Nonequilibrium Modeling of Hydrate Dynamics in ReservoirMohammad T. Vafaei, Bjørn Kvamme,* Ashok Chejara, and Khaled Jemai
Department of Physics and Technology, University of Bergen, Allegaten 55, N-5007 Bergen, Norway
ABSTRACT: Gas hydrates in reservoirs are generally not in thermodynamic equilibrium, and there may be several competingphase transitions involving hydrate. Calculations of hydrate phase transitions based on a modified statistical−mechanical modelfor hydrate (Kvamme, B.; Tanaka, H. J. Phys. Chem. 1995, 99, 7114−7119) are used to illustrate differences in properties ofhydrates formed from different phases. In more general terms, hydrates in porous media are discussed in terms of the Gibbsphase rule. It is argued that phase transitions involving hydrates in porous media can rarely reach any state of equilibrium due tosituations of over specified systems with reference to requirements for equilibrium. As a consequence of this, a strategy fornonequilibrium description of hydrates in reservoirs is proposed. This involves the formulation of kinetic expressions for allpossible hydrate formations and dissociations as competing pseudoreactions. This involves hydrate formation on a water/carbondioxide interface, from water solution and from carbon dioxide adsorbed on mineral surfaces, as well as all different possiblehydrate dissociation possibilities. The basic idea is that the direction of free energy minimum, under constraints of mass and heattransport, will control the progress of phase transitions in nonequilibrium systems. A new hydrate reservoir simulator based onthis concept is introduced in this study. The reservoir simulator is developed from a platform previously developed for carbondioxide storage in aquifers, RetrasoCodeBright (RCB). The main tools for generating kinetic models have been phase fieldtheory simulations, with thermodynamic properties derived from molecular modeling. The detailed results from these types ofsimulations provides information on the relative impact of mass transport, heat transport, and thermodynamics of the phasetransition, which enable qualified simplifications for implementation into RCB. The primary step was to study the effect ofhydrate growth or dissociation with a single kinetic rate on the mechanical properties of the reservoir. Details of the simulatorand numerical algorithms are discussed, and relevant examples are shown.
1. INTRODUCTIONThe international focus on natural gas hydrates is increasing fora number of different reasons. Unlike conventional fossil fuels,hydrates are distributed worldwide and can be a source ofenergy for countries that have little or no conventional fossilfuel resources. Japan is a good example; test production at thefirst offshore pilot plant will take place in 2012, and fullproduction is planned from 2018. Production of permafrosthydrates has been going on since 1970 in Russia. Theworldwide amount of hydrates is uncertain, but the UnitedStates Geological Survey (USGS) estimated 10 years ago that itmight amount to more than twice all known conventional fossilfuels. Hydrates in porous media are generally not stable, as willbe discussed in more detail later in this paper. This implies that,basically, the demands for trapping/sealing formations to keephydrate in place are similar to oil and gas sealings (cap rock). Inthe case of offshore hydrates, deficiencies in sealing above thehydrate layers in the form of fractures and faults will lead tocontact between hydrate and groundwater. The result isdissociating hydrate and methane seepage to the oceanabove. Since methane is a more aggressive greenhouse gasthan CO2, the global methane fluxes from hydrates and freemethane is a global environmental concern. Besides, in the longrun, these leakage situations may lead to dement instabilitiesand, in the worst case, cause landslides. In some permafrostregions, the ice above is the only trap for the gas hydrate.Decreasing permafrost ice is therefore a global concern.There are several scenarios for methane production from
natural gas hydrate reservoirs. Among these, four methods aredescribed here, which are believed to be more feasible,
specifically with respect to economy. The depressurizationmethod in which the hydrate stability condition is shifted bypressure reduction resulting in hydrate dissociation and releaseof methane. It is currently considered as the most feasibleprocess considering expenses and production rate and has beeninvestigated by many research groups through simulationstudies. Thermal stimulation is another method, which is basedon moving out from the stability region by temperatureincrease. It is considered to be costly because of the hugeamount of energy waste to the surroundings. The third methodis to use inhibitors such as methanol or brine to shift theequilibrium curve and dissociate hydrate, which is also costly.The fourth method is injection of CO2 into the methanehydrate reservoirs. CO2-hydrate is more stable than CH4-hydrate over substantial regions of pressure and temperature.CO2 and CH4 will both form hydrate structure I, which consistof a ratio of 3 to 1 of large cavities (24 water moleculessurrounding enclathrated molecule) to small cavities (20water). Under normal conditions, CO2 will only fill the largecavities and the mole fraction of CO2 is close to 0.12, whileCH4 hydrate has a mole fraction of CH4 close to 0.14. A mixedhydrate in which CO2 occupies the large cavities and CH4occupies the small cavities will be more stable than methanehydrates over all regions of temperature and pressure whereeither hydrate can exist. Injection of CO2 into a natural gashydrate reservoir can release the natural gas by direct
Received: February 28, 2012Revised: May 11, 2012Published: May 14, 2012
conversion of the in situ hydrate, which is a solid statetransformation and slow.2,3 Hydrate cannot attach directly tomineral surfaces because of incompatibility with hydrogenbonding and partial charges on atoms in the mineral surfaces.Any hydrates in porous media will therefore be surrounded byfluids that separate them from the mineral surfaces. InjectedCO2 may therefore also form hydrates directly with free waterin the pore volume. Released heat will contribute todissociation of in situ natural gas hydrate. This heat release issubstantially higher than the heat release due to solid statereformation discussed by Kvamme et al.4 The absolute smallestdistance separating hydrate from the mineral surfaces is in theorder of 3 nm, based on molecular dynamics simulations. Thepermeability would be extremely small if natural hydratesoccurred with such a high pore filling. Reported filling fractionsfor natural gas hydrates, as well as synthetic laboratoryproduced hydrates in sediments, are normally substantiallylower. Most reported natural gas hydrate reservoirs have below50% of pore volume filled with hydrate. If the remainingvolume is liquid water, then solid state conversion of the CH4hydrate over to CO2 hydrate is still one of the processes.Formation of new CO2 hydrate will, however, be much faster,and therefore, normally the most important mechanisminvolved in combined storage of CO2 in hydrate form andproduction of natural gas hydrates. The heat released from theformation of new CO2 hydrate will assist in dissociating the insitu CH4 hydrate.During more than two decades, different research groups
worldwide have applied different theoretical modeling andsimulation approaches to studies of methane production fromhydrate reservoirs. The progress has been considerable, butthere are still many uncertainties and unresolved challenges inthe physical description of these complex systems. A limitedreview of past modeling efforts of hydrate exploration is givenin this section. Special emphasis has been on studies that haveresulted in new models and/or new simulator for simulation ofmethane production from hydrates.Holder and colleagues5 considered a system of stratified
