine. LB models have been applied to predict pore space evolution as a consequence of precipitation and/or dissolution for several years, but the detailed evolution of the moving boundaries has received limited attention. Bouzidi et al. (2001) present a moving boundary scheme consisting of a combination of the bounce-back rule and interpolations for boundaries that do not lie half-way between lattice nodes in order to represent curved surfaces. Lallemand and Luo (2003) extend the model proposed by Bouzidi et al. (2001) and apply the boundary condition to solid objects moving in a fluid phase. Verhaeghe et al. (2005) use a linear kinetic boundary condition and interpolations/extrapolations to model evolution of the boundary due to dissolution. Kang et al. 2002, 2003, 2004, 2010 have implemented a LB simulator for reactive transport at the pore scale, and their first description of a moving boundary routine is given by Kang et al. (2004). Their growth routine is random in the choice of which fluid nodes change to solid nodes. This randomness is removed in a later update of the method (2010). Verhaeghe et al. (2006) introduce a flux boundary condition to the method described in 2005. Arnout et al. (2008) build on the work done by Verhaeghe et al. (2005, 2006), and in this work the true position of the interface is approximated by two simplified formulas. In our experience, the above-mentioned methods will be affected by the underlying mathematical grid. The reason is that they either do not include sub-grid resolution of the boundary (Kang et al.), they do not use information about the position and/or direction of the surface (Verhaeghe et al.), and they do not take into account the effect of surface curvature which is essential for growth phenomena. In order to calculate the correct distance from fluid nodes to the interface, the position of the interface and the interface normal is needed. Surface position and normal can be estimated with different interface tracking methods. Examples of some commonly used methods for fixed grids are volume of fluid (VOF) methods, particle tracking methods, and level set methods. Here we follow the suggestion by Verhaeghe et al. (2006) that a normal can be calculated from the volume fraction field by using a VOF method. In their work, they suggested a more
123
Improved Lattice Boltzmann Models for Precipitation and Dissolution
595
simple approach when studying dissolution phenomena. When studying growth phenomena, we experienced that this method was affected by the grid orientation. In this paper, we study models for precipitation in the pore space, and the evolution of the solid boundary should not be dependent on the structure’s orientation in the underlying grid, if there are no external forces. Thus, in this paper, we consider only diffusion as a transport mechanism and ignore other external forces, such as gravity, external pressure drops, electromagnetic forces, etc. The VOF method is used to estimate the surface position and normal. The paper is structured as follows; Sect. 2 describes the numerical model, including the LB model, the boundary conditions and the three different moving boundary routines. Section 3 describes the simulations and presents the results, while summary and conclusions are given in Sect. 4. Some mathematical details are collected in an Appendix.
2 Numerical Model 2.1 LB Model for the Diffusion Equation Evolution of the concentration of a chemical species in the absence of an external velocity field (u = 0) is described by the diffusion equation (1), where c is the concentration of the chemical species and D is a constant diffusion coefficient: ∂c (1) = D∇ 2 c. ∂t It can be shown that the LB equation (LBE) recovers the macroscopic diffusion equation (Wolf-Gladrow 1995). The LBE (2) describes evolution of distribution functions (gα ) for virtual particles moving with different velocities (eα ) in different discrete directions (α). eq An equilibrium distribution function (gα ) is given by the macroscopic system variables eq (concentration c and fluid velocity u). The form of gα when u = 0 is given by Eq. (3), where ωα are weight coefficients for the different discrete directions. The relaxation time τ is related to the diffusion coefficient D through Eq. (4), where cs is the lattice sound speed. gα (x + eα δt, t + δt) = gα (x, t) − eq gα
1 eq gα (x, t) − gα (ρ, u) , τ
= ωα c, 1 D τ = 2 + . cs 2
(2) (3) (4)
The concentration of the species is found by summing over distribution functions in all the discrete directions: gα . (5) c= α
We use a two-dimensional lattice with nine velocity vectors commonly known as the D2Q9 model. The velocity √ vectors and weight coefficients are given by Eqs. (6)–(8), and the lattice sound speed cs is 1/ 3. ex = [0, 1, 1, 0, −1, −1, −1, 0, 1],
(6)
e y = [0, 0, 1, 1, 1, 0, −1, −1, −1], 4 1 1 1 1 1 1 1 1 , , , , , , , , . w= 9 9 9 9 9 36 36 36 36
(7) (8)
123
596
J. Pedersen et al.
Fig. 1 Visualization of distributions used in the boundary condition. At time t, the gα distribution pointing toward the surface is known. At time t + 1, this distribution is streamed to a virtual node inside the wall (x + 1); g˜ α (x + 1, t + 1) = gα (x, t). At this timestep, the unknown distribution at the fluid node (x) is given in terms of the g˜ α -distribution inside the wall; gα¯ (x, t + 1) = f (g˜ α (x + 1, t + 1)). For simplicity, we use the following notations: g˜ α = g˜ α (x + 1, t + 1) and gα¯ = gα¯ (x, t + 1)
2.2 Flux Boundary Condition Generally, a boundary condition within the LB framework gives the unknown distributions leaving the surface, gα¯ , as a function of the known incoming distributions entering the surface, g˜ α , where α and α¯ denote the opposite directions (i.e., eα¯ = −eα ) that form a link between a pair of wall an fluid nodes. Figure 1 shows the distributions entering and leaving the surface for two consecutive time steps, t and t + 1, for one link. At time t, the gα distribution pointing toward the surface is known. At time t + 1, this distribution is streamed to a virtual node inside the wall (x + 1); g˜ α (x + 1, t + 1) = gα (x, t). At this timestep, the unknown distribution at the fluid node (x) is given in terms of the g˜ α -distribution inside the wall; gα¯ (x, t + 1) = f (g˜ α (x + 1, t + 1)). For example, when precipitation is being considered only part of the incoming distribution is reflected from the surface, while the rest is absorbed to the surface. For simplicity, we use the following notations: g˜ α = g˜ α (x + 1, t + 1) and gα¯ = gα¯ (x, t + 1). For a surface that is constantly changed by dissolution and precipitation of minerals, a flux boundary condition can be implemented to give the unknown distributions leaving the surface, gα¯ , as a function of the known incoming distributions entering the surface, g˜ α (Verhaeghe et al. 2006; Bouzidi et al. 2001): gα¯ − g˜ α = 6ωα eα¯ · JR n,
(9)
where JR is the macroscopic chemical flux on a surface that lies half-way between the grid points, and eα¯ is the velocity vector pointing away from the surface. The concentrations are related through a similar expression (Verhaeghe et al. 2006): gα¯ + g˜ α = 2ωα c.
123
(10)
Improved Lattice Boltzmann Models for Precipitation and Dissolution
597
The implemented boundary condition naturally extends to general rate equations (Hiorth et al. 2013) and keeps its structural form for all LB lattice types. Hence, it is simple to implement for complex pore-scale geometries. In the simulations presented in this paper, we use a linear rate law to compare the proposed methods, but more complicated rate laws are also applicable (Hiorth et al. 2013). We have added a curvature-dependent surface tension term, following the Gibbs–Thompson approach (see e.g., Lasaga 1998), to set a low wavelength cutoff for surface growth. The chemical flux is given by JR = −k c − ceq − γ κ, (11) where k is the rate constant, ceq is the equilibrium concentration, γ is a constant, and κ is the surface curvature. The rate constant is usually determined experimentally, while the equilibrium concentration is determined from chemical equilibrium constants. Surface tension can be added to any complex rate law to include the desired wavelength cutoff (see e.g., Lasaga 1998). By solving Eq. (10) for c, inserting that into Eq. (11) and then inserting the flux expression (JR ) into Eq. (9) yield the following moving boundary condition gα¯ =
kα 1 − kα 6ωα γ κ 2ωα ceq + g˜ α − eα¯ · n, 1 + kα 1 + kα 1 + kα
(12)
where kα = 3keα¯ · n. Equation (12) equals the bounce-back condition (zero flux BC) in the limit k, γ → 0, and is valid for a boundary that is located half-way between the computational nodes. For low reaction rates, the system is approximated to diffusive equilibrium, and the concentration in the pore space is approximately constant. Hence the system reduces to a constant flux boundary condition. For high reaction rates (when γ = 0) the boundary condition reduces to a Dirichlet boundary condition, gα¯ + g˜ α = 2ωα ceq , and is independent of the surface normal. 2.3 Three Moving Boundary Methods Dissolution and precipitation constantly change the position and shape of the solid–fluid interface, and it is no longer constrained to lie half-way between the computational nodes. To allow sub-grid resolution of the interface, the boundary condition in Eq. (12) must be modified, and we need a method for keeping track of the fluid–solid boundary. Here, we use a VOF approach (Pilliod and Puckett 2004) to track the fluid–solid interface. Each computational node is given a solid fraction, a, ranging from zero (only fluid) to one (only solid); nodes crossed by the fluid–solid boundary have 0 < a < 1. In each time step, the solid fraction is updated according to the material balance in the fluid phase (Verhaeghe et al. 2006):
−1
a = Vm gα¯ − g˜ α , (13) links
Vm−1
accounts for the difference in densities between the fluid and the solid phase. where The model is described for only one chemical species, but can easily be extended to several mineral phases and chemical components by assigning one set of distribution functions gαi for each chemical component i and one solid fraction field a j for each mineral phase j. In this work, we have studied three moving boundary methods with increasing complexity. The two first methods are simplifications of the full VOF method (method 3). Method 1 The boundary condition is given by Eq. (12) which is valid for a surface positioned half-way between the grid nodes. Hence, we assume the interface is always in the
123
598
J. Pedersen et al.
middle between fluid and solid nodes and ignores the effect of a sub-grid surface resolution. The scalar eα · n is set to unity for all α. Method 2 The surface normal is estimated using the ELVIRA VOF method (see Appendix) to obtain a surface-dependent scalar eα · n in the boundary condition (12). The inclusion of a VOF method increases the computational cost of the simulation. Method 3 This method implements both an estimate of n and a sub-grid resolution of the fluid–solid interface position using ELIVRA VOF. A sub-grid resolution of the interface requires an interpolation scheme to obtain the outgoing distribution functions in fluid nodes that have solid neighbors as described in Bouzidi et al. (2001) (see Appendix).
3 Results 3.1 Simulation Setup The three methods described above are used to simulate the growth of a solid seed centered in the middle of a circular container with a fixed outer boundary. The interface of the initial seed is given by López et al. (2010); x(s) = [0.1 + 0.02cos(8πs)]cos(2πs),
(14)
y(s) = [0.1 + 0.02cos(8πs)]sin(2πs).
(15)
The largest radius of the initial seed is 0.12, and the radius of the outer boundary is 1. We have used a grid size of 0.02. The outer boundary is circular to facilitate rotations of the initial seed within the box. Simulations have been performed with the initial seed rotated 0◦ , 19◦ , and 45◦ . 19◦ is chosen to assure that the solid seed is not symmetric with respect to the underlying grid, and that the axes of symmetry do not coincide with any of the discrete LB directions. This case should thus be representative for the random geometries encountered in simulations on realistic pore networks. The three seeds are shown in Fig. 2. The growth depends on the rate of reaction on the solid surface and on the rate of transport of species to and from the surface, which in our case is by diffusion. We follow the approach by Kang et al. (2003) and study the growth as function of the dimensionless Peclét–Damköhler number, PeDa, defined as the rate of reaction over rate of diffusion; PeDa =
(a) 0o
(b) 19o
kL , D
(16)
(c) 45o
Fig. 2 Initial seeds used in the surface growth simulations. The seed in a is given by Eqs. (14)–(15), while the seeds in b and c are rotated with respect to a
123
Improved Lattice Boltzmann Models for Precipitation and Dissolution
599
where k is the chemical reaction rate, L is the linear size of the system, and D is the diffusion coefficient. PeDa corresponds to the square of the Thiele modulus (see e.g., Aris (1975)), named after Thiele (1939) for his work on reaction-diffusion problems in catalysts. In the low PeDa limit, the transport away from the surface (through diffusion) is much faster than the chemical reactions on the surface, and the growth is comparable to constant flux growth. When PeDa increases, the rate of reaction dominates on increasingly smaller scales and diffusion driven instabilities with increasingly small wavelengths will occur. Hence, grid independence is expected to be more difficult to obtain for larger Peclét–Damköhler numbers. The initial concentration field is a stationary solution with c = 0.6 and reaction rate k = 0.014 on the seed surface, and with c = 1.0 and reaction rate k = 14 on the circular outer boundary (values in lattice units). The boundary condition on the circular outer boundary is unchanged throughout the growth simulations, and the equilibrium concentration, ceq , at the evolving surface is set to 0.6 in all simulations. Simulations have been performed with and without surface tension. 3.2 Simulation Results For low PeDa numbers, the growth patterns are virtually independent of grid orientation for all the three methods, even without surface tension, see Fig. 4. In the limit of PeDa → 0, the flux onto the surface is constant and we have checked how the three methods behave for the constant flux growth of a circle, see Fig. 3. (Note the use of a circle as the initial seed for these simulations. All other simulations use the seeds shown in Fig. 2.) Methods 2 and 3 give the same results since the flux is constant and no interpolation is needed. Hence, this is a test on the importance of estimating the surface normal at low PeDa numbers. Methods 2 and 3 grow more or less like a perfect circle, while method 1 shows some deviation. Figure 4 shows contours (a = 0.5) for the growing solid seed for PeDa = 100 and 10 for the three methods. Blue contours indicate the initial seed rotated 0◦ relative to the underlying grid, red contours indicate 45◦ rotation, and black contours indicate 19◦ rotation (for illustration purposes, the contours are all rotated back to 0◦ ). The black contour in the center is the initial seed. Method 3 shows a smaller grid dependence than methods 1 and 2 at PeDa = 100, while all three methods are more or less grid independent at PeDa = 10. Lower PeDa numbers
(a) Method 1
(b) Method 2 and 3
Fig. 3 Contours from constant flux growth. Initial circle in black. Black dotted curves represent perfect circles
123
600
J. Pedersen et al.
(a) Method 1, PeDa=100
(b) Method 2, PeDa=100
(c) Method 3, PeDa=100
(d) Method 1, PeDa=10
(e) Method 2, PeDa=10
(f) Method 3, PeDa=10
Fig. 4 Simulated interface positions for PeDa = (10, 100) using methods 1–3. Blue, red, and black contours, respectively, represent seeds oriented 0◦ , 45◦ , and 19◦ with respect to the grid, see Fig. 2. The contours are rotated until the initial seeds overlap in order to show the deviation between contours of different methods, which indicate that the underlying mathematical grid affects the results
(a) Method 1, PeDa=100
(b) Method 2, PeDa=100
(c) Method 3, PeDa=100
Fig. 5 Results from simulations with surface tension (γ = 0.01) for methods 1–3 with PeDa = 100. Blue 0◦ , red 45◦ , black 19◦ (only in c). For each method, three simulations have been performed with varying orientation of the initial seed (see Fig. 2) for PeDa = 100 and surface tension strength γ = 0.01. The simulated contours have then been rotated until the initial seeds overlap, in order to show the deviation between the contours at later times
mean slower growth and the system is stable for the given system size. At PeDa = 100, the interface growth is unstable at scales larger than the grid resolution and grid orientation effects become visible. To improve the surface growth model for high PeDa numbers surface tension was included, and Fig. 5 shows the results for PeDa = 100 with surface tension strength γ = 0.01.
123
Improved Lattice Boltzmann Models for Precipitation and Dissolution 0.5
0.07
method 3, γ=0, 0 vs 45 method 3, γ=0, 0vs 19 method 2, γ=0, 0 vs 45 method 1, γ=0, 0 vs 45
0.4
method 3, γ =0.01, 0 vs 45 method 3, γ =0.005, 0vs 45 method 2, γ =0.01, 0 vs 45 method 1, γ =0.01, 0 vs 45
0.06 0.05
e
e
0.3
601
0.04
0.2 0.03 0.1 0
0.02 0
10
20
30
40
3
10 timesteps (a) No surface tension
50
0.01
0
10
20
30
40
50
60
3
0
10 timesteps (b) With surface tension
0.01
Fig. 6 Relative error (e) between contours
With the inclusion of surface tension, the growth is now less dependent on the grid orientation for all three methods, but method 3 stands out as the better one. This observation is verified by the error plots in Fig. 6 showing a relative error e = E/A0 where E is the area between the contours for the 0◦ structure and the 19◦ /45◦ structure given by E = 2 AC − (A0 + Arot ). AC is the area of the composite figure obtained from the contour of the 0◦ structure and the 19◦ /45◦ structure (rotated back to 0◦ ), A0 is the area of the 0◦ solid structure and Arot is the area of the rotated solid structure (19◦ /45◦ ). Figure 6a shows the error for PeDa = 100 when surface tension is not included. The relative error starts at approximately 5 % and increases with the growing solid to approximately 40 % for methods 1 and 2 and 30 % for method 3. Figure 6b shows the relative error for PeDa = 100 when surface tension is included, and the error is now reduced to a maximum of approximately 6 % for method 2 and below 2 % for method 3. Hence, the error is reduced by 1/10 with the inclusion of a surface tension strength of γ = 0.01. It is also worth noting that the error of method 3 is less than half the error of methods 1 and 2. Another observation is that the error decreases initially before it starts to increase. This can be due to the relatively poorer resolution of the initial solid structure compared to the later structures. This might be explained by statistical effects since the initial structure covers less grid nodes compared to the larger structures at later times.
