Article pubs.acs.org/EF

Nonequilibrium Modeling of Hydrate Dynamics in Reservoir Mohammad 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 competing phase transitions involving hydrate. Calculations of hydrate phase transitions based on a modiﬁed statistical−mechanical model for hydrate (Kvamme, B.; Tanaka, H. J. Phys. Chem. 1995, 99, 7114−7119) are used to illustrate diﬀerences in properties of hydrates formed from diﬀerent phases. In more general terms, hydrates in porous media are discussed in terms of the Gibbs phase rule. It is argued that phase transitions involving hydrates in porous media can rarely reach any state of equilibrium due to situations of over speciﬁed systems with reference to requirements for equilibrium. As a consequence of this, a strategy for nonequilibrium description of hydrates in reservoirs is proposed. This involves the formulation of kinetic expressions for all possible hydrate formations and dissociations as competing pseudoreactions. This involves hydrate formation on a water/carbon dioxide interface, from water solution and from carbon dioxide adsorbed on mineral surfaces, as well as all diﬀerent possible hydrate dissociation possibilities. The basic idea is that the direction of free energy minimum, under constraints of mass and heat transport, will control the progress of phase transitions in nonequilibrium systems. A new hydrate reservoir simulator based on this concept is introduced in this study. The reservoir simulator is developed from a platform previously developed for carbon dioxide storage in aquifers, RetrasoCodeBright (RCB). The main tools for generating kinetic models have been phase ﬁeld theory simulations, with thermodynamic properties derived from molecular modeling. The detailed results from these types of simulations provides information on the relative impact of mass transport, heat transport, and thermodynamics of the phase transition, which enable qualiﬁed simpliﬁcations for implementation into RCB. The primary step was to study the eﬀect of hydrate growth or dissociation with a single kinetic rate on the mechanical properties of the reservoir. Details of the simulator and numerical algorithms are discussed, and relevant examples are shown.

1. INTRODUCTION The international focus on natural gas hydrates is increasing for a number of diﬀerent reasons. Unlike conventional fossil fuels, hydrates are distributed worldwide and can be a source of energy for countries that have little or no conventional fossil fuel resources. Japan is a good example; test production at the ﬁrst oﬀshore pilot plant will take place in 2012, and full production is planned from 2018. Production of permafrost hydrates has been going on since 1970 in Russia. The worldwide amount of hydrates is uncertain, but the United States Geological Survey (USGS) estimated 10 years ago that it might amount to more than twice all known conventional fossil fuels. Hydrates in porous media are generally not stable, as will be discussed in more detail later in this paper. This implies that, basically, the demands for trapping/sealing formations to keep hydrate in place are similar to oil and gas sealings (cap rock). In the case of oﬀshore hydrates, deﬁciencies in sealing above the hydrate layers in the form of fractures and faults will lead to contact between hydrate and groundwater. The result is dissociating hydrate and methane seepage to the ocean above. Since methane is a more aggressive greenhouse gas than CO2, the global methane ﬂuxes from hydrates and free methane is a global environmental concern. Besides, in the long run, these leakage situations may lead to dement instabilities and, in the worst case, cause landslides. In some permafrost regions, 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 are described here, which are believed to be more feasible, © 2012 American Chemical Society

speciﬁcally with respect to economy. The depressurization method in which the hydrate stability condition is shifted by pressure reduction resulting in hydrate dissociation and release of methane. It is currently considered as the most feasible process considering expenses and production rate and has been investigated by many research groups through simulation studies. Thermal stimulation is another method, which is based on moving out from the stability region by temperature increase. It is considered to be costly because of the huge amount of energy waste to the surroundings. The third method is to use inhibitors such as methanol or brine to shift the equilibrium curve and dissociate hydrate, which is also costly. The fourth method is injection of CO2 into the methane hydrate reservoirs. CO2-hydrate is more stable than CH4hydrate over substantial regions of pressure and temperature. CO2 and CH4 will both form hydrate structure I, which consist of a ratio of 3 to 1 of large cavities (24 water molecules surrounding enclathrated molecule) to small cavities (20 water). Under normal conditions, CO2 will only ﬁll the large cavities and the mole fraction of CO2 is close to 0.12, while CH4 hydrate has a mole fraction of CH4 close to 0.14. A mixed hydrate in which CO2 occupies the large cavities and CH4 occupies the small cavities will be more stable than methane hydrates over all regions of temperature and pressure where either hydrate can exist. Injection of CO2 into a natural gas hydrate reservoir can release the natural gas by direct Received: February 28, 2012 Revised: May 11, 2012 Published: May 14, 2012 3564

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conversion of the in situ hydrate, which is a solid state transformation and slow.2,3 Hydrate cannot attach directly to mineral surfaces because of incompatibility with hydrogen bonding and partial charges on atoms in the mineral surfaces. Any hydrates in porous media will therefore be surrounded by ﬂuids that separate them from the mineral surfaces. Injected CO2 may therefore also form hydrates directly with free water in the pore volume. Released heat will contribute to dissociation of in situ natural gas hydrate. This heat release is substantially higher than the heat release due to solid state reformation discussed by Kvamme et al.4 The absolute smallest distance separating hydrate from the mineral surfaces is in the order of 3 nm, based on molecular dynamics simulations. The permeability would be extremely small if natural hydrates occurred with such a high pore ﬁlling. Reported ﬁlling fractions for natural gas hydrates, as well as synthetic laboratory produced hydrates in sediments, are normally substantially lower. Most reported natural gas hydrate reservoirs have below 50% of pore volume ﬁlled with hydrate. If the remaining volume is liquid water, then solid state conversion of the CH4 hydrate 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 mechanism involved in combined storage of CO2 in hydrate form and production of natural gas hydrates. The heat released from the formation of new CO2 hydrate will assist in dissociating the in situ CH4 hydrate. During more than two decades, diﬀerent research groups worldwide have applied diﬀerent theoretical modeling and simulation approaches to studies of methane production from hydrate reservoirs. The progress has been considerable, but there are still many uncertainties and unresolved challenges in the physical description of these complex systems. A limited review of past modeling eﬀorts of hydrate exploration is given in this section. Special emphasis has been on studies that have resulted in new models and/or new simulator for simulation of methane production from hydrates. Holder and colleagues5 considered a system of stratiﬁed hydrate and gas surrounded by impermeable layer from bottom and top to build a 3D model for studying hydrate dissociation eﬀect on gas production due to depressurization. Several assumptions were made to simplify the calculations. They considered the dissociation to happen only at the interface between hydrate and gas phase, and only conduction was considered to ﬁnd temperature distribution in the gas phase. They concluded that hydrate can contribute signiﬁcantly to gas production from such reservoirs.5 Burshears et al.6 developed a two-phase, 3D numerical model to study gas production from a system 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 and equilibrium conditions in the gas−hydrate interface. Water ﬂow was present in their model but only heat conduction was considered.6 Yousif et al.7 developed a one-dimensional model to simulate isothermal depressurization of hydrate in Berea sandstone samples. The model considered three phases of water, gas, and hydrate and used the kinetics model of Kim and Bishnoi8 for dissociation of hydrate. It also considered the water ﬂow due to hydrate dissociation, which according to experimental and numerical results was considerable. The variations in porosity and permeability of the gas phase were taken into account. Finally, they validated their model with experimental data.7 Xu and Ruppel9 developed an analytical