hydrate and gas surrounded by impermeable layer from bottomand top to build a 3D model for studying hydrate dissociationeffect on gas production due to depressurization. Severalassumptions were made to simplify the calculations. Theyconsidered the dissociation to happen only at the interfacebetween hydrate and gas phase, and only conduction wasconsidered to find temperature distribution in the gas phase.They concluded that hydrate can contribute significantly to gasproduction from such reservoirs.5 Burshears et al.6 developed atwo-phase, 3D numerical model to study gas production from asystem of hydrate formation and dissociation in the reservoir.Their model consisted of water and any mixture of methane,ethane, and propane. They considered radial flow andequilibrium conditions in the gas−hydrate interface. Waterflow was present in their model but only heat conduction wasconsidered.6 Yousif et al.7 developed a one-dimensional modelto simulate isothermal depressurization of hydrate in Bereasandstone samples. The model considered three phases ofwater, gas, and hydrate and used the kinetics model of Kim andBishnoi8 for dissociation of hydrate. It also considered thewater flow due to hydrate dissociation, which according toexperimental and numerical results was considerable. Thevariations in porosity and permeability of the gas phase weretaken into account. Finally, they validated their model withexperimental data.7 Xu and Ruppel9 developed an analytical
formulation to solve the coupled momentum, mass, and energyequations for the gas hydrate system consisting of twocomponents (water and methane gas) and three phases (gashydrate, free gas, water, and dissolved methane). They madeseveral assumptions such as ignoring capillary effects andignoring kinetics of the phase transition between hydrate andaqueous phase.9 Swinkels and Drenth10 used an in-house 3Dthermal reservoir simulator and modified the PVT tool to studyhydrate production scenarios, as well as heat flow andcompaction in the reservoir and hydrate cap. They representedthe reservoir fluid by a gaseous phase, a hydrate phase, and anaqueous phase. Their system consisted of three components,including two hydrocarbons and a water component. Heat wasconsidered as an extra component for all phases internally inthe simulator. The Joule−Thomson cooling effect was alsoconsidered automatically.10 Davie and Buffet11 proposed anumerical model for calculation of gas hydrate volume anddistribution in marine sediments. They considered the organicmaterial from sedimentation as the main source of carbonsupply to the hydrate stable region. The rate of sedimentation,the quantity and quality of organic material, and biologicalproductivity were considered as key parameters in this model.Hydrate formation and dissociation were controlled bythermodynamic conditions and driving force from equilibriumconcentration. A first order model with a large rate constantwas used to account for the reaction rate. They compared theirresults with observations from practical field. Their model wasdependent on the available data of sedimentation rate andorganic content.11 Goel et al.12 developed a single phase flowmodel to predict the performance of an in situ gas hydrate. Thegas withdrawal rate was assumed to be constant, and thereservoir was assumed to be infinite. It used an nth orderkinetics rate coupled with gas flow in the porous media. Themodel was simple and had several limitations due tosimplifications.12 Moridis13 has proposed a new module forTOUGH2 simulator named EOSHYDR2. TOUGH2 is a 3D,multicomponent, multiphase flow simulator for subsurfacepurposes. EOSHYDR2 is designed to model hydrate behaviorin both sediments and laboratory conditions. It is able toconsider up to four phases of gas, liquid, ice and hydrate, andup to nine components of water, CH4-hydrate, CH4 as nativeand from hydrate dissociation, a second native and dissociatedhydrocarbon, salt, water-soluble inhibitors, and heat as apseudocomponent. It includes both equilibrium and kineticmodels for hydrate formation and dissociation. It uses hydratereaction model of Kim and Bishnoi8 for kinetic studies. Hestudied four test cases of CH4 production and concluded thatboth depressurization and thermal stimulation can produce asubstantial amount of hydrate, and he suggested that thecombination of both can be desirable. He used just theequilibrium approach because of a lack of enough suitable datanecessary for the parameters of the kinetic model, whilementioning that slower processes such as depressurizationfollow kinetic dissociation. Finally, he discussed severaluncertainties mostly due to lack of knowledge in this area ofresearch.13 Later on, using the same module, Kowalsky andMoridis14 made a comparison study between kinetic andequilibrium approach and concluded that based on the accuracyof the available kinetic models; this approach is important forsimulation of short-term and core scale systems whileequilibrium approach can be used for large scale simulationsof production processes. Hong et al.15,16 presented a 2Dcylindrical model to study gas production from hydrate
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reservoirs. They used both analytical and numerical approaches.The numerical model considered the equations for gas andwater two-phase flow, conductive and convective heat flow, andtreated hydrate decomposition kinetically using the Kim andBishnoi8 model. In the analytical approach, they did notconsider the fluid flow. They studied the effect of hydrate zoneon the gas production from gas layer and also the importance ofthe intrinsic kinetics of hydrate decomposition with respect toequilibrium assumptions. They concluded that the kineticequation was affected only when it was considered 5 orders ofmagnitude lower than the values found by Kim and Bishnoi8 ina PVT cell.15,16 Xu17 has modeled dynamic gas hydrate systemsconsidering two main elements. The first element provides amodel to account for dynamic phase transitions of marine gashydrates by considering the effects of fluid pressure, temper-ature, and salinity changes on hydrate stability, solubility, andfluid densities and enthalpies. This allows describing coex-istence of three phases, namely hydrate, gas, and liquid solutionat equilibrium. The model consists of three components (water,gas, and salt) and four phases (free gas, liquid solution, gashydrate, and solid halite). It considers thermodynamicequilibrium for and among individual phases. The secondelement of the model consists of the numerical tool to solve forfluid flow and transport equations in porous media. It considersfour components and up to five phases. The response of themodel to the pressure drop and temperature rise scenarios atseafloor are studied.17 Ahmadi et al.18 developed a 1D model tostudy methane production from hydrate dissociation in aconfined reservoir using depressurization method. Theyconsidered heat of dissociation, conduction, and convectionin both gas and hydrate phase. They considered equilibriumconditions at the dissociation front and neglected water flow inthe reservoir and the Joule−Thomson cooling effect.18 Sun andMohanty19 developed a 3D simulator using the Kim andBishnoi8 kinetic model to study formation and dissociation ofhydrate in porous media. They considered four components(hydrate, methane, water, and salt) and five phases (hydrate,gas, aqueous, ice, and salt precipitate). Water freezing and icemelting are considered, assuming equilibrium phase transitions.Mass transport including fluid flow and molecular diffusion plusheat transport including conduction and convection are solvedimplicitly.19 Phirani et al.20 upgraded the simulator to accountfor CO2 as a new component and CO2 hydrate as a new phase.The purpose was to study CO2 flooding of methane hydratebearing sediments. They used the model of Kim and Bishnoi8