4 Summary and Conclusions Three different implementations of moving boundary conditions in a LB model are presented with the purpose of minimizing the grid effects. The implementations differ in the amount of information they include about the solid–fluid interface in terms of a normal vector and a sub-grid resolution of the interface. In the limit of low PeDa, we can assume that the system is in diffusive equilibrium, and the LB boundary conditions (12) reduce to a constant flux boundary condition since the concentration in the pore space is approximately constant. For this case, grid independence is achieved by including the surface normal in the boundary condition; the exact location of the interface is not needed. This is confirmed by the good behavior of methods 2 and 3 in the limit of low PeDa numbers. On the other hand, for large PeDa we get a boundary condition that is independent of the surface normal as boundary condition (12) reduces to a Dirichlet boundary condition: gα +gα = 2wα ceq . The system now becomes diffusion limited and we observe that all three methods show the same tendency of stronger dependence on
123
602
J. Pedersen et al.
grid orientation for higher PeDa numbers, i.e., larger errors when the grid is rotated. To overcome this effect, surface tension is added in the boundary condition to introduce a lower length scale in the simulations, which is larger than the size of the grid and thus regularizes the surface growth. So, even though we do not need the surface normal for the Dirichlet boundary condition, we need the curvature to regularize the growth. Hence, for all PeDa, we need detailed information about the surface. From our simulation results, method 3 stands out as the best method with an error less than half the error in methods 1 and 2. The error at PeDa = 100 is below 2 %. On the other hand, method 1 is the most computationally efficient, and the error for low PeDa is not too large. For large scale simulations where computational speed is critical compared to the accuracy of the surface description, this method would be a good choice. The methods are presented for a single-solid phase and a single-chemical component. However, the methods can easily be extended to include several mineral phases and chemical components. This is the topic for further work. Acknowledgments The authors would like to thank the three anonymous reviewers for their valuable comments and suggestions to improve the manuscript, and the Research Council of Norway, BP Norge AS and the Valhall co-venturers, including Hess Norge AS, ConocoPhillips and the Ekofisk co-venturers, including Total E& P Norge AS, ENI Norge AS, Statoil Petroleum AS, and Petoro AS for financial support
Appendices ELVIRA VOF Method The ELVIRA VOF method is a piecewise linear interface construction method, i.e., the interface is represented by linear functions in each boundary cell (0 < a ≤ 1). Six candidate slopes from backward, central, and forward differences of the column sums in the x- and y-direction for each boundary cell are calculated m˜ bx =
1
ai, j+l − ai−1, j+l ,
(17)
1 1 ai+1, j+l − ai−1, j+l , 2
(18)
l=−1
m˜ cx =
l=−1
m˜ xf =
1
ai+1, j+l − ai, j+l ,
(19)
ai+l, j − ai+l, j−1 ,
(20)
1 1 ai+l, j+1 − ai+l, j−1 , 2
(21)
l=−1 y
m˜ b =
1 l=−1
y
m˜ c =
l=−1
y
m˜ f =
1
ai+l, j+1 − ai+l, j .
(22)
l=−1
Note that the m˜ y slopes (20)–(22) are with respect to the coordinate system rotated 90◦ from the original coordinate system.
123
Improved Lattice Boltzmann Models for Precipitation and Dissolution
603
A linear function y = mx ˜ + b is estimated to each of the six candidate slopes from the correct solid fraction in the cell. From the six candidate linear functions, we choose the one that yields least error in estimated solid fraction in the eight neighboring cells (23). The linear function that minimizes (23) is chosen as the representation of the interface in cell (i, j). 2 E= a − a est . (23) neighbors
When a linear function has been found in the VOF routine, the position of the surface is known and the distance from the interface to the nearest fluid node along the discrete lattice directions, qα , can easily be calculated. Curvature Calculation The curvature of the interface is needed in the surface tension term in the boundary condition (12), and it is estimated using the HF method (Cummins et al. 2005). In the HF method, a height function is given by the sum of solid fractions along rows or columns. The sum is taken along the direction in which the component of the estimated surface normal is largest, i.e., if n x < n y the height function, HF, is defined as HF(i, j) =
j+3
a(i, j).
(24)
j−3
The curvature, κ, is defined in cell (i, j) if 3 < HF < 4. The curvature is then given by ∂ 2 HF ∂x2
κ= 2 3/2 , 1 + ∂HF ∂x
(25)
where the derivatives are estimated using a central differences scheme. In boundary cells with HF ≤ 3 or HF ≥ 4, we assign the curvature from the neighbor cell along a direct link with the same condition for the normal components as cell (i, j), i.e., if n x < n y in cell (i, j) the curvature in cell (i, j − 1) or (i, j + 1) is assigned to cell (i, j). If none of these two cells have a defined curvature, the curvature from a cell along the other direction, in this case the x-direction, is assigned to cell (i, j). Interpolation Scheme The boundary condition (12) can be updated to hold for any position of the fluid–solid interface. The position of the surface is given by the distance from the surface to the fluid node closest to the surface (fn) along each discrete direction, qα . We follow the procedure described by Bouzidi et al. (2001) and introduce a point P such that for qα < 1/2 the distance from P to the surface and back to fn is unity (see Fig. 7), and for qα > 1/2 the distance from fn to the surface and back to P is unity (see Fig. 7). Calculation of the BC is different for the two cases qα < 1/2 and qα > 1/2. For qα < 1/2 we first interpolate to find the outgoing distribution at point P (26), then this distribution undergoes the BC and travels to fn (27); gα (P) = gα (fn, t) + (1 − 2qα ) [gα (fn + e¯ α , t) − gα (fn, t)] , kα 1 − kα 6ωα γ κ 2ωα ceq + gα (P) − eα¯ · n. gα¯ (fn) = 1 + kα 1 + kα 1 + kα
(26) (27)
123
604
J. Pedersen et al.
Fig. 7 Sketch over symbols and distances used in the interpolation routine. Cells are marked with black lines and nodes by crosses. The length of the two arrows is 1
For qα > 1/2 the distribution at fn undergoes the BC and travels to point P (28), then we interpolate to find the new distribution at fn (29); kα 1 − kα 6ωα γ κ 2ωα ceq + gα (fn) − eα¯ · n, 1 + kα 1 + kα 1 + kα 2qα − 1 gα¯ (fn) = gα¯ (P) + [gα¯ (fn + eα¯ ) − gα¯ (P)] . 2qα gα¯ (P) =
(28) (29)
The distribution functions after collision but before streaming are denoted gα , and distribution functions after both collision and streaming are denoted gα¯ . A linear interpolation scheme has been used for the work presented in this paper.
References Al-Shalabi, E.W., Sepehrnoori, K., Delshad, M.: Mechanisms behind low salinity water flooding in carbonate reservoirs. In: Presented at SPE Western Regional and AAPG Pacific Section Meeting, 2012 Joint Technical Conference, , Monterey, CA, USA, 19–25 April 2013 Aris, R.: Mathematical Theory of Diffusion and Reaction in Permeable Catalyst. Oxford University Press, London (1975) Arnout, S., Verhaeghe, F., Blanpain, B., Wollants, P.: Lattice Boltzmann model for diffusion-controlled indirect dissolution. Comput. Math. Appl. 55, 1377–1391 (2008) Arns, C.H., Knackstedt, M.A., Pinczewski, V., Garboczi, E.J.: Computation of linear elastic properties from microtomographic images: methodology and agreement between theory and experiments. Geophysics 67, 1396–1405 (2002) Arns, C.H., Knackstedt, M.A., Pinczewski, V., Martys, N.S.: Virtual permeametry on microtomographic images. J. Pet. Sci. Eng. 45, 41–46 (2004) Ashan, R., Madland, M.V., Bratteli, F., Hiorth, A.: A study of sulphate ions—effects on aging and imbibition capillary pressure curve. In: Presented at the International Symposium of the Society of Core Analysts held in Aberdeen, Scotland, UK, 27–30 August 2012 Bouzidi, M., Firdaouss, M., Lallemand, P.: Momentum transfer of a Boltzmann-lattice fluid with boundaries. Phys. Fluids 13(11), 3452–3459 (2001) Cummins, S.J., Francois, M., Kothe, D.B.: Estimating curvature from volume fractions. Comput. Struct. 83, 425–434 (2005)
123
Improved Lattice Boltzmann Models for Precipitation and Dissolution
605
Hiorth, A., Jettestuen, E., Cathles, L.M., Madland, M.V.: Precipitation, dissolution, and ion exchange processes coupled with a lattice Boltzmann advection diffusion solver. Geochim. Cosmochim. Acta 104, 99–110 (2013) Jin, G., Torres-Verdin, C., Radaelli, F., Rossi, E.: Experimental validation of pore-level calculations of static and dynamic petrophysical properties of clastic rocks. SPE paper 109547. In: Proceedings of the 2007 SPE Annual Technical Conference and Exhibition, Anaheim, CA, 11–14 November 2007 Kang, Q., Zhang, D., Chen, S., He, X.: Lattice Boltzmann simulation of chemical dissolution in porous media. Phys. Rev. E 65, 036318 (2002) Kang, Q., Zhang, D., Chen, S.: Simulation of dissolution and precipitation in porous media. J. Geophys. Res. 108 (2003). doi:10.1029/2003JB002504 Kang, Q., Zhang, D., Lichtner, P.C., Tsimpanogiannis, I.N.: Lattice Boltzmann model for crystal growth from supersaturated solution. Geophys. Res. Lett. 31 (2004). doi:10.1029/2004GL021107 Kang, Q., Lichtner, P.C., Viswanathan, H.S., Abdel-Fattah, A.I.: Pore scale modeling of reactive transport involved in geologic CO2 sequestration. Transp. Porous Media 82, 197–213 (2010) Knackstedt, M.A., Arns, C.H., Limaye, A., Sakellariou, A., Senden, T.J., Sheppard, A.P., Sok, R.M., Pinczewski, W.V.: Digital Core Laboratory: Properties of Reservoir Core Derived from 3D Images. SPE paper 87009, Kuala Lumpur (2004) Lallemand, P., Luo, L.-S.: Lattice Boltzmann method for moving boundaries. J. Comput. Phys. 184, 406–421 (2003) Lasaga, A.C.: Kinetic Theory in the Earth Sciences. Princeton University Press, Princeton (1998) López, J., Gómez, P., Hernández, J.: A volume of fluid approach for crystal growth simulation. J. Comput. Phys. 229, 6663–6672 (2010) Morrow, N., Buckley, J.: Improved oil recovery by low-salinity waterflooding. J. Pet. Technol. 63(5), 106–112 (2011) Øren, P.E., Bakke, S.: Process based reconstruction of sandstones and prediction of transport properties. Transp. Porous Media 46, 311–314 (2002) Øren, P.E., Bakke, S., Held, R.: Direct pore-scale computation of material and transport properties for North Sea reservoir Rocks. Water Resour. Res. 43, W12S04 (2007) Pilliod Jr, J.E., Puckett, E.G.: Second-order accurate volume-of-fluid algorithms for tracking material interfaces. J. Comput. Phys. 199, 465–502 (2004) Tang, G.Q., Morrow, N.R.: Salinity, temperature, oil composition, and oil recovery by waterflooding. SPE Reserv. Eng. 12(4), 269–276 (1997) Thiele, E.W.: Relation between catalytic activity and size of particle. Ind. Eng. Chem. 31, 916–920 (1939) Verhaeghe, F., Arnout, S., Blanpain, B., Wollants, P.: Lattice Boltzmann model for diffusion-controlled dissolution of solid structures in multicomponent liquids. Phys. Rev. E 72, 036308 (2005) Verhaeghe, F., Arnout, S., Blanpain, B., Wollants, P.: Lattice-Boltzmann modeling of dissolution phenomena. Phys. Rev. E 73, 036316 (2006) Webb, K.J., Black, C.J.J., Al-Ajeel, H.: Low salinity oil recovery—log-inject-log. In: Presented at SPE/DOE Symposium on Improved Oil Recovery, Tulsa, Oklahoma, 17–21 April 2004 Wolf-Gladrow, D.: A lattice Boltzmann equation for diffusion. J. Stat. Phys. 79, 1023–1032 (1995) Zhang, P., Austad, T.: Waterflooding in chalk: relationship between oil recovery, new wettability index, brine composition and cationic wettability modifier. In: Presented at SPE Europec/EAGE Annual Conference, Madrid, Spain, 13–16 June 2005
123
Paper 4: A dissolution model that accounts for coverage of mineral surfaces by precipitation in core oods. J. Pedersen, E. Jettestuen, M. V. Madland, T. Hildebrand-Habel, R. I. Korsnes, J. L. Vinningland and a. Hiorth. Submitted to Advances in Water Resources. Accepted after
major revision.
A dissolution model that accounts for coverage of mineral surfaces by precipitation in core floods Janne Pedersena,c,∗, Espen Jettestuena,c , Merete V. Madlandb,c , Tania Hildebrand-Habeld , Reidar I. Korsnesb,c , Jan Ludvig Vinninglanda,c , Aksel Hiortha,b,c a IRIS,
P.O. Box 8046, N-4068 Stavanger, Norway of Stavanger, N-4036 Stavanger, Norway c The National IOR Centre of Norway d Norwegian Petroleum Directorate, P.O. Box 600, N-4003 Stavanger, Norway b University
Abstract In this paper, we propose a model for evolution of reactive surface area of minerals due to surface coverage by precipitating minerals. The model is used to interpret results from an experiment where a chalk core was flooded with MgCl2 for 1072 days, and calcite dissolves while magnesite precipitates. The model successfully describes the long-term behavior of the measured effluent concentrations, and the more or less homogeneous distribution of magnesite found in the core after 1072 days. Further, the model predicts that precipitating magnesite minerals form as larger crystals or aggregates of smaller size crystals, and not as a thin flakes or a monomolecular layer. To match the simulations to the experimental data after approximately one year of flooding, a rate constant that is four orders of magnitude lower than reported by powder experiments had to be used. We argue that a static rate constant is not sufficient to describe a chalk core flooding experiment lasting for nearly three years. The model is a necessary extension of standard rate equations in order to describe long term core flooding experiments where there is a large degree of textural alteration. Keywords: Dissolution, Precipitation, Surface coverage, Lattice Boltzmannn, Reactive flow
1. Introduction When a non-equilibrium pore fluid contacts individual minerals at the pore walls of natural rocks, the system will tend to minimize its free energy. This process can lead to changes of the pore surface such as a change in surface charge and potential due to surface complexation, and/or alteration of the mineral ∗ Corresponding
author Email address:
[email protected] (Janne Pedersen)
Preprint submitted to Advances in Water Resources
June 30, 2015
assemblies in the pore space due to dissolution and precipitation. Reactive flow modeling [46] couples fluid transport to geochemical calculations to predict compositional changes to both the rock and the pore fluid. Helgeson and coworkers were the first to model water-rock interactions as a coupled system of dissolution and precipitation reactions [17, 18]. Their model was applied to chemical weathering, which is still one of the major fields for application of reactive flow models [45, 33]. However, the so-called reaction path models pioneered by Helgeson et al. did not include explicit treatment of real-time kinetics. Their approach can be used for description of batch or closed systems, but these systems do not represent natural systems where transport is an important driving force [46]. Modern multi-component reactive flow models date back to the mid 1980s. Lichtner [28] presented a continuum model for transport in porous media. Reacting species are split into primary and secondary species corresponding to a particular form of the stoichiometric reactions. Together with transport equations for the primary species, this result in a system of coupled nonlinear partial differential equations which completely describe the time-evolution of the system. Yeh and Tripathi [59] give a review of early reactive flow models that couple linear partial differential equations describing transport to a set of non-linear algebraic equations describing chemical equilibria. Steefel and Van Cappelen [45] introduced a new approach to waterrock interactions, which replaces the assumption of partial equilibrium with a complete calculation of dissolution and precipitation rates. They address the reactive surface area and changes to this due to nucleation, crystal growth, dissolution and Ostwald ripening. They introduce the crystal size distribution as a tool to keep track of minerals’ volume fraction and reactive surface area. Further, they study the role of precursors when the model is applied to weathering of granite. In recent years, reactive flow modeling has become an important tool in several Earth science disciplines, such as carbon capture and storage (CCS) [see e.g. 12, 19, 37, 55, 58, 54, 22, 35, and references therein], mineralization [e.g. 49, 29, 11, 62, 61] and water injection into hydrocarbon reservoirs [2, 3, 52, 60, 15, 14, 20, 6]. Before a new enhanced oil recovery (EOR) injection strategy can be employed on field scale, standard practice has been to perform core scale experiments. In this context, a core is a cylindrical sample of porous rock (approx. 7 cm long and 4 cm in diameter) either extracted from the reservoir or a representative outcrop. Core scale experiments are desirable because it is possible to perform the experiments at realistic conditions, i.e. high temperature, stress, pore pressure, and with realistic fluids [5, 31]. The disadvantage with core scale experiments is that in many respects, the core acts as a black box. During the experiments it is usually not possible to get quantitative information about what happens inside the pore space, and therefore it can be a challenge to determine the exact mechanisms for the release of additional oil and to observed permeability and porosity changes. However, by examining the core with scanning electron microscopy (SEM), monitoring pressure drop and the strain over the core, and analyzing the effluent and combining these observations with models, much can still be learned of the underlying processes. 2
It is possible to simulate these processes on the pore scale. Several groups have formulated reactive flow simulators that can do these simulations [24, 27, 16, 20], and due to recent advances in imaging techniques, it is also possible to acquire high resolution images of the exact pore geometries. However, even if the core is only 7cm long, it contains > 108 pores, and since the time step length is usually closely linked to the spatial discretization it is almost impossible (or at least impractical) to simulate a core scale experiment where typically several pore volumes are flooded through the core, and the simulation parameters are not known and have to be fitted to the observed data in order to extract information about e.g. rate constants. In this paper we use a lattice Boltzmann model to simulate a core scale experiments. We adapt a relatively simple geometry for the core (a cylinder), where the pore surface of the core is mapped onto the cylinder surface, and the mass of the grains are mapped onto the cylinder. This approach has several advantages. First of all, it is possible to simulate core scale experiments within a short time frame which then makes it possible to history match important parameters than can be used for upscaling. Secondly, as there is a fluid/pore surface boundary it is possible to incorporate different mineral growth and dissolution mechanisms and study how well they capture the experimental data. Finally, once some of the important mechanisms are identified, a pore-scale simulation can be performed with the same numerical model in order to test hypotheses or mechanisms suggested by the effective (cylindrical) model. In this paper we propose a surface coverage model that reduces the dissolution rate of an existing mineral phase due to coverage of the reactive surface area by a precipitating mineral. The model is applied to a 1072 days core flooding experiment [39] where the chemical analysis clearly indicates that the carbonate core that consisted of more or less 100% calcite was converted to magnesite. Numereous experiments have shown that one mineral assembly can completely be replaced by another mineral phase during reactive flow [see e.g. 41, 9, and refecences therein]. Coupled dissolution/precipitation processes can be responsible for such replacement. As proposed by Putnis and Putnis [42] coupled dissolution/precipitation reactions can also explain replacement reactions that preserve morphology even though the primary solid surface is covered by the precipitating mineral phase. The generation of porosity in the precipitates due to differences in molar volume or solubility of the two mineral phases can assure an ongoing dissolution of the primary solid phase. Surface coverage has been observed in previous pore-scale simulation studies [23, 10], and has recently been studied in detail by Chen et al. [9]. They use the lattice Boltzmann method with a volume of pixel interface tracking to model a coupled dissolution-precipitation reaction. A thorough investigation of the effect of parameters such as dissolution and precipitation rate constants, solubility and molar volume ratio is presented, and four types of coupled dissolution/precipitation processes are proposed based on these investigations. In the first type, generated precipitates are so rare that the primary solid phase can dissolve completely. In type 2, more precipitates are generated, but the porosity in the precipitated layer is too high to inhibit dissolution of the primary solid 3
phase. Type 3 represents the case where the precipitated layer at some point becomes dense enough to inhibit further dissolution, hence the primary solid phase will only partly dissolve. In type 4, precipitation is so fast that it quickly armours the entire surface of the primary solid phase, completely suppressing dissolution of it. In this paper, we gain more insight into surface coverage processes by comparing to experimental data on the core scale. The proposed model is motivated by SEM studies of flooded cores, and the porosity generation/armoring taking place when solids re-equilibrate in the presence of a fluid phase [42]. It has the capability of simulating different surface coverage scenarios ranging from no coverage, i.e. reactive surface area equals total surface area, to full coverage where the precipitating mineral forms a monomolecular layer on the dissolving mineral. Possible generation of microporosity in the screening layer is also included in the model. In this paper we present a surface coverage model with one dissolving mineral (calcite) and one precipitating mineral (magnesite), but it can quite easily be extended to several dissolving and precipitating minerals. The rest of this paper is organized as follows: section 2 describes the numerical model, while simulation results and discussions are presented in section 3 and 4, followed by a short summary and conclusions in section 5.