formulation to solve the coupled momentum, mass, and energy equations for the gas hydrate system consisting of two components (water and methane gas) and three phases (gas hydrate, free gas, water, and dissolved methane). They made several assumptions such as ignoring capillary eﬀects and ignoring kinetics of the phase transition between hydrate and aqueous phase.9 Swinkels and Drenth10 used an in-house 3D thermal reservoir simulator and modiﬁed the PVT tool to study hydrate production scenarios, as well as heat ﬂow and compaction in the reservoir and hydrate cap. They represented the reservoir ﬂuid by a gaseous phase, a hydrate phase, and an aqueous phase. Their system consisted of three components, including two hydrocarbons and a water component. Heat was considered as an extra component for all phases internally in the simulator. The Joule−Thomson cooling eﬀect was also considered automatically.10 Davie and Buﬀet11 proposed a numerical model for calculation of gas hydrate volume and distribution in marine sediments. They considered the organic material from sedimentation as the main source of carbon supply to the hydrate stable region. The rate of sedimentation, the quantity and quality of organic material, and biological productivity were considered as key parameters in this model. Hydrate formation and dissociation were controlled by thermodynamic conditions and driving force from equilibrium concentration. A ﬁrst order model with a large rate constant was used to account for the reaction rate. They compared their results with observations from practical ﬁeld. Their model was dependent on the available data of sedimentation rate and organic content.11 Goel et al.12 developed a single phase ﬂow model to predict the performance of an in situ gas hydrate. The gas withdrawal rate was assumed to be constant, and the reservoir was assumed to be inﬁnite. It used an nth order kinetics rate coupled with gas ﬂow in the porous media. The model was simple and had several limitations due to simpliﬁcations.12 Moridis13 has proposed a new module for TOUGH2 simulator named EOSHYDR2. TOUGH2 is a 3D, multicomponent, multiphase ﬂow simulator for subsurface purposes. EOSHYDR2 is designed to model hydrate behavior in both sediments and laboratory conditions. It is able to consider up to four phases of gas, liquid, ice and hydrate, and up to nine components of water, CH4-hydrate, CH4 as native and from hydrate dissociation, a second native and dissociated hydrocarbon, salt, water-soluble inhibitors, and heat as a pseudocomponent. It includes both equilibrium and kinetic models for hydrate formation and dissociation. It uses hydrate reaction model of Kim and Bishnoi8 for kinetic studies. He studied four test cases of CH4 production and concluded that both depressurization and thermal stimulation can produce a substantial amount of hydrate, and he suggested that the combination of both can be desirable. He used just the equilibrium approach because of a lack of enough suitable data necessary for the parameters of the kinetic model, while mentioning that slower processes such as depressurization follow kinetic dissociation. Finally, he discussed several uncertainties mostly due to lack of knowledge in this area of research.13 Later on, using the same module, Kowalsky and Moridis14 made a comparison study between kinetic and equilibrium approach and concluded that based on the accuracy of the available kinetic models; this approach is important for simulation of short-term and core scale systems while equilibrium approach can be used for large scale simulations of production processes. Hong et al.15,16 presented a 2D cylindrical model to study gas production from hydrate 3565

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the heat of reaction, eﬀects of water and gas permeabilities, and eﬀective porosity with possibility of variation with time. They found that the production process was sensitive with respect to temperature, pressure, and core permeability. They compared their results with experimental data as well.22 Liu et al.23 developed a 1D model to study hydrate dissociation by depressurization in porous media. They used a moving front to separate the hydrate zone from gas zone. They considered conductive and convective heat transfer and mass transfer in gas and hydrate zones and energy balance at moving front. They considered equilibrium at the front and concluded that the assumption of stationary water phase results in overprediction of dissociation front location and underprediction of gas production in the well. They also found that reservoir permeability and well pressure can highly aﬀect the production rate.23 Gamwo and Liu24 have presented a detailed theoretical description of the open source reservoir hydrate simulator HydrateResSim developed previously by Lawrence Berkeley National Laboratory (LBNL). They have also applied it to a system of three components (methane, water, and hydrate) and four phases (aqueous, gas, hydrate, and ice). Darcy’s law is used for multiphase ﬂow of mass in porous media and local thermal equilibrium is considered in the code. It considers both kinetic and equilibrium approaches, using Kim and Bishnoi8 as the kinetic model of hydrate dissociation. The studied model has been validated using the TOUGH-HYDRATE simulator. They concluded that HydrateResSim is a suitable freeware, open source code for simulating methane hydrate behavior in the reservoir. They found out that equilibrium approach usually overpredicts the hydrate dissociation compared to kinetic approach.24 From the presented literature review it is clear that the hydrate dissociation process in the reservoir is mostly treated as an equilibrium reaction and in fewer cases as kinetics. The majority of kinetic approaches are based on the kinetic model of Kim and Bishnoi,8 developed according to laboratory experiments as follows:

reservoirs. They used both analytical and numerical approaches. The numerical model considered the equations for gas and water two-phase ﬂow, conductive and convective heat ﬂow, and treated hydrate decomposition kinetically using the Kim and Bishnoi8 model. In the analytical approach, they did not consider the ﬂuid ﬂow. They studied the eﬀect of hydrate zone on the gas production from gas layer and also the importance of the intrinsic kinetics of hydrate decomposition with respect to equilibrium assumptions. They concluded that the kinetic equation was aﬀected only when it was considered 5 orders of magnitude lower than the values found by Kim and Bishnoi8 in a PVT cell.15,16 Xu17 has modeled dynamic gas hydrate systems considering two main elements. The ﬁrst element provides a model to account for dynamic phase transitions of marine gas hydrates by considering the eﬀects of ﬂuid pressure, temperature, and salinity changes on hydrate stability, solubility, and ﬂuid densities and enthalpies. This allows describing coexistence of three phases, namely hydrate, gas, and liquid solution at equilibrium. The model consists of three components (water, gas, and salt) and four phases (free gas, liquid solution, gas hydrate, and solid halite). It considers thermodynamic equilibrium for and among individual phases. The second element of the model consists of the numerical tool to solve for ﬂuid ﬂow and transport equations in porous media. It considers four components and up to ﬁve phases. The response of the model to the pressure drop and temperature rise scenarios at seaﬂoor are studied.17 Ahmadi et al.18 developed a 1D model to study methane production from hydrate dissociation in a conﬁned reservoir using depressurization method. They considered heat of dissociation, conduction, and convection in both gas and hydrate phase. They considered equilibrium conditions at the dissociation front and neglected water ﬂow in the reservoir and the Joule−Thomson cooling eﬀect.18 Sun and Mohanty19 developed a 3D simulator using the Kim and Bishnoi8 kinetic model to study formation and dissociation of hydrate 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 ice melting are considered, assuming equilibrium phase transitions. Mass transport including ﬂuid ﬂow and molecular diﬀusion plus heat transport including conduction and convection are solved implicitly.19 Phirani et al.20 upgraded the simulator to account for CO2 as a new component and CO2 hydrate as a new phase. The purpose was to study CO2 ﬂooding of methane hydrate bearing sediments. They used the model of Kim and Bishnoi8 for kinetics of dissociation and constant rates for hydrate formation.20 Uddin21 developed a general kinetic model of hydrate formation and decomposition based on Kim and Bishnoi8 model to study CO2 sequestration in methane hydrate reservoirs along with CH4 production. The model consisted of ﬁve components of water in aqueous phase, CH4 and CO2 in gas phase, and CO2-hydrate and CH4-hydrate in solid phase. The gas hydrate model was coupled with a compositional thermal reservoir simulator (CMG STARS) to study dynamics of hydrate formation and decomposition in the reservoir. It was concluded that the eﬀect of kinetic rate constant on hydrate decomposition was signiﬁcant in case it was lowered a few orders of magnitude. Reservoir permeability has also a great impact on the decomposition rate.21 Nazridoust and Ahmadi22 developed a computational hydrate module for FLUENT code. It provides the possibility to study hydrate dissociation for complex 3D geometries. They used the kinetics rate of dissociation proposed by Kim and Bishnoi,8 and they included

−

dnH = kdA s(fe − f ) dt

(1)

If the phase transition as a whole is dominated by the mass transport, the form of the equation may be adequate enough. Therefore, we have also used similar form of eq 1 as a basis for transferring results from advanced theories (see section 3) into reservoir simulator use. However, then, it is acceptable only after veriﬁcation (rate versus square root of time indicate Fick’s law) of the mass transport dominance. The use of eq 1 for ﬁtting laboratory experiments from PVT cell and then using these data for a reservoir situation might be more questionable, since it has never been proven that these data are transferable to any other experimental apparatus or real situation of hydrate phase transitions in nature. This also raises the question of how to evaluate the impact of kinetics if the rate equation is not appropriate. For pressure reduction as production method one could argue that the heat transport might have a strong impact. Some even use the heat transport rate model alone to model the kinetic rate in this case. The diﬀerent heat transport models applied are normally not documented well enough in terms of theory and/or parameters to judge whether they are appropriate or not. Heat transport to a system undergoing dynamically a phase transition that involves ﬂow is fairly complex, so it is unclear whether any reasonable model exists. Another critical question related to analysis of the impact of 3566

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kinetics is the reference case. As the system by itself cannot reach equilibrium, the P, T projections are just two dimensions of a general variable set consisting of concentration in all surrounding phases, including impact of minerals. Hydrate will dissociate toward undersaturated water, regardless of whether the system is inside stability with reference to P and T or not. Also, it will sublimate toward undersaturated gas. With hydrate saturations between 18% and 50% (which are typical ranges for reservoirs considered as “rich” enough), induced ﬂow will exchange ﬂuids surrounding the hydrate, and the impact of these ﬂuid changes on hydrate dissociation have never been evaluated. In summary, there are many uncertain issues around the kinetic models used as a basis for concluding that kinetic eﬀects are not important, as well as clearly the assumption on the reference system (read: P, T deﬁnes equilibrium regardless of concentration induced nonequilibrium). The fact that hydrates in porous media are in nonequilibrium does require kinetic models in descriptions of all relevant competing phase transitions (all phase transitions that have negative free energy changes of suﬃcient magnitude to overcome nucleation barriers). There might, however, be a time-dependent local situation where one process, or a few processes, dominates and is fast enough to qualify for some quasi-equilibrium approximation. However, those are exceptions and do not need to be in the P, T projection of the multidimensional thermodynamic dependency of the system. In this paper, a diﬀerent approach according to nonequilibrium nature of hydrate phase transitions in the reservoir will be presented. A new reservoir hydrate simulator will be introduced, which is developed on a former reactive transport reservoir simulator namely RetrasoCodeBright (RCB).25 The module is designed so that it can easily work according to the nonequilibrium thermodynamic package that is being developed in this group. The reactive nature of the code gives the possibility to distinguish between diﬀerent phase transition scenarios and treat them on a competitive basis. Therefore, it is possible to evaluate the amount of components that are used to form hydrate or produced from hydrate dissociation and include these results in the overall heat and mass balance equations. In addition, it gives the possibility to visualize the eﬀect of hydrate formation or dissociation on the solid phase properties in the form of direct changes in porosity and permeability. At this stage, kinetic models of hydrate formation and dissociation from phase ﬁeld theory simulations26,27 are used to examine the performance of the module through example cases.