for kinetics of dissociation and constant rates for hydrateformation.20 Uddin21 developed a general kinetic model ofhydrate formation and decomposition based on Kim andBishnoi8 model to study CO2 sequestration in methane hydratereservoirs along with CH4 production. The model consisted offive components of water in aqueous phase, CH4 and CO2 ingas phase, and CO2-hydrate and CH4-hydrate in solid phase.The gas hydrate model was coupled with a compositionalthermal reservoir simulator (CMG STARS) to study dynamicsof hydrate formation and decomposition in the reservoir. It wasconcluded that the effect of kinetic rate constant on hydratedecomposition was significant in case it was lowered a feworders of magnitude. Reservoir permeability has also a greatimpact on the decomposition rate.21 Nazridoust and Ahmadi22
developed a computational hydrate module for FLUENT code.It provides the possibility to study hydrate dissociation forcomplex 3D geometries. They used the kinetics rate ofdissociation proposed by Kim and Bishnoi,8 and they included
the heat of reaction, effects of water and gas permeabilities, andeffective porosity with possibility of variation with time. Theyfound that the production process was sensitive with respect totemperature, pressure, and core permeability. They comparedtheir results with experimental data as well.22 Liu et al.23
developed a 1D model to study hydrate dissociation bydepressurization in porous media. They used a moving front toseparate the hydrate zone from gas zone. They consideredconductive and convective heat transfer and mass transfer in gasand hydrate zones and energy balance at moving front. Theyconsidered equilibrium at the front and concluded that theassumption of stationary water phase results in overpredictionof dissociation front location and underprediction of gasproduction in the well. They also found that reservoirpermeability and well pressure can highly affect the productionrate.23 Gamwo and Liu24 have presented a detailed theoreticaldescription of the open source reservoir hydrate simulatorHydrateResSim developed previously by Lawrence BerkeleyNational Laboratory (LBNL). They have also applied it to asystem of three components (methane, water, and hydrate) andfour phases (aqueous, gas, hydrate, and ice). Darcy’s law is usedfor multiphase flow of mass in porous media and local thermalequilibrium is considered in the code. It considers both kineticand equilibrium approaches, using Kim and Bishnoi8 as thekinetic model of hydrate dissociation. The studied model hasbeen validated using the TOUGH-HYDRATE simulator. Theyconcluded that HydrateResSim is a suitable freeware, opensource code for simulating methane hydrate behavior in thereservoir. They found out that equilibrium approach usuallyoverpredicts the hydrate dissociation compared to kineticapproach.24
From the presented literature review it is clear that thehydrate dissociation process in the reservoir is mostly treated asan equilibrium reaction and in fewer cases as kinetics. Themajority of kinetic approaches are based on the kinetic modelof Kim and Bishnoi,8 developed according to laboratoryexperiments as follows:
− = −nt
k A f fdd
( )Hd s e (1)
If the phase transition as a whole is dominated by the masstransport, the form of the equation may be adequate enough.Therefore, we have also used similar form of eq 1 as a basis fortransferring results from advanced theories (see section 3) intoreservoir simulator use. However, then, it is acceptable onlyafter verification (rate versus square root of time indicate Fick’slaw) of the mass transport dominance. The use of eq 1 forfitting laboratory experiments from PVT cell and then usingthese data for a reservoir situation might be more questionable,since it has never been proven that these data are transferableto any other experimental apparatus or real situation of hydratephase transitions in nature. This also raises the question of howto evaluate the impact of kinetics if the rate equation is notappropriate. For pressure reduction as production method onecould argue that the heat transport might have a strong impact.Some even use the heat transport rate model alone to modelthe kinetic rate in this case. The different heat transport modelsapplied are normally not documented well enough in terms oftheory and/or parameters to judge whether they areappropriate or not. Heat transport to a system undergoingdynamically a phase transition that involves flow is fairlycomplex, so it is unclear whether any reasonable model exists.Another critical question related to analysis of the impact of
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kinetics is the reference case. As the system by itself cannotreach equilibrium, the P, T projections are just two dimensionsof a general variable set consisting of concentration in allsurrounding phases, including impact of minerals. Hydrate willdissociate toward undersaturated water, regardless of whetherthe system is inside stability with reference to P and T or not.Also, it will sublimate toward undersaturated gas. With hydratesaturations between 18% and 50% (which are typical ranges forreservoirs considered as “rich” enough), induced flow willexchange fluids surrounding the hydrate, and the impact ofthese fluid changes on hydrate dissociation have never beenevaluated. In summary, there are many uncertain issues aroundthe kinetic models used as a basis for concluding that kineticeffects are not important, as well as clearly the assumption onthe reference system (read: P, T defines equilibrium regardlessof concentration induced nonequilibrium).The fact that hydrates in porous media are in nonequilibrium
does require kinetic models in descriptions of all relevantcompeting phase transitions (all phase transitions that havenegative free energy changes of sufficient magnitude toovercome nucleation barriers). There might, however, be atime-dependent local situation where one process, or a fewprocesses, dominates and is fast enough to qualify for somequasi-equilibrium approximation. However, those are excep-tions and do not need to be in the P, T projection of themultidimensional thermodynamic dependency of the system.In this paper, a different approach according to non-
equilibrium nature of hydrate phase transitions in the reservoirwill be presented. A new reservoir hydrate simulator will beintroduced, which is developed on a former reactive transportreservoir simulator namely RetrasoCodeBright (RCB).25 Themodule is designed so that it can easily work according to thenonequilibrium thermodynamic package that is being devel-oped in this group. The reactive nature of the code gives thepossibility to distinguish between different phase transitionscenarios and treat them on a competitive basis. Therefore, it ispossible to evaluate the amount of components that are used toform hydrate or produced from hydrate dissociation andinclude these results in the overall heat and mass balanceequations. In addition, it gives the possibility to visualize theeffect of hydrate formation or dissociation on the solid phaseproperties in the form of direct changes in porosity andpermeability. At this stage, kinetic models of hydrate formationand dissociation from phase field theory simulations26,27 areused to examine the performance of the module throughexample cases.