2. Reactive flow model The numerical model is based on the model described in Hiorth et al. [20], and consists of a pore-scale lattice Boltzmann (LB) flow model coupled to a full geochemical model. In the following sections we briefly describe the LB and geochemical models. We use the LB pore-scale model as a core-scale model, matching physical parameters of a core flooding experiment such as mass, surface area and Peclét number. We present a scaling of the rate equations such that the LB pore-scale model can simulate flow on the core-scale. Further, we present a surface coverage model that updates the reactive surface areas of the minerals due to both dissolution/precipitation and surface coverage.
2.1. The lattice Boltzmann method We use standard lattice Boltzmann methods (LBM) to solve the Navier-Stokes equation and the advection-diffusion equation [47, 48]. The flow equation will not be described here, but an overview of the flow simulator we use can be found in Hiorth et al. [20] and more detailed descriptions can be found in Succi [47], Sukop and Thorne [48]. The lattice Boltzmann equation (LBE) that describes the advection-diffusion equation is given as: gαi (x + eα δt, t + δt) − gαi (x, t) =
1 i gα (x, t) − gαeq,i (x, t) , τD
(1)
where gαi is the distribution function for particles of species i moving along 4
direction α with velocity eα , x is the position vector, t is the time and δt is the time step, τD is the relaxation time associated with the diffusion process, given in terms of the diffusion coefficient, D (τD = D/c2s + 1/2), and gαeq is the equilibrium distribution function; # " 2 u·u eα · u (eα · u) eq,i . (2) + − g α = ω α Mi 1 + c2s 2c4s 2c2s where ωα are weight coefficients for the different velocities (directions), cs is the lattice sound speed and u is the macroscopic velocity calculated from the flow equation. The macroscopic concentration, Mi , is computed by X Mi = gαi . (3) α
The presented simulations are performed in 3D and we use the d3q15 discretization of the momentum space for both the LB flow equation and the √ LB advection-diffusion equation. In this model the lattice sound speed is cs = 1/ 3 and the weight coefficients are 2/9 for the stationary state, 1/9 for direct links, and 1/72 for diagonal links. The chemical reactions are taken care of in the boundary condition at the rock-water interface. We use a rate equation suggested by Morse and Berner [38], Steefel and Van Cappelen [45] and Lasaga [26]: n dMj Aj j Aj , = J = sgn(1 − Ωj )(k1j + k2j aH ) 1 − Ωm j dt V M V
(4)
j where Mj is the concentration of mineral j [mol/l], JM is the chemical flux 2 2 [mol/m /s], Aj is the surface area of mineral j [m ], V is the pore volume [l], Ωj is the saturation index for mineral j, k1j and k2j are rate constants [mol/m2 /s], and m and n are exponents indicating the order of reaction. The macroscopic chemical flux enters directly into a LB bounce back boundary condition as suggested by Bouzidi et al. [7], Lallemand and Luo [25] and Verhaeghe et al. [51];
gαi¯ − g˜αi =
2ωα lb J (eα¯ · n) , c2s i
(5)
where gαi¯ are the distributions reflected from the surface, g˜αi are the incoming distributions and Jilb is the macroscopic chemical flux (in LB units). The fluidsolid boundaries are constant throughout the simulations. Since we are comparing LB simulation results to experimental results, we will be converting between dimensionless LB units and physical units. Conversions between the two are done according to;
5
Lphys
=
LLB δx
(6)
tphys i Mphys
= =
tLB δt i MLB δc
(7) (8) (9)
where L represents any length in the system, t is the time and M i is the concentration of basis species i. δx is the physical space interval related to the number of grid points δx = Lphys /Nx , δt is the physical time interval of one iteration and can be related to either diffusivity or velocity (both with units Lp /tq ). We have related the time interval to the velocity through δt = (uLB /uphys )δx. δc can be chosen arbitrarily.
2.2. Scaling of rate equations As can be seen from eq. (4), the chemical reaction rates are proportional to the specific surface area (A/V ) of each mineral. If the solid-liquid boundaries are kept constant, the pore volume is fixed and only the surface area can change due to dissolution and precipitation. The change in moles of mineral j is given by dmj j = Aj JM , (10) dt where the two factors on the right hand side can change with time. The flux, j JM , changes as the core equilibrates with the flooding water, while the surface area of individual minerals changes due to dissolution and precipitation. The reactive surface area, i.e. the surface area that is accessible to the flow, can also possibly be reduced by surface coverage of precipitating minerals that give rise to a screening effect. Hence, the area Aj will be treated below as a dynamic reactive surface area.
6
(a)
(b)
Figure 1: a) Physical core plug, and b) cylinder used for simulations. For simulation purposes we have mapped the mass and surface area of the physical core plug onto the surface of a hollow circular cylinder (see figure 1). In the LB formulation, the surface area of each solid voxel is given by the number of nearest neighbor fluid voxels. Hence, to obtain the correct dissolution and precipitation rates, we assign a mass and a surface area for each solid voxel, starting from an even distrobution of the mass and surface area of the physical core plug. We then introduce a rescaling factor to the rate equation, such that the correct surface area is used to calculate dissolution and precipitation rates. The rescaling factor is given by rs,j =
Aplug,j , Alb,j δx2
(11)
where Aplug,j is the surface area of mineral j in the physical core plug and Alb,j is the LB surface area of mineral j for the given geometry. The total LB flux of species i is then given by X Jilb = rs,j Jjlb,i . (12) j
As water flows through the inside of the cylinder in the simulations, while the physical core consists of a complex flow path system, the pore volume can also differ from the simulation geometry to the physical core, but this will only be compensated for in the initialization of the mineral mass in the LB geometry mlb,i wn,init =
Aplug Alb wn Vlb . δx lb V Atot plug Sj Mjmol δcδx3
7
(13)
We can calculate the rescaling factor, rs,j , more rigorously by matching the LB and physical chemical rates. The LB rate for species i at a wall node (in physical units) is given by ∆miwn ∆t
=
X
wall links
=
X
wall links
(gαi¯ − g˜αi )δx3 δc δt
(14)
2ωα − 2 (eα · n)δx2 · Jilb cs
δcδx δt
,
where the sum is over all wall links, i.e. all possible pairs of wall nodes and neighboring fluid nodes. The number of links depends on the discretization of momentum space. We have substituted for the chemical flux boundary condition (5). From this expression we recognize the LB surface area of the wall node as Alb wn =
X
wall links
and the LB rate becomes
2ωα (n · eα¯ ), c2s
∆miwn 2 lb = Alb wn δx · Ji ∆t
δcδx δt
(15)
.
(16)
This should be equal to the physical rate X plug ∆miwn = Awn,j Jji , ∆t j
(17)
where Aplug wn,j is the surface area of mineral j in a wall node wn in the physical core plug. Hence the rescaling factor is given as by eq. (11).
2.3. Surface coverage model The model is motivated from previous SEM investigations of flooded and unflooded chalk cores. Investigations of chalk cores after flooding with MgCl2 solution at 130o C revealed that the dominating mineral replacement reaction involves dissolution of calcite and precipitation of magnesite. Figure 2 shows textural alterations observed in flooded cores. Precipitated magnesite is marked with ’M’. Figure 2a is an image of an unflooded core, while Figure 2b illustrates formation of an encrusting secondary mineral phase covering the primary grains, partly screening them from the injected fluid. The precipitates in figure 2b are marked with ’M’ followed by a question mark. The reason for this is uncertainty to whether they consist of pure magnesite. Heterogeneous nucleation of nano-sized magnesite crystals is widespread in the chalk matrix and oriented attachment of magnesite nanocrystals may produce larger aggregates (Figs. 2c,d). Interaction of nanocrystals and aligned crystallographic orientation can result
8
in growth of single euhedral magnesite crystals. Euhedral crystal growth was found to occur especially in intrafossil pores where crystals have more space to grow (Fig. 2e). Analysis of a chalk core after MgCl2 injection for about 1.5 years showed that the calcite by magnesite replacement is a continuous process that proceeds from core inlet towards the outlet [63].
Figure 2: SEM micrographs illustrating overgrowth in chalk cores after injection of MgCl2 solution in comparison to natural, unflooded chalk (M = magnesite). (a) No overgrowth on calcite grains in unflooded chalk. (b) Encrusting secondary mineral phase on calcite grains. (c,d) Precipitation of nano-sized magnesite crystals and formation into larger aggregates. (e) Crystal growth resulting in single euhedral magnesite crystals in intrafossil pores. The proposed surface coverage model contains several parameters that can be adjusted to study different types of coverage as motivated from the SEM studies. The parameters are listed in table 1. The basic features of the model are: i) precipitating magnesite minerals can reduce surface area of existing calcite minerals, ii) different rates for precipitation onto calcite/magnesite minerals, iii) dissolution/precipitation can be favored in specific sites corresponding to crystallographic defects such as steps and kinks, iv) dissolution of calcite surfaces that is covered by magnesite can take place with a separate (low) rate constant. This will happen if microporosity is generated in the precipitation process. With this model the secondary minerals can form either as a thin layer of precipitates (see Figure 2b), or as larger crystals as in figure 2e. If magnesite precipitates as larger aggregates as in figure 2c,d, there is a possibility for water to reach the calcite surface through (micro) porosity in the aggregates. 9
Table 1: Overview of model parameter. Parameter kr
dadef fmc
kslow kcov amag
Description Rate constant for magnesite precipitation on calcite minerals (nucleation) relative to that on magnesite minerals (crystal growth) Fraction of surface area that contains defects Parameter describing the surface coverage process; zero if all mass accumulates in a single point and one if the mass distributes as a monolayer Rate constant for dissolution in non-defect areas relative to that in defect areas Rate constant for dissolution of covered area relative to that of uncovered area Area on the calcite surface occupied by 1 mol of magnesite
Range >0
0-1 0-1
>0 >0 ≤ 6.02 · 105 m2* /mol
* We have assumed that the area occupied by one molecule is maximum 1nm2 , hence amag ≤ (10−9 )2 · 6.02 · 1023 m2 /mol
We account for this by allowing a low rate of dissolution through the covered area. Hence, the model is capable of describing behavior in accordance with observations made by SEM studies of cores flooded with magnesium rich brines. 2.3.1. Calcite dissolution The calcite dissolution rate is on the form X ∆mcal wn = (kj Aj ) J cal , ∆t j
(18)
where we have incorporated the volume V into the chemical flux J cal from eq. (4). Aj is the surface area of different sites on the calcite surface, and kj is a corresponding multiplier for the rate constant. Three types of sites have been used in the calcite dissolution rate; i) sites that dissolve slowly (kslow Aslow ), ii) sites that dissolve faster due to crystal defects (Adef ), and iii) sites that are covered by magnesite precipitates (kcov Acov ). The corresponding mulitiplier for the defect sites is assumed to be one, hence the calcite dissolution rate constant is assumed to be that for the defect sites. The calcite dissolution rate takes the following form ∆mcal wn = (Adef + kcov Acov + kslow Aslow )J cal . (19) ∆t Before any magnesite has precipitated, the calcite surface is split into two types of sites, the slowly dissolwing sites, and the defect sites dissolving with 10
a higher rate (kslow < 1). It has been suggested that calcite dissolution takes place by three different mechanisms depending on the degree of undersaturation; i) dissolution of pre-existing steps (low undersaturation), ii) defect-assisted dissolution (medium undersaturation), and iii) etch pit nucleation at defect-free surfaces (high undersaturation) [50, 57]. The proposed dissolution model accounts for the first two of these mechanisms, i.e. dissolution at pre-existing steps which correspond to the slowly dissolving sites, and defect-assisted dissolution which corresponds to dissolution at crystal defect sites. By accounting for more than one dissolution mechanism, we allow for different rate constants in different parts of the core, even though the dimensionless numbers, m and n in the rate law (see eq. (4)), which values depend on the dissolution mechanism, are kept constant throughout the simulations. As magnesite precipitates onto the calcite surface, it will cover parts or all of the underlying calcite surface. According to Putnis [41], porosity can be produced in the precipitated layer if either the molar volume of the precipitated mineral is lower than the original mineral, or the amount of dissolved material is more than precipitated material due to difference in solubility of the dissolving and the precipitating mineral. Putnis [41] also mention that if no porosity is produced, the coupled dissolution-precipitation process results in a solid layer of precipitates armouring the original surface and prevent further dissolution. In the special case of calcite dissolution and magnesite precipitation, production of microporosity is expected due to a lower molar valume of magnesite compared to calcite. Hence, the multiplier for the calcite surface covered by magnesite is expected to be non-vanishing, i.e. kcov > 0. Initially the different areas are calculated according to Ainit def
=
cal mol cal Acal Mcal minit dadef , init dadef = S
(20)
Ainit cov Ainit slow
= =
0, init Acal init − Adef ,
(21) (22)
and the changes in the areas are given by ∆Adef ∆Acov
= =
−∆Acov + dadef ∆Acal , fmc amag ∆mcov ,
(23) (24)
∆Aslow
=
mol S cal Mcal ∆mslow ,
(25)
where Acal = Adef + Acov + Aslow is the total surface area used in calculation of the rescaling factor, and ∆mcov = kcov Acov J cal ∆t and ∆mslow = kslow Aslow J cal ∆t are the second and third term from eq. 19. The area occupied by one mole of magnesite on the calcite surface is given by amag , and fmc is a geometrical factor describing the 3D mineral growth (see Table 1). The first 11
term in eq. (23) gives the change in defect area due to surface coverage, and the second term gives the change due to change in mineral mass (dissolution). 2.3.2. Magnesite precipitation The magnesite precipitation rate is constructed in the same manner as the calcite dissolution rate, i.e. with different sites for precipitation and corresponding multipliers. Two types of sites are used in the precipitation rate; i) precipitation onto magnesite (Amag ), and ii) precipitation onto calcite (kr Adef ). In this context, precipitation onto magnesite represents crystal growth, while precipitation onto calcite represents heterogeneous nucleation. It is assumed that the rate constant for magnesite precipitation is that for precipitation onto magnesite (crystal growth), hence the multiplier for this term is one. kr is the multiplier for precipitation onto calcite. Further, due to the fact that nucleation is energetically favorable on crystal dislocations [see e.g. 44, and references therein], we only allow for magnesite precipitation on the defect part of the calcite surface, hence Adef is the calcite area used in the precipitation rate. The resulting precipitation rate expression becomes ∆mmag wn = (kr Adef + Amag )J mag . (26) ∆t Since we assume that the core initially consists of 100% calcite, the initial magnesite area is zero, Ainit mag = 0. The change in the defect area is given by eq. (23), and the change in magnesite area is cal mag mol ∆mmag ∆Amag = fmc amag ∆mcal Mmag mag + (1 − fmc )∆mmag , mag + S
where ∆mcal mag is the mole number of magnesite precipitated onto calcite and given by the first term in eq. (26), and ∆mmag mag is the mole number of magnesite precipitated onto magnesite given by the second term in eq. (26). The total magnesite area is given by Amag .
