Microscale modeling of natural gas hydrates in reservoirs Muhammad Qasim
Dissertation for the degree philosophiae doctor (PhD) at the University of Bergen
2012
i
Dedication
I would like to dedicate this thesis to three very important personalities in my life: To my father and late mother as a small gratitude for their invaluable efforts in raising and guiding me to reach this level. To my wife Imrana Shahzadi without whose support I would not have been able to complete this work.
ii
Acknowledgements Working on the Ph.D. has been a wonderful and often overwhelming experience. It has been the real learning experience in grappling with how to write papers, give talks, work in a group and independently, stay up until the birds start singing, and stay focus. This dissertation would not have been possible without the guidance and the help of several individuals who in one way or another contributed and extended their valuable assistance in the preparation and completion of this study. First and foremost, my utmost gratitude goes to my supervisor Professor Bjørn Kvamme, for his invaluable guidance and immense help during my PhD. To work with him was extraordinary not only because is he highly knowledgeable in his field of expertise, but also he displayed extreme patience in answering my questions. His positive and friendly attitude always makes me comfortable to discuss problems with him and always provided prompted feedbacks. I am thankful to him for introducing me to the field of natural gas hydrates. I would take this opportunity to thank my cosupervisor Professor Tatiana Kuznetsova, who helped me in dealing with challenges related to using the computational resources available in the department and the Cray XE6 distributed memory system called Hexagon. She also supported me in practical issues in initial times of my PhD which helped me in settling in my workplace. I sincerely acknowledge the grant and support from Research Council of Norway through the following projects: SSC-Ramore, “Subsurface storage of CO2 - Risk assessment, monitoring and remediation”, Research Council of Norway, project number: 178008/I30, FME-SUCCESS, Research Council of Norway, project number: 804831, PETROMAKS, “CO2 injection for extra production”, Research Council of Norway, project number: 801445 and Gas hydrates on the NorwegianBarents Sea-Svalbard margin (GANS, Norwegian Research Council Project No. 175969/S30).
iii Here, I also want to acknowledge number of my colleagues who supported me in my work one way or another. I am pleased to thank my colleague Khuram Baig, who shared the office with me and was deeply involved during my work and introduced me to the existing Phase Field Theory simulation. I also wish to thank Ashok Chejara, Mohammad Taghi Vafaei, Bjørnar Jensen and Khaled Jemai for providing a very good environment. My deepest gratitude goes to my family for their unflagging love and support. I am thankful to my late mother, father, brothers and sisters for their trust on me and encouragement. Here I wish to give a special thanks to my wife Imrana Shahzadi and my two daughters Ameena and Aleeza, whose love always provided me an extra energy and strength.
iv
List of Publications Reviewed publications in scientific journals M. Qasim, B. Kvamme, and K. Baig, Phase field theory modeling of CH4/CO2 gas hydrates in gravity fields, International Journal of Geology, Volume 5, Issue 2, 2011, pp. 48-52 (Attached as Paper 2) M. Qasim, K. Baig, B. Kvamme and J. Bauman, Mix Hydrate formation by CH4-CO2 exchange using Phase Field Theory with implicit Thermodynamics, International Journal of Energy and Environment, Volume 6, Issue 5, 2012, pp. 479-487 (Attached as Paper 6) B. Kvamme, K. Baig, M. Qasim and J. Bauman, Thermodynamic and Kinetic Modeling of Phase Transitions for CH4/CO2 Hydrates, Submitted to International Journal of Energy and Environment (Attached as Paper 5) B. Kvamme, M. Qasim, K. Baig and P. H. Kivelä , J. Bauman, Hydrate phase transition kinetics from Phase Field Theory with implicit hydrodynamics and heat transport, Submitted to Journal of Chemical Physical (Attached as Paper 7)
Publications in conference proceedings with review system P. H. Kivelä, K. Baig, M. Qasim and B. Kvamme, Phase Field Theory Modeling of Methane Fluxes from Exposed Natural Gas Hydrate Reservoirs, American Institute of Physics Conference Proceedings, Volume 1504, Issue 1, 2012, pp. 351-363 (Attached as Paper 1) M. Qasim, B. Kvamme and K. Baig, Phase Field Theory Modeling of CH4/CO2 Gas Hydrates in Gravity Fields, In Proceedings from the European Conference of Chemical Engineering, Puerto De La Cruz, Tenerife, November 30-December 2, 2010, pp. 164-167 M. Qasim, B. Kvamme and K. Baig, Modeling Dissociation and Reformation of Methane and Carbon Dioxide Hydrate using Phase Field Theory with implicit hydrodynamics, In Proceedings from the 7th international conference on gas hydrate (ICGH7), Edinburgh, Scotland, July 17, 2011- July 21, 2011, 9 pages (Attached as Paper 3)
v M. Qasim, K. Baig and B. Kvamme, Phase Field Theory modeling of Phase transitions involving hydrate, In Proceedings from the 9th international conference on Heat and Mass transport, Harvard, Cambridge, USA, January 25-27, 2012, pp. 222-228 (Attached as Paper 4) B. Kvamme, K. Baig, M. Qasim and J. Bauman, Thermodynamic and Kinetic Modeling of Phase Transitions for CH4/CO2 Hydrates, In Proceedings from the 9th international conference on Heat and Mass transport, Harvard, Cambridge, USA, January 25-27, 2012, pp. 229-235
Oral and Poster Presentations K. Baig, M. Qasim, P. H. Kivelä, and B. Kvamme, Phase Field Theory Modeling of Methane Fluxes from Exposed Natural Gas Hydrate Reservoirs, Oral presentation at the 7th International Conference of Computational Methods in Sciences and Engineering (ICCMSE), Rhodes, Greece, September 29 – October 04, 2009 M. Qasim, B. Kvamme and K. Baig, Phase Field Theory Modeling of CH4/CO2 Gas Hydrates in Gravity Fields, Oral presentation at the European Conference of Chemical Engineering, Puerto De La Cruz, Tenerife, November 30-December 2, 2010 M. Qasim, B. Kvamme and K. Baig, Modeling Dissociation and Reformation of Methane and Carbon Dioxide Hydrate using Phase Field Theory with implicit hydrodynamics, Poster presentation at the 7th international conference on gas hydrate (ICGH7), Edinburgh, Scotland, July 17, 2011- July 21, 2011 M. Qasim, Microscale Modeling of natural gas hydrates dynamics in reservoirs, Oral presentation in PhD and Research fellow seminar in Oslo, September 1, 2011 M. Qasim, K. Baig and B. Kvamme, Phase Field Theory modeling of Phase transitions involving hydrate, Oral presentation at the 9th international conference on Heat and Mass transport, Harvard, Cambridge, USA, January 25-27, 2012 B. Kvamme, K. Baig, M. Qasim and J. Bauman, Thermodynamic and Kinetic Modeling of Phase Transitions for CH4/CO2 Hydrates, Oral presentation at the 9th international conference on Heat and Mass transport, Harvard, Cambridge, USA, January 25-27, 2012
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Contents DEDICATION....................................................................................................................................... I ACKNOWLEDGEMENTS ................................................................................................................ II LIST OF PUBLICATIONS ............................................................................................................... IV CHAPTER 1:
INTRODUCTION ................................................................................................. 1
1.1
STRUCTURE OF HYDRATE ........................................................................................................ 1
1.2
HISTORY AND THE NEGATIVE AND POSITIVE SIDES OF HYDRATES IN NATURE AND INDUSTRY . 6
1.3
KINETICS OF GAS HYDRATE FORMATION AND DISSOCIATION................................................. 10
1.4
CO2 SEQUESTRATION IN CH4 HYDRATE ............................................................................... 13
CHAPTER 2: 2.1
2.2
THERMODYNAMICS ............................................................................................................... 18 2.1.1
Free energy ................................................................................................................ 18
2.1.2
Gibbs Phase Rule ....................................................................................................... 20
2.1.3
Hydrate Thermodynamics .......................................................................................... 22
2.1.4
Fluid thermodynamics ................................................................................................ 23
2.1.5
Aqueous Thermodynamics.......................................................................................... 25
PHASE FIELD THEORY ........................................................................................................... 26 2.2.1
CHAPTER 3: 3.1
Three Components PFT ............................................................................................. 28 NUMERICAL IMPLEMENTATION ............................................................... 30
PHASE FIELD CODE ............................................................................................................... 30 3.1.1
Extensions of the code ................................................................................................ 32
3.1.2
Hydrodynamics Implementation................................................................................. 33
3.1.3
Heat Transport Inclusion ........................................................................................... 33
CHAPTER 4: 4.1
PHASE FIELD THEORY AND THERMODYNAMICS ................................ 17
SUMMARY OF PAPERS ................................................................................... 37
PHASE FIELD THEORY MODELING OF METHANE FLUXES FROM EXPOSED NATURAL GAS HYDRATE
RESERVOIRS .................................................................................................................................... 37
vii 4.2
PHASE FIELD THEORY MODELING OF CH4/CO2 GAS HYDRATES IN GRAVITY FIELDS ............. 39
4.3
MODELING DISSOCIATION AND REFORMATION OF METHANE AND CARBON DIOXIDE HYDRATE USING
PHASE FIELD THEORY WITH IMPLICIT HYDRODYNAMICS ................................................................. 40 4.4
PHASE FIELD THEORY MODELING OF PHASE TRANSITIONS INVOLVING HYDRATE ................. 41
4.5
THERMODYNAMIC AND KINETIC MODELING OF PHASE TRANSITIONS FOR CH4/CO2 HYDRATES 41
4.6
MIX HYDRATE FORMATION BY CH4-CO2 EXCHANGE USING PHASE FIELD THEORY WITH IMPLICIT
THERMODYNAMICS ......................................................................................................................... 42 4.7
HYDRATE PHASE TRANSITION KINETICS FROM PHASE FIELD THEORY WITH IMPLICIT HYDRODYNAMICS
AND HEAT TRANSPORT ..................................................................................................................... 42
4.8
OVERVIEW AND MODIFIED CODE ........................................................................................... 44
CHAPTER 5:
CONCLUSIONS .................................................................................................. 48
CHAPTER 6:
FUTURE WORK ................................................................................................ 51
6.1
COMPUTATIONAL IMPROVEMENTS ........................................................................................ 51
6.2
INFLUENCE OF WATER SOLUBLE COMPONENTS ON AQUEOUS THERMODYNAMICS ................ 51
6.3
BUBBLE MERGING ................................................................................................................. 52
6.4
IMPROVEMENT IN HYDRODYNAMIC MODEL .......................................................................... 52
6.5
HYDRATE SEALING ............................................................................................................... 52
6.6
HEAT TRANSPORT DURING HYDRATE PRODUCTION ............................................................... 53
Chapter 1: Introduction This chapter introduces some basics related to natural gas hydrates as well as carbon dioxide hydrates. The content is based on reviews of some of all the available literature on the classes of hydrates formed by small hydrocarbons as well as some small inorganic components like for instance carbon dioxide and hydrogen sulfide. The introduction is narrowed down to a focus on aspects that are relevant for the background and priorities of the scientific work in this PhD project. Sections 1.1 and 1.2 are devoted to some basic knowledge of structures, historical perspective, geo and environmental Hazards, potential energy source and the current interests in hydrates. Modeling of hydrate phase transition is the core of this PhD study and a brief overview of hydrate kinetic modeling in the past is given in section 1.3. Some of the examples used to illustrate the kinetic theories applied in this work are related to storage of CO2, with special focus on exchange of in situ CH4 hydrate in reservoirs with injected CO2. The last section of this chapter gives a brief review of this concept. The content given in this chapter is based on more general knowledge from a recently updated book on natural gas hydrate from Sloan and Koh [1], the book from Makogon [2] and a PhD thesis [3] and some other sources which will otherwise be referred to during the text.