2. THEORYWhen a clathrate hydrate comes into contact with an aqueoussolution containing its own guest molecules (CH4 or CO2), thenumber of the degrees of freedom available to the new systemof three phases (aqueous, gas, hydrate) will be decreasedcompared to the two-phase system (aqueous, gas) as a result ofthe Gibbs phase rule.
π= − +F N 2 (2)
In this equation, F is the degree of freedom, N is the numberof components, and π is the number of phases. For the two-phase system comprising two components, water and gas, therewill be two degrees of freedom while, in a three-phase systemwith hydrate as an additional phase, there will be only onedegree of freedom. The Gibbs phase rule is simply conservationof mass under the constraints of equilibrium between coexisting
phases. The number of independent thermodynamic variablesspecified for the system must equal the number of degrees of
Table 1. Potential Phase Transition Scenarios for a Systemof Hydrate in Reservoir
i δinitial
phase(s) driving forcefinal
phase(s)
1 −1 hydrate outside stability in terms of P and/or T
gas, liquidwater
2 −1 hydrate sublimation (gas under saturatedwith water)
gas
3 −1 hydrate outside water under saturated withmethane
liquidwater
4 +1 gas + liquidwater
hydrate more stable thansurrounding fluids
hydrate
5 +1 surfacereformation
nonuniform hydrate rearranges dueto mass limitations
hydrate
6 +1 aqueousphase
liquid super saturated with methanewith reference to hydrate
hydrate
7 +1 adsorbed water and methane adsorbed onmineral surfaces
hydrate
Figure 1. Water chemical potential in empty SI (dash), empty SII(dash-dot), water as ice or liquid water (solid).
Figure 2. Predicted hydrate equilibrium (solid line), as estimated usingthe parameters from Kvamme and Tanaka1 for empty hydratechemical potential (SI), liquid water chemical potential, and theharmonic oscillator method for evaluation of the cavity partitionfunction. Fugacity coefficients are estimated using eq 2 from Duan etal.30 Stars are measured hydrate equilibrium data from Deaton andFrost34 and Unruh and Katz.35
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freedom if the system should be able to reach equilibrium. As aresult, when both temperature and pressure are specified, as isthe case in local reservoir conditions, the system is over-determined and will be unable to reach three-phase equilibrium.Despite this fact, it is quite common to ignore theconcentration dependencies of the different co-existing phaseson phase transitions. A minimum requirement for a phasetransition is negative free energy change, but in a non-equilibrium system that is not sufficient. All gradients inindependent thermodynamic variables also have to point in thedirection of lower free energy if the phase transition should gounconditionally. A practical example can be found in nature byall the seeps of hydrates that dissociates toward groundwater,which is undersaturated with methane. Injection of CO2 intoaquifers which contain cold zoneswhere pressure andtemperature conditions are suitable for CO2 hydrateformationwill lead to the formation of hydrate films, which
subsequently will dissociate in contact with groundwater fromtop as a result of undersaturation with CO2. As such, a kind ofstationary situation of formation of new CO2 hydrate frombelow and dissociation from top is expected. See alsoKvamme28 for examples of saturation limits of aqueoussolutions toward hydrate as function of temperature andpressure. The statements about equilibrium approach inKowalsky and Moridis14 (on page 1851 of the paper) thatthe system of four phases always exists at equilibrium issomewhat confusing and cannot describe a realistic hydratereservoir, since it could happen only in a unique pointconsisting of a unique temperature, pressure, and composition.In hydrate systems, we are usually dealing with a range oftemperatures, pressures, and compositions that can beexplained only if the system is considered nonequilibrium.The typical case during the production of hydrate is threephases (gas, water, hydrate) and two components, which makethe system overdetermined because temperature and pressureare defined locally in every point of a real reservoir and everygrid point of a reservoir model. If the system crosses the iceformation condition due to cooling, degrees of freedom will be0 and the system even more overdetermined. Figure 1 inKowalsky and Moridis14 has no other meaning than a possiblelimit to kinetics in a two thermodynamic variable projection ofthe nonequilibrium system, which is generally characterized bytemperature, pressure, and concentrations in all coexistingphases as thermodynamic variables. Examples of two hydratephase transitions that are thus excluded are hydrate formationfrom aqueous solution and hydrate dissociation towardundersaturated water. Keeping this in mind, also thecomparison between equilibrium and kinetics is out ofthermodynamic balance since the reference (read: equilibrium)situation is not correct. However, an additional problem is thatthe kinetic model of Kim and Bishnoi8 is empirical and doesnot contain necessary theoretical basis to be transferableoutside the experimental apparatus and experimental conditionsfor which the model was developed. Given the applied stirringrates (and corresponding hydrodynamically induced stresses)in the experimental cell, the kinetic rates from the model withthe original mode parameters are likely to be very differentfrom the hydrodynamic stress involved in reservoir flowsituations related to production.A system that cannot reach equilibrium will, however, always
tend toward the minimum free energy as a consequence of thecombined first and second law and when the hydrate is insideits pressure−temperature stability region, this means that itsfree energy is lower than that of the aqueous solution.29 In caseof hydrate formation and dissociation in the reservoir, thesystem will be even more complex because hydrate can formfrom or dissociate toward phases with different free energies,which will produce different phases. Consequently, the degreeof freedom will decrease further and there will be noequilibrium condition and competing phase transition reactionsof hydrate formation, dissociation, and reformation amongdifferent phases will rule the system. The main factor in such asystem is therefore the minimum free energy of the system andkinetics of competing reactions. Assume, for example, a systemof water, methane, and hydrate in porous media. According tothis system, three phases of gas, aqueous, and hydrate areevident. Methane and water are considered as the onlycomponents available in the system. The degree of freedomfor such a system is 1, while it can reduce to −1 if twoadditional phases of adsorbed CH4 and adsorbed water are also
Figure 3. Estimated water hydrate chemical potential (solid line) forthe experimental conditions of Deaton and Frost34 and Unruh andKatz35 (corresponding to the stars in Figure 2). The dashed line is fortemperatures along the isobar 61 bar.36 The dash-dot line is fortemperatures along the isobar 101 bar.36
Figure 4. Estimated chemical potential of water in hydrate using theextended statistical−mechanical model for hydrate due to Kvammeand Tanaka1 with chemical potential for dissolved CO2 in aqueousphase (solid line). The dashed line is pure water liquid chemicalpotential1 disregarding pressure changes effect on liquid waterchemical potential.