Table 1. Potential Phase Transition Scenarios for a System of Hydrate in Reservoir δ

1

−1

hydrate

2

−1

hydrate

3

−1

hydrate

4

+1

5

+1

6

+1

7

+1

gas + liquid water surface reformation aqueous phase adsorbed

driving force outside stability in terms of P and/ or T sublimation (gas under saturated with water) outside water under saturated with methane hydrate more stable than surrounding ﬂuids nonuniform hydrate rearranges due to mass limitations liquid super saturated with methane with reference to hydrate water and methane adsorbed on mineral surfaces

ﬁnal phase(s) gas, liquid water gas liquid water hydrate hydrate hydrate hydrate

Figure 1. Water chemical potential in empty SI (dash), empty SII (dash-dot), water as ice or liquid water (solid).

2. THEORY When a clathrate hydrate comes into contact with an aqueous solution containing its own guest molecules (CH4 or CO2), the number of the degrees of freedom available to the new system of three phases (aqueous, gas, hydrate) will be decreased compared to the two-phase system (aqueous, gas) as a result of the Gibbs phase rule. F=N−π+2

initial phase(s)

i

Figure 2. Predicted hydrate equilibrium (solid line), as estimated using the parameters from Kvamme and Tanaka1 for empty hydrate chemical potential (SI), liquid water chemical potential, and the harmonic oscillator method for evaluation of the cavity partition function. Fugacity coeﬃcients are estimated using eq 2 from Duan et al.30 Stars are measured hydrate equilibrium data from Deaton and Frost34 and Unruh and Katz.35

(2)

In this equation, F is the degree of freedom, N is the number of components, and π is the number of phases. For the twophase system comprising two components, water and gas, there will be two degrees of freedom while, in a three-phase system with hydrate as an additional phase, there will be only one degree of freedom. The Gibbs phase rule is simply conservation of mass under the constraints of equilibrium between coexisting

phases. The number of independent thermodynamic variables speciﬁed for the system must equal the number of degrees of 3567

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subsequently will dissociate in contact with groundwater from top as a result of undersaturation with CO2. As such, a kind of stationary situation of formation of new CO2 hydrate from below and dissociation from top is expected. See also Kvamme28 for examples of saturation limits of aqueous solutions toward hydrate as function of temperature and pressure. The statements about equilibrium approach in Kowalsky and Moridis14 (on page 1851 of the paper) that the system of four phases always exists at equilibrium is somewhat confusing and cannot describe a realistic hydrate reservoir, since it could happen only in a unique point consisting of a unique temperature, pressure, and composition. In hydrate systems, we are usually dealing with a range of temperatures, pressures, and compositions that can be explained only if the system is considered nonequilibrium. The typical case during the production of hydrate is three phases (gas, water, hydrate) and two components, which make the system overdetermined because temperature and pressure are deﬁned locally in every point of a real reservoir and every grid point of a reservoir model. If the system crosses the ice formation condition due to cooling, degrees of freedom will be 0 and the system even more overdetermined. Figure 1 in Kowalsky and Moridis14 has no other meaning than a possible limit to kinetics in a two thermodynamic variable projection of the nonequilibrium system, which is generally characterized by temperature, pressure, and concentrations in all coexisting phases as thermodynamic variables. Examples of two hydrate phase transitions that are thus excluded are hydrate formation from aqueous solution and hydrate dissociation toward undersaturated water. Keeping this in mind, also the comparison between equilibrium and kinetics is out of thermodynamic balance since the reference (read: equilibrium) situation is not correct. However, an additional problem is that the kinetic model of Kim and Bishnoi8 is empirical and does not contain necessary theoretical basis to be transferable outside the experimental apparatus and experimental conditions for which the model was developed. Given the applied stirring rates (and corresponding hydrodynamically induced stresses) in the experimental cell, the kinetic rates from the model with the original mode parameters are likely to be very diﬀerent from the hydrodynamic stress involved in reservoir ﬂow situations related to production. A system that cannot reach equilibrium will, however, always tend toward the minimum free energy as a consequence of the combined ﬁrst and second law and when the hydrate is inside its pressure−temperature stability region, this means that its free energy is lower than that of the aqueous solution.29 In case of hydrate formation and dissociation in the reservoir, the system will be even more complex because hydrate can form from or dissociate toward phases with diﬀerent free energies, which will produce diﬀerent phases. Consequently, the degree of freedom will decrease further and there will be no equilibrium condition and competing phase transition reactions of hydrate formation, dissociation, and reformation among diﬀerent phases will rule the system. The main factor in such a system is therefore the minimum free energy of the system and kinetics of competing reactions. Assume, for example, a system of water, methane, and hydrate in porous media. According to this system, three phases of gas, aqueous, and hydrate are evident. Methane and water are considered as the only components available in the system. The degree of freedom for such a system is 1, while it can reduce to −1 if two additional phases of adsorbed CH4 and adsorbed water are also

Figure 3. Estimated water hydrate chemical potential (solid line) for the experimental conditions of Deaton and Frost34 and Unruh and Katz35 (corresponding to the stars in Figure 2). The dashed line is for temperatures along the isobar 61 bar.36 The dash-dot line is for temperatures along the isobar 101 bar.36

Figure 4. Estimated chemical potential of water in hydrate using the extended statistical−mechanical model for hydrate due to Kvamme and Tanaka1 with chemical potential for dissolved CO2 in aqueous phase (solid line). The dashed line is pure water liquid chemical potential1 disregarding pressure changes eﬀect on liquid water chemical potential.

freedom if the system should be able to reach equilibrium. As a result, when both temperature and pressure are speciﬁed, as is the case in local reservoir conditions, the system is overdetermined and will be unable to reach three-phase equilibrium. Despite this fact, it is quite common to ignore the concentration dependencies of the diﬀerent co-existing phases on phase transitions. A minimum requirement for a phase transition is negative free energy change, but in a nonequilibrium system that is not suﬃcient. All gradients in independent thermodynamic variables also have to point in the direction of lower free energy if the phase transition should go unconditionally. A practical example can be found in nature by all the seeps of hydrates that dissociates toward groundwater, which is undersaturated with methane. Injection of CO2 into aquifers which contain cold zoneswhere pressure and temperature conditions are suitable for CO 2 hydrate formationwill lead to the formation of hydrate ﬁlms, which 3568

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fraction of guest molecule k in the hydrate type i will be calculated according to eq 7.