1.1 Structure of hydrate Natural gas hydrates are crystalline solids composed of water and gas at high pressure and low temperature conditions within the upper hundred meters of the subseabed sediments [4]. Macroscopically they look very similar to ice or snow but with many different properties and therefore sometimes gas hydrate are known as ice that burns.
2 It is worth noting that gas hydrates are not chemical compounds since the enclathrated molecules are not bonded to the lattice and water molecules are held together by hydrogen bonds and mainly non-polar attractions to enclathrated molecules. These enclathrated gas molecules are called guest molecules and obviously have to fit into the water cavities (host) in terms of volume as illustrated in Figure 1.1. Note that in this figure the spheres inside the lattice have sizes of approximately methane (nearly spherical) relative to the distance between the oxygen positions in the cavity. The size of the small blue spheres should be approximately 75% of the spheres inside the cavity so it is a fairly dense structure. Volume of water in the hydrate is roughly 10% larger than in liquid water. There may be number of guest molecules for example methane (CH4), ethane, propane and carbon dioxide. Methane is the most dominant guest found in naturally existing (insitu) gas hydrate and that is why it is considered to be a potential source of energy. This aspect will be discussed in more details in section 1.2 and also in section 1.4 which discusses one possible method to produce these hydrate resources.
Figure 1.1: Structure - I clathrate lattice with encaged methane model molecules approximated as spheres. (Prepared by Bjørn Kvamme[5])
Gas hydrate mostly exist in two cubic crystalline structures, structure I (sI) and structure II (sII)[1]. Sloan and Koh [1] also propose that there may also rarely be
3 found a third type hexagonal structure H, denoted as sH. These structures vary in composition and types of cavities that constitutes the hydrate structure depending on the size of the guest molecule [6]. Structure I is a cubic crystalline in shape and formed with guest molecules having diameter between 4.2 and 6 Å, such as methane, carbon dioxide, ethane and hydrogen sulfide. One unit cell in it consists of 46 water molecules and the cubic unit cell have average lengths of 12.01 Å which is normally treated as constant for the limited range of temperatures relevant for industrial hydrate occurrences. It has two small and six large cages in one unit cell. The small cage has the shape of a 12 sided
cavity
with
12
pentagonal
faces
in
each
side
and
called
pentagonal dodecahedron (512). The large cage has the shape of a 14 sided cavity (with
12
pentagonal
faces
and
2
hexagonal)
faces
and
called
tetrakaidecahedron (51262) as shown in Figure 1.2. Structure II can be formed by small molecule gases like nitrogen and hydrogen (diameter < 4.2 Å) and/or by large guest molecules like propane or iso-butane (6Å < diameter < 7Å). It is again cubical in its crystal structure. One unit cell is made of 16 small and 8 large cages. The average sides of the unit cell is 17.3 Å. Small cage has pentagonal dodecahedron (512) structure. Large cage has 16 sided cavity (with 12 pentagonal faces and 4 hexagonal faces) and called hexakaidecahedron (51264). Structure H is formed with larger guest molecules like iso-pentane (7Å < diameter < 9Å) when accompanied by smaller molecules such as nitrogen, methane or hydrogen sulfide. Its crystal structure is hexagonal. One unit cell consists of 3 small, 2 medium and 1 large cages. Small cage in this case is also pentagonal dodecahedron (512).
Medium cage is called irregular dodecahedron (4 5663), it has 12 sided cavity (with
3 square faces, 6 pentagonal faces and 3 hexagonal faces). Large cage is called icosahedrons (51268) and it consists of 20-sided cavity (with 12 pentagonal faces and 8 hexagonal faces). This Structure was first discovered in 1987 by Ripmeester et al. [7].
4
Figure 1.2: Schematic of a unit cell of gas hydrate structure I, consisting of a 3:1 ratio of large to small cavities. [8]
Within the scope of the work in this thesis all hydrate systems are structure I hydrate. It is even further narrowed down to CO2 and/or CH4 as the only guest molecules. CH4 enters both small and large cavities but is essentially outcompeted by CO2 in occupation of the large cavities in terms of mixed hydrate since CO2 have a larger stabilization impact on these cavities. CO2 is strictly not very favorable for entering the small cavities due to the size and shape compared to available volume in the symmetric small cavity. There is some evidence [9] that small amounts of CO2 may enter the small cavities but available experimental evidence for this from different groups is not entirely consistent as these finding are fairly recent. In addition we should keep in mind that our focus is on modeling leading to simplified kinetic models for hydrates in porous media related to production of hydrates using injection of CO2. Whether CO2 filling in small cavities will really be significant under the dynamic flow conditions during those types of scenarios is still unverified. So within the scope of this work, which is more fundamental on the general modeling side, CO2 in small cavities is neglected. Necessary modification to include some minor fraction of CO2 filling in small cavities will primarily enter the equilibrium thermodynamics for the hydrate, as well as the corresponding impact on chemical potentials for water and guest in hydrate outside of equilibrium.
5 Since structure I is the primary hydrate structure in this work some more details on this structure is given so as to provide a better platform for understanding the upcoming chapters. The two small cages are located at the center of the unit cell and have one water molecule at each of its 20 vertices. Each large cavity adds up 24 water molecule and overall there are 46 water molecules per unit cell. The small to large cavity ratio is 1:3. Normally, not all cavities are found to be filled with guest, but in case if all the cavities are filled with guest molecule the mole percent of water would be 85 %. With such high water content it is natural to assume the hydrate properties may depend very little on guest molecules. The volume of the guest molecules prevent the hydrogen bonded water lattice to collapse. But the guest molecules also provide some attraction energy towards the water molecules and contribute to stabilization of the lattice via the canonical partition functions for the different filled cavities in the hydrate structure. For most guest molecules this is limited short range van der Waal type of interactions. For some guest molecules like for instance H2S also electrostatic interactions between water and H2S is significant. There is, however, a limit to the degree to polarity of guest molecules that can be enclathrated without competing on hydrogen bonding with waters. For this reason alcohols are not hydrate formers and components like methanol are frequently use to prevent hydrate formation. Hydrates are distinct from ice and liquid in properties like for instance thermal conductivity, thermal expansion and electrical conductivity. The latter have been used commercially as a supplement to seismic techniques for detection and characterization of hydrates in nature. The most striking property of
hydrate is that they can exist on temperatures higher than 0 if the pressure is high
enough and this is the reason that water following hydrocarbon flow have been a problem in the oil and gas industry, in which there are many situations of low
temperatures and high pressures. The melting or dissociation of hydrates varies significantly with pressure due to the impact of guest molecules inside the Clathrate, and corresponding molecules in fluid phases after dissociation. This is visualized in more details through the cavity partition functions in the statistical mechanical model for hydrate equilibrium and fluid thermodynamics in Chapter Chapter 1:. On the other hand the melting point of ice depend very little on pressure since it only
6 depends on water to water interactions and corresponding changes in hydrogen bonds from 4 ice to an average number of hydrogen bonds of typically around 3.5 in liquid water at conditions from 0 degrees Celsius to 15 degrees Celsius. A given volume of water increases by 9% upon freezing into ice, and on the other hand as described by Lee, et al., [10], hydrates have even larger expansion and a given water volume increases up to 26-32% during the phase transition. There are also variations of properties between different structures of hydrate. Details may be found elsewhere [1].