3. Results The effluent curves for the 1072 days core flooding experiment reported by Nermoen et al. [39] are presented in Figure 3a. The dashed lines in the figure represent the injected concentrations. The tested chalk core is from the Liége quarry in Belgium, and was flooded with 0.219 mol/l MgCl2 at 130o C and exposed to an effective hydrostatic stress level of 10.4 MPa (0.7 MPa pore pressure and 11.1 MPa confining pressure). The flooding rate was measured in pore volumes per day (PV/day), and varied between 1 PV/day and 3 PV/day, as indicated by intervals I-IV in the figure. The flooding rate sequence was 1 PV/day - 3 PV/day 1 PV/day - 3 PV/day. By comparing effluent concentrations to the composition of the injection water, it was found that Mg2+ is retained in the core, Ca2+ ions are produced from the core, while Cl– acts as a non-reactive tracer. These ion 12
changes are interpreted as signatures of dissolution of dominantly calcite and precipitation of magnesite. In the following we present simulation results obtained with the proposed surface coverage model. The circular cylinder used to represent the core is contained within 40x8x8 voxels, and is the same for all simulations. The length of the core was 7 cm, and the initial Peclét number was 11. The time interval and grid size in the LB simulations are δt = 122.22 s and δx = 1.94 mm, respectively. In order to obtain a resolution of the effluent profile that allows us to discuss different properties, from this point we will only plot the effluent concentration for Ca2+ . However, the Mg2+ concentration can easily be calculated as the sum of Ca2+ and Mg2+ equals the injected Mg2+ concentration in the simulations.
3.1. Limiting cases of the surface coverage model By using the proposed model in the limit of no coverage (kr = dadef = 1, fmc = 0) one can reproduce approximately the initial level of the calcium effluent concentration by using literature rate constants (see Figure 3b). It is assumed that calcite is the only dissolving mineral, and that magnesite is the only precipitating mineral. The term with k2 in eq. (4) is significant only for solutions with low pH. The pH is above 6.5 throughout the experiment, hence the term with k2 has no significance. The values used for k1 for these minerals are kcalcite = 3.88·10−6 mol/m2 /s [40] and kmagnesite = 2.74 · 10−11 mol/m2 /s [43]. Despite the match of the initial calcium concentration, the zero-coverage model is only capable of describing the effluent profile for approximately one week. As seen in Figure 3c, large discrepancies between the experimental and the numerically simulated curves are seen through the rest of the experiment. Even when lowering the values for the rate constants (to improve the fit), as demonstrated in Figure 3d, this model is not adequate to describe the experimental data. A better fit of the mean concentrations in each interval is achieved, and the relative hight in the jumps between the intervals are improved, but the slopes within each interval are poorly matched. In the opposite limit of full coverage, magnesite forms as a monomolecular layer on the calcite surface (kr >> 1, dadef = fmc = 1). In this case simulations show that the whole calcite surface is covered by magnesite after approximately 2.5 days, hence in this limit the surface coverage is clearly too fast to account for the experimental data.
3.2. Discussion of the model parameters Before we present the best match of the model to the experimental data in the next section, we give a discussion of the different model parameters. The model contains several tunable parameters (see table 1), hence it is important to understand the importance of each. First we discuss the value of the rate constants for calcite kcal and magnesite kmag . Because the magnesite rate constant is several orders of magnitude lower than the calcite rate constant, magnesite precipitation is the rate determining 13
0.08
1
Cl−
Cl− injected
Mg2+
Mg2+ injected
0.6
0.4
−
0.1
concentration [mol/L]
0.2
I
II
III
IV
exp sim
0.07
Ca2+ injected 0.8
2+
Ca
Cl concentration [mol/L]
Ca2+, Mg2+ concentrations [mol/L]
0.3
0.06 0.05 0.04 0.03 0.02
0.2
0.01 0 0
200
400
600 days
800
1000
0 0
0 1200
1
2
3
(a)
7
0.06
I
II
III
exp sim
0.07 concentration [mol/L]
concentration [mol/L]
6
0.08 exp sim
0.07
IV
0.04 0.03 0.02 0.01 0 0
5
(b)
0.08
0.05
4 days
0.06 0.05
I
II
III
IV
0.04 0.03 0.02 0.01
200
400
600 days
800
1000
0 0
1200
(c)
200
400
600 days
800
1000
1200
(d)
Figure 3: (a) Experimental effluent profiles for Ca2+ , Mg2+ and Cl– from the 1072 days core flooding test reported by Nermoen et al. [39]. Dashed lines represent the injected concentrations. (b) Experimental effluent profile for Ca2+ reported by Nermoen et al. [39] compared to simulation results of calcite dissolution / magnesite precipitation using the same rate equation and rate constant as reported in previous work [20] - first week of experiment. (c) Experimental effluent profile for Ca2+ reported by Nermoen et al. [39] compared to simulation results of calcite dissolution / magnesite precipitation using the same rate equation and rate constant as reported in previous work [20] - 1072 days. (d) Experimental effluent profile for Ca2+ reported by Nermoen et al. [39] compared to simulation results of calcite dissolution / magnesite precipitation using the same rate equation as reported in previous work [20], but with lower rate constants (kcal = 5.145 · 10−6 mol/m2 /s and kmag = 1.00 · 10−13 mol/m2 /s, and n = 2 for magnesite).
14
process in the beginning of the experiment, and the value of the calcite rate constant has minor impact as long as it is much higher than the magnesite rate constant. Initially there is no magnesite, so precipitation will take place through heterogeneous nucleation on the calcite surface. Hence, the term kr Adef J mag in the precipitation rate (eq. 26) is the dominating term. Adef is given by dadef Acal , where Acal initially is calculated from the mass of the core and a specific surface area of calcite of Scal = 2m2 /g. Thus, The term determining the initial reactin rate is kr dadef J mag , where the magnesite rate constant, kmag is part of the flux. The match to the experimental curve can be obtained by altering either of the parameters kr , dadef and kmag . E.g. if a larger part of the calcite surface is available for magnesite nucleation (dadef ), then the same rate can be obtained with a lower rate constant. However, some of these parameters will also affect the behavior of the effluent curve at later times. The behavior of the system at a specific time depends on the evolution of the system before this time, hence it is difficult to give a unique description of the impact of the different parameters at a particular time. Despite this fact, we have performed simulations where the parameters were varied one at a time, to study the effect of each parameter. dadef and kmag have similiar effect on the effluent curve, where a high value will give a high initial level of the effluent curve followed by a fast decrease. kr was varied in inverse ratio to kmag to see the effect that kr has on the effluent curve other than on the initial concentration level. Large values kr ≥ 1 gave similar results, while a value of kr = 10−3 gave different behavior in the beginning of the experiment, but similar to kr ≥ 1 from approximately the middle of interval II. In this case, the curve increses more slowly due to the preferred precipitation onto magnesite as soon as some magnesite has been formed on the calcite surface. In other words, the dominating term in the precipitation rate is changed from kr Adef J mag to Amag J mag . However, this process is slow, since the magnesite surface area is small in the beginning. The area on the calcite surface occupied by one mole of magnesite, amag , was kept constant, while the paramter determining whether magnesite grows into smaller or larger crystals, fmc was varied. The main finding from these simulations was that a value of fmc ≥ 0 assures that the effluent concentration does not vanish long before the end of the experiment. When magnesite does not cover the calcite surface (fmc=0 ), the calcite dissolutiomn rate remains high far into the experiment due to a high reactive surface area. In the calcite dissolution rate equation (19), we have varied the multipliers kslow and kcov , keeping the calcite rate constant unchanged. kslow had an effect when it was much lower than zero, so that the value of the term kslow Aslow is comparable to the term Adef (Adef was lower than Aslow in these simulations). However, the form of the effluent curve was not changed dramatically. The value of kcov has most of its impact at the later stage of the experiment, when large parts of the calcite surface is covered by magnesite. It was seen that kcov could be tuned to match the effluent curve in interval IV very well, but the match in interval II and III were not satisfactory. Even though th emodel contains several tunable parameters, it turned out to 15
0.08 exp sim
concentration [mol/L]
0.07 0.06 0.05
I
II
III
IV
0.04
B
0.03 0.02
A
0.01 0 0
200
400
600 days
800
1000
1200
Figure 4: Ca2+ effluent profile for the best fit to the experimental data. Values of model parameters: kr = fmc = 1, dadef = 0.3, kslow = 10−4 , kcov = 10−3 , amag = 6.02 · 105 m2 /mol. Rate constants: kcal = 3.88 · 10−6 mol/m2 /s, kmag = 5 · 10−13 mol/m2 /s. be difficult to find a match to the entire experimental effluent curve. However, by focusing on later parts of the experiment, we were able to obtain a good match to the experimental data. The best match to the entire experiment, and the match to the later stages of the experiment are presented in the next sections.
3.3. Best match between simulations and the experiment Figure 4 shows the best match of the whole experimental effluent curve from the simulation study, obtained from manual fitting. The values used for the model parameters are kr = fmc = 1, dadef = 0.3, kslow = 10−4 , kcov = 10−3 and amag = 6.02 · 105 m2 /mol. A literature rate constant was used for calcite (same as in Figure 3b,c), while a reduced rate constant was used for magnesite (kmag = 5 · 10−13 mol/m2 /s). The model matches the experimental data very well in interval I, except from the peak value during the very first days of the experiment. The sudden changes in concentration caused by modified flooding rates also compares well with the experiment. In intervals II and III there are clear discrepancies between simulation and experiment, but in interval IV the simulation again yield values very close to the experimental data. Despite the difference between the simulated and the experimental curve in interval II, the model is capable of producing an inflection point (point B in Figure 4) at approximately the same time as in the experiment. A model with zero coverage (ref. Figure 3d) yields an effluent curve that is monotonically decreasing after gaining the initial level. Hence, a model including surface coverage is capable of capturing more of the complexity of the effluent curve than a model without surface coverage. 16
We interpret the inflection point at approximately 260 days (point B) as the point where the rate determining (slowest) process changes from magnesite precipitation to calcite dissolution. The rate constants for magnesite is 5 orders of magnitude lower than that of calcite, hence magnesite precipitation is the rate determining process at the start of the experiment. Since the magnesite surface area is increasing, the Ca2+ effluent curve is expected to increase as long as magnesite dominates the reaction rate (rate determining), and decrease when the calcite dissolution rate takes over as the rate determining process. This is a consequence of the decreasing calcite surface area and the increasing magnesite area. Also, the core compacts approximately 25% during the experiment, due to hydrostatic stress. 80% of the axial compaction happens in interval I and II. Evolution of radial compaction is unfortunately not measured because of a failure of the radial strain gauge after 113 creep days. We have performed simulations starting from interval III, where calcite dissolution is the rate determining process and axial compaction is less extensive. From the above study it was found that the term kslow Aslow in eq. (19) has little effect on the effluent concentration, hence the rate equation was reduced to ∆mcal wn = (Acal + kcov Acov )J cal , (27) ∆t leaving us with two terms; the global calcite dissolution rate, and the dissolution rate at parts of the surface covered by magnesite. Note that Acal is the surface area of calcite in contact with the pore fluid, and that it changes due to surface coverage, as well as dissolution. Figure 5 shows two simulations where each of the terms in the above rate equation is dominating. In both cases the effective calcite dissolution rate constant is kcal = 1.8 · 10−10 mol/m2 /s, which gives the match to the experimental data at the start of interval III (at 369 days). The simulation results are not sensitive to the value of the magnesite rate constant, kmag , for values down to kmag ∼ 10−9 mol/m2 /s. As the value decreases further, magnesite precipitation will take over as the rate determining process. The value of the remaining parameters were kr = 1 and fmc = 1, for both simulations. Figure 5a shows the simulation results for the best match to the experimental data when kcov Acov is the dominating term in the dissolution rate equation. This result does not give a particularly better match than when simulating the whole experiment. With surface coverage (Acal term as this is reduced by surface coverage) as the dominating factor on the dissolution rate, we find a much better match to the experimental curve in interval III, as seen in Figure 5b. In addition to adjusting the calcite dissolution rate constant to match the concentration level at the start of interval III, we have adjusted the value of the area occupied by one mole of magnesite on the calcite surface (amag ) to obtain the best match. A value of amag = 500 m2 /mol was used in the simulation presented in Figure 5b. The initial surface areas for calcite and magnesite in the two above simulations, i.e. where the dominating processes are dissolution at sites covered by magnesite (kcov Acov term in rate equation) and surface coverage (Acal term in 17
0.08
0.08 exp sim
0.06 0.05
I
II
III
IV
0.04 0.03 0.02 0.01 0 0
exp sim
0.07 concentration [mol/L]
concentration [mol/L]
0.07
0.06 0.05 0.04 0.03 0.02
III
0.01 200
400
600 days
800
1000
0 0
1200
(a)
200
400
IV 600 days
800
1000
1200
(b)
Figure 5: a) Ca2+ effluent profile from simulation started at 369 days into the experiment, with kcov Acov as the dominating dissolution term. b) Ca2+ effluent profile from simulation started at 369 days into the experiment, with overgrowth dominating the dissolution rate of calcite. rate equation) respectively, are shown in Figure 6 by the blue (calcite) and red (magnesite) crosses . The mineral mole numbers after 369 days were calculated from the experimental effluent profiles, and surface areas were then calculated from A = S · M mol · m, where S is the specific surface area [m2 /g], M mol is the molar weight [g/mol] and m is the mole number. For the simulation with kcov Acov as the dominating term in the dissolution rate (Figure 5a), the initially covered area equals the initial calcite area (Acov = Acal ), hence the whole calcite surface is covered by magnesite. In the case where surface coverage is dominating the dissolution rate, we started with no coverage (Acov = 0). A sensitivity analysis was performed to investigate the dependency of the simulated effluent profile in Figure 5b on the initial surface area configurations. Three different initial configurations were used; i) even distribution (as used for simulations in Figure 5), ii) "‘smooth"’ front distribution, and iii) sharp front distribution. The three sets of distributions are presented in Figure 6. In the even distribution and the smooth front distribution runs, the initially covered area was set to zero. In the sharp front run the covered area was set to zero for x > xf , where xf is the position of the front, and equal to the calcite area for x < xf , i.e. the calcite surface is fully covered by magnesite. This means that surface coverage is dominating the dissolution rate for x > xf and kcov Acov dominates for x < xf . The results from this sensitivity analysis towards initial configurations are presented in Figure 7. There is little difference between the results from the even distribution and the smooth front distribution, and only a small difference between these and the sharp front distribution. This means that even though kcov Acov control the dissolution rate for x < xf , the surface coverage process taking place for x > xf is dominating the overall dissolution. With all initial configurations, the simulations show that dissolution is more
18
2
Initial cross section surface area [m ]
2.5
calcite even distribution magnesite even distribution calcite smooth front distribution magnesite smooth front distribution calcite sharp front distribution magnesite sharp front distribution
2
1.5
1
0.5
0 0
0.2
0.4 0.6 Dimensionless length
0.8
1
Figure 6: Initial cross section calcite and magnesite surface areas along the core for simulations started at 369 days into the experiment. Three different initial configurations were used to test the sensitivity towards initial configuration. Crosses are for even distributions, dotted lines for smooth front distributions, and solid lines for sharp front distributions. Blue is used for calcite, and red is used for magnesite.
0.04 exp even initial distribution smooth front initial distribution sharp front initial distribution
concentration [mol/L]
0.035 0.03 0.025 0.02
III
0.015
IV
0.01 0.005 0
400
600
800 days
1000
1200
Figure 7: Simulated effluent curves for three different initial configurations (see Figure 6) when overgrowth is the dominating mechanism on the evolution of the calcite dissolution rate.
19
or less homogeneous through the core. In the paper by Nermoen et al. [39] one can find SEM analysis on different parts of the plug after flooding. The plug was sliced into 6 slices, and similar magnesium weight percents are found for slice 2 and 5. This indicates a more or less homogeneous distribution of magnesite.
4. Discussion 4.1. Model limitations We obtain a much better fit to the experimental data when starting the simulation at interval III, compared to simulation from the start of the experiment. There are some limitations to the model that can possibly explain the poor match to the full experiment in interval II: • In the experiment the core compacts approximately 10% in the axial direction and approximately 25% in total due to effective hydrostatic stress (a slightly higher stress was applied in the axial direction to assure proper measurement of axial strain) [39]. Compaction is not included in our model. Compaction can lead to e.g. the generation of fresh calcite surfaces available for magnesite nucleation, which can increase the precipitation rate and surface area generation. From the experimental data presented by Nermoen et al. [39], it can be seen that more than 80% of the axial compaction happens in intervals I and II. The evolution of the radial compaction is not obtained throughout the experiment, because the radial strain gauge failed after 133 creep days. • In the simulations, we assume that the core initially consist of 100% calcite. Measurements of the CaCO3 concentration in Liége chalk give values between 91-95% [21, 36, 63]. Other components might be important to the overall behavior of the rock. • We use bulk saturation indices to represent the value on the mineral surfaces. In diffusion limited systems, this could lead to errors, but because the diffusion is fast in our system, we believe it is a good approximation to use the bulk value. However, to really investigate this effect one should perform pore scale simulations where gradients from the pore surface could be studied in detail. We will pursue this as a continuation of this work. • The pore volume is kept constant in the simulations. During the experiment, the pore volume can change due to dissolution/precipitation and compaction. However, the porosity of the plug only changes from 41.32% to 40.02% from the beginning to the end of the experiment [39]. • The fitting of the parameters were performed manually. • The core was represented with a simple cylinder. With this approach the pore space of the core with a complex flow pattern is simplified with
20
the flow through a hollow cylinder. The core might be subject to heterogeneities which are not captured by the cylinder geometry. The entire mass and surface area of the core was mapped onto the cylinder. If parts of the core were not subjected to flow, this would effectively reduce the mass and surface area seen by the water phase. Further, a lower surface area would result in a lower dissolution rate. When the core compacts, the pore space might alter completely, and new flow paths may form. Hence, during the time of compaction, the surface area seen by the water phase might increase. This is one example of how compaction can produce an increased dissolution/precipitation (nucleation) rate. The permeability of the core changes most prominently in interval I and II [39], and only small variations are seen in interval III and IV. Hence, changes in permeability should have little effect on the simulation results from interval III and IV. Since interval III and IV are less dominated by compaction, we have performed simulations from the start of interval III. At this point we also believe that the possible influence from non-carbonate content is less than in the start of the experiment.