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considered. It is while temperature and pressure are provided inthe reservoir and are not adjustable. All potential phasetransition scenarios due to hydrate dissociation and reformationin this system are summarized in Table 1.The change in the free energy for any of the processes
mentioned in Table 1 will be calculated according to eq 3.
δ μ μ μ μΔ = − + −G x x[ ( ) ( )]ii i i i
wH,
wH,
wp
CHH,
CHH,
CHp
4 4 4 (3)
In this equation, H represents hydrate phase, i represents anyof the seven phase transition scenarios, p represents liquid, gas,and adsorbed phases, x represents composition, and μrepresents chemical potential. δ is 1 for hydrate formation orreformation and −1 for dissociation. Keeping in mind that thephases are not in equilibrium, it might be useful to also visualizethat the hydrates created along the pathways in Table 1 mustalso have different fillings and corresponding different freeenergies, which, by definition, implies that each hydrate formingprocess results in a unique phase. The chemical potential forwater in the hydrate of type i can be estimated using a modifiedversion of the statistical−mechanical model1 μw
H,i:
∑ ∑μ μ ν = − +T P x T P h( , , ) ( , ) ln(1 )i
jj
kkji
wH, H
wH,0
(4)
where superscript H,0 denotes empty clathrate, νj is the fractionof cavity of type j per water molecules, and hkj
i is the canonicalpartition function for guest molecule of type k in cavity type jfor hydrate of type i. Estimated chemical potentials1 for emptyclathrates of structures I and II (SI and SII) are plotted inFigure 1, together with estimates for ice and liquid waterchemical potential. The canonical partition functions can beexpressed as
β μ= − + Δh gexp( ( ))kji
kji
kjinclusion
(5)
where μkji is the chemical potential of guest molecule k in cavity
j for type i hydrate. The second term in the exponent is the freeenergy change of inclusion of the component k in cavity type j,which is independent of the specific hydrate type i. This latterterm can be evaluated in the classical fashion by integrating theBoltzmann factors of the interactions between the waters in thelattice and the guest molecule over the cavity volume andnormalized1 or alternatively evaluated using a harmonicoscillator approach. At equilibrium, the chemical potential fora guest in a cavity will be equal to the chemical potential for thesame component in the equilibrium phase (gas, liquid, oradsorbed). In a nonequilibrium situation, these equilibriumchemical potentials can be used in a series expansion intemperature, pressure, and concentrations over to the realsituation. In view of eqs 5 and 4, the chemical potentials forwaters, as well as guest molecules, will be different betweendifferent hydrates, depending on how the hydrate was formed(read: from where the different molecules in the hydratecome). Also, the mole fractions of water and guest moleculeswill be different for each hydrate, since the filling fractions aregiven by
θ =+ ∑
h
h1kji kj
i
k kji
(6)
In eq 6, θkji is the filling fraction of guest molecule k in cavity
type j for hydrate type i. Therefore, xki, which is the mole
fraction of guest molecule k in the hydrate type i will becalculated according to eq 7.
νθ
νθ=
∑
+ ∑ ∑x
1ki j j kj
i
j k j kji
(7)
For a given hydrate i to grow unconditionally, Gibbs freeenergy change according to eq 3 must be negative and allgradients in free energy change (temperature, pressure, andconcentrations) must be negative. Practically, this implies forwater, as one example, that the chemical potential of water inhydrate must be lowered from empty clathrate chemicalpotentials to equal or lower than the chemical potentials ofice or liquid water by means of guest stabilization through guestinclusion and chemical potentials of guests through eqs 4 and 5.The final result is nonuniform hydrates due to all the differenthydrate phase transitions that are possible inside a porousmaterial. The kinetics of different phase transition is an implicitset of dynamic equations that couples the mass transport, heattransport, and the thermodynamics of the phase transition itself(the impact of free energy change).To illustrate, we may use the statistical mechanical model of
Kvamme and Tanaka1 with all the original parameters.However, instead of the virial equation for estimation ofchemical potential for guest molecules, we apply eq 2 fromDuan et al.30 in order to be able to stretch estimates beyondmoderate gas condition for CO2. The results are plotted inFigures 2 and 3. Note that these estimates are pure prediction,using very simple models for the water−methane interactionsfrom open literature, and no attempts have been made toempirically adjust the model or parameters. For the case ofillustration, these estimates are considered to be fair enough,and we now apply the same model with chemical potential ofCO2 from aqueous solution. Rearranging of the gas/liquidequilibrium criteria for pure CO2 in equilibrium with a CO2
fluid phase gives eq 8:
μ γ
μϕ
+
= +
∞ ∞
⎡⎣⎢⎢
⎤⎦⎥⎥
T P x RT
T P RTT P
x
( , , ) ln
( , ) ln( , )
CO CO
COidealgas CO
CO
2 2
2
2
2 (8)
where the superscript infinite denotes infinite dilution of CO2
in aqueous phase as reference state for the activity coefficient.Fitting the compiled experimental data for CO2 solubility at278.22 K from Valtz et al.31 and neglecting dissociation ofdissolved CO2 into bicarbonate and carbonate ions, we obtainthe following fit:
μγ
μγ
+
≈
+
≈ − + −
∞∞
∞∞
x
RTx
xx
x x
(278.22 K, )ln (278.22 K, )
(278.22 K, )
RTln (278.22 K, )
9.4149 12.67 5.3643
COCO
COCO
CO CO2
2
2
2
2
2 2 (9)
in which we have neglected small Poynting corrections forpressure. Equation 9 can be rearranged into the total chemicalpotential of CO2 in aqueous solution at 278.22 K:
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μ
= − + −
+
x
R x x
x
(278.22 K, )
278.22 [ 9.4149 12.67 5.3643
ln( )]
CO
CO CO2
CO
2
2 2
2 (10)
Application of eq 10 together with free energies of inclusionfrom Kvamme and Tanaka1 in eqs 4 and 5 gives the plot inFigure 4 for chemical potentials of water in hydrate formedfrom CO2 in aqueous solution. These results can be comparedto the chemical potentials of free water in the same plot to findcoexistence limit for hydrate forming from solution. However,more important is the differences in the free energy of thedifferent hydrates. Differences in hydrate water chemicalpotential between hydrate forming from fluid CO2 as separatephase and from solution will reflect back in filling fractions (eqs6 and 7) and also in the chemical potentials that enter the freeenergy difference as driving force for phase transition in eq 3.Heat transport is fast in these systems, which are dominated