1.2 History and the negative and positive sides of hydrates in nature and industry Natural gas hydrates were probably first discovered by Sir Humphrey Davy [1] in 1810. For the first time he commented briefly on chlorine (oxymuriatic gas) in the Bakerian lecture to the Royal Society in 1810 that in water it can freeze more readily than pure water. But Makogon and Gordejev [1] suggest that Priestley in 1778 might have discovered Clathrate hydrate before Davy’s discovery of Clathrate hydrate. Joseph Priestley performed an experiment in his laboratory leaving the window open before departing. That was a winter night, and next morning when he returned he
observed that would impregnate water and cause it to freeze and refreeze. While marine acid air and flour acid air would not. In this experiment, made
hydrate with water below the water freezing point. Davy’s experiments with chlorine was done at temperatures above the water freezing point, this way he is at least the first who discovered ‘warm ice’. Both experiments did not attract much enthusiasm towards hydrate. That was the reason gas hydrates remain only laboratory curiosity in which gas and water are transformed into a solid, until 1934. The second period starts from 1934 and still continues. It started when American chemist E.G. Hammerschmidt [1] first identified the problem as ice (hydrate) plugging in gas pipeline above water freezing point from left over water in a pipeline after construction. This discovery was vital in causing increased interest in gas hydrate. After this discovery American gas association started a thorough study of
7 hydrate at US Bureau of Mines. These plugging are more expected in cold places or deep water installments where pressure is high, and during pipeline transport of hydrocarbons on seafloor, but also in other situations of oil and gas processing. Examples include low temperatures after gas expansion and low temperature separators. Lowest temperature in the Troll processing plant at Kollsnes is -22 Celsius at 69 bars. These low temperatures are used to kick out as much as possible of (valuable) components larger than methane. Gas containing even more of these valuable components (particularly unsaturated components like ethane and propene that readily can be made into plastic) is even processed down to -70 Celsius like for instance Snøhvit gas. Since the early start of the second hydrate period in 1934, often called the industrial hydrate period, substantial efforts have been put into understanding the gas hydrates. For example Deaton and Frost [1] experimentally investigated the formation of hydrates from methane, ethane and propane. Others investigated the effects of inhibitors on hydrates as it was very important at that time, as well as now, for the oil and gas industry to prevent blocking due to hydrate plugs. In particular, techniques of chemical inhibitors like anti-agglomerants, kinetic and thermodynamic hydrate inhibitors were introduced to prevent hydrate plugging in pipelines. During the early days thermodynamic inhibitors like alcohols and salts dominated the hydrate problem prevention strategies while the kinetic related strategies have been devoted much attention during the last two decades. The other main research includes the efforts from D. Katz and R. Kobayashi [1], who devoted most of their professional lives to studies of hydrates. In the same period a state of the art equilibrium thermodynamic model was proposed by van der Waals and Platteeuw in 1959. Most of the hydrate research during this period was mainly due to concerns of gas hydrate as a problem for the oil and gas industry. The next period in hydrate research as mentioned by Sloan and Koh [1] starts from 1967 till now when a group of Russian researchers led by Makogon discovered first major natural gas hydrates deposit (Messoyakha field) in the permafrost regions. Details may be found elsewhere [1, 2]. This hydrate deposit is estimated to have one
8 third of entire gas reservoir in world, which then leads to a number of later discoveries. A world atlas, given by Hester and Brewer [11] shows sites both offshore and onshore for hydrate deposits around the world. Many of these occurrences are uncertain but even with a most conservative estimate, the energy in these hydrate deposits is likely to be significant compared to all other fossil fuel deposits. Another promising fact about hydrate is it can provide energy with little CO2 emission into the environment. With the ever increasing need for energy it was natural to turn the attention to exploit the energy stored in gas hydrate reservoirs around the world. Despite of hydrate being a source of energy, there are other reasons for interest in hydrates for the hydrate community. One of the very interesting facts is that CO2 make a more stable sI hydrate than a CH4 hydrate for substantial regions of pressure and temperature. CH4-CO2 mixed hydrate is even more stable than either pure CH4 hydrate or pure CO2 hydrate for all ranges of pressure and temperature. The greenhouse gas CO2 is a major source of global warming. CO2 hydrate offers a possible technology for capturing and disposal of CO2 in deep Ocean at conditions that allow CO2 hydrate to form. The hydrate will not be unconditionally stable since Gibbs phase rule makes full equilibrium in the three phase system of two active components (water and CO2). Nevertheless – the leakage flux due to hydrate dissociation towards ocean water is substantially lower than dissolution from liquid CO2 into ocean water. According to Shindo et al. [12] and Ohsumi [13], Ocean deposits of CO2, either as CO2 lakes at depths where CO2 is heavier than seawater or at intermediate depths is a promising alternative for long term storage of CO2. The work from Kuznetsova and Kvamme [14] further comments that these storage sites are characterized by high pressure as well as low temperature, conditions that will favor rapid formation of CO2 hydrate on the interface between CO2 and seawater and this hydrate will significantly reduce the dissolution of CO2 into the surrounding water. There will still be CO2 flux because the surrounding ocean currents and diffusion will dilute the water in contact with the hydrate below the
9 CO2 concentration that can keep the hydrate film intact. Equilibrium is not possible due to Gibbs phase rule and also due to breaking and reformation of the hydrate film due to stress from ocean currents. This aspect will be more discussed in the later parts of this chapter. Another interesting option is to inject CO2 into CH4 hydrate, which represents a win-win situation in which energy is produced while CO2 is stored safely. This concept have been tested in a large scale pilot in Prudoe Bay in Alaska but the result will not be completely published before December 2012 although the National Energy
Technology
Laboratory
(NETL)
web
pages
for
the
project
http://www.netl.doe.gov/technologies/oil-gas/FutureSupply/MethaneHydrates/ projects/DOEProjects/MH_06553HydrateProdTrial.html is continuously updated. See section 1.4 for more details. Besides these positive things, there are also negative features related to gas hydrate. One is already discussed as a problem for oil and gas industry. The vast quantities of hydrates in marine sediments pose a risk as a geo hazard and have been implicated in past climate change events [15]. The stability of gas hydrate is greatly dependent on the local temperature and pressure conditions but also by concentrations of all components in surrounding phases. Because of global warming and temperature change on the ocean surface water, hydrate both onshore and offshore can dissociate. The disappearance of huge hydrate beneath the earth surface can results in landslides. Gas hydrate dissociation is thought to have contributed to slides at water depths of 1000 to 1300 m off the east coast of the United States and the Storegga slide off the east coast of Norway as mentioned by Locat and Lee [16] and Huhnerbach and Masson [17]. Thermodynamically hydrate cannot be stable in porous sediments due to Gibbs Phase rule, which simply counts the number of independent variables needed to conserve masses of all components in all phases under the constraints of thermodynamic equilibrium. Since temperature and pressure is always given locally in a reservoir any number of degrees of freedom (number of components minus number of phases plus two) different than two means that thermodynamic equilibrium is never possible. And that will be the case for all
10 hydrates in sediments so any deficiency in trapping (shale and clay) will lead to hydrate dissociation through contact with under saturated phases. The dissociating methane is substantial risk for the environment. Note that methane is 25 times more potent as a greenhouse gas than carbon dioxide, but there’s far less of it in the atmosphere—about 1,800 parts per billion. When related climate affects are taken into account, methane’s overall climate impact is nearly half that of carbon dioxide. But with increasing global temperature the threat can be more noticeable.
1.3 Kinetics of gas hydrate formation and dissociation One of the challenges in understanding hydrate formation and dissociation is related to the non-equilibrium nature of hydrates. In nature as well as in processing of hydrocarbons, or mixtures containing other hydrate formers like for instance CO2, the local conditions of temperature and pressure is often given by external conditions, like for instance fluid mechanics in flowing systems through pipelines or equipment. Except for homogeneous hydrate formation from aqueous solution of hydrate former then Gibbs phase can generally not be fulfilled and the system cannot reach equilibrium because it is over determined. This is discussed in more detail in section 1.2 above. The problem related to the assumption of equilibrium has been for example discussed by Kvamme et al. [18] and [19]. This important aspect related to in situ hydrate stability will be more discussed in chapter Chapter 1:. Hydrate formation may be divided in two stages which are well defined in terms of physics, nucleation process and growth stage. Hydrate nucleation is a process during which small clusters of water and guest grow together but also disperse again in attempt to attain a critical crystal size of hydrate. This critical size is the turning point for which the net gain of the hydrate growth outcompete the penalty of pushing on the surroundings to get space. This process involves tens to thousands of molecules and this is why it is very difficult to observe the dynamics of the nucleation process. Once they reach a critical size they will grow monotonically provided that the hydrate cores are not dominated by competing hydrate cores with more favorable free energy for hydrate growth. The lower the free energy change the
11 more rapid hydrate formation although the mass transport part of the kinetics often dominate the overall kinetic rate. A third stage, which is not as well defined in terms of physics, is the onset of massive growth. This is normally called induction and the time needed for crystal nuclei to grow to this stage is called the induction time. The technology applied to monitor crystal growth in terms of resolution in volume and time will therefore have some impact on when the induction is observed. The induction time will be effected by the local temperature and pressure conditions as well as available mass. Deviations from equilibrium are either super-saturated if the free energy benefit is favorable in terms of hydrate formation or sub-saturated if the conditions favor hydrate dissociation. The higher the super saturation is the lower the induction time is normally. But generally speaking the system can be super saturated in some independent thermodynamic variables and sub saturated in other independent thermodynamic variables, which implies competing phase transitions of hydrate formation and hydrate dissociation. Of course, the absolute necessity for nucleation to occur is the saturation of water with guest. But normally the solubility of guest molecules is limited and therefore, the formation is mostly observed on the vapor-water interface. There are two hypotheses about nucleation presented by Sloan and Fleyfel [20] and by Kvamme [21]. The first hypothesis suggests that water molecule form clusters around guest molecules and then these clusters combine to form unit cells. When the size of the collective clusters reaches critical size, then it grows. On the contrary the work by Kvamme [21] proposes that gas molecules form first partial and then complete cages around the adsorbed species. The clusters then join and grow on the vapor side of the surface until critical size is achieved. Further, nucleation is generally divided in two types, homogeneous and heterogeneous nucleation. By homogenous nucleation means if all the hydrate components are extracted from a single phase like for instance hydrate formation from liquid solution. Hydrate formation on a gas/water interface, as mentioned above, is an example of heterogeneous nucleation in which the component come from two different phases. But heterogeneous nucleation can also happen on solid surfaces from adsorbed water and adsorbed hydrate formers, or from adsorbed water and hydrate former from a separate phase outside. After nucleation stage have reached
12 resulted in hydrate crystals of critical size (see discussion below) crystals will grow unconditionally except for completion of mass with other hydrate crystals which have reached levels of lower free energy. In those cases less stable hydrate cores may be consumed at the cost further growth of the more stable ones. There are also other exceptions related to non-equilibrium which will not be discussed in detail. Unconditional growth in a non-equilibrium system is only possible if the impact of all gradients in independent thermodynamic variables (temperature, pressure, concentrations) in free energy change is negative. . In order to illustrate the mechanisms behaving the two stages it can be useful to look at very simple theories like classical nucleation theory. If the cluster (nucleus) is considered a complete sphere then bulk free energy change involved in the phase transition Δ , can be formulated as:
4 4 3
(1.1)
is the free energy change per unit volume and is the interfacial tension. The
terms and are the surface free energy which is positive and decreases and the volume free energy which is negative and increases. This implies passes through a maximum at some radius that corresponds to the critical size. This
nucleus with maximum radius is called critical nucleus. The critical size is the turning point when the free energy gains of the phase transition turns over to dominate the total free energy change over the free energy penalty of pushing away “old phases” in order to give space for the hydrate phase. At stages before the critical nucleus size the competition between the penalty term and the free energy term results in nuclei growth as well as nuclei decay. As for kinetic rates corresponding flux of hydrate growth in classical theory is given by a mass transport flux multiplied by a Boltzmann factor of the total free energy change. Associated heat transport requires a heat transport model but the consistent enthalpy changes are trivially obtained from (1.1) through classical thermodynamic relationship.