4.2. Parameter values To match the concentration level at the start of interval III, we used a value of kcal that is a factor ∼ 104 lower than the literature rate of 3.88 · 10−6 mol/m2 /s. There might be several reasons for this discrepancy: i) We believe that this can be explained by all easily dissolving sites (crystal defect, locations with higher roughness, etc.) in the calcite structure already being dissolved. These sites might have a lower activation energy, which will result in a higher reaction rate constant according to the Arrhenius equation. Less dissolvable sites, i.e. more homogeneous parts of the calcite grains, might have a higher activation energy and hence a lower reaction rate constant as used in the simulations. ii) The literature rate was measured in bulk powder experiments, which give rates up to two orders of magnitude higher than measured with atomic force microscopy (AFM) or vertical scanning interferometry (VSI) [see e.g. 4]. This discrepancy has been ascribed to differences in step densities in powders and on cleaved surfaces typically used for AFM and VSI measurements [13]. These high step density areas might also contribute significantly to the surface area. Further, laboratory measured dissolution rates of several minerals have been found to be several orders of magnitude faster than those observed on the field scale [34, 8, 53, 32, 30]. Thus, differences between the system studied here, and the system used to measure the literature rate constant might explain the discrepancy in the value of the rate constant. iii) Mg2+ ions in solution can have an inhibitory effect on the calcite dissolution rate by adsorbing on the surface and acting as barriers to step retreat [see e.g. 56, and references therein]. The good match of the slope in interval III was obtained by lowering the value of amag compared to the monolayer-growth model. A value of amag = 500 m2 /mol was used in the simulation, while, for comparison, in the monolayer21
growth model the value used was amag = 6.02 · 105 m2 /mol. The relatively low value of amag compared to the monolayer model is interpreted as preferred growth into either aggregates of nano-sized crystals (Figure 2c,d) or into single euhedral crystals (Figure 2e) rather than a layered growth.
4.3. Comparison to other experimantal data The simulations reproduce the homogeneous distribution of magnesite in the core after 1072 days. From a similar experiment where Liège chalk was flooded with MgCl2 for 516 days, a transformation front from calcite to magnesite was found after 516 days [63]. A difference between this experiment and the one reported by Nermoen et al. [39] is that in this experiment, the flooding rate was kept constant at 1 PV/day throughout the test. It is expected that a higher flooding rate will result in a smoothening of the front observed by Zimmermann et al. [63], because the residence time in each cross section of the core is lower, which will result in less reactions near the inlet and more reactions further into the core. This is equivalent to a lower reaction rate constant, which has been shown to give a more uniform distribution of precipitates [1]. Since we only have mineralogical data at the end of the experiment, we do not know how the mineral profiles evolve with time. However, from the sensitivity study towards initial mineral configurations, our model gives a homogeneous distribution of magnesite independent of the mineral configuration at 369 days.
5. Summary and conclusions We have presented a general framework in which one can use the LB method to interpret single phase reactive flow core flooding experiments. The core is represented by a cylinder where the initial mass and specific surface area match the experimental values. Further, the surface areas of different minerals evolve with time as a consequence of dissolution and precipitation processes. We have clearly demonstrated that it is necessary to extend "standard" rate equations to take into account effects of surface coverage to explain long term core flooding experiments. Simulation results are compared to experimental water chemistry data from a Liège chalk core flooded with MgCl2 solution for 1072 days [39]. The best match to the experimental data was obtained in a simulation that was initiated 369 days after the start of the experiment. Coverage of the calcite surface by precipitating magnesite crystals is responsible for the good match to the experimental data. A dissolution rate of calcite in the parts of the surface covered by magnesite can not give a good match to the data, however this can still be an important mechanism at times beyond the duration of this experiment. We conclude that a dynamically changing dissolution rate is necessary to explain the experimental data reported by Nermoen et al. [39]. The surface coverage mechanism, where the surface area of calcite decreases due to precipitation of magnesite minerals on the calcite surface, gives an evolution of the dissolution rate that fits the experimental data well.
22
The value of the parameter amag indicates that magnesite preferentially grows into either aggregates of nano-sized crystals or into single euhedral crystals rather than as a layer growth. The simulation predicts that calcite dissolution and magnesite precipitation is more or less homogeneous through the core. SEM-EDS measurements of the flooded plug showed that the amount of magnesium in the core is homogeneous. This indicates a homogeneous distribution of magnesite in the plug. Thus, the simulation predicts the behavior of the experiment. To match the experimental effluent profile after approximately one year, we had to use a rate constant that is four orders of magnitude lower than what is reported in bulk powder experiments. We argue that as sites with the lowest activation energy, e.g. crystal defect, locations with higher roughness, etc., are dissolved, the rate constant decreases due to higher activation energies for the remaining sites. We conclude that static rate constants are not sufficient to describe chalk core flooding experiments lasting for as long as three years. The outcome of this work will be used to develop a pore scale surface coverage model that will be run on realistic chalk geometries obtained from micro imaging. Even though simulations in realistic chalk geometries are limited to a small segment of a core, this can be used to verify model parameters such as amag from this work, and to study the long term evolution of porosity and permeability in chalk due to coupled dissolution and precipitation reactions.
Acknowledgments The authors acknowledge the Norwegian Research Council (200600/S60, 230303), the Ekofisk and Valhall lisences and the industry partners of The National IOR Centre of Norway; ConocoPhillips Scandinavia AS, BP Norge AS, Det Norske Oljeselskap AS, Eni Norge AS, Maersk Oil Norway AS, DONG Energy A/S, Denmark, Statoil Petroleum AS, GDF SUEZ E&P NORGE AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS, for financial support.
23
Appendix A: List of symbols Acov Adef Aj Alb,j Aplug,j Aslow Alb tot Alb wn aH amag α cs D dadef δc δt δx δmcal mag ∆mmag mag eα fmc gαi gαeq,i JM J lb k1 k2 kcal
Surface area of calcite covered by magnesite Surface area of defects on calcite Surface area of mineral j Surface area of mineral j in the LB geometry (cylinder) Surface area of mineral j in the physical core Surface area of slowly dissolving calcite sites (1 − Adef when there is no magnesite) Total LB surface area of the cylinder LB surface area in a wall node wn Activity of hydrated hydrogen ion Area on the calcite surface occupied by 1 mol of magnesite Discrete lattice direction Lattice sound speed Diffusion coefficient Fraction of calcite surface area that contains defects Concentration interval Time interval Grid spacing Change in moles of magnesite due to magnesite precipitation onto calcite Change in moles of magnesite due to magnesite precipitation onto magnesite Discrete lattice velocity geometrical factor describing the 3D mineral growth. Zero if all mass accumulates in one single point, and one if the mass distributes asa monolayer LB distribution function for chemical species i along lattice direction α Equilibrium distribution function for species i along lattice direction α Chemical flux Chemical flux in LB units (dimensionless) Rate constant for in rate law Rate constant in rate law Rate constant (k1 ) for calcite (assumed to be for the defect part of the calcite surface).
24
kcov kmag kr
kslow M M mol m mj n n Ωj ωα rs,j Sj t τD u V x
Multiplier for calcite dissolution rate constant at areas covered by magnesite Rate constant (k1 ) for magnesite precipitation onto magnesite Multiplier for rate constant for magnesite precipitation onto calcite. Assumed that the rate for magnesite precipitation onto calcite can differ from that onto magnesite. Multiplier for calcite dissolution rate at slowly dissolving sites. kslow ≤ 1. Concentration Molecular weight [g/mol] Exponent in rate law indicating order of reaction Moles of mineral j Exponent in rate law indicating order of reaction Surface normal Saturation state of mineral j Weight coefficient for direction α rescaling factor for mineral j Specific surface area of mineral j [m2 /mol] Time Relaxation time associated with diffusion Velocity Pore volume Position vector
Subscripts and superscripts α cal cov def init lb phys plug slow wn
Discrete lattice direction Value is for calcite Value is for calcite area covered by magnesite Value is for area of defects Initial value of the variable Indcates LB value Indicates physical value Value is for the physical core plug Value is for slowly dissolving calcite sites Value is for a wall node
References [1] Mesoscopic modeling of multi-physiochemical transport phenomena in porous media. Advances in Mechanical Engineering, page DOI: 10.1155/2010/142879, 2010. [2] P. O. Andersen, S. Evje, and H. Kleppe. A model for spontaneous imbibi-
25
tion as a mechanism for oil recovery in fractured reservoirs. Transp. Porous Med., 101(2):299–331, 2014. [3] Pål Østebø Andersen, Steinar Evje, Merete V. Madland, and Aksel Hiorth. A geochemical model for interpretation of chalk core flooding experiments. Chemical engineering science, 84:218–241, 2012. [4] R. S. Arvidson, I. E. Ertan, J. E. Amonette, and A. Luttge. Variation in calcite dissolution rates: A fundamental problem? Geochim. et Cosmochim. Acta, 67:1623–1634, 2003. [5] T. Austad, S. Strand, M. V. Madland, T. Puntervold, and R. I. Korsnes. Seawater in chalk: An EOR and compaction fluid. SPE Reservoir Evaluation & Engineering, 11(4):648–654, 2008. [6] P. Birkle, K. Pruess, T. Xu, R. A. H. Figueroa, M. D. Lopez, and E. C. Lopez. Using laboratory experiments and reactive chemical transport modeling for designing waterflooding of the Agua Fria reservoir, Poza RicaAltamira field, Mexico. SPE Reservoir Evaluation & Engineering, 11(6): 1029–1045, 2008. [7] M. Bouzidi, M. Firdaouss, and P. Lallemand. Momentum transfer of a Boltzmann-lattice fluid with boundaries. Physics of fluids, 13(11), 2001. [8] S. L. Brantley. Reaction kinetics of primary rock-forming minerals under ambient conditions. Treatise on Geochemistry, 5:73–117, 2003. [9] L. Chen, Q. Kang, B. carey, and W.-Q. Tao. Pore-scale study of diffusionreaction processes involving dissolution and precipitation using the lattice boltzmann method. International Journal of Heat and mass Transfer, 75: 483–496, 2014. [10] L. Chen, Q. Kang, Q. Tang, B. A. Robinson, Y.-L. He, and W.-Q. Tao. Pore-scale simulation of multicomponent multiphase reactive transport with dissolution and precipitation. International Journal of heat and Mass Transfer, 85:935–949, 2015. [11] G. Chi and C. Xue. An overview of hydrodynamic studies of mineralization. Geoscience Frontiers, 2(3):423–438, 2011. [12] B.-Y. Choi, S.-T. Yun, K.-H. Kim, H.-S. Choi, G.-T. Chae, and P.-K. Lee. Geochemical modeling of CO2 -water-rock interactions for two different hydrochemical types of CO2 -rich springs in Kangwon district, Korea. J. Geochem. Explor., page http://dx.doi.org/10.1016/j.gexplo.2014.02.009, 2014. [13] P. M. Dove and F. M. Platt. Compatible real-time reaction rates for in situ imaging of mineral-water interactions using scanning force microscopy. Chem. Geol., 127:331–338, 1996.
26
[14] S. Evje and A. Hiorth. A model for interpretation of brine-dependent spontaneous imbibition experiments. Advances in Water Resources, 34(12): 1627–1642, 2011. [15] S. Evje, A. Hiorth, M. V. Madland, and R. I. Korsnes. A mathematical modelrelevant for weakening of chalk reservoirs due to chemical reactions. Networks and Heterogeneous Media, 4(4):755–788, 2009. [16] F. Flukiger and D. Bernard. A new numerical model for pore scale dissolution of calcite due to CO2 saturated water flow in 3d realistic geometry: Principles and first results. Chemical Geology, 265(1-2):171–180, 2009. [17] H. C. Helgeson. Evaluation of irreversible reactions in geochemical processes involving minerals and aqueous solutions - i. thermodynamic relations. Geochim. Cosmochim. Acta, 32:853–877, 1968. [18] H. C. Helgeson, R. M. Garrels, and F. T. MacKenzie. Evaluation of irreversible reactions in geochemical processes involving minerals and aqueous solutions - ii. applications. Geochim. Cosmochim. Acta, 33:455–482, 1969. [19] H. Hellevang, Van T. Pham, and P. Aagaard. Kinetic modelling of CO2 water-rock interactions. International Journal of Greenhouse Gas Control, 15:3–15, 2013. [20] Aksel Hiorth, Espen Jettestuen, Lawrence M. Cathles, and Merete V. Madland. Precipitation, dissolution, and ion exchange processes coupled with a lattice Boltzmann advection diffusion solver. Geochimica et Cosmochimica Acta, 104:99–110, 2013. [21] M. Hjuler and I. Fabricius. Engineering properties of chalk related to diagenic variations in the upper cretaceous onshore and offshore chalk in the north sea area. Journal of Petroleum Science and Engineering, 68:151–170, 2009. [22] J. W. Johnson, J. J. Nitao, C. I. Steefel, and K. G. Knaus. Reactive transport modeling of geologic CO2 sequestration in saline aquifers: the influence of intra-aquifer shales and the relative effectiveness of structural, solubility, and mineral trapping during prograde and retrograde sequestration. 1st Nat. Conf. Carbon Sequestration, Washington, DC, 2001. [23] Q. Kang, P. C. Lichtner, H. S. Viswanathan, and A. I. Abdel-Fattah. Pore scale modeling of reactive transport involved in geologic CO2 sequestration. Transp. Porous Med., 82:197–213, 2010. [24] Q. J. Kang, D. X. Zhang, S. Y. Chen, and X. Y. He. Lattice boltzmann simulation of chemical dissolution i porous media. Phys. Rev. E, 65(3): 036318, 2002. [25] P. Lallemand and L. Luo. Lattice Boltzmann method for moving boundaries. J. Comp. Phys., 184:406–421, 2003. 27
[26] A. C. Lasaga. Kinetic theory in the Earth sciences. Princeton University Press, New Jersey, 1998. [27] L. Li, C. A. Peters, and M. A. Celia. Upscaling geochemical reaction rates using pore-scale network modeling. Advances in Water Resources, 29:1351– 1370, 2006. [28] P. C. Lichtner. Continuum model for simultaneous chemical reactions and mass transport in hydrothermal systems. Geochim. Cosmochim. Acta, 49: 779–800, 1985. [29] Y. Liu and T. Dai. Numerical modeling of pore-fluid flow and heat transfer in the Fushan iron ore district, Hebei, China: Implications for hydrothermal mineralization. J. Geochem. Explor., page http://dx.doi.org/10.1016/j.gexplo.2014.02.023, 2014. [30] A. Lüttge. Etch pit coalescence, surface area, and overall mineral dissolution rates. Am. Mineral, 90(11-12):1776–1783, 2005. [31] M. V. Madland, A. Hiorth, E. Omdal, M. Megawati, T. Hildebrand-Habel, R. I. Korsnes, S. Evje, and L. M. Cathles. Chemical alterations induced by rock-fluid interactions when injecting brines in high porosity chalks. Transp. Porous Med., 87:679–702, 2011. [32] K. Maher, D. J. DePaolo, and C. F. Jin J. Rates of silicate dissolution in deap-sea sediment: In situ measurement using u-234/u-238 of pore fluids. Geochim. Cosmochim. Acta, 68(22):4629–4648, 2004. [33] K. Maher, C. I. Steefel, A. F. White, and D. A. Stonestrom. The role of reaction affinity and secondary minerals in regulating chemical weathering rates at the Santa Cruz Soil Chronosequence, California. Geochim. Cosmochim. Acta, 73(10):2804–2831, 2009. [34] M. E. Malmstrom, G. Destouni, S. A. Banwart, and B. H. E. Stromberg. Resolving the scale-dependence of mineral weathering rates. Environ. Sci. Technol., 34(7):1375–1378, 2000. [35] B. J. O. L. McPherson and P. C. Lichtner. CO2 sequestration in deep aquifers. 1st Nat. Conf. Carbon Sequestration, Washington, DC, 2001. [36] M. Megawati, A. Hiorth, and M. V. Madland. The impact of surface charge on the mechanical behavior of high-porosity chalk. Rock Mech. Rock Eng., 46:1073–1090, 2013. [37] A. B. Mitiku, D. Li, S. Bauer, and C. Beyer. Geochemical modeling of CO2 -water-rock interactions in a potential storage formation of the North German sedimentary basin. Applied Geochemistry, 36:168–186. [38] J. W. Morse and R. A. Berner. Dissolution kinetics of calsium carbonate in seawater: II. A kinetic origin for lysocline. Am. J. Sci., 272:840–851, 1972. 28
[39] A. Nermoen, R. I. Korsnes, A. Hiorth, and M. V. Madland. Porosity and permeability development in compacting chalks during flooding of nonequilibrium brines: Insights from long-term experiment. Journal of Geophysical Research - Solid Earth, 2014. DOI: 10.1002/2014JB011631. [40] L. N. Plummer, T. M. L. Wigley, and D. L. Parkhurst. The kinetics of calcite dissolution in CO2 -water systems at 5 to 60 degC and 0.0 to 1.0 atm CO2 . Am. J. Sci., 278:179–216, 1798. [41] A. Putnis. Mineral replacement reactions: from macroscopic observations to microscopic mechanisms. Mineralogical Magazine, 66(5):689–708, 2002. [42] A. Putnis and C. V. Putnis. The mechanism of reequilibration of solids in the presence of a fluid phase. Journal of Solid State Chemistry, 180(5): 1783–1786, 2007. [43] G. D. Saldi, G. Jordan, J. Schott, and E. H. Oelkers. Magnesite growth rates as a function of temperature and saturation state. Geochim. Cosmochim. Acta, 73:5646–5657, 2009. [44] K. Sangwal. Growth kinetics and surface morphology of crystals grown from solutions: recent observations and their interpretations. Prog. Crystal Growth and Charact., 36(3):163–248, 1998. [45] C. I. Steefel and P. Van Cappelen. A new kinetic approach to modeling water-rock interaction: the role of nucleation, precursors, and ostwald ripening. Geochim. Cosmochim. Acta, 54:2657–2677, 1990. [46] C. I. Steefel, D. J. DePaolo, and P. C. Lichtner. Reactive transport modeling: An essential tool and new research approach for the earth sciences. Earth and Planetary Science Letters, 240:539–558, 2005. [47] Sauro Succi. The Lattice Boltzmann Equation for Fluid Dynamics and Beyaond. Oxford University Press, New York, 2001. [48] M.C. Sukop and D. T. Thorne. Lattice Boltzmann Modelling: An Introduction for Geoscientists and Engineers. Springer, 2006. [49] T. Sun and L. Liu. Delineating the complexity of cu-mo mineralization in a porphyry intrusion by computational and fractal modeling: A case study of the chehugou deposit in the chifeng district, Inner Mongolia, China. J. Geochem. Explor., page http://dx.doi.org/10.1016/j.gexplo.2014.02.015, 2014. [50] H. H. Teng. Controls by saturation state on etch pit formation during calcite dissolution. Geochim. Cosmochim. Acta, 68(2):253–262, 2004. [51] F. Verhaeghe, S. Arnout, B. Blanpain, and P. Wollants. Lattice-Boltzmann modeling of dissolution phenomena. Phys. Rev. E, 73:036316, 2006.