considered. It is while temperature and pressure are provided in the reservoir and are not adjustable. All potential phase transition scenarios due to hydrate dissociation and reformation in 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. H, i H, i p ΔGi = δ[x wH, i(μwH, i − μwp ) + xCH (μCH − μCH )] 4 4

4

xki =

∑ νj ln(1 + ∑ hkji) j

k

(4)

where superscript H,0 denotes empty clathrate, νj is the fraction of cavity of type j per water molecules, and hikj is the canonical partition function for guest molecule of type k in cavity type j for hydrate of type i. Estimated chemical potentials1 for empty clathrates of structures I and II (SI and SII) are plotted in Figure 1, together with estimates for ice and liquid water chemical potential. The canonical partition functions can be expressed as hkji = exp( − β(μkji + Δgkjinclusion))

(5)

where μikj is the chemical potential of guest molecule k in cavity j for type i hydrate. The second term in the exponent is the free energy change of inclusion of the component k in cavity type j, which is independent of the speciﬁc hydrate type i. This latter term can be evaluated in the classical fashion by integrating the Boltzmann factors of the interactions between the waters in the lattice and the guest molecule over the cavity volume and normalized1 or alternatively evaluated using a harmonic oscillator approach. At equilibrium, the chemical potential for a guest in a cavity will be equal to the chemical potential for the same component in the equilibrium phase (gas, liquid, or adsorbed). In a nonequilibrium situation, these equilibrium chemical potentials can be used in a series expansion in temperature, pressure, and concentrations over to the real situation. In view of eqs 5 and 4, the chemical potentials for waters, as well as guest molecules, will be diﬀerent between diﬀerent hydrates, depending on how the hydrate was formed (read: from where the diﬀerent molecules in the hydrate come). Also, the mole fractions of water and guest molecules will be diﬀerent for each hydrate, since the ﬁlling fractions are given by θkji

=

(7)

∞ ∞ μCO (T , P , x ⃗) + RT ln γCO 2

2

⎡ ϕ (T , P ) ⎤ CO2 idealgas ⎥ = μCO (T , P) + RT ln⎢ 2 xCO2 ⎢⎣ ⎥⎦

(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 at 278.22 K from Valtz et al.31 and neglecting dissociation of dissolved CO2 into bicarbonate and carbonate ions, we obtain the following ﬁt: ∞ (278.22 K, x ⃗) μCO

∞ (278.22 K, x ⃗) + ln γCO 2 RT ∞ (278.22 K, x ⃗) μCO ∞ 2 (278.22 K, x ⃗) ≈ + ln γCO 2 RT 2

hkji 1 + ∑k hkji

i 1 + ∑j ∑k νθ j kj

For a given hydrate i to grow unconditionally, Gibbs free energy change according to eq 3 must be negative and all gradients in free energy change (temperature, pressure, and concentrations) must be negative. Practically, this implies for water, as one example, that the chemical potential of water in hydrate must be lowered from empty clathrate chemical potentials to equal or lower than the chemical potentials of ice or liquid water by means of guest stabilization through guest inclusion and chemical potentials of guests through eqs 4 and 5. The ﬁnal result is nonuniform hydrates due to all the diﬀerent hydrate phase transitions that are possible inside a porous material. The kinetics of diﬀerent phase transition is an implicit set of dynamic equations that couples the mass transport, heat transport, 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 of chemical potential for guest molecules, we apply eq 2 from Duan et al.30 in order to be able to stretch estimates beyond moderate gas condition for CO2. The results are plotted in Figures 2 and 3. Note that these estimates are pure prediction, using very simple models for the water−methane interactions from open literature, and no attempts have been made to empirically adjust the model or parameters. For the case of illustration, these estimates are considered to be fair enough, and we now apply the same model with chemical potential of CO2 from aqueous solution. Rearranging of the gas/liquid equilibrium criteria for pure CO2 in equilibrium with a CO2 ﬂuid phase gives eq 8:

(3)

In this equation, H represents hydrate phase, i represents any of the seven phase transition scenarios, p represents liquid, gas, and adsorbed phases, x represents composition, and μ represents chemical potential. δ is 1 for hydrate formation or reformation and −1 for dissociation. Keeping in mind that the phases are not in equilibrium, it might be useful to also visualize that the hydrates created along the pathways in Table 1 must also have diﬀerent ﬁllings and corresponding diﬀerent free energies, which, by deﬁnition, implies that each hydrate forming process results in a unique phase. The chemical potential for water in the hydrate of type i can be estimated using a modiﬁed version of the statistical−mechanical model1 μH,i w : μwH, i (T , P , x ⃗ H) = μwH,0 (T , P) −

i ∑j νθ j kj

2 ≈ −9.4149 + 12.67xCO2 − 5.3643xCO 2

(6)

(9)

in which we have neglected small Poynting corrections for pressure. Equation 9 can be rearranged into the total chemical potential of CO2 in aqueous solution at 278.22 K:

In eq 6, θikj is the ﬁlling fraction of guest molecule k in cavity type j for hydrate type i. Therefore, xki, which is the mole 3569

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To analyze this hydrate system, all impossible (ΔG > 0) and unlikely (|ΔG| < ε) cases must be disregarded. Taking into account the mass transport limited cases, all realistic phase transition scenarios will be determined. The purpose is to develop a hydrate reservoir simulator that has the possibility to consider the free energy changes of all phase transition scenarios and take into account the eﬀect of the realistic ones on the ﬂow and properties of the porous media through advanced kinetic models.

μCO (278.22 K, x ⃗) 2

2 = 278.22R[− 9.4149 + 12.67xCO2 − 5.3643xCO 2

+ ln(xCO2)]

(10)

Application of eq 10 together with free energies of inclusion from Kvamme and Tanaka1 in eqs 4 and 5 gives the plot in Figure 4 for chemical potentials of water in hydrate formed from CO2 in aqueous solution. These results can be compared to the chemical potentials of free water in the same plot to ﬁnd coexistence limit for hydrate forming from solution. However, more important is the diﬀerences in the free energy of the diﬀerent hydrates. Diﬀerences in hydrate water chemical potential between hydrate forming from ﬂuid CO2 as separate phase and from solution will reﬂect back in ﬁlling fractions (eqs 6 and 7) and also in the chemical potentials that enter the free energy 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 mass transport.32,33 The local progress of the diﬀerent phase transitions and resulting phase distributions at any given time is therefore a complex function of diﬀerences in thermodynamics and diﬀerences in mass transport associated with each possible phase transition. On a reservoir scale, the enthalpy changes associated with the diﬀerent hydrate phase transitions need to be transferred to the energy ﬂuxes in the reservoir simulator. Because the chemical potential for guest molecules in hydrate is expanded from equilibrium, a direct evaluation of the temperature dependency will be more complicated than numerical estimation using the thermodynamic relationship: μ ∂⎡⎣ RTk ⎤⎦ P,N⃗