13 During growth mass and heat transfer become increasingly important, especially on aqueous-vapor interface where transport across hydrate is rate limiting unless hydrate films are broken by shear forces during flow. Modeling of hydrate kinetics has historically been very much dominated by the industrial funding of hydrate projects. Experimental activities have dominated and the modeling has often been empirical fitting to some function of super-cooling conditions. Other directions have been using different empirical concepts from chemical engineering, which practically relates mass transport constants to a sum over different stages believed to be mass transport limiting. Thermodynamic driving forces are often modeled as fugacity differences between equilibrium and real state. A more detailed, but not necessarily complete, review may be found in chapter 3 in the book from Sloan and Koh [1]. Chemical additions for hydrate prevention in pipelines or process equipment are mainly 1) thermodynamic inhibitors, 2) kinetic inhibitors or 3) anti-agglomerants. These will not be discussed in detail here as this is outside the main focus of the thesis so readers are directed to Sloan and Koh [1] and references therein for more details. Dissociation of hydrates in porous media for natural gas production can be promoted by several different means, which all have in common that energy for breaking bonds must be provided. The dissociation is an endothermic process and therefore heat must be provided from external source to break hydrogen bonds between water molecules and interaction forces between guest and water molecules of the hydrate lattice to decompose. The methods mainly applied for dissociation of in situ natural gas hydrates are depressurization, thermal stimulation ad inhibitors injection. Further details are easily available in literature.
1.4 CO2 sequestration in CH4 hydrate As briefly discussed before, due the massive in-situ hydrates deposits worldwide it is tempting to produce this energy source. The main production methods so for applied to produce CH4 are: depressurization, thermal stimulation and inhibitors injection. All these methods bring the hydrates to thermodynamically unstable states with
14 reference to corresponding fluid phases in order to trigger dissociation for CH4 production. But due to the related economical and potential environmental threats they are so far not used for commercial production of CH4, with the exception of Messoyakha field in Russia as mentioned earlier. An alternative novel method is currently being investigated in several laboratories worldwide and has also been tested in a pilot plant project in Prudoe Bay. This method not only provide a great solution to produce CH4 from in-situ hydrate to fulfill the global energy need, but also help to reduce the climate emissions of greenhouse gases. As per earlier discussion in this chapter both CH4 and CO2 form structure I hydrate and CH4 can be recovered if CH4 hydrate brought into contact with CO2 as predicted by Svandal [22] and [23]. This method has been verified experimentally by number of researcher like for example by Lee et al. [24] and Baldwin et al. [25]. As discussed earlier in this thesis, CO2 is, as an approximation, considered to penetrate only into the large cavities of structure I hydrate. Large cavities are in 3:1 ratio with small cavities in structure I. This means, theoretically speaking, that if all large cavities are filled with CO2 it implies that 75% of CH4 has been recovered existed in the hydrate. Also, recall that CO2 form more stable hydrate over a wide range of temperature and pressure, and mixed hydrate containing CH4 in small cavities are more stable over all regions of temperature and pressure. This therefore provides a long term storage solution of the greenhouse gas CO2. Beside storage in hydrates, the storage of CO2 in aquifer reservoirs has already been established as a feasible alternative for reducing CO2 emissions into the atmosphere. Injection of produced CO2 from Sleipner oil and gas field into the Utsira formation was the first industrial CO2 aquifer storage project. See for instance Xu et al. [26] for more details. But potential for hydrate formation during aquifer storage of CO2 exists in the northern parts of North Sea and Barents Sea. Suitable reservoirs for aquifer storage of CO2 that contains regions of pressure and temperature conditions which are within the CO2 hydrate stability region. Kinetic rates for CO2 hydrate towards under saturated water are therefore important in reservoir modeling of CO2 storage in those regions which is main focus of few papers presented in this PhD dissertation. In the same regions CH4 hydrate exists and therefore CO2 can be stored and produce CH4 as
15 well. As mentioned currently in May 2012 ConocoPhillips has completed working on a field trial on the Alaska North Slope together with the US Department of Energy (DOE project MH-06553) as a major funding agency for the project. The data achieved after ConocoPhillips field trial (mentioned before in section 1.2) hopefully will come up with answers to some of the many questions.
16
Chapter 2: Phase Field Theory and Thermodynamics As discussed in previous chapter, natural gas hydrates are thermodynamically unstable in reservoirs (Gibbs phase rule). In the total set of independent thermodynamic variables there may be several competing phase transitions of significance for the total distribution of different co-existing phases. For a given phase transition (hydrate formation or dissociation) the minimum criteria is that free energy change is negative and at least exceeds the barrier due to the work of creating a new phase (penalty of giving space), which is proportional to the interface free energy. But a given phase transition will only proceed unconditionally as long as the impact of all gradients to free energy changes from independent thermodynamic variables implies lower free energy. The non-equilibrium situation also implies that hydrate formed from different phases will have different free energies since the chemical potentials of guest molecules will be different. In some cases also water will have different chemical potentials as water molecules may come from liquid water, extracted from gas or from water adsorbed on solid surfaces. The above implication of non-equilibrium requires a kinetic concept based on a universal reference state for free energy as well as a concept for minimizing free energy under constraints of mass- and heat transport. Using ideal gas as reference for thermodynamics solves the first challenge. In density functional theory (DFT) phase transition kinetics is proportional to changes in structures over the phase transition boundaries. Since structure is directly linked to free energy it might be more convenient to use changes in free energy directly. This is the basis for Phase Field theory (PFT). Molecular dynamics simulations and other theoretical methods can link these two theories even tighter through the shape of the interface and corresponding concentration profiles of the interface, as well as through estimates of interface free energies. Phase Field theory (PFT) can be considered as free energy minimization under the constraints of mass and heat transport dynamics. Molecular structures are uniquely
18 linked to corresponding free energies via statistical mechanics see for detail [27]. As PFT uses free energy changes directly as the driving forces for kinetic progress of the phase transition therefore it is required to use appropriate description of nonequilibrium thermodynamics. If the dissociation rate is faster than the capacity of the surrounding fluids to dissolve the released gas a separate gas phase will form on the interface during dissociation and eventually release from the interface as bubbles. An implicit coupling to hydrodynamics is therefore needed. In this work we have applied Navier-Stokes equations to extend the original NVT ensemble over to NPT ensemble and corresponding development of local density for the different phases in each volume element. A brief description of the Navier Stokes equations, which is implicitly used in PFT in this work, is given in the papers enclosed with the thesis. Some of the background for PFT without hydrodynamics and heat transport is collected from the PhD work [3] and from the work by Kvamme [28] as well as other publications referred to during the different sections of this chapter.
2.1 Thermodynamics Accurate and consistent model for free energy of the different co existing phases is a critical basis for PFT modeling. Gas hydrates in reservoirs are thermodynamically unstable due to the interaction with surrounding fluids (aqueous, gas) and mineral surfaces (Gibbs phase rule). Therefore, this section is devoted to present the thermodynamic functions required for PFT model described in section 2.2. This section will initially explain the free energy and Gibbs phase rule in sections 2.1.1 and 2.1.2 respectively. The description of thermodynamic properties for hydrates, fluids and aqueous phases are described in sections 2.1.3, 2.1.4 and 2.1.5 respectively.
2.1.1 Free energy The 2nd law of thermodynamics tells that any isolated system will strive towards maximum entropy. The combined 1st and 2nd law of thermodynamics gives the changes in internal energy for phase i:
19
'
!" # $% &% %()
(2.1)
Where summation runs over all components, S is the entropy, µ the chemical potential and N the number of moles of components. The equality is for reversible changes, which is only a theoretical possibility, while the less than case is for all real and irreversible changes. Transformation of the natural variables is accomplished through Legendre transforms by subtracting * + on both sides to get Helmholtz
free energy,
'
!" # $% &% %()
(2.2)
Free energy can, in a simplified sense, be considered as the “available” energy level under the constraints of losses associated to entropy generation. The term !"
represent the technical work, or shaft work, since the work involved in pushing fluids internally in the systems is subtracted. The last term on the right hand side is called chemical work and is the work related to extracting or inserting particles. Removing a molecule from the system involves releasing the molecule from the interaction energy of the surroundings and also involves an entropy contribution related to reorganization of the system. Free energy is an extensive state quantity, so to get the total for an entire system, which may consist of more than one phase; one just adds the contributions from the different phases. At constant volume and temperature the equations (2.1) and (2.2) give, '
# $% &%
%()
(2.3)
In the original formulation applied by Kvamme et al. [29] and [30], Svandal [3] and Buanes [31] and papers therein the system is isothermal, which allows reversible and irreversible processes related to phase transitions proceed until a minimum total free energy is achieved locally and globally in the system. This will be given by:
20
,-,.% 0 ,-,.%
,-,.% / '
(2.4)
This means that differences in the free energy between two phases can be seen as a driving force, and the system will strive towards minimum free energy. The final limit of free energy minimum can easily be verified to be the situation where chemical potential of each component is the same in all co-existing phases, if the number of degrees of freedom is so that full equilibrium can be reached. If, on the other hand, the number of independent thermodynamic constraints on the system does not match number of degrees of freedom then Gibbs phase rule is not fulfilled and the system can locally result in competing processes of different directions (hydrate growth and hydrate dissociation).