29
[52] L. Wei. Sequential coupling of geochemical reactions with reservoir simulations for waterflood and EOR studies. SPE Journal, 17(2):469–484, 2012. [53] A. F. White and S. L. Brantley. The effect of time on the weathering of silicate minerals: why do weathering rates differ in the laboratory and field? Chem. Geol., 202(3-4):479–506, 2003. [54] S. P. White, R. G. Allis, J. Moore, T. Chidsey, C. Morgan, W. Gwynn, and M. Adams. Simulation of reactive transport of injected CO2 on the colorado plateau, utah, usa. Chemical Geology, 16(4):399–413, 2005. [55] Y. Xiao, T. Xu, and K. Pruess. The effects of gas-fluid-rock interactions on CO2 injection and storage: insights from reactive transport modeling. Energy Procedia, 1(1):1783–1790, 2009. [56] M. Xu and S. R. Higgins. Effects of magnesium ions on near-equilibrium calcite dissolution: Step kinetics and morphology. Geochim. et Cosmochim. Acta, 75:719–733, 2011. [57] M. Xu, X. Hu, K. G. Knauss, and S. R. Higgins. Dissolution kinetics of calcite at 50−70o : an atomic force microscopy study under near-equilibrium conditions. Geochim. Cosmochim. Acta, 74:4285–4297, 2010. [58] T. F. Xu, J. A. Apps, and K. Pruess. Numerical simulation of CO2 disposal by mineral trapping in deep aquifers. Applied Geochemistry, 19(6):917–936, 2004. [59] G. T. Yeh and V. S. Tripathi. A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components. Water Resour. Res., 25:93–108, 1989. [60] Y. Zaretskiy, S. Geiger, and K. Sorbie. Direct numerical simulation of pore-scale reactive transport: applications to wettability alteration during two-phase flow. International Journal of Oil Gas and Coal Technology, 5 (2-3):142–156, 2012. [61] C. Zhao. Dynamic and Transient Infinite Elements: Theory and Geophysical, Geotechnical and Geoenvironmental Applications. Springer, Berlin, 2009. [62] C. Zhao, B. E. Hobbs, and A. Ord. Convective and Advective Heat Transfer in Geological Systems. Springer, Berlin, 2008. [63] U. Zimmermann, M. V. Madland, A. Nermoen, T. Hildebrand-Habel, S. R. A. Bertolino, A. Hiorth, R. I. Korsnes, J-N. Audinot, and P. Grysan. Challenging the chalk: SEM, nano-SIMS and geochemical data from flooded chalk and its consequenses for enhanced oil recovery applications. In press in AAPG Bulletin, DOI 10.1306/12221412196, 2015.
30
Paper 5: Lattice Boltzmann simulations of advection driven ow by CO2 diusion into water. J. Pedersen, E. Jettestuen, H. A. Rabenja manantsoa, Rune Time, Ørjan Tveteraas and A. Hiorth. In preparation.
Lattice Boltzmann simulations of advection driven flow by CO2 diffusion into water Janne Pedersen,∗,† Espen Jettestuen,† Herimonja A. Rabenjafimanantsoa,‡ Rune Time,‡ Ørjan Tveteraas,¶ and Aksel Hiorth†,‡ International Research Institute of Stavanger, P.O. Box 8046, N-4068 Stavanger, Norway, University of Stavanger, Petroleum Department, N-4036 Stavanger, Norway, and BP Norge AS, N-4065 Stavanger, Norway E-mail:
[email protected]
Abstract Convective mixing of CO2 gas into water is studied using the lattice Boltzmann (LB) method. The boundary condition for the CO2 -water system is determined by the partial pressure of the CO2 gas phase and the chemical composition of the water close to the gas water interface. A pressure decay experiment where CO2 gas at initially 5 bar overlies a water column, diffuses into the water phase and leads to convective currents in the water, is simulated. A pH indicator was used in the experiment to observe the convective currents. The concentration patterns predicted from simulations are in good agreement with visual observations during the experiment, made possible by the use of a pH indicator. We argue that pH indicator with a high transition pH will give the best information about the rate of reactions taking place at the CO2 -water interface, and ∗
To whom correspondence should be addressed International Research Institute of Stavanger ‡ University of Stavanger ¶ BP Norge AS †
1
about the onset of instability. On the other hand, a pH indicator with a low transition pH will give the best information about the structure of the instability and the longterm importance of advection. Our simulations suggest that the time at which the entire water volume reaches the low pH color, scales as the geometric mean of diffusive and buoyancy time scales.
Introduction Storage of CO2 in saline aquifers has received a great deal of attention in the last decades as it might be a way to permanently store CO2 . When CO2 is injected into a saline aquifer it moves upward because of the buoyancy effect. During the migration, CO2 will mix into the water by diffusion. The CO2 molecule has a higher molecular weight than the water molecule, and a situation will arise behind the CO2 plume, where denser water overlays lighter water; as a consequence of this, instability will occur that give rise to natural convective currents. The net effect is an enhanced mass transfer between the CO2 gas and water interface. Depending on the size and the availability of the aquifer, a significant amount of CO2 can be dissolved into the aqueous phase in the case of long term storage. The investigation of CO2 mixing as dissolution-diffusion-convection (DDC) may also have practical interest for short term application in the case of CO2 injection for enhanced oil recovery (EOR) in fractured reservoirs. When CO2 , usually as a supercritical gas, is injected into a fractured reservoir the gas will mainly follow high permeable flow paths and be transported only by diffusion and DDC into tighter matrix blocks. The speed of the recovery mechanism is thus controlled by the speed of the mixing process. Other researchers have studied the interaction of CO2 and water, both numerically and experimentally. In particular Chen and Zhang 1 performed LB simulations in an idealized fractured porous medium and investigated the effect of Rayleigh number on convection varying both body force and fracture width. Yang and Gu 2 studied the mass transfer from a gaseous CO2 phase into a reservoir brine at elevated pressures and temperatures. They 2
performed pressure decay experiments and expressed the mass transfer of CO2 into brine as function of gas pressure. Yang and Gu did not take into account DDC in their modeling and had to use a diffusion coefficient almost two orders of magnitude higher than the (expected) molecular diffusion coefficient. Pruess and Nordbotten 3 studied long term evolution of CO2 plume in a saline aquifer, they included the effect of DDC in their modeling, and pointed out the importance of fine grid simulation to capture this effect. Farajzadeh et al. 4 performed pressure decay experiments and simulated the experiments using the Finite Element Method (FEM) software package Comsol multiphysics. Both the gas phase and the water phase was simulated. The water phase was described by Navier-Stokes equations and the advectiondiffusion equation, and the gas phase was described by the diffusion equation. They predict an instability at the CO2 -water interface that results in natural convection taking place. From the predicted concentration profiles it seems like the instability never separates from the top layer, and that it dies out with time, contrary to what is observed in this work. In this paper we focus on models applicable on the lab scale. However, a good interpretation of lab experiments is absolutely necessary for a reliable up scaling of the process. We use a lattice Boltzmann (LB) algorithm to model the dissolution of CO2 into water, and study the pressure decay curves in terms of varying Prandtl number and Rayleigh number. The system is similar to that used by Farajzadeh et al., 4 but the use of the lattice Boltzmann model for this particular system is new. The LB method is chosen because, in the future, we would like to extend this work to study CO2 mixing inside the porous media by extending the work in Hiorth et al. 5 The model is used to predict times for onset of instabilities (convection) and parameter space for which an instability will occur. We are particularly interested in pH profiles, as this is comparable to experimental observations and hence gives an additional verification parameter for simulations. A study of the information that can be retrieved from an experiment as function of pH indicator is performed, and will be useful for designing experiments in the future. The outline of this paper is as follows: In the next section we introduce the simulation
3
setup. In section we present the mathematical formulation of the CO2 -water DDC problem, both the continuum formulation and the LB formulation. Section presents the main results, and in chapter we summarize and draw some conclusions from the study. Some calculations are put in an appendix for the interested reader. Appendix gives thorough calculations on the chemistry of the CO2 -water system, appendix describes the imposed instability on the CO2 -water interface, and in appendix we have tested parts of the simulation model againts analytical calulations.
Simulation setup Simulations have been performed in two dimensions for a container with height 20 cm and width 4 cm. The height of the water column is 11.2 cm, and the height of the gas cap is 9.8 cm. The initial pressure in the gas phase is 5 bar and the temperature is assumed to be 21o C. The pressure in the gas phase is assumed to be uniform, and we assume the height of the water column to be constant, because the volume change of CO2 -water mixture is small at the pressure under consideration. 6 We employ a quadratic grid, and hence the ratio between grid points in the height (y) and width (x) direction equals the height to width ratio of the water column, i.e. 2.8. Since simulations have been performed at different resolutions to handle high Rayleigh numbers, the resolution for each simulation will be given as the results are presented. The instability was initiated in the middle of the box by an imposed sinusoidal perturbation of the top layer. Numerical generated dispersion relations showed that the fastest growing initial wavelength perturbation was given by the width of the box, for Ra ∼ 106 . Choosing a given perturbation makes comparison between different grid size and variables more easy (see Appendix for details).
4
Mathematical formulation Continuum formulation The CO2 -water convective system is described by the Navier Stokes equation and the Advectiondiffusion equation coupled by the gravitational force:
∂~u ~ 1 ~ 2~u + c · M mCO2 ~g , + ~u · ∇ ~u = − ∇p + ν ∇ ∂t ρ ρ ∂c 2 + ~u · (∇c) = D∇ c, ∂t
(1) (2)
where ~u [m/s] is the fluid velocity, p [Pa] is the pressure, ν [m2 /s] is the water kinematic viscosity, c [mol/m3 ] is the concentration of CO2 in the water, M mCO2 [kg/mol] is the molecular weight of CO2 , ~g [m/s2 ] is the gravitational acceleration, ρ [kg/m3 ] is the water density and D [m2 /s] is the diffusion coefficient of CO2 in water. We have incorporated the hydrostatic part of the density (ρ~g · ~x) into the pressure and assumed that the fluid is incompressible (ρ = constant) and that the viscosity is constant (Boussinesq approximation). The boundary conditions for the velocity are no-slip on the rigid walls (3) and free-slip at the CO2 -water interface (4).
~u · ~n = 0
~u × ~n = 0
no-slip,
~u · ~n = 0 (∇ × ~u) × ~n = 0 free-slip.
(3) (4)
~n is the outward surface normal. For the CO2 concentration the boundary condition on the rigid walls is the no-flux condition (5). The boundary condition at the CO2 -water interface is
5
given by a rate equation for dissolution of CO2 into water. We have chosen to use an infinite rate constant, which means that there exist instantaneous equilibrium between water and CO2 at the interface (6).
~n · ∇c = 0
rigid wall,
c = ceq PCO2
ceq PCO2
CO2 -water interface.
(5) (6)
is the equilibrium concentration of CO2 at the CO2 (g) -water interface. This
concentration is a function of the partial pressure of CO2 , which is given by the equation of state (EOS). Since the pressure is much lower than the critical pressure for CO2 the difference between the Peng-Robinson EOS and the ideal gas EOS is negligible, thus we simply use the ideal gas EOS for mathematical simplicity. In the Appendices an expression for ceq is derived:
ceq =
K1 PCO2 (g) , K2
(7)
where K1 and K2 are equilibrium constants. Since ceq is dependent on the partial pressure of CO2 , which decreases as CO2 dissolves into the water phase, the boundary condition will be time dependent (dynamic). The final pressure in the gas phase can be estimated from a mass balance consideration for CO2 (see Appendices for details): eq PCO 2 init PCO 2
=
1 , K1 Vw 1 + (101)−1 RT K 2 Vg
(8)
where R is the universal gas constant, T is the absolute temperature and Vw and Vg are the volumes of water and gas respectively. The expression for equilibrium pressure has been
6
used to verify simulation results in appendix . A relation between CO2 concentration and pH is also found in appendix . The result is plotted in figure 1. This relation will be used to produce pH patterns from the simulation results. 7
pH
6
5
4
3 -5
-4
-3 log10(mtot CO )
-2
-1
2
Figure 1: pH as function of CO2 concentration in solution.
The Lattice Boltzmann method The Lattice Boltzmann Method was first proposed by McNamara and Zanetti 7 as a numerical study of the Navier-Stokes equation. Since then the method has become widely used for simulation of porous media fluid dynamics, because of its simplicity in dealing with complex geometries and boundary conditions. In the Lattice Boltzmann formulation fluid flow and advective-diffusive transport of chemical species are described by two sets of single particle distribution functions.
1 [fi (~x, t) − fieq (ρ, ~u)] , τ 1 [gi (~x, t) − gieq (ρ, ~u)] . gi (~x + ~ei δt, t + δt) = gi (~x, t) − τD
fi (~x + ~ei δt, t + δt) = fi (~x, t) −
(9) (10)
fi are the single particle distribution functions for the i’th discrete lattice direction used to 7
describe fluid flow, and τ is the relaxation time, which is given by the kinematic viscosity (ν) of the fluid and the lattice sound speed (cs ). τ describes how fast the distributions are moving towards the equilibrium distributions (fieq ) given by the fluid density and velocity fields. Similarly gi are the single particle distribution functions used to describe the advectivediffusive transport of chemical species. The relaxation time for this process, τD , is given by the diffusion coefficient (D) and the lattice sound speed.
1 ν + , c2s 2 D 1 = 2+ . cs 2
τ = τD
(11) (12)
The equilibrium distribution functions are chosen as functions of the macroscopic variables ρ and ~u in such a way that the Lattice Boltzmann Equation recovers the Navier Stokes equation for fluid flow and the Advection-diffusion equation for transport of chemical species:
fieq gieq
# ~ei · ~u (~ei · ~u)2 ~u · ~u − 2 , = ωi ρ 1 + 2 + cs 2c4s 2cs # " 2 ~u · ~u ~ei · ~u (~ei · ~u) − 2 . = ωi n 1 + 2 + cs 2c4s 2cs "
(13) (14)
ωi are weight coefficients for the different discrete directions and ~ei are the discrete lattice velocities. In the D2Q9 model that we employ, the momentum space is represented by 9 discrete velocity vectors of unit length, as shown in Figure 2. Further, the speed of sound is √ cs = 1/ 3, and the weights are ωi = 4/9 for i = 0, ωi = 1/9 for i = 1 − 4 and ωi = 1/36 for i = 5 − 8. 8
Figure 2: Discrete velocities of unit length in the D2Q9 model. The fluid density (ρ), the concentration of the chemical species (n) and the macroscopic velocity are found from the distribution functions:
ρ =
X
fi ,
(15)
gi ,
(16)
fi~ei .
(17)
i
n =
X i
ρ~u =
X
In order to implement a body force in the LB formulation we use the altered velocity method in which one simply changes the equilibrium velocity (~ueq ) at time t such that the macroscopic velocity at time t + 1 becomes: 8
~u = ~u0 + ~aδt =
1X fi~ei + ~a. ρ i
(18)
By adding an extra term to the equilibrium velocity (~ueq = ~ueq u1 ) and working through the 0 +~ expressions for the equilibrium distributions it is found that ~u1 equals τ~a. The acceleration is given by the force per unit volume (F~ ) divided by the density (ρ). In the altered velocity method the macroscopic velocity is given as the average of the velocity at times t and t + 1. 9
ρ~ueq =
X
fi~ei + τ F~ ,
(19)
1 fi~ei + F~ . 2
(20)
i
ρ~u =
X i
The lattice boundary conditions for the fluid is the bounce-back condition that gives noslip (3) on the rigid walls and the specular reflection that gives the free-slip (4) condition on the CO2 -water interface. The boundary conditions are described by the following relations for the boundary nodes,
fi = f−i f4,7,8 = f2,6,5
bounce-back,
(21)
specular reflection,
(22)
where we use the notation −i to denote the direction opposite to direction i. For the chemical species we used the bounce-back scheme (23) on the rigid walls. On the CO2 -water interface we used a boundary condition adopted from Hiorth et al. 9 For an infinite rate constant for dissolution of CO2 into water this boundary condition is given by (24) where gieq = ωi ceq when the velocity at the boundary is approximated to zero. ceq is the equilibrium concentration defined in equation (7).
gi = g−i
rigid wall,
gi = −g−i + 2gieq CO2 -water interface.