by water, typically 2−3 orders of magnitude faster than masstransport.32,33 The local progress of the different phasetransitions and resulting phase distributions at any given timeis therefore a complex function of differences in thermody-namics and differences in mass transport associated with eachpossible phase transition. On a reservoir scale, the enthalpychanges associated with the different hydrate phase transitionsneed to be transferred to the energy fluxes in the reservoirsimulator. Because the chemical potential for guest molecules inhydrate is expanded from equilibrium, a direct evaluation of thetemperature dependency will be more complicated thannumerical estimation using the thermodynamic relationship:
∂
∂= −
μ
⎡⎣ ⎤⎦
TH
RT
RT kP,N2
k
(11)
for any component k in a given phase. The line above Hindicates partial molar enthalpy. For each component (waterand hydrate formers), eq 11 gives the relationship betweenpartial molar enthalpy and the chemical potentials, and thechemical potentials are directly estimated outside equilibrium,so numerical differentiation of the left side of equ 11 is theeasiest way to get the partial molar enthalpy for water and allhydrate formers inside the hydrate in a general nonequilibriumsituation. Summing the contributions to enthalpy from thepartial molar enthalpies and mole fractions of all components ineach phase provides the necessary enthalpy information for theconvective terms of the energy balances in the reservoirsimulator.In this paper, we will limit the number of competing phase
transitions to a minimum and focus mainly on describing thesimulator and the specific method for implementing non-equilibrium thermodynamics in the code. More complexscenarios will be studied in the future, as the code will developand logistics will be worked out further.Up to this point, the purpose of this paper was to provide
enough evidence through equations and figures to show theimportance of a kinetic approach in the simulation of hydrate inreservoir and to remind that there are many unattended issuesin this area that leave uncertainties in comparisons betweenequilibrium and kinetic approach. The rest of this paperintroduces the simulation approach that provides the possibilityto consider, as much as possible, the kinetic theories presentedin the previous sections.
To analyze this hydrate system, all impossible (ΔG > 0) andunlikely (|ΔG| < ε) cases must be disregarded. Taking intoaccount the mass transport limited cases, all realistic phasetransition scenarios will be determined.The purpose is to develop a hydrate reservoir simulator that
has the possibility to consider the free energy changes of allphase transition scenarios and take into account the effect ofthe realistic ones on the flow and properties of the porousmedia through advanced kinetic models.
3. SIMULATION3.1. Kinetic Model. The results from phase field theory
simulations26,32,33,37−39 have been modified to be used in the
kinetic model. Phase field simulations are based on theminimization of Gibbs’ free energy on the constraint of heatand mass transport. Extensive research has been going on in thesame group on application of phase field theory in prediction ofhydrate formation and dissociation kinetics, which is still inprogress.4,26−29,32,33,37−40 In this study, the simulation resultsfor hydrate formation kinetics from such studies have beenextrapolated and used as the reaction rate constant of the Kimand Bishnoi8 kinetic model in the numerical tool. Assumingthat the formation process happens from gas phase anddissociation is toward aqueous phase, the diffusivity of CO2 inliquid and gas phase would be important. According to theavailable data in literature, diffusivity of CO2 in liquid phase isaround 4 orders of magnitude lower than gas phase.41,42
Therefore, dissociation rate has been chosen accordingly in thisstudy.
3.2. Numerical Tool. For reservoirs where hydrate mayform in some regions, hydrate formation involves roughly 10%volume increase compared to water. In parallel to this, there arechemical reactions that can, for example, supply extra CO2through dissolution of carbonates in regions of low pH, andalso regions of high pH where transported ions may precipitateand even extract CO2 from water and hydrate. The formedhydrate will not be in equilibrium due to Gibbs’ phase rule, as
Table 2. Equations and Independent Variables
equation variable name
equilibrium of stresses displacementsbalance of liquid mass liquid pressurebalance of gas mass gas pressurebalance of internal energy temperature
Table 3. Constitutive Equations
constitutive equation variable name
Darcy’s Law liquid and gas advective fluxFick’s law vapor and gas non-advective fluxFourier’s law conductive heat fluxretention curve liquid phase degree of saturationmechanical constitutive model stress tensorphase density liquid densitygas law gas density
Table 4. Equilibrium Restrictions
equilibrium restrictions variable name
Henry’s law air dissolved mass fractionpsychometric law vapor mass fraction
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mentioned in the previous sections. Neither will it attach to themineral surfaces due to incompatibilities of hydrogen bonds inhydrate and interactions with atomic partial charges on themineral surfaces. For this reason, there is a need for a logisticsystem that can handle competing processes of formation anddissociation. A reactive transport simulator can handle that.Implicit geomechanics is needed in order to handle competingphase transitions, which are very rapid (seconds) and aredynamically coupled to geochemical reactions, which can be
fairly fast (hours to days). For this purpose, a reactive transportreservoir simulator, RetrasoCodeBright (RCB),43 is used in thisstudy and extended with hydrate phase transitions as“pseudoreactions”. RCB is capable of realistic modeling of thereaction rates for mineral dissolution and precipitation, at leastto the level of available experimental kinetic data. In contrast tosome oil and gas simulators, the simulator has flow descriptionranging from diffusion to advection and dispersion25,43−48 and,as such, is able to handle flow in all regions of the reservoir,including the low permeability regimes of hydrate filled regions.The mathematical equations for the system are highly
nonlinear and solved numerically. The numerical approach canbe viewed as divided into two parts: spatial and temporaldiscretizations. The finite element method is used for thespatial discretization, while finite differences are used for thetemporal discretization. The Newton−Raphson method isadopted for the iterative scheme.25,43−48
A brief overview of independent variables, constitutiveequations, and equilibrium restrictions are given in Tables2−4, respectively, while the details are presented else-where.43−47
3.2.1. Independent Variables. The governing equations fornon-isothermal multiphase flow of liquid and gas throughporous deformable saline media have been established.Variables and corresponding equations are tabulated in Table 2.