2.1.2 Gibbs Phase Rule A phase can be defined as a form of matter which is homogeneous in its physical
state and also in chemical composition. A phase can be characterized by 0 2
independent thermodynamic variables in terms of mole numbers of 0 components in
the system and two other independent thermodynamic variables. In our cases these two variables are conveniently chosen as temperature and pressure. As discussed before; in real world gas hydrate reservoir systems consists of more than two phases. Hydrate and mineral surfaces are not compatible due to different distributions of partial charges on surface atoms. As a minimum two adsorbed types of phases bridge hydrate and mineral water phases, one adsorbed phase towards mineral and another adsorbed phase towards the hydrate surface. These two water interfaces are by definition unique phases since the structures, densities and composition are different from other coexisting phases. In addition to this there is the hydrate phase itself, possible (normally) free liquid water or ice water phase and gas (or fluid) phase containing hydrate formers and possibly also components which do not participate in hydrate phase transitions. The number n discussed above refer only to components that participate in phase transitions. Other components and ions are to be considered as “inert” with respect to phase transition balances but may still influence the
21 properties of different phases. Ions in the water phase, as one example, will lower the chemical potential of liquid water and as such actually serve as a hydrate inhibitor that shifts the stability regions of hydrate. The minimum number of phases in a hydrate reservoir is as such minimum 4 and often 5. With only methane and water as components distributing over these phases the number of degrees of freedom is then 0 or -1 and in the extreme case of no free liquid or ice water +1. In either case local pressure or temperature is given by local hydrostatics and possible flow and geothermal gradient. So 2 independent thermodynamic properties are constraints on the hydrate system in porous media and equilibrium cannot be reached. With more than 2 guests equilibrium is only possible if chemical potential of each component is the same in every phase it exists. Gibbs' phase rule is simply an expression for the number of variable which needs to be defined/imposed on a system to ensure conservation of mass under the constraints of equilibrium. The degrees of freedom, or the number of independent variables needed to characterize the system equilibrium is given as: 20
2
(2.5)
Here is the number of phases. In case of two components (CH4/H2O) and inside hydrate stability region there are three phases (aqueous liquid, fluid and hydrate
phase) and therefore the degree of freedom is 1 which implies the system is uniquely defined with respect to equilibrium when temperature or pressure is fixed. But in contrast in most common situation in nature both pressure and temperature is defined and so the system is thermodynamically over determined. This means the system will not be able to establish complete equilibrium and the combination of 1st and 2nd law of thermodynamics will dictate this system to approach a state of minimum free energy. The minimum free energy will control the content of CH4 dissolved in the aqueous solution to achieve the equilibrium between the hydrate and the aqueous solution. And as mentioned above the presence of minerals adds more phases to the system.
22
2.1.3 Hydrate Thermodynamics The hydrate thermodynamics presented here is based on the extended adsorption theory as proposed by Kvamme and Tanaka [32]. The expression for chemical potential of water in hydrate is, 4, 4 $3 $3
# 6 78 0 91 # ; 8
,?
@
'A ; B
; -, < CDE *&, + , , , , -F ,
(1)
which is an integration over the system volume, while the subscripts ), G represents the three components, ' is molar density depending on relative compositions, phase and flow. The bulk free energy density described as CDE 0 *&- < A1 B *&-IJ * + , , , , - < *&-K * + , , , , - .
(2)
The phase field parameter switches on and off the solid and liquid contributions J and K through the function *&- 0 & *10 B 15& < 6& , -, and note that *0- 0 0 and *1- 0 1. This function was derived from density functional theory studies of binary alloys and has been adopted also for our system of hydrate phase transitions. The binary alloys are normally treated as ideal solutions. The free energy densities of solid and liquid is given by OPQ
J 0 NJ '
(3)
,
K (4) . K 0 NK ' The thermodynamics for the hydrate system is discussed in more detail in the thermodynamics section below, with derivations of the free energies NJ for hydrate and NK for liquid state. Hydrate #PQ density ' is calculated using the formulation by Sloan et al. [13]
K for fluid phase is calculated The liquid density ' as
KRED Q
0 S
/+
K,RED Q ' 0
+ UVWXY>Z T
(5)
,
where represents the molar volume of ith component. The molar volume is calculated using gas law 0
[ ] < \ ^ ][ _,`,
>a?
,
(6)
where represents the pressure and is compressibility factor calculated using SRK equation of state. For simplicity to avoid partial molar volume at infinite dilution the density of
liquid in aqueous phase is calculated as bcDdeDf
K
,
0 S <
/+
K,bcDdeDf '
0
1
(7)
bcDdeDf
K
,
where is the average molar volume of pure water. The function *&- 0 & , *1 B & , -⁄4 ensures a double well form of the CDE with a free energy scale 0 i1 B > l m < > n with jk
jk
*0- 0 *1- 0 0, where is the average molar volume of water. In order to derive a kinetic model we assume that the system evolves in time so that its total free energy decreases monotonically [12]. The usual equations of motion are supplemented with appropriate convection terms as explained by Tegze et al [14]. Given that the phase field is not a conserved quantity, the simplest form for the time evolution that ensures a minimization of the free energy is o op o> op
< *q. s-& 0 B *&, + , , , -
tR t
,
tR
< *q. s- 0 s. i *&, + , , , -s l t
,
where q is the velocity, 0 *1 B 0 i1 B
> l m jk
n jk
+ - u`
(8)
(9)
and
are the mobilities
associated with coarse-grained equation of motion which in turn are related to their microscopic counter parts. Where 0 J < *K B J -*&- is the diffusion coefficient. The details are given elsewhere [11]. An extended phase field model is formulated to account for the effect of fluid flow, density change and gravity. This is achieved by coupling the time evolution with the Navier Stokes Equations. The phase and concentration fields associates hydrodynamic equation as described by conti [1517]
'
wq oj op
ov op
0 B' ;. q ,
< '*q. ;-q 0 'q < ;. .
(10)
Where q is the gravitational acceleration. ' is the #PQ density of the system in hydrate (' ) and liquid K ). Further (' 0 x < y.
(11)
is the generalization of stress tensor [15-18], x represents non-dissipative part and Π represents the dissipative part of the stress tensor. The expression for chemical potential of water in hydrate is {# 0 {,# B S ( |[ 71 < S F ,
(12)
This equation is derived from the macro canonical ensemble under the constraints of constant amount of water, corresponding to an empty lattice of the actual structure. Details of the derivation are given elsewhere [19] and will not be repeated here. {,# is the chemical potential for water in an empty hydrate structure and is the cavity partition function of component G in cavity type ). The first sum is over cavity types, and the second sum is over components G going into cavity type ). Here ( is the number of type ) cavities per water molecule. Fluid Thermodynamics The free energy of the fluid phase is assumed to have NK 0 ∑ /+ { RED Q ,
(13)
{RED Q 0 {∞ < |[* - ,
(14)
where { RED Q is the chemical potential of the ith component. The solubility of water is assumed to follow the Raoult’s law. The lower concentration of water in the fluid phase and its corresponding minor importance for the thermodynamics results in the following form of water chemical potential with some approximation of fugacity and activity coefficient:
where {∞ chemical potential of water at infinite dilution and is the mole fraction of water in the fluid phase. The chemical potential for the mixed fluid states is approximated as
{ RED Q
0
Ju},_D~d {
< |[* - ,
(15)
where ) represents CH4 or CO2. The details are available in Svandal et al. [20]. Aqueous Thermodynamics The free energy of the aqueous phase assumed as bcDdeDf
NK 0 ∑ /+ {
,
(16)
bcDdeDf
the chemical potential {
of aqueous phase has the general form derived from excess thermodynamics {# 0 { ∞ < |[* % ∞ - < * B -.
(17)
{ ∞ is the chemical potential of component ) in water at infinite dilution, % ∞ is the activity coefficient of component ) in the aqueous solution in the asymmetric convention (% ∞ approaches unity in the limit of becoming infinitely small). The chemical potentials at infinite dilution as a function of temperature are found by assuming equilibrium between fluid and aqueous bcDdeDf phasesA{RED Q 0 { I. This is done at low pressures where the solubility is very low, using experimental values for the solubility and extrapolating the chemical potential down to a corresponding value for zero concentration. The activity coefficient can be regressed by using the model for equilibrium to fit experimental solubility data. The chemical potential of water can be written as: _
{ 0 { < |[*1 B -% < * B -,
remain constant in the system. The values for temperature and pressure are taken at Nyegga cold seeps located on the edge of the Norwegian continental slope and the northern flank of the Storegga slide, on the border between two large oil/gas prone sedimentary basins – the Møre basin to the south and the Vøring basin to the north [2122]. The temperature and pressure condition is well inside the stability region of the guest molecules. The standard value of 9.8 m/s2 is assumed for gravity along the Y-axis of 2D geometry.
(18)
_
where { is pure water chemical potential. The strategy for calculating activity coefficient is given in [20]. RESULTS AND DISCUSSIONS The phase field model is implemented on a 2D geometry. This structure is used for the exchange of methane with carbon dioxide. Figure 1 show the circle of methane hydrate with blue color which is placed in the center surrounded by liquid carbon dioxide. The size of system is (500×500) grids with diameter of 200 grids for circular hydrate. One grid is equal to one angstrom (Å) and temperature (273.21 K) and pressure (63.90 bar)
Figure 1. Simulation at time zero, showing the initial picture of CH4 hydrate and liquid CO2 with 500x 500 grid points and a hydrate radius of 200 grid points, color codes represent φ = 0 and 1 for methane hydrate and CO2 liquid phases respectively. The simulation is run to 91.809E+06 total time steps this corresponds to the time of 91.809 ns Total number of grid points System area in m2 No. of time steps Total time in seconds CH4 mole fraction in hydrate CO2 mole fraction in Liquid
500×500 2.5E-15 91.809E+06 91.809E-09 0.14 1.0
Table 1. Simulation setup.