10
(23) (24)
Matching LB parameters and physical parameters Next we will describe how lattice parameters can be related to physical parameters in order to compare with experimental results. The physical length (Lphys ) of the system is divided into nx cells of physical length δxphys = Lphys /nx . Similarly the physical time (tphys ) is divided into nt intervals of length δtphys = tphys /nt . In the LB system the equations are: δxLB = LLB /nx and δtLB = tLB /nt . In our simulations the LB cell length and LB time interval are set to unity: δxLB = δtLB = 1. It follows that the number of cells and number of time intervals equals the LB length and the LB time respectively, so the physical length and time can now be expressed by the LB length and time and the physical cell length and time interval length:
δxphys = Lphys /LLB ,
(25)
δtphys = tphys /tLB .
(26)
This means that the physical value of every parameter with dimension Lp Tq can be found by multiplying the LB value with δxpphys · δtqphys . The length scale δxphys is set by choosing the grid size in the simulation. Then the time scale, δtphys , can be determined from any known parameter with dimension Lp Tq . In this case we have two parameters of this kind that are essential for description of the system, the viscosity of water and the diffusion coefficient of CO2 in water, both with dimension L2 T-1 . The value of the water viscosity, ν, is 1 · 10−6 m2 /s, and the diffusion coefficient of CO2 in water, D, is 2 · 10−9 m2 /s. As these two values are not equal we have two unlike definitions of δt, which means that we can only achieve the physical value for one of these parameters. In order to get around this problem and not having to choose which parameter that gets its physical value, or having to choose two unphysical values as a compromise we rather turn to 11
the dimensionless form of the problem. The dimensionless form of equations (1-2) is:
r P r ~ ∗ 2 ∗ ∂~u∗ ∗ ~ ∗ ∗ ∗ ∗ ~ u = −∇ p + + ~ u · ∇ ∇ ~u + c∗ n~g , ∂t∗ Ra r 1 ∂c∗ ∗ ∗ ∗ + ~ u · (∇ c ) = (∇∗ )2 c∗ . ∂t∗ P rRa
(27) (28)
Two dimensionless numbers, the Prandtl number (Pr) and the Rayleigh number (Ra) enter the equations. They are defined as: ν , D αg0 l03 Ra = , νD Pr =
(29) (30) (31)
where ν is the kinematic viscosity, D is the diffusion coefficient, g0 is the gravitational acceleration, l0 is the system length scale and the constant α is given by
α = c0 M mCO2 /ρ.
(32)
Here c0 is the equilibrium concentration of CO2 at 5 bar, M mCO2 is the molecular weight of CO2 and ρ is the density of water. The starred quantities in equations (27-28) are the p dimensionless variables rescaled by the system length scale (l0 ), the time scale t0 = l0 /αg0 , water density (ρ) and the equilibrium concentration of CO2 at 5 bar (c0 ).
Using the physical values of ν and D, 1 · 10−6 m2 /s and 2 · 10−9 m2 /s respectively the physical value of the Prandtl number is 500. With gravitational acceleration 9.81m/s2 and l0 = 11.2cm, the Rayleigh number becomes Ra = 5 · 1010 . In the next section we will 12
investigate how the pressure decay curve scales as a function of Prandtl number and Rayleigh number. Pr is varied in the simulations by varying the diffusion coefficient. Different Ra are obtained by varying the gravitational acceleration.
Results Simulations at different Pr and Ra 0
0 -0.1
-0.1
-0.2 -0.2
log(P/Pi)
log(P/Pi)
-0.3 -0.3 -0.4
-0.4 -0.5 -0.6
-0.5 -0.7 -0.6
-0.8
-0.7
-0.9 0
10
20
30
40
50
60
70
80
90
0
10
20
30
t1/2 [min1/2] Ra=0.0e0, Pr=200 Ra=2.6e5, Pr=200
40
50
60
70
80
t1/2 [min1/2]
Ra=2.6e6, Pr=200
Ra=2.6e6, Pr=10 Ra=2.6e6, Pr=100
Ra=2.6e6, Pr=200
(b) Constant Ra (Ra = 2.6 · 104 ).
(a) Constant Pr (Pr=200).
Figure 3: Simulated pressure decay curves. Grid size 100x280 for all the simulations.
In figure 3 we have plotted the pressure in the gas phase for different Pr and Ra numbers. The black line in figure 3a shows the pressure development during the simulations when the only force is mixing by diffusion. When Ra > 2.6 · 104 , at some point the pressure deviates from this black line. We call this the first instability and the corresponding time this occurs for tc . We see that when Ra increases the first instability happens earlier due to a higher gravitational force and/or lower viscous force. tc vs. Ra is plotted in figure 4. In figure 3b we have plotted the pressure decay curve for constant Ra, but varying Pr. We see that the instability occur for similar pressure (log(P/Pi ) ≈ -0.02) but varying tc . This is because the Pr number controls the mass transfer between the gas phase and the fluid phase, and is not that important for the transport in the fluid phase.
13
2.5 2
log10(tc [min])
1.5 1 0.5 0 -0.5 -1 -1.5 5
6
7
8 log10(Ra)
9
10
11
Figure 4: Logarithm of time (min) at which the instability occur vs. logarithm of Rayleigh number for Pr=200. The dotted part of the line is extrapolated from the other points.
Comparison with lab data In this section we will compare simulation results to lab data from a pressure decay experiment in a cylindrical container of length 20 cm and diameter 4 cm. The dimensionless numbers were Pr = 500 and Ra = 5 · 1010 . For this value of the Prandtl number, two dimensional simulations are expected to give a good representation of the convective system. 10 Figure 5 shows several pressure decay curves (5a) and their derivatives (5b), including the experimental curves and simulations at different Ra. From these curves we find that the best match to the experimental data is obtained for the simulation with Ra approximately a factor 240 lower than the value of Ra in the experiment, i.e. Ra = 2.1 · 108 . We see from Figure 5b that the pressure decay is too fast for the simulation with a higher Ra (Ra = 2.1 · 109 ). In Schmalzl et al. 10 it is found that both the flow characteristics of the convective system and the system mean velocity is well represented in a 2D simulation. We therefore explain the deviation in Ra to match the experimental pressure decay with the rate of mass transfer between the gas and the water phase, which can be different in two and three dimensions despite similar flow patterns and velocities in the water phase. From the simulations, in addition to the pressure in the gas phase, we have logged the
14
0
0.02
-0.1
0
-0.2 dlog(P/Pi)/d(t1/2)
-0.02
log(P/Pi)
-0.3 -0.4 -0.5
-0.04 -0.06 -0.08
-0.6
-0.1
-0.7 -0.8
-0.12 0
10
20
30
40
50
60
70
80
0
t1/2 [min1/2]
10
20
30
40
50
60
70
80
t1/2 [min1/2]
Experimental: Ra=5.0e10, Pr=500 Simulated: Ra=2.1e07, Pr=200 Simulated: Ra=2.1e08, Pr=200 Simulated: Ra=2.1e09, Pr=200
Experimental: Ra=5.0e10, Pr=500 Simulated: Ra=2.1e07, Pr=200 Simulated: Ra=2.1e08, Pr=200 Simulated: Ra=2.1e09, Pr=200
(a) Pressure decay curves
(b) Derivative of pressure decay curves
Figure 5: Pressure decay curves for (Pr, Ra) = (200, 2.1 · 107 ), (Pr, Ra) = (200, 2.1 · 108 ), (Pr, Ra) = (200, 2.1 · 109 ) and for the experimental results. Grid sizes were 150x420 for Ra = 2.1 · 107 , 300x840 for Ra = 2.1 · 108 and 900x2520 for Ra = 2.1 · 109 . concentration of CO2 in the water phase. Concentration patterns for the simulation with Pr = 200 and Ra = 2.1 · 108 at different times are given in figure 6. We see that the first instability, a mushroom plume, is already prominent well before 1 minute. Also several secondary instabilities become apparent as time passes. In three dimensions the area where most of the transport from the gas phase to the water phase takes place can be represented by a disk (with the diameter of the mushroom stem) of radius r, while in the two dimensional simulation we simulate a cross section which translated to three dimensions effectively gives a channel of width 2r and the length of the container, L. The ratio between these two areas scale as ∼ L/r. If this ratio equals the ratio in Ra between experiment and simulation (∼ 240), it gives a mushroom stem diameter of 2.5 nodes (L = 300 nodes in the simulation). This fits well with the simulation data, where the mushroom stem span in the order of 1-10 nodes. Concentration profiles can not be measured in an experiment, but the use of a pH indicator and a transparent container will give information about concentrations (pH). Brom Thymol blue was used in the experiment. It has a transition pH range that is 6-7.6. The low pH color is yellow, and the high pH color is blue. If we choose the transition pH to be 6.0, the pH profiles for the simulation shown in Figure 6 become as shown in Figure 7. After
15
(a) t = (b) t = (c) t = (d) t = (e) t = (f) t = (g) t = (h) t = (i) t = (j) t = (k) t = (l) t = 5.9 min 11.8 min 17.7 min 23.6 min 29.5 min 35.4 min 41.3 min 47.2 min 53.1 min 59.0 min 64.9 min 70.8 min
(m) t = (n) t = (o) t = (p) t = (q) t = (r) t = (s) t = (t) t = 76.7 min 82.6 min 88.5 min 94.4 min 100.3 min106.2 min112.1 min118.0 min
Figure 6: Concentration patterns for Pr = 200 and Ra = 2.1 · 108 at different times. approximately 17 minutes the pH is below 6 in the whole volume.
(a) t = 5.9 min
(b) t = 11.8 min
(c) t = 17.7 min
Figure 7: pH pattern for Pr = 200 and Ra = 2.1 · 109 at different times. Figure 8 shows pH patterns from the experiment at three different times. The first photo is taken after 4 min, and one can already see traces (thin lines) of low pH falling down from the top interface. The second photo is taken after 11.3 min, and now one can clearly see the low pH regions. The CO2 has reached the bottom of the PVT cell at this point. The third photo is taken after 13.5 min and the PVT cell consist almost completely of low pH. The boxed area in the third photo looks like a mushroom plume, as seen in the simulations (figure 6b). The pH profiles obtained from the simulation shows that the first mushroom instability 16
causes the pH to decrease in a large part of the container, and low pH water has almost reached the bottom of the container after 5.9 min. After 11.8 min it shows that low pH water is distributed more or less throughout the entire container, and somewhere between 11.8 min and 17.7 min the whole container consist of low pH water. The pictures taken during the experiment shows that low pH water has started to mix into the fresh water phase after 4 min. After 11.3 min, as in the simulation, low pH water is scattered around the entire container. The container consist of almost only low pH water after 13.5 min. As seen from the figures, the simulated pH profiles do not exactly match the profiles from the experiment. This is believed to be caused by the sharp transition at pH 6 for the simulation results. There might be a more gradual transition in the experiment.
Figure 8: Photos from the experiment at three different times. The first photo is taken after 4 min and it is possible to see some traces of low pH from the top and down. The second photo is taken after 11.3 min and one sees that CO2 has now reached the bottom of the PVT cell. The third photo is taken after 13.5 min, and the PVT cell is almost filled with CO2 . Also one can clearly see patterns of falling CO2 rich water similar to those seen in the simulations.
The use of pH idicators We have studied the information obtainable during an experiment with use of different pH indicators. By use of different pH indicators, different information can be taken from the experiment. E.g. with a pH indicator with a high transition pH one would get the best 17
information about the rate of the chemical reactions taking place at the CO2 -water interface and the onset of instability. On the other hand, by using a pH indicator with a lower transition pH one will get more information about the structure of the instability (depends on visibility in the experiment) and about the long term importance of advection, see Figure 9.
(a) Transition pH 6. t=0.7 min.
(b) Transition pH 4. t=0.7 min.
(c) Transition pH 6. t=5.9 min.
(d) Transition pH 4. t=5.9 min.
Figure 9: pH profiles at times t=0.7 min and t=5.9 min for Pr = 200 and Ra = 2.1 · 109 , with transition pH 6 and 4. For better resolution (a) and (b) shows only the top water layer, while (c) and (d) show the full water column.
Figure 10 shows the time at which the water will have fully changed color (and visual observations become superfluous) as function of transition pH for for different Pr and Ra numbers. The y-axis has been scaled with the time scales for the different processes; 11
18
τ0 = t/τν τν = L2 /ν τD = L2 /D τb = Vb /(αg0 )
where τν is a viscous time scale, τD a diffusive time scale and τb a buoyancy time scale. τb is the time needed for the acceleration αg0 to act in such a way as to induce a vertical velocity change ∆u = Vb . In order for the Rayleigh number to take the form of equation 31, Vb must equal the viscous velocity Vν = ν/L. From our simulations it seems like the dimensionless time, t0 = t/τν , scales as t0 ∼
√
τD τb .
We interpret this as the geometric mean of the diffusive and buoyancy time scales, which indicates an equal importance of buoyancy (gravitational) forces and viscous/diffusive forces for the convection process for the Ra and Pr values considered in this work. The diffusive time scale can say something about how fast the instability broadens through diffusion, while the buoyancy time scale can say something about how fast the instability is pulled towards the bottom of the container. The viscous time scale is accounted for through Vb = Vν . The equal importance of the two time scales (diffusion and buoyancy) can be interpreted as follows; a thin convective path caused by a high gravitational force and a low viscosity (high Ra) transports the same amount of CO2 rich water as does a broad convective path caused by a lower gravitational force and a higher viscosity (low Ra). In Figure 10 we have also indicated the range of transition for some pH indicators. It can be used as a pointer to which indicator one would use in order to get the desired information from an experiment.
19
14
t0/(τDτb)1/2
12
Screened methyl orange (second transition)
Pr200RaE09 Pr200RaE08 Pr200RaE06 Pr100RaE06 Pr10RaE06
10
8
6 Bromocreosol green 4
4.5
5 5.5 transition pH
Bromthymol blue 6
Figure 10: Time (dimensionless) at which the entire water volume has changed color from high pH color to low pH color, as function of transition pH for different Pr and Ra. The transition pH for some pH indicators are marked in the plot.
Summary and Conclusions We have studied the effect of Rayleigh number and Prandtl number on CO2 -water mixing through dissolution-diffusion-convection with the lattice Boltzmann method. The use of the LB method for the particular system under consideration is new. The simulations show the importance of advection contrary to diffusion. We have found that instabilities start to develop somewhere between Ra = 2.6 · 104 and Ra = 2.6 · 105 in the two dimensional simulations. Pressure decay experiments have been used in the literature to determine diffusion coefficients. 12,13 These methods were developed for gases diffusing into liquids where the molecular weight of the gas molecules are lower than the molecular weight of the liquid molecules. However, because of the simplicity of the method it is also desirable to use it for the CO2 -water system. We want to stress the importance of advection for this system. Simulations at Rayleigh numbers several decades below the physical Rayleigh number shows one or more secondary instabilities, meaning that advection is still a dominating process, after several
20
hours of experiment time. Two dimensional simulations are compared to an experiment performed in a cylinder. We find that a Rayleigh number a factor ∼ 240 lower than the experimental value gives the best fit between simulated and experimental pressure decay, and explain this with a higher mass transfer rate from the gas phase to the water phase in two dimensions, due to a larger mass transfer area at the CO2 -water interface. We have calculated pH profiles from the simulated concentration profiles, and compared to experimental pH profiles. Similar structures are found in the water. However, we suggest that even more effort is put into documenting visual observations of pH profiles. The study illustrates the importance of the pH indicator used in the experiment to verify the physical model, since different models can explain similar results. 2,4 Hence, it is important with optimal documentation in the form of pictures and possibly also recorded movie. We also suggest that a pH indicator with a lower transition pH is used to gain more information about the long term behavior of the system.
Acknowledgement The authors thank the Norwegian Research Council (200600/S60) and the Ekofisk and Valhall lisences for financial support.
Appendices Description of the chemistry of the CO2-water system The equilibrium between CO2 and water is described by the chemical reactions given in Table 1.