∂T

=−

H̅k RT 2

3. SIMULATION 3.1. Kinetic Model. The results from phase ﬁeld theory simulations26,32,33,37−39 have been modiﬁed to be used in the Table 2. Equations and Independent Variables equation

variable name

equilibrium of stresses balance of liquid mass balance of gas mass balance of internal energy

displacements liquid pressure gas pressure temperature

Table 3. Constitutive Equations constitutive equation

variable name

Darcy’s Law Fick’s law Fourier’s law retention curve mechanical constitutive model phase density gas law

liquid and gas advective ﬂux vapor and gas non-advective ﬂux conductive heat ﬂux liquid phase degree of saturation stress tensor liquid density gas density

Table 4. Equilibrium Restrictions (11)

for any component k in a given phase. The line above H indicates partial molar enthalpy. For each component (water and hydrate formers), eq 11 gives the relationship between partial molar enthalpy and the chemical potentials, and the chemical potentials are directly estimated outside equilibrium, so numerical diﬀerentiation of the left side of equ 11 is the easiest way to get the partial molar enthalpy for water and all hydrate formers inside the hydrate in a general nonequilibrium situation. Summing the contributions to enthalpy from the partial molar enthalpies and mole fractions of all components in each phase provides the necessary enthalpy information for the convective terms of the energy balances in the reservoir simulator. In this paper, we will limit the number of competing phase transitions to a minimum and focus mainly on describing the simulator and the speciﬁc method for implementing nonequilibrium thermodynamics in the code. More complex scenarios will be studied in the future, as the code will develop and 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 the importance of a kinetic approach in the simulation of hydrate in reservoir and to remind that there are many unattended issues in this area that leave uncertainties in comparisons between equilibrium and kinetic approach. The rest of this paper introduces the simulation approach that provides the possibility to consider, as much as possible, the kinetic theories presented in the previous sections.

equilibrium restrictions

variable name

Henry’s law psychometric law

air dissolved mass fraction vapor mass fraction

kinetic model. Phase ﬁeld simulations are based on the minimization of Gibbs’ free energy on the constraint of heat and mass transport. Extensive research has been going on in the same group on application of phase ﬁeld theory in prediction of hydrate formation and dissociation kinetics, which is still in progress.4,26−29,32,33,37−40 In this study, the simulation results for hydrate formation kinetics from such studies have been extrapolated and used as the reaction rate constant of the Kim and Bishnoi8 kinetic model in the numerical tool. Assuming that the formation process happens from gas phase and dissociation is toward aqueous phase, the diﬀusivity of CO2 in liquid and gas phase would be important. According to the available data in literature, diﬀusivity of CO2 in liquid phase is around 4 orders of magnitude lower than gas phase.41,42 Therefore, dissociation rate has been chosen accordingly in this study. 3.2. Numerical Tool. For reservoirs where hydrate may form in some regions, hydrate formation involves roughly 10% volume increase compared to water. In parallel to this, there are chemical reactions that can, for example, supply extra CO2 through dissolution of carbonates in regions of low pH, and also regions of high pH where transported ions may precipitate and even extract CO2 from water and hydrate. The formed hydrate will not be in equilibrium due to Gibbs’ phase rule, as 3570

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Table 6. Initial and Boundary Conditions boundary variable

value

pressure at the top (MPa) pressure at the bottom (MPa) temp. at the top (K) temp. at the bottom (K) initial mean stress at the top (MPa) initial mean stress at the top (MPa) CO2 injection pressure (MPa)

1.0 4.0 273.35 284.15 2.33 8.76 4.0

Table 7. Reservoir and Hydrate Properties

Figure 5. RCB solves the integrated equations sequentially in one time step.

Table 5. Chemical Species in the Media phase

species H2O, HCO3−, OH−, H+, CO2 (aq), CO32− CO2 (g)

value

Young’s modulus (GPa) Poisson’s ratio zero stress porosity in aquifer zero stress permeability in aquifer (m2) zero stress porosity in cap rock zero stress permeability in cap rock (m2) zero stress porosity in fracture zero stress permeability in fracture (m2) Van Genuchten’s gas entry pressure (at zero stress) (kPa) Van Genuchten’s exponent thermal conductivity of dry medium (W/m K) thermal conductivity of saturated medium (W/m K) solid phase density (kg/m3) rock speciﬁc heat (J/kg K) CO2 hydrate molecular weight (g/mol) CO2 hydrate density (kg/m3) CO2 hydrate speciﬁc heat (J/kg K) CO2 hydrate reaction enthalpy (J/mol) CO2 hydrate kinetic formation rate constant (mol/Pa m2 s) CO2 hydrate kinetic dissociation rate constant (mol/Pa m2 s)

0.5 0.25 0.3 1.0 × 10−13 0.03 1.0 × 10−17 0.5 1.0 × 10−10 196 0.457 0.5 3.1 2163 874 147.5 1100 1376 51858 1.441 × 10−12 1.441 × 10−16

fairly fast (hours to days). For this purpose, a reactive transport reservoir simulator, RetrasoCodeBright (RCB),43 is used in this study and extended with hydrate phase transitions as “pseudoreactions”. RCB is capable of realistic modeling of the reaction rates for mineral dissolution and precipitation, at least to the level of available experimental kinetic data. In contrast to some oil and gas simulators, the simulator has ﬂow description ranging 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 can be viewed as divided into two parts: spatial and temporal discretizations. The ﬁnite element method is used for the spatial discretization, while ﬁnite diﬀerences are used for the temporal discretization. The Newton−Raphson method is adopted for the iterative scheme.25,43−48 A brief overview of independent variables, constitutive equations, and equilibrium restrictions are given in Tables 2−4, respectively, while the details are presented elsewhere.43−47 3.2.1. Independent Variables. The governing equations for non-isothermal multiphase ﬂow of liquid and gas through porous 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 necessary constitutive and equilibrium laws. Tables 3 and 4 present a