Figure 2. Structural phase parameter of the dissociating CH4 hydrate and exchange of CO2 at
different times. Between phases = 0 and 1 are interface values. Figure 2 shows that the methane initially starts to dissociate into the surrounding CO2 liquid. This is due to the driving force in terms of the change in chemical potential of methane in liquid phase and hydrate phase. CO2 is assumed to only enter the large cavities of structure I due to its size. The CO2 starts penetrating into the methane hydrate after some amount of methane has been released into the liquid phase. The hydrate size is reduced because of methane dissociation until 9.025 nano seconds and then it increased until 91.809 nano seconds due to reformation with CO2 penetration. This phenomenon is best explained by assistance of figure 3 which illustrates the reformation process of CO2 hydrate, where the kinetics of the liquid CO2 from its liquid phase transformation to solid phase at different stages is plotted. A thick interface between liquid and solid phase is highlighted with black circle in figure 3.
Figure 4. CO2 concentration inside hydrate as a function of radial distance at different times 0.0 ns, 6.4 ns, 10 ns, 22.5 ns, 40.0 ns and 77.84 ns. To observe the movement of methane from solid phase to liquid, the velocity on the interface is determined by tracking the & values which is used to calculate the dissociation rate until 91.809 ns using the following equation [3]: 0
vZ
i
l.
.+#
(19)
where is the dissociation rate (moles/m s), is hydrate radius shrinkage rate (m/s), '#PQ is 2
density of hydrate (kg/m ), is molar weight of the guest (kg/moles) and is Hydrate number. The calculated flux profile is plotted in figure 5. The initial value of flux was high due to the initial relaxation of a system into a physically realistic interface. The actual dependency of formation rate on driving forces is illustrated in figure 5. 3
Figure 3. CO2 concentration as a function of distance and times 0 ns, 6.4 ns, 10 ns, 22.5 ns, 40.0 ns and 77.84 ns. The concentration of CO2 in the hydrate as a function of radial distance is shown in figure 4. The totally converted liquid CO2 is shown by point A on right side of figure 4. This point was used to make an estimate of converted CO2 which was approximated about 0.12 mole fraction. The system formed an interface with liquid phase between A to B resulting into an initial relaxation of the system into a physically realistic interface.
Figure 5. Carbon dioxide flux as a function of time. The rate is decreasing gradually after this relaxed point on the curve. One reason for this is the decrease in thermodynamic driving force, which is proportional to the increasing chemical potential inside the hydrate, which is still filled with methane. The noise seen on the calculated curves is due to grid resolution effects. The interface in this simulation perfectly follows a power law
which is proportional to square root of time (α t1/2), which according to Fick's law indicates a diffusion controlled process. The total number of CO2 molecules penetrated into methane hydrate is 1.5308E+03. The simulated results have been extrapolated to 3.1536E+20 nano seconds from 91.809 nano seconds according to this observation and the maximum time in figure 6 is equal to 104 years. This is done to see the possible trend on a long time scale.
liquid CO2 is increasing. Higher concentration of methane was observed to have diffused into the liquid phase by forming a transition zone at the end of the simulation. Some methane molecules have diffused rapidly into the liquid phase because some of the vertices of water cages in the hydrate has dissociated shown on the right side of figure 8.
Figure 8. Methane concentration inside the hydrate as a function of radial distance at different times. Figure 6. Extrapolation of carbon dioxide flux up to 10000 years. Due to the length of the time scale the values were plotted in the figure with 100 years of time intervals. After 10000 years the reformation rate of CO2 is 2.198E-12 moles/m2s which corresponds to 6.932E-05 moles/m2yr.
The interface became stable after some time, as illustrated in fig. 8. This can be explained through the definition of solubility which is a measure of how much solute (methane) will dissolve in all the solvent (carbon dioxide). The methane will not completely dissolve in carbon dioxide because of the polar solvents molecules separate the molecules of other polar substance. There is no thermodynamic equilibrium between the wall of the hydrate and that of the fluid in sharp stable interface. This sharp interface is shown in figure 9 which illustrate the concentration of methane in liquid and solid phases.
Figure 7. Methane concentration as function of length. Figure 7 illustrate the change of methane concentration with respect to position in simulation box while methane diffuses into liquid phase and CO2 penetrating into the large cavities of methane hydrate. At time t=0 nano seconds there is no change in methane concentration but as the time passes to t=0.4 interface between liquid and solid phase is developed which represents the presence of methane in liquid and solid phases. The interface thickness is increasing as the time is increasing, which is highlighted with black circles in figure 7, and also the methane concentration in
Figure 9. Methane concentration as a function of length at different times. Figure 10 shows the diffusion of methane from hydrate phase to liquid phase. This is sampled by tracing the movement of methane concentration from the interface between liquid and solid.
Figure 10. Methane dissociation rate. Initially the value of methane flux is higher because of initial relaxation of the system as discussed in the case of carbon dioxide flux. To show the actual dependency of dissociation on driving forces the close look on the curve is shown in figure 10. The flux is gradually decreasing after this relaxation is because of the decrease in thermodynamic driving forces which is proportional to the increasing methane concentration in the surrounding liquid. The value of methane at the end of simulation is 9.297 moles/m2s. This high value is because system is not reached at equilibrium. The total number of 2.5545E+03 methane molecules dissociated after 91.809. The interface allows almost perfectly the power law which is proportional to square root of time (α t1/2) showing a diffusion control process.
Figure 12. Methane and carbon dioxide concentration profile in solid and liquid phase. The CH4 and CO2 profiles are going opposite directions which is due to dissociation of methane from hydrate to liquid phase and CO2 reformation which refilling the large cavities of the methane hydrate. This is best illustrating by figure 13. Over the period of simulations, the comparison of CH4 dissociation and CO2 reformation resulted in approximately 60 percent of methane exchanged while CO2 is stored as CO2 hydrate. Theoretically, the maximum exchange could be 75 % corresponding to CO2 filling all the large cavities.
The simulation is extrapolated to 3.1536E+20 nano seconds from 91.809 nano seconds which is equal to 10000 years. The flux of the methane after 10000 years is 2.954E-14 moles/m2s which corresponds to 9.316E-07 moles/m2yr.
Figure 13. Methane and carbon dioxide concentration profile inside the hydrate at different times.
Figure 11. Extrapolation of methane flux up to 10000 years. The comparison between CH4 and CO2 is shown in figure 12 where CH4 and CO2 concentration profile were combined to illustrate different phases achieved during the exchange process.
CONCLUSION Phase field simulation with more appropriate description of density dependencies in the phase field part as well as in the hydrodynamic parts [8] has been applied to model the exchange of CH4 with CO2 from natural gas hydrate at conditions corresponding to hydrates in Nyegga. The kinetic data attained are examples of important results which will be useful in the modeling and optimization for the production of methane from
hydrate reservoir as well as sequestration of CO2. As expected it was observed that the mole fraction of CO2 in the hydrate phase increases, while that of CH4 decreases with increasing time. Within the limited simulations time approximately 60% of initial CH4 in hydrate was exchanged. REFERENCES [1] Kvenvolden K. A, Rogers B. W. Gaia’s breath—global methane exhalations. Marine and Petroleum Geology 2005;Vol.22: 579-590. [2] MacDonald I. R, Guinasso N. L, Sassen Jr. R, Brooks J. M, Lee L, Scott K. T. Gas hydrate that breaches the sea floor on the continental slope of the Gulf of Mexico. 1994:699-702. [3] Rehder G, Kirby S. H, Durham W. B, Stern L. A, Peltzer E. T, Pinkston J, Brewer P. G. Dissolution rates of pure methane hydrate and carbon-dioxide hydrate in undersaturated seawater at 1000-m depth. Geochimica et Cosmochimica Acta 2004; 68(2):285-292. [4] Egorov A. V, Crane K, Vogt P. R, Rozhkov A. N, Shirshov P. P. Gas hydrates that outcrop on the sea floor: stability models. Geo-Marine Letters 1999;Vol.19:68-75. [5] Saji A, Yoshida H, et al. Fixation of carbondioxide by clathrate-hydrate. Energy Conversion and Management 1992;33(5-8):643-649. [6] Yamasaki A, Teng H, et al. CO2 hydrate formation in various hydrodynamics conditions. Gas Hydrates: Challenges for the Future 2002; 912:235-245. [7] Lee J. W, Chun M. K, et al. Phase equilibria and kinetic behavior of CO2 hydrate in electrolyte and porous media solutions: Application to ocean sequestration CO2. Korean Journal of Chemical Engineering 2002; 19(4):673-678. [8] Qasim M, Kvamme B, Baig K. Phase field theory modeling of CH4/CO2 gas hydrates in gravity fields. International Journal of Geology 2011; 5(2):48-52. [9] Caldeira K, Wickett M. E. Ocean model predictions of chemistry changes from carbon dioxide emissions to the atmosphere and ocean. J. Geophys. Res. 2005; 110:12. [10] Wheeler A. A, Boethinger W. J, McFadden G. B, Phase field model for isothermal phase transitions in binary alloys. Physical Review A 1992; Vol. 45:7424-7439. [11] Kvamme B, Svandal A, Buanes T, KuznetsovaT. Phase field approaches to the kinetic modeling of hydrate phase transitions. AAPG Memoir 2009 ;Volume 89:758-769
[12] Svandal A. Modeling hydrate phase transitions using mean field approaches. University of Bergen 2006:1-37. [13] Sloan E. D, Koh C. A. Clathrate hydrates of natural gases (3rd ed.). Boca Raton, FL: CRC Press, 2008. [14] Tegze G, Gránásy L. Phase field simulation of liquid phase separation with fluid flow. Material science and engineering 2005; vol.413-414:418422. [15] Conti M. Density change effects on crystal growth from the melt. Physical Review 2001; vol.64:051601. [16] Conti M, Fermani M. Interface dynamics and solute trapping in alloy solidification with density change. Physical Review 2003; vol.67:026117. [17] Conti M. Advection flow effects in the growth of a free dendrite. Physical Review 2004; vol.69:022601. [18] Baig K, Qasim M, Kivelä P. H, Kvamme B. Phase field theory modeling of methane fluxes from exposed natural gas hydrate reservoirs. American Institute of Physics 2010, in press. [19] Kvamme B, Tanaka H. Thermodynamic stability of hydrates for ethane, ethylene and carbon dioxide. J. Chem. Phys. 1995; vol. 99:7114-7119. [20] Svandal A, Kuznetsova T, Kvamme B. Thermodynamic properties and phase transitions in the H2O/CO2/CH4 system. Fluid Phase Equilibria 2006; vol.246:177-184. [21] Bunz S, Mienert J, Berndt C. Geological controls on the storegga gas-hydrate system of the mid-Norwegian continental margin. Earth and Planetary Science Letters 2003; 209(3-4):291-307. [22] Chen Y, Haflidason H, Knies J. Methane fluxes from pockmark area in Nyegga, Norwegian Sea. In: International Geological Conference, August 2008.