3.2.2. Constitutive Equations and Equilibrium Restrictions.Associated with this formulation, there is a set of necessaryconstitutive and equilibrium laws. Tables 3 and 4 present a
Figure 5. RCB solves the integrated equations sequentially in one timestep.
Figure 6. Details of the simulation model.
Table 5. Chemical Species in the Media
phase species
aqueous H2O, HCO3−, OH−, H+, CO2 (aq), CO3
2−
gas CO2 (g)
Table 6. Initial and Boundary Conditions
boundary variable value
pressure at the top (MPa) 1.0pressure at the bottom (MPa) 4.0temp. at the top (K) 273.35temp. at the bottom (K) 284.15initial mean stress at the top (MPa) 2.33initial mean stress at the top (MPa) 8.76CO2 injection pressure (MPa) 4.0
Table 7. Reservoir and Hydrate Properties
property value
Young’s modulus (GPa) 0.5Poisson’s ratio 0.25zero stress porosity in aquifer 0.3zero stress permeability in aquifer (m2) 1.0 × 10−13
zero stress porosity in cap rock 0.03zero stress permeability in cap rock (m2) 1.0 × 10−17
zero stress porosity in fracture 0.5zero stress permeability in fracture (m2) 1.0 × 10−10
Van Genuchten’s gas entry pressure (at zero stress) (kPa) 196Van Genuchten’s exponent 0.457thermal conductivity of dry medium (W/m K) 0.5thermal conductivity of saturated medium (W/m K) 3.1solid phase density (kg/m3) 2163rock specific heat (J/kg K) 874CO2 hydrate molecular weight (g/mol) 147.5CO2 hydrate density (kg/m3) 1100CO2 hydrate specific heat (J/kg K) 1376CO2 hydrate reaction enthalpy (J/mol) 51858CO2 hydrate kinetic formation rate constant (mol/Pa m2 s) 1.441 × 10−12
CO2 hydrate kinetic dissociation rate constant(mol/Pa m2 s)
1.441 × 10−16
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summary of the constitutive laws and equilibrium restrictionsthat should be incorporated in the general formulation. Thedependent variables that are computed using each law are alsoincluded.RCB is a coupling of a reactive transport code Retraso with a
code for simulation of multiphase flow of material and heat,namely, CodeBright. CodeBright45 contains an implicitalgorithm for solution of material flow, heat-flow, andgeomechanical model equations.44−46 The Retraso extensionof CodeBright involves an explicit algorithm for updating thegeochemistry,25,48 as shown in the Figure 5. This new coupledtool RCB is capable of handling both saturated and unsaturatedflow, heat transport, and reactive transport in both liquid andgas. It is a user-friendly code for flow, heat, geomechanics, andgeochemistry calculations. It offers the possibility to justcompute the chosen unknowns of user’s interest such as hydro-mechanical, hydro-chemical-mechanical, hydro-thermal, hydro-thermal-chemical-mechanical, thermo-mechanical, etc.It can handle problems in one, two, and three dimensions.43
An important advantage of RCB is the implicit evaluation ofgeomechanical dynamics. According to the Figure 5, flow, heat,and geomechanics are solved initially in CodeBright modulethrough Newton−Raphson iteration, and then, the flowproperties are updated according to the effects of reactivetransport on porosity and salinity in a separate Newton−Raphson procedure but for the same time step.48 This makes itpossible to study the implications of fast kinetic reactions suchas hydrate formation or dissociation more realistically. Hydrateformation happens through nucleation and growth as physicallywell-defined processes and an additional stage called induction,which may be interpreted as the onset of massive growth.Hydrate formations in systems with little impact of surfaces andhydrodynamics can have very long induction times but ourexperience from hydrate formation in porous media shows littleor no induction time.27
3.2.3. Modifications in RCB. The Retraso part of the codehas a built-in state of the art geochemical solver and, inaddition, capabilities of treating aqueous complexiation(including redox reactions) and adsorption. The density ofCO2 plumes, which accumulate under traps of low permeabilityshale or soft clay, depends on depth and local temperature ineach unique storage scenario. The difference in density and thedensity of the groundwater results in a buoyancy force forpenetration of CO2 into the cap rock. Even if the solubility ofwater into CO2 is small, dissolution of water into CO2 may alsolead to out-drying of clay. Mineral reactions between CO2 andshale minerals are additional effects that eventually may lead toembrittlement. Linear geomechanics may not be appropriatefor these effects. Clay is expected to exhibit elastic nonlinearcontributions to the geomechanical properties. Different typesof nonlinear models are already implemented in the Code-Bright part of the code, and the structure of the code makes iteasy to implement new models derived from theory and/orexperiments. The current version of RCB has been extendedfrom ideal gas into handling of CO2 according to the SRKequation of state.49 This equation of state is used for densitycalculations as well as the necessary calculations of fugacities ofthe CO2 phase, as needed in the calculation of dissolution ofthe CO2 into the groundwater.49−51 The Newton−Raphsonmethod used in the original RCB has been also modified toimprove the convergence of the numerical solution whileincreasing the range of working pressure in the system.50,51
The following modifications are additional in this study. Toaccount for nonequilibrium thermodynamics of hydrate and todetermine the kinetic rates of different competing scenarios ineach node and each time step according to the temperature,pressure, and composition of the system, CO2 and CH4hydrates are added into the simulator as pseudomineralcomponents with kinetic approach for hydrate formation anddissociation. Hydrate formation and dissociation can directly beobserved through porosity changes in the specific areas of theporous media. Porosity reduction indicates hydrate formationand porosity increase indicates hydrate dissociation. Theconsequences on the intrinsic permeability are implementedthrough Kozeny’s model, as has been used in the codeoriginally.43 The code also supports several models forcalculating relative permeability, which are described in detailelsewhere.43 The surface area of hydrate is calculated in asimilar approach to ordinary minerals assuming sphericalparticles.43 Temperature, pressure, and concentrations arethree factors that influence hydrate formation or dissociation.The kinetic rate used in this study is calculated fromextrapolated results of phase field theory simula-tions.26,27,29,32,33,37−40 In the next stage, it will be replaced bya thermodynamic code, which is already in the final stages, toaccount for all different competing reactions in a non-equilibrium approach. Hydrate reaction is considered basedon the eq 12, which is good enough for illustration purposes.For more rigorous analysis the numbers in this equation shouldcontain empty cavities, but there are many model parameterswhich are even more uncertain. So, for a model case, thesenumbers are used. The energy balance for the gas phase ismodified from ideal gas to real gas according to the eq 13 usingSRK equation of state to calculate fugacity coefficient andderivatives. The energy balance for solid phase is also modifiedaccording to hydrate reaction enthalpy of Table 7.