Figure 6. Details of the simulation model.

aqueous gas

property

mentioned in the previous sections. Neither will it attach to the mineral surfaces due to incompatibilities of hydrogen bonds in hydrate and interactions with atomic partial charges on the mineral surfaces. For this reason, there is a need for a logistic system that can handle competing processes of formation and dissociation. A reactive transport simulator can handle that. Implicit geomechanics is needed in order to handle competing phase transitions, which are very rapid (seconds) and are dynamically coupled to geochemical reactions, which can be 3571

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summary of the constitutive laws and equilibrium restrictions that should be incorporated in the general formulation. The dependent variables that are computed using each law are also included. 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 implicit algorithm for solution of material ﬂow, heat-ﬂow, and geomechanical model equations.44−46 The Retraso extension of CodeBright involves an explicit algorithm for updating the geochemistry,25,48 as shown in the Figure 5. This new coupled tool RCB is capable of handling both saturated and unsaturated ﬂow, heat transport, and reactive transport in both liquid and gas. It is a user-friendly code for ﬂow, heat, geomechanics, and geochemistry calculations. It oﬀers the possibility to just compute the chosen unknowns of user’s interest such as hydromechanical, hydro-chemical-mechanical, hydro-thermal, hydrothermal-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 of geomechanical dynamics. According to the Figure 5, ﬂow, heat, and geomechanics are solved initially in CodeBright module through Newton−Raphson iteration, and then, the ﬂow properties are updated according to the eﬀects of reactive transport on porosity and salinity in a separate Newton− Raphson procedure but for the same time step.48 This makes it possible to study the implications of fast kinetic reactions such as hydrate formation or dissociation more realistically. Hydrate formation happens through nucleation and growth as physically well-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 and hydrodynamics can have very long induction times but our experience from hydrate formation in porous media shows little or no induction time.27 3.2.3. Modiﬁcations in RCB. The Retraso part of the code has a built-in state of the art geochemical solver and, in addition, capabilities of treating aqueous complexiation (including redox reactions) and adsorption. The density of CO2 plumes, which accumulate under traps of low permeability shale or soft clay, depends on depth and local temperature in each unique storage scenario. The diﬀerence in density and the density of the groundwater results in a buoyancy force for penetration of CO2 into the cap rock. Even if the solubility of water into CO2 is small, dissolution of water into CO2 may also lead to out-drying of clay. Mineral reactions between CO2 and shale minerals are additional eﬀects that eventually may lead to embrittlement. Linear geomechanics may not be appropriate for these eﬀects. Clay is expected to exhibit elastic nonlinear contributions to the geomechanical properties. Diﬀerent types of nonlinear models are already implemented in the CodeBright part of the code, and the structure of the code makes it easy to implement new models derived from theory and/or experiments. The current version of RCB has been extended from ideal gas into handling of CO2 according to the SRK equation of state.49 This equation of state is used for density calculations as well as the necessary calculations of fugacities of the CO2 phase, as needed in the calculation of dissolution of the CO2 into the groundwater.49−51 The Newton−Raphson method used in the original RCB has been also modiﬁed to improve the convergence of the numerical solution while increasing the range of working pressure in the system.50,51

The following modiﬁcations are additional in this study. To account for nonequilibrium thermodynamics of hydrate and to determine the kinetic rates of diﬀerent competing scenarios in each node and each time step according to the temperature, pressure, and composition of the system, CO2 and CH4 hydrates are added into the simulator as pseudomineral components with kinetic approach for hydrate formation and dissociation. Hydrate formation and dissociation can directly be observed through porosity changes in the speciﬁc areas of the porous media. Porosity reduction indicates hydrate formation and porosity increase indicates hydrate dissociation. The consequences on the intrinsic permeability are implemented through Kozeny’s model, as has been used in the code originally.43 The code also supports several models for calculating relative permeability, which are described in detail elsewhere.43 The surface area of hydrate is calculated in a similar approach to ordinary minerals assuming spherical particles.43 Temperature, pressure, and concentrations are three factors that inﬂuence hydrate formation or dissociation. The kinetic rate used in this study is calculated from extrapolated results of phase ﬁeld theory simulations.26,27,29,32,33,37−40 In the next stage, it will be replaced by a thermodynamic code, which is already in the ﬁnal stages, to account for all diﬀerent competing reactions in a nonequilibrium approach. Hydrate reaction is considered based on the eq 12, which is good enough for illustration purposes. For more rigorous analysis the numbers in this equation should contain empty cavities, but there are many model parameters which are even more uncertain. So, for a model case, these numbers are used. The energy balance for the gas phase is modiﬁed from ideal gas to real gas according to the eq 13 using SRK equation of state to calculate fugacity coeﬃcient and derivatives. The energy balance for solid phase is also modiﬁed according to hydrate reaction enthalpy of Table 7. 8(CH4 /CO2 ) + 46H 2O ↔ Hydrate

H = H (id.gas) − RT 2

d(ln ϕ) dT

(12)

(13)

3.3. Model Description. A system of CO2 injection into the reservoir presented in Figure 6 is selected to show the performance of the simulator. It consists of a 2D model of 300 m × 1000 m located at the depth of 100 m. The pressure gradient in the reservoir is 1 MPa/100 m, and the temperature gradient is 3.6 °C/100 m. The model is discretized into 1500 elements with dimensions of 10 m × 20 m. The cap rock starts at the depth of 270 m and goes down to 310 m. There is a fracture in the middle of the cap rock. CO2 is injected at constant pressure of 4 MPa at the speciﬁed location on Figure 6. Equation 14 is used to describe CO2 hydrate equilibrium conditions in the model. This is based on the model developed by Kvamme and Tanaka,1 while the SRK equation of state is used to calculate the fugacity of the liquid phase. P = 9.968156693851430 × 10−7T 6−1.721355993747740 × 10−3T 5+1.237931459591990T 4−4.745780290305340 × 102T 3+1.022898518566810 × 105T 2 −1.175309918126070 × 107T +5.624214942384240 × 108

(14)

In this equation, P is calculated in MPa and T is in Kelvin. 3572

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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.