Paper 4
Phase Field Theory modeling of Phase transitions involving hydrate M. Qasim, K. Baig and B. Kvamme In Proceedings from the 9th international conference on Heat and Mass transport, Harvard, Cambridge, USA, January 25-27, 2012, pp. 222-228
Recent Advances in Fluid Mechanics, Heat & Mass Transfer and Biology
PHASE FIELD THEORY MODELING OF PHASE TRANSITIONS INVOLVING HYDRATE MUHAMMAD QASIM, KHURAM BAIG, BJØRN KVAMME* Department of Physics and Technology University of Bergen Allégaten 55, 5007 Bergen NORWAY
[email protected], http://www.uib.no Abstract: - Natural gas hydrates are thermodynamically unstable in reservoirs due to contacts with the undersaturated phases and these contacts may both lead to dissociation and formation. This phenomenon allows forming CO2 hydrate and producing CH4 through injection of CO2 into CH4 hydrate. A thin layer of water between CH4 hydrate and CO2 phase has to be considered in reservoir. The nucleation on water-CO2 interface is expected to be very slow, therefore to speed the CO2 hydrate formation a small region of CO2 hydrate is considered on the water-CO2 interface. The exchange of CO2 and CH4 is expected to become faster due to introduction of water layer between CH4 hydrate and CO2 phase. The kinetic rates of hydrate formation and dissociation towards these undersaturated phase are vital in the understanding of natural hydrates in sediments and the impact of contact with surrounding fluid phases and adsorbed phase (on mineral surfaces). Common to this exchange process and the dissociations toward undersaturated phases is that the kinetic rates will depend on how fast the processes happen. This effect comes into play due to the impicit implementation of hydrodynamics. A better picture of free energy is always important as it is the driving force to decide the faith of flow and hydrate formation or dissociation. This is achieved by implementing the non-thermodynamic model implicitely in the phase field simulation. This will allow specially giving a proper effect of free energy gradients.
Key-Words: - Phase field theory, Natural gas hydrate, Hydrodynamics, Dissociation, Hydrate, Exchange point of view the reason is simply that water structure on hydrate surfaces are not able to obtain optimal interactions with surfaces of calcite, quarts and other reservoir minerals. The impact of this is that hydrates are separated from the mineral surfaces by fluid channels. The sizes of these fluid channels are not known and are basically not even unique in the sense that it depends on the local fluxes of all fluids in addition to the surface thermodynamics. Stability of natural gas hydrate reservoirs therefore depends on sealing or trapping mechanisms similar to ordinary oil and gas reservoirs. Many hydrate reservoirs are in a dynamic state where hydrate is leaking from top by contact with groundwater/seawater which is under saturated with respect to methane. Dissociating hydrate degasses as bubbles if dissociation rate is faster than dilution in surrounding fluids and/or surrounding fluid is supersaturated. The kinetic rate depends on mass transport dynamics as well as thermodynamic driving force. Phase field theory is a power full tool to quantify this balance and provide a theoretical
1 Introduction Gas hydrates are ice-like substances of water molecules encaging gas molecules (mostly methane) that form under high pressure and low temperature conditions within the upper hundred meters of the sub-seabed sediments [1]. These gas hydrates are widely distributed in sediments along continental margins, and harbor enormous amounts of energy. Massive hydrates that outcrop the sea floor have been reported in the Gulf of Mexico [2]. Hydrate accumulations have also been found in the upper sediment layers of Hydrate ridge, off the coast of Oregon and a fishing trawler off Vancouver Island recently recovered a bulk of hydrate of approximately 1000kg [3]. Håkon Mosby Mud Volcano of Bear Island in the Barents Sea with hydrates is openly exposed at the ocean floor [4]. These are only few examples of the worldwide evidences of unstable hydrate occurrences that leaks methane to the oceans and eventually may be a source of methane increase in the atmosphere. Hydrates of methane are not thermodynamically stable at mineral surfaces. From a thermodynamic
ISBN: 978-1-61804-065-7
222
Recent Advances in Fluid Mechanics, Heat & Mass Transfer and Biology
basis for development of simplified models for reservoir modeling tools. Gas hydrates are a potential source of energy as well as a potential solution for the reduction of CO2 emissions. CO2 hydrate is more stable than CH4 hydrate over substantial regions of pressure and temperature and a mixed hydrate of structure I in which CO2 fills large cavities and CH4 fills small cavities are stable over all regions of temperatures and pressures. Gas hydrates have great capacity to store gases [5-7] and several investigations of potential for using hydrate phase for storage and transport have been conducted. The primary goal of this work is to develop a kinetic model for the replacement of CH4 with CO2 in natural gas hydrate reservoirs using modified Phase Field Theory [8] and non-thermodynamic model [14]. This process is a favorable way to store a greenhouse gas (CO2) for long period of time and enables the ocean floor to remain stabilized even after recovering the methane gas [9]
+ "#$%&
,
−
,
,
,
(1)
',
,
which is an integration over the system volume, while the subscripts i, j represents the three components, ρ is molar density depending on relative compositions, phase and flow. The bulk free energy density described as
"#$%& = , -
.
+ 1−. "1 , ,
/"0 , , .
,
,
+
(2)
The phase field parameter switches on and off the solid and liquid contributions f3 and f4 through the function p ϕ = ϕ 10 − 15ϕ + 6ϕ , and note that p 0 = 0 and p 1 = 1. This function was derived from density functional theory studies of binary alloys and has been adopted also for our system of hydrate phase transitions. The binary alloys are normally treated as ideal solutions. The free energy densities of solid and liquid is given by
2 Phase Field Model Phase field theory model follows the formulation of Wheeler et al. [10], which historically has been mostly applied to descriptions of the isothermal phase transition between ideal binary-alloy liquid and solid phases of limited density differences. The hydrodynamics effects and variable density were incorporated in a three components phase field theory by Kvamme et al. [11] through implicit integration of Navier stokes equation following the approach of Qasim et al.[8]. The phase field parameter is an order parameter describing the phase of the system as a function of spatial and time coordinates. The phase field parameter is allowed to vary continuously from 0 to 1 on the range from solid to liquid. The solid state is represented by the hydrate and the liquid state represents fluid and aqueous phase. The solidification of hydrate is described in terms of the scalar phase field , , where , , represents the molar fractions of CH4, CO2 and H2O respectively with obvious constraint on conservation of mass ∑ = 1. The field is a structural order parameter assuming the values = 0 in the solid and = 1 in the liquid [12]. Intermediate values correspond to the interface between the two phases. The starting point of the three component phase field model is a free energy functional [11],
ISBN: 978-1-61804-065-7
+∑,
F=
"0 = 90 "1 = 91
;
Where µ^
Y_` a`
4 , \6 ln4] 6
(17)
4T, P6 chemical potential of water in ideal gas and y^ is the mole fraction of water in the fluid
9
phase and can be calculated as: ]
3 e 4 , \, 3f 6\>[g 4 6 h 4 , \, ]i6
(18)
The vapour pressure can be calculated using many available correlations but one of the simplest is given in24 as a fit to the simple equation: ln4\6 jk
jl
jm
(19)
The temperature of the system is obviously available and VA 52.703, VB 3146.64 and VC 5.572. Further, the fugacity and the activity coefficient are approximated to unity merely because of the very low water content in
fluid phase and its corresponding minor importance for the thermodynamics of the system. Hydrate formation directly from water in gas is not considered as significant within the systems discussed in this work. A separate study reveals that hydrate formation from water dissolved in carbon dioxide may be feasible from a thermodynamic point of view25 but more questionable in terms of mass transport in competition with other hydrate phase transitions. The water phase is close to unity in water mole fraction. Raoult’s law is therefore accurate enough for our purpose. The chemical potential for the mixed fluid states considered as: P=Q R
R.'[>,wQSZ
ln4] 6 ln h 4 , \, ]i6
(20)
Where i represents CH4 or CO2. The fugacity coefficients of component i in the mixture is calculated using the classical SRK equation of state(EOS),26
ln h 4EE6 4y 16 ln4y E6
C E 44CC6 4EE6 6 ln z1 { E y
(21)
Where Z is the compressibility factor of the phase and is calculated using the following cubic SRK EOS: y M y I 4C E EI 6y CE 0
(22)
10
Where, C
|K\ I I
E
}\
K 0.427480
I 1I \1
} 0.086640
1 \1
| 1 40.48508 1.55171 0.15613 I61 S
I
Where ω is the accentric factor of components. For mixture, the mixing rule with modification proposed by Soave26 is used using the following formulations:
4|K6< ] ] 4|K6 ; 4|K6 4|K6 4|K6 1
(23)
Where k is the binary interaction parameter. (Coutinho et al.27) has proposed number of values for k for CO2/CH4
system. Here we selected an average value k k 0.098 for unlike pairs of molecules and it is zero for alike pairs of
molecules.