21
Table 1: Chemical reactions describing the CO2 -water system. The logK values are given at 25o C and 5 bar. For practical reasons we have used the notation H2 CO3 +CO2(aq) → H2 CO3 , as described by Garrels and Christ. 14 CO2 (g) + H2 CO3 HCO-3 OH+
H2 O H+
⇀ ↽ ⇀ ↽ ⇀ ↽ ⇀ ↽
HCO-3 HCO-3 CO23 H2 O
+ + +
H+ H+ H+
logK1 logK2 logK3 logK4
= = = =
-7.87 -6.39 -10.32 14.05
Carbonic acid is formed and therefore the pH of the solution will be slightly acidic and the CO23 concentration will be low. The total concentration of CO2 in aqueous form is then:
mtot CO2 (aq) = mH2 CO3 + mHCO3− ,
(33)
where mi denotes the concentration of species i. If we assume that activity coefficients are close to unity mH2 CO3 and mHCO3− are given by two mass action expressions:
mH2 CO3 = mHCO3− =
mHCO3− mH + K2
,
K1 PCO2 , mH +
(34) (35)
where PCO2 is the partial pressure of CO2 in atm. PCO2 can be found from an equation of state (EOS). The ideal gas law was used in this case. The concentration of H+ is given by charge balance:
mH + = mHCO3− + mOH − ,
(36)
where mOH − = (K4 mH + )−1 . At equilibrium mOH − > (K1 K4 )−1 ≈ 10−6 , so we neglect mOH − in equation (36) when at equilibrium. We then find an expression for the total CO2 concentration in the water at equilibrium: 22
ceq =
q K1 PCO2 (g) + K1 PCO2 (g) . K2
(37)
The second term in equation (37) is more than two orders of magnitude lower than the first term, and hence it can be neglected. The equilibrium pressure in the gas phase can be estimated analytically by considering mass balance for CO2 . The initial amount of CO2 in the gas phase must be equal to the amount of CO2 in the water at equilibrium plus the amount of CO2 that is left in the gas phase at equilibrium. We have the following relations:
ninit CO2 /g = nend CO2 /g =
init PCO V 2 g
RT
eq PCO V 2 g
RT
,
(38)
,
(39)
K1 −2 eq 10 PCO2 Vw . K2
(40)
−1 nend CO2 /w ≈ (1.01)
Vg and Vw are the volume of the gas phase and the liquid phase respectively, R is the universal gas constant, T is the absolute temperature and the CO2 pressure is given in Pa. The analytical expression for the equilibrium pressure is eq PCO 2 init PCO 2
=
1 1+
K1 Vw (101)−1 RT K 2 Vg
.
(41)
When activity coefficients are assumed to be close to unity the pH of the solution is defined as pH = −log10 mH + . Combining equations (33-35) with equation (37) and the approximation mH + ≈ mHCO3− we find an equation for mH + at equilibrium as function of CO2 pressure: 1 2 K1 m + + mH + − PCO2 = 0. K2 H K2 23
(42)
The solution of this is found straightforward and the equilibrium pH is expressed as:
eq
pH
= −log10
"
K2 2
−1 +
s
4K1 PCO2 1+ K22
!#
.
(43)
The equilibrium pH is plotted as function of CO2 pressure in figure 11a. When the solution is not in equilibrium with the CO2 gas, equation (35) is no longer valid. Equations (33-34) give an expression for mHCO3− as function of mtot CO2 , and charge balance is given by equation (36). We can no longer neglect the OH- -term as this term becomes important when pH → 7. The equation for mH + when the solution is out of equilibrium with the gas becomes a third order equation: 1 1 1 3 2 m + + mH + − mCO2 tot + = 0. mH + − K2 H K2 K4 K4
(44)
The only positive root of this third order equation is:
mH + = 2
p −Qcos
θ a2 − , 3 3
where
Q = R = θ = a0 = a1 =
3a1 − a22 , 9 9a1 a2 − 27a0 − 2a32 , 54 ! R arccos p , −Q3 K2 /K4 , 1 , −K2 mtot + CO2 K2 K4
a2 = K 2 .
24
(45)
The pH is plotted versus CO2 concentration in figure 11b. We have used this relation to plot
7
7
6
6
pH
pH
pH profiles in solution.
5
4
5
4
3
3 -5
-4
-3
-2 -1 log10(PCO [atm])
0
1
-5
2
-4
-3 log10(mtot CO )
-2
-1
2
(a) equilibrium
(b) non-equilibrium
Figure 11: (a) Equilibrium pH as function of CO2 pressure. (b) pH as function of CO2 concentration in solution.
Imposed instability The LB code was run on different grid sizes to check the numerics. It was found that the size of numerical errors was significant in that instabilities appeared at some grid sizes but not at others. Therefore it was necessary to impose an initial instability on the top boundary such that the numerical errors become negligible. This was done by multiplying the gas pressure with a sinusoidal instability at each time step: 3πxδx +π . PCO2 (x) → PCO2 (x) · 1 + 0.1sin W
(46)
W is the width of the container and δx is the space interval between each node, such that xδx is the distance from the left wall. We have studied different wavelengths of the imposed instability and the effect of Prandtl number and Rayleigh number on growth of the instability. Figure 12 shows concentration profiles close to the onset of instability (critical time) for simulations with varying wavelength on the imposed instability and varying Pr and Ra. Figure 13 shows the time of onset of 25
instability as function of wavelength for the different combinations of Pr and Ra. It was expected that the Prandlt number could affect the growth of the instability since it is a measure of the resistance in the fluid. A high resistance is expected to give less coalescence of neighboring concentration maxima. However, simulation series at (Pr, Ra) = (100, 2.6·105 ) and (Pr, Ra) = (200, 2.6 · 105 ) showed similar concentration patterns. Pr = 200 is the highest value of the Prandlt number that we have been able to run in the simulations. The Rayleigh number was also expected to affect the results, since a higher Rayleigh number is given by a higher force in the vertical direction. If that force is large enough the acceleration becomes large enough that the concentration maxima falls down faster than they can diffuse into the neighboring maxima. From figure 12 it is clear that the Rayleigh number affects the coalescence, and also the wavelength of the fastest growing instability. The interpretation is that it is the relation between the characteristic times for the force and for diffusion that determine the most unstable wavelength and the critical time for onset of instability.
(Semi) Analytical calculations on diffusion The DDC process does not have an analytical solution, but a semi analytical solution can be found for the case where diffusion is the only mass transfer mechanism. By assuming that the dissolution can be described using Henry’s law the CO2 -water system can be described by Fick’s second law (diffusion equation) in one spatial dimension (47). ∂ 2C ∂C −D 2 =0 ∂t ∂x
(47)
The initial condition is described by
C=
Pi H
x=0, t=0 x>0, t=0
0
26
,
(48)
(a) λ ≈ 0.031L (b) λ ≈ 0.092L (c) λ ≈ 0.174L (d) λ ≈ 0.337L (e) λ ≈ 0.674L (f) λ ≈ 0.031L t ≈ 145 min t ≈ 145 min t ≈ 145 min t ≈ 102 min t ≈ 73 min t ≈ 30 min
(g) λ ≈ 0.092L (h) λ ≈ 0.174L (i) λ ≈ 0.337L (j) λ ≈ 0.674L (k) λ ≈ 0.031L (l) λ ≈ 0.092L t ≈ 28 min t ≈ 21 min t ≈ 16 min t ≈ 16 min t ≈ 14.5 min t ≈ 9.2 min
(m) λ ≈ 0.174L (n) λ ≈ 0.337L (o) λ ≈ 0.674L t ≈ 7.5 min t ≈ 8.0 min t ≈ 9.7 min
Figure 12: CO2 concentration profiles in water at onset of instability for different wavelengths of the imposed instability at different Pr and Ra. Subfigure (a)-(e) are for (Pr, Ra) = (100, 2.6 · 105 ), subfigure (f)-(j) are for (Pr, Ra) = (100, 2.6 · 106 ) and subfigure (k)-(o) are for (Pr, Ra) = (200, 2.1 · 107 ). A simulation series was also performed for (Pr, Ra) = (200, 2.6·105 ), and the concentration profiles look the same as for the (Pr, Ra) = (100, 2.6·106 ) case (subfigures (a)-(f)).
300
250
tc [min]
200
150
100
50
0 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
λ [m] Pr=100, Ra=2.6e+05 Pr=200, Ra=2.5e+05
Pr=100, Ra=2.6e+06 Pr=200, Ra=2.1e+07
Figure 13: Wavelength versus time for onset of instability.
27
and the boundary conditions are stated at x = 0 (CO2 -water interface) and at x = L (bottom of container). Since there is no flux through the bottom of the container the boundary condition at x = L is ∂C = 0. ∂x x=L
(49)
The boundary condition at x = 0 is given by conservation of mass. The number of moles of CO2 lost by the gas phase must equal the number of moles dissolved in the water. The flux of CO2 out of the gas phase is found by differentiating the equation of state, i.e. the ideal gas law, while the flux of CO2 into the water phase is given by Fick’s first law. Mass conservation then gives
where α =
hg H RT D
∂C ∂C = α , ∂x x=0 ∂t x=0
(50)
is constant. Equation (50) gives the boundary condition at the CO2 -water
interface. The boundary value problem given by equations (47-50) can by solved by Laplace transform. In the Laplace space the equations take the following form:
D
d2 C¯ ¯ − C(x, 0), = Cs dx2
∂ C¯ Pi ¯ , = α Cs − ∂x x=0 H ∂ C¯ = 0, ∂x x=L ¯ s). The solution of this system is where C¯ = C(x,
28
(51)
√s √s √s Pi e D x−2 D L + e− D x ¯ s) = h √s C(x, p p i . H s + α1 Ds + e−2 D L s − α1 Ds
(52)
√s √s √s Pi e D x−2 D L + e− D x √s P¯ (s) = h p p i . s + α1 Ds + e−2 D L s − α1 Ds
(53)
The pressure in the gas phase is obtained by using the Laplace transform of Henry’s law; ¯ s)|x=0 : P¯ (s) = H C(x,
Equations (52-53) give the solution of the boundary value problem in Laplace space. In order to find the solution in the real variables, C(x, t), the inverse Laplace transforms of (52-53) must be found. In this case we must turn to numerical methods. An algorithm 15 based on a method by de Hoog et al has been implemented in order to model concentration profiles and pressure decay over time. These results has then been used to verify results from the Lattice Boltzmann code. 1
0.8 analytical simulated
analytical simulated
0.9
0.6
c/c0
P/Pi
0.8 0.4
0.7 0.2
0.6
0.5
0 0
2500 Time [h]
5000
0
(a) pressure decay
2
4
6 8 Distance [cm]
10
12
(b) concentration profile
Figure 14: (a) Simulated (200x200 nodes) and semi-analytical pressure decay curve for D = 2 · 10−9 m2 /s. (b) Simulated (200x200 nodes) and semi-analytical concentration profile. Figure 14a shows the pressure decay curve for the system when diffusion is the only transport mechanism. Both the semi-analytical solution and the simulated solution are plotted, and the results are more or less identical. The normalized equilibrium pressure also eq init equals the value predicted by equation (8), PCO /PCO = 0.521. The grid size used in the 2 2
simulation was 200x200. 29
Figure 14b shows the concentration profile after 60 hours. Both the semi-analytical solution and the simulated solution are plotted.
References 1. Chen, C.; Zhang, D. Pore-scale simulation of density-driven convection in fractured porous media during geological CO2 sequestration. Water Resour. Res. 2010, 46 . 2. Yang, C.; Gu, Y. Accelerated Mass Transfer of CO2 in Reservoir Brine due to Density Driven Natural Convection at High Pressures and Elevated Temperatures. Ind. Eng. Chem. Res. 2006, 45, 2430–2436. 3. Pruess, K.; Nordbotten, J. Numerical Simulation Studies of the Long-term Evolution of a CO2 Plume in a Saline Aquifer with a Sloping Caprock. Transp. Porous Med. 2011, 90, 135–151. 4. Farajzadeh, R.; Zitha, P. L. J.; Bruining, H. Enhanced Mass Transfer of CO2 into Water: Experiment and Modeling. Ind. Eng. Chem. Res. 2009, 48, 6423–6431. 5. Hiorth, A.; Jettestuen, E.; Cathles, L. M.; Madland, M. V. Precipitation, dissolution, and ion exchange processes coupled with a lattice Boltzmann advection diffusion solver. Geochim. Cosmochim. Acta 2013, 104, 99–110. 6. Tegetmeier, A.; Dittmar, D.; Fredenhagen, A.; Eggers, R. Density and Volume of Water and Triglyceride Mixtures in Contact with Carbon Dioxide. Chem. Eng. Process. 2000, 39, 399–405. 7. McNamara, G. R.; Zanetti, G. Use of the Boltzmann Equation to Simulate Lattice-Gas Automata. Phys. Rev. Lett. 1988, 61 . 8. Buick, J. M.; Greated, C. A. Gravity in a lattice Boltzmann model. Phys. Rev. E 2000, 61, 5307–5320. 30
9. Hiorth, A.; Lad, U. H. A.; Evje, S.; Skjæveland, S. M. A Lattice Boltzmann-BGK algorithm for a diffusion equation with Robin boundary condition - application to NMR relaxation. International Journal for Numerical Methods in Fluids 2009, 59, 405–421. 10. Schmalzl, J.; Breuer, M.; Hansen, U. On the validity of two-dimensional numerical approaches to time-dependent thermal convection. Europhys. Lett. 2004, 67, 390–396. 11. Costa, V. A. F. A time scale-based analysis of the laminar convective phenomena. International Journal of Thermal Sciences 2002, 41, 1131–1140. 12. Renner, T. A. Measurement and Correlation of Diffusion Coefficients for CO2 and Rich Gas Applications. SPE Reservoir Engineering 1988, 3, 517–523. 13. Riazi, M. R. A new method for experimental measurement of diffusion coefficients in reservoir fluids. Journal of Petroleum Science and Engineering 1996, 14, 235–250. 14. Garrels, R. M.; Christ, C. L. Solutions, minerals and equilibria; Harper & Row: New York, 1965. 15. Hollenbeck, K. J. INVLAP.M: A matlab function for numerical inversion of Laplace transforms by the de Hoog algorithm. http://www.isva.dtu.dk/staff/karl/invlap.htm, 1998.
31
Abstract 1: A study on the eect of pore geometry on mineral changes. J. Pedersen, E. Jettestuen, J. L. Vinningland, M. Madland, L. M. Cathles III and A. Hiorth. Presented at the Goldschmidt conference, Prague, Czech Republic, August 14-19, 2011.
A study on the effect of pore geometry on mineral changes J. PEDERSEN1, E. JETTESTUEN1*, J.L. VINNINGLAND1, M. MADLAND2, L.M. CATHLES III3 AND A. HIORTH1,2 1
International Research Institute of Stavanger (IRIS), P. O. Box 8046, 4068 Stavanger, Norway (*correspondence:
[email protected]) 2 The University of Stavanger, 4036 Stavanger, Norway 3 Cornell University, Ithaca, New York, USA Injection of seawater in 130°C offshore chalk oil fields will induce chemical alterations as the seawater is not in equilibrium with the formation. This has been confirmed by core experiments. Furthermore, the core experiments reveal that the chemical alteration induces enhanced compaction and enhanced oil production. Many conceptual models have been suggested for this phenomenon, but quantitative geochemical simulations on the pore scale is missing. We investigate the spatial distribution of the chemical alterations induced by seawater and MgCl2 flooding at 130°C; the net chemical change in the pore space is compared with the net chemical change observed in the core experiments. A full geochemical solver integrated with a lattice Boltzmann (LB) model is used to model the pore scale reactive flow. The LB model solves the fluid flow and the advection-diffusion of the chemical species, while the geochemical model gives the interaction between the aqua chemical species and the pore mineral-surfaces. The geochemical solver has been compared successfully with PHREEQC and effluent from core experiments. A simulation on a carbonate sample is shown in Fig.1.
Figure 1: 0.219M MgCl2 is flooded from the left at 130°C, the pore space is initial in equilibrium with distilled water. After 2 pore volumes of flooding magnesite is precipitated nonuniformly in the pore space (blue (opaque) to red (solid) colours). Only calcite is present initially.
Abstract 2: Models for evolution of reactive surface area during dissolution and precipitation. J. Pedersen, E. Jettestuen, T. Hildebrand-Habel, J.
L. Vinningland, M. V. Madland, R. I. Korsnes and A. Hiorth. Presented at European Geosciences Union General Assembly, Vienna, Austria, 27 April - 02 May, 2014.
Geophysical Research Abstracts Vol. 16, EGU2014-12775, 2014 EGU General Assembly 2014 © Author(s) 2014. CC Attribution 3.0 License.
Models for evolution of reactive surface area during dissolution and precipitation Janne Pedersen (1), Espen Jettestuen (1), Tania Hildebrand-Habel (1), Jan Ludvig Vinningland (1), Merete Vadla Madland (2), Reidar Inge Korsnes (2), and Aksel Hiorth (1) (1) International Research Institute of Stavanger, P.O. Box 8046, N-4068 Stavanger, Norway, (2) University of Stavanger, Petroleum Department, N-4036 Stavanger, Norway
During water flooding of a reservoir, minerals can dissolve and/or precipitate if the injected water is out of equilibrium with the formation. A net mass transfer between solid and fluid will result in a dynamically changing pore space, which in turn may change the permeability and/or the porosity of the reservoir. When secondary minerals precipitate from solution they will form on top of the primary minerals on the pore walls, and hence the reactive surface area of the individual minerals constituting the porous medium will change. This will in turn affect the dissolution/precipitation rates. In this work we study three different models for the evolution of reactive surface areas during flooding of a core with a brine that is in disequilibrium with the rock. The three models differ in the way secondary minerals are distributed on the solid surface: In model I the whole surface area of both primary and secondary minerals is reactive; hence there will be no screening effect when secondary minerals form. In model II secondary minerals form as a monolayer on the primary minerals that will screen primary minerals from the flow. Model I and II represent two extremes regarding the screening effect; namely no screening (model I) and full screening (model II). Model III is motivated from observations of unflooded and flooded chalk using scanning electron microscopy (SEM). In this model secondary minerals are assumed to form preferentially near crystal defects, being e.g. dislocations in crystallographic pattern or grain contacts. Also the rate of precipitation for forming minerals varies for precipitation onto primary (dissimilar) minerals and precipitation onto secondary (similar) minerals. The three models are implemented into a lattice Boltzmann (LB) based geochemical simulator, and simulation results are compared to results from a 3 years long core flooding experiment with outcrop chalk flooded with 0.219M MgCl2 at 130◦ C, as well as SEM studies. Before flooding 98wt% of the core consisted of calcite, and after the flooding less than 20wt% was calcite and ∼80wt% was magnesite. When simulating this experiment, the three models give very different effluent profiles. Model I and III fits the experimental curve quite well. We will discuss the three models in light of the experimental results.