+ ↔8(CH /CO ) 46H O Hydrate4 2 2 (12)
ϕ= −H H RTT
d(ln )d
(id.gas) 2(13)
3.3. Model Description. A system of CO2 injection intothe reservoir presented in Figure 6 is selected to show theperformance of the simulator. It consists of a 2D model of 300m × 1000 m located at the depth of 100 m. The pressuregradient in the reservoir is 1 MPa/100 m, and the temperaturegradient is 3.6 °C/100 m. The model is discretized into 1500elements with dimensions of 10 m × 20 m. The cap rock startsat the depth of 270 m and goes down to 310 m. There is afracture in the middle of the cap rock. CO2 is injected atconstant pressure of 4 MPa at the specified location on Figure6. Equation 14 is used to describe CO2 hydrate equilibriumconditions in the model. This is based on the model developedby Kvamme and Tanaka,1 while the SRK equation of state isused to calculate the fugacity of the liquid phase.
In this equation, P is calculated in MPa and T is in Kelvin.
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Figure 7. Thermodynamically suitable area for CO2 hydrate formation according to initial conditions of the reservoir.
Figure 8. Gas pressure (MPa) after 200 and 615 days.
Figure 9. Liquid phase flux (m/s) after 200 and 615 days.
Figure 10. Gas phase flux (m/s) after 200 and 615 days.
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Tables 5−7 present the information regarding availablespecies in different phases, initial and boundary conditions, andmaterial properties.
4. RESULTS AND DISCUSSIONStorage of carbon dioxide in reservoirs that contain zones ofcarbon dioxide hydrate stability with respect to temperatureand pressure results in special flow paths, since the hydratesformed are not in unconditional equilibrium. This implies thatthe hydrate may dissociate again toward under saturated water
in regions of significant exchange of aqueous phase due to flow.In regions of lower flow rates, more stationary situations oflocal saturations may result in hydrate formation anddissociation, which maintains very low permeability and inducesconstraints of directions of flow. Relevant aquifer storagereservoirs with hydrate zones are available offshore Norway. Anexample is the Snøhvit injection project offshore north Norway.The basic data of location, seafloor depth, and seafloortemperature indicates that the upper few hundred meters areinside formation region for CO2 hydrate. Very limited detailed
Figure 11. Porosity after 615 days. White area refers to porosity of 0.03 in cap rock.
Figure 12. Comparison of gas phase flux with hydrate existence (top) and without hydrate existence (bottom).
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data are available in the open literature on this reservoir, and anexample is used to illustrate the new hydrate reservoirsimulator.The simulation results processed by RCB are visualized using
GiD visual window.52 Figure 7 presents the suitablethermodynamic conditions for CO2 hydrate formationaccording to initial conditions of the reservoir.Figures 8−10 show gas pressure, liquid phase flux, and gas
phase flux, respectively, after 200 and 615 days. Pressureincrease can be observed below the cap rock, in Figure 8, whichis parallel to gas phase flux pattern in Figure 10. Gas phase flowupward is limited due to cap rock low porosity and permeabilityand stretches toward the fracture. Figure 11 shows porositychange after 615 days. It is presented in three different scalesfor better illustration. Porosity change due to hydrate formationprocess becomes visible after around 550 days which is anindication that gas flow has reached the upper aquifer wherethermodynamic conditions are suitable for hydrate formation.Formation of hydrate results in increase of hydrate concen-tration as a pseudomineral component in the system, which willbe added to total concentration of solid phase leading to changeof porosity. It should be mentioned that porosity here refers tothe void volume available to reservoir fluids.Figure 12 presents a comparison between gas phase flux
pattern with and without presence of hydrate. In the top panelof the figure, hydrate formation results in porosity change,while in the bottom panel, hydrate formation is excluded fromsimulation. The more limited expansion of gas phase above thefracture in the top panel compared to the bottom one can beexplained by presence of hydrate.
5. CONCLUSION
The nonequilibrium nature of hydrate in porous media hasbeen investigated in this paper. On the basis of the statistical−mechanical model of Kvamme and Tanaka,1 differences inproperties of hydrates formed from different phases isillustrated and discussed. All possible phase transitionsinvolving hydrate are formulated and corresponding simplifiedkinetic models are derived on the basis of phase field theorysimulations. The resulting set of phase transition kinetic modelsis implemented into a reservoir simulator as pseudoreactions.The platform used to develop this new hydrate simulator is theRetrasoCodeBright reactive transport simulator. The imple-mentation is illustrated by a carbon dioxide injection exampleinto a formation containing hydrate stability regions.
NotesThe authors declare no competing financial interest.
■ ACKNOWLEDGMENTS
We acknowledge the grant and support from Research Councilof Norway through the following projects: SSC-Ramore,“Subsurface storage of CO2Risk assessment, monitoringand remediation”, Project No. 178008/I30. FME-SUCCESS,Project No. 804831. PETROMAKS, “CO2 injection for extraproduction”, Research Council of Norway, Project No. 801445.
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