}< ] }
4AA6 and 4BB6 in equation (21) are calculated as:
(24)
11
4CC6
2 4K|6 4K|6<
4EE6
1.1.2
} }
The chemical potential S
[QZQ>
3S S
ST1,
(27)
for components c (carbon dioxide) and m (methane) dissolved into the aqueous
phase is described by nonsymmetric excess thermodynamics: 4 6 ln43 e 6 4\ \ 6
(28)
is the chemical potential of component i in water at infinite dilution, e is the activity coefficient of component
in the aqueous solution and ∞ is the partial molar volume of the component i at infinite dilution. The chemical
potentials at infinite dilution as a function of temperature are found by assuming equilibrium between fluid and aqueous phases P=Q R
[QZQ>
. This is done at varying low pressures where the solubility is very low and the
gas phase is close to ideal gas using experimental values for the solubility and extrapolating the chemical potential
down to a corresponding value for zero concentration. The Henry’s constants are calculated for CH4 and CO2 using the expression proposed by Sander.28
4 6
%∆ L L : @A) ! % +; B 5 5
(29)
Where is the reference temperature, which is equal to 298.15K. ∆ is the enthalpy of dissolution and it is
12
represented by the Clausius-Clapeyron equation
29
as:
ln ∆>=0 41⁄ 6
(30)
30 31 The values of 4d ln k_H 6/d41 ⁄ T6 and k H are given by Zheng et al. and by Kavanaugh et al. for CO2 and CH4
respectively which is shown in Table 1.
Table 1: Values of parameters.
Constants
kH
(M/atm)
4d ln k_H 6/d41 ⁄ T6 (K)
CO2
CH4
0.036
0.0013
2200
1800
The activity coefficient at infinite dilution e is calculated as: e
¡ 4 6
(31)
Where, ¡ %
"(¢
Where ¡ is the fugacity of component i, while is calculated from.32 The activity coefficient can be regressed
by using the model for equilibrium to fit experimental solubility data. The chemical potential of water can be written as:
wQSZ = Q R
where
wQSZ = Q R
4 6 ln41 36e 4\ \ 6
(32)
is pure water chemical potential and is the molar volume of water. The strategy for
calculating activity coefficient is given by Svandal et al.32
13
1.2
Thermodynamics outside of equilibrium
The Phase Field Theory (PFT) model presented in this work have dynamically varying local densities, temperatures and concentrations and the constraints on the system is the pressure. Unlike our earlier PFT models9,15 , which where at constant temperature, the calculations of non-equilibrium thermodynamic9 properties are implemented implicit calculations into the Phase Field Theory (PFT) model in this version since the free energies of all co existing phases changes dynamically with local conditions and possibly competing phase transitions. We use examples based on conversion of CH4 hydrate into CO2 hydrate or mixed CO2-CH4 hydrate, which have two primary mechanisms. The first one is that CO2 creates a new hydrate from free water in the porous media and the released heat dissociate the in situ CH4 hydrate. This mechanism is primarily dominated by mass transport rates through fluid phases. A second mechanism is the direct conversion of the in situ CH4 hydrate with the CO2, which is a much slower solid state phase transition. The first mechanism implies heat release as well as heat consumption and complex coupled behavior of mass and heat transport around the hydrate core is expected. Also note that fluid thermodynamics and aqueous thermodynamics outside of equilibrium is trivial in contrast to hydrate, for which the thermodynamic model is derived from statistical mechanics based on equilibrium between hydrate and fluid phases. For this reason we expand the thermodynamic properties of hydrate by a first order Taylor-expansion. This is considered accurate enough since the rate limiting kinetic contributions are expected to be in the mass transport according to earlier studies10,11,31 . First of all note that the mass is conserved inside the phase field theory. The thermodynamics have to be developed in terms of gradients in all directions (P, T, molefractions) without conservation of mole-fractions in order to obtain the appropriate relative local driving forces and also avoid double conservation constraints in the free enrgy minimalization. In this paper we limit ourselves to three components, where CH4 is the additional component to CO2 and water. This can be directly extended to more components though straightforward extensions of the equilibrium and supersaturation thermodynamics, and appropriate adjustment of the PFT. As the thermodynamic changes are outlined here the primary additional change in the PFT model is in the free energy of the thermal fluctuations10,33,34 as function of concentrations, which is mathematically trivial. Supersaturations of fluid phases is straightforward and not different from what we have published before10,6 and in the first part of this paper.
14
The Gibbs free energy of the hydrate phase is written as a sum of the chemical potentials of each component,10,6 N 3S S S
(33)
where S and 3S is chemical potential and mole fraction of component r respectively. N is the free energy of hydrate.
In the earlier work due to Svandal et al.10 a simple interpolation in mole-fractions was used between pure CH4 hydrate and pure CO2 hydrate, which was considered as sufficient to theoretically illustrate the exchange concept under phase field theory. This will of course not reproduce the absolute minimum in free energy for a mixed hydrate in which CH4
occupies portions of the small cavities and increases stability over pure CO2 hydrate. The expression for free energy gradients with respect to mole fraction, pressure and temperature is: ¨N © ¨3S ¥,ª,5,«
N£¤¥ N£¦ § S
(¬
¨N ¨N © 4\[1g \£¦ 6 § © 4 [1g £¦ 6 ¨\ 5,ª,«® ¨ ¥,ª,«®
43S[1g 3S£¦ 6 §
(34)
Here N£¤¥ is the free energy of hydrate away from equilibrium and the superscripts ¯° and K±² represent the
corresponding states at equilibrium and actual states respectively. We are now seeking gradients in all directions, independent of mole-fraction conservation (sum of mole-fractions are conserved inside PFT). So in terms of supersaturation in mole-fractions these have to be evaluated as orthonormal gradient effects. In simple terms that means: ¨3³ 0, µ ¶ · § ´ 1, µ · ¨3S
(35)
Where µ and · both represent any of the components of the hydrate: water, methane, and carbon dioxide. This is just
means that the mole fractions are all independent. Using equation (33) we simply take the derivative with respect to one
of the mole fractions (· ¸, ± or ¹) and the mole fraction derivatives are obtained using equation (35) for mole fraction
independence, resulting in:
15
¨N ¨1 ¨< ¨ ¨3S 31 3< 3 S ¨3S ¨3S ¨3S ¨3S ¨3S
(36)
It was previously shown10 that the chemical potential of a guest molecule can be approximated to a high degree of accuracy and in gradient terms: º C43º 6 E,
¨º »0, · ¶ ¼ ¨3S
(37)
Where and · both represents any of the components of the hydrate (CO2, CH4 & water). For the gradient due to a
guest molecule, these simplifications lead to:
¨N ¨º 3º º ¨3º ¨3º
(38)
¨N ¨S 3S ¨3 ¨3
(39)
01 º ∆/º 4º 6
(40)
For water, the form has two more terms:
S
The chemical potential of a guest in the hydrate º can be written as (Kvamme and Tanaka1):
01 Where ∆/º is the Gibbs free energy of inclusion of guest molecule in cavity ½, º the cavity partition function of
component in cavity ½, the universal gas constant is and is temperature. The derivative of equation (40) with respect to an arbitrary molecule r is:
01 ¨4 4º 66 ¨º ¨∆/º ¨3S ¨3S ¨3S
(41)
16
The first term of equation (41), the stabilization energy is either evaluated as the Langmuir constant or using harmonic oscillator approach.1 In either case it is assumed to be approximately of temperature and pressure. Omitting the first term of (41) and approximating impacts of guest-guest interactions to be zero we arrive at: ¨º ¨º ¨3S º ¨3S
(42)
The validity of omitting guest-guest interactions may be questionable for some systems even though it is omitted in most hydrate equilibrium codes or empirically corrected for. Extensions for corrections to this can be implemented at a later stage.
The chemical potential of water: ¿, 4 , \¿ 6 À Á1 º Â , \, 2®
(43)
º
¿, Where is the chemical potential of water in an empty hydrate structure, the first sum is taken over both small and
a large cavity, the second one is over the components in the cavity ½. Here À is the number of type- ½ cavities per water molecule. Hydrate structure I contains 3 large cavities and 1 small cavity per 23 water molecules, À=
M
IM
and À>
L
IM
.
The paper by Kvamme & Tanaka1 provides the empty hydrate chemical potential as polynomials in inverse temperature,
the Gibbs free energies of inclusion, and chemical potential of pure water equation with respect to an arbitrary molecule · results in:
¨ ∑º º ¨ ¨3S À Ã Ä ∑ ¨3S 1 º º
wQSZ
4 6. The derivative for the above
(44)
From equations (42) and (44), the derivative of the partition function can be evaluated from the equation that relates the filling fraction to the partition function:
17
º
2º 1 ∑ 2
(45)
Where 2º is the filling fraction of the components in the cavity ½. But it is easiest to recast everything in terms of
mole fraction because of the basic assumption of mole fraction independence: 3º À 3
(46)
3º À 3 ∑ 3
(47)
2º
Since mass conservation is not used, the usual form of 1 35 is not considered. This is substituted into equation (45)
and we get:
º
Now we can take the derivative with respect to an arbitrary component · and using equation (47) we get: I ¨º º ¨3º º ¨3 ¨3 ÁÀ  ¨3S 3º ¨3S 3º ¨3S ¨3S
(48)
The first thing that must be dealt with the cavity mole fractions as a function of total mole fraction of a component: 3º 3º
(49)
Since the derivative of one mole fraction with respect to another is independent, the mole fraction in the cavity is also independent: ¨3º 0, ¶ · · · ¹ § : ;Å 1, · ¨3S
(50)
18
If · ¹, then the derivative has to be zero because the mole fraction of the guest are independent of the mole fraction
of water. Now equation (48) is simplified by using equation (49) and equation (50): I º À ¨º ¨3 3º
I ¨º º ¨3º º ¨3w ¨3w 3º ¨3w 3º ¨3w
(51)
(52)
Where Æ is an arbitrary guest molecule, is also a guest molecule. These can be the same or different. If and Æ are
the same molecule, this gradient still exist and the “cross terms” are still able to be found even if there is independency in the mole fractions.
R«Ç# R«Ç
is calculated by starting with the equation (49) which is the basic definition of the mole
fraction of the cavities and how they relate to the total mole fraction of the component. The total methane mole fraction 3< , is the sum of the mole fraction in the large cavities 3