3573

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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).

in regions of signiﬁcant exchange of aqueous phase due to ﬂow. In regions of lower ﬂow rates, more stationary situations of local saturations may result in hydrate formation and dissociation, which maintains very low permeability and induces constraints of directions of ﬂow. Relevant aquifer storage reservoirs with hydrate zones are available oﬀshore Norway. An example is the Snøhvit injection project oﬀshore north Norway. The basic data of location, seaﬂoor depth, and seaﬂoor temperature indicates that the upper few hundred meters are inside formation region for CO2 hydrate. Very limited detailed

Tables 5−7 present the information regarding available species in diﬀerent phases, initial and boundary conditions, and material properties.

4. RESULTS AND DISCUSSION Storage of carbon dioxide in reservoirs that contain zones of carbon dioxide hydrate stability with respect to temperature and pressure results in special ﬂow paths, since the hydrates formed are not in unconditional equilibrium. This implies that the hydrate may dissociate again toward under saturated water 3574

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data are available in the open literature on this reservoir, and an example is used to illustrate the new hydrate reservoir simulator. The simulation results processed by RCB are visualized using GiD visual window. 52 Figure 7 presents the suitable thermodynamic conditions for CO 2 hydrate formation according 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. Pressure increase can be observed below the cap rock, in Figure 8, which is parallel to gas phase ﬂux pattern in Figure 10. Gas phase ﬂow upward is limited due to cap rock low porosity and permeability and stretches toward the fracture. Figure 11 shows porosity change after 615 days. It is presented in three diﬀerent scales for better illustration. Porosity change due to hydrate formation process becomes visible after around 550 days which is an indication that gas ﬂow has reached the upper aquifer where thermodynamic conditions are suitable for hydrate formation. Formation of hydrate results in increase of hydrate concentration as a pseudomineral component in the system, which will be added to total concentration of solid phase leading to change of porosity. It should be mentioned that porosity here refers to the 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 panel of the ﬁgure, hydrate formation results in porosity change, while in the bottom panel, hydrate formation is excluded from simulation. The more limited expansion of gas phase above the fracture in the top panel compared to the bottom one can be explained by presence of hydrate.

REFERENCES

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5. CONCLUSION The nonequilibrium nature of hydrate in porous media has been investigated in this paper. On the basis of the statistical− mechanical model of Kvamme and Tanaka,1 diﬀerences in properties of hydrates formed from diﬀerent phases is illustrated and discussed. All possible phase transitions involving hydrate are formulated and corresponding simpliﬁed kinetic models are derived on the basis of phase ﬁeld theory simulations. The resulting set of phase transition kinetic models is implemented into a reservoir simulator as pseudoreactions. The platform used to develop this new hydrate simulator is the RetrasoCodeBright reactive transport simulator. The implementation is illustrated by a carbon dioxide injection example into a formation containing hydrate stability regions.

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AUTHOR INFORMATION

Corresponding Author

* Phone: (+47) 55580000/(+47) 55583310. Fax: (+47) 55583380. E-mail: [email protected] Notes

The authors declare no competing ﬁnancial interest.

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ACKNOWLEDGMENTS We acknowledge the grant and support from Research Council of Norway through the following projects: SSC-Ramore, “Subsurface storage of CO2Risk assessment, monitoring and remediation”, Project No. 178008/I30. FME-SUCCESS, Project No. 804831. PETROMAKS, “CO2 injection for extra production”, Research Council of Norway, Project No. 801445. 3575

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Barcelona, Spain, 2005. See also: http://www.h2ogeo.upc.es/ software/retraso/RETRASO_manual%20nuevo/ManualRCB.pdf (44) Olivella, S.; Carrera, J.; Gens, A.; Alonso, E. E. Non-isothermal multiphase ﬂow of brine and gas through saline media. Transp. Porous Media 1994, 15, 271−293. (45) Olivella, S.; Gens, A.; Carrera, J. CodeBright User’s Guide; Universitat Politecnica de Catalunya and Instituto de Ciencias de la Tierra, CSIS: Barcelona, E.T.S.I. Caminos, Canales y Puertos, 1997. (46) Olivella, S.; Gens, A.; Carrera, J.; Alonso, E. E. Numerical formulation for a simulator (CODE_BRIGHT) for the coupled analysis of saline media. Eng. Computations 1996, 13 (7), 87−112. (47) Liu, S.; Lågeide, L. A.; Kvamme, B. Sensitivity study on storage of CO2 in saline aquifer formation with fracture using reactive transport reservoir simulator RCB. J. Porous Media 2011, 14 (11), 989−1004. (48) Saaltink, M.; Benet, I.; Ayora, C. RETRASO, A FORTRAN Code for solving 2D Reactive Transport of Solutes, Users’s guide; Universitat Politecnica de Catalunya and Instituto de Ciencias de la Tierra, CSIC: Barcelona, E.T.S.I. Caminos, Canales y Puertos, 1997. (49) Kvamme, B.; Liu. S. Reactive transport of CO2 in saline aquifers with implicit geomechanical analysis. Proceedings of the Greenhouse Gas Control Technologies (GHGT) Conference, Washington, DC, 2008. (50) Hellevang, H.; Kvamme, B. An explicit and eﬃcient algorithm to solve kinetically constrained CO2−water−rock interactions. Proceedings of the 3rd WSEAS International Conference on Mathematical Biology and Ecology, Gold Coast, Queensland, Australia, 2007. (51) Kvamme, B.; Liu, S. Simulating long term reactive transport of CO2 in saline aquifers with improved code RetrasoCodeBright. 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG), Goa, India, 2008. (52) GiDThe Personal Pre- and Post-Processor. CIMNE International Center for Numerical Methods in Engineering: Barcelona, Spain, 2012. Available online: http://gid.cimne.upc.es/ (accessed: May 20, 2012)

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Nonequilibrium Modeling of Hydrate Dynamics in Reservoir Mohammad 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 competing phase transitions involving hydrate. Calculations of hydrate phase transitions based on a modiﬁed statistical−mechanical model for hydrate (Kvamme, B.; Tanaka, H. J. Phys. Chem. 1995, 99, 7114−7119) are used to illustrate diﬀerences in properties of hydrates formed from diﬀerent phases. In more general terms, hydrates in porous media are discussed in terms of the Gibbs phase rule. It is argued that phase transitions involving hydrates in porous media can rarely reach any state of equilibrium due to situations of over speciﬁed systems with reference to requirements for equilibrium. As a consequence of this, a strategy for nonequilibrium description of hydrates in reservoirs is proposed. This involves the formulation of kinetic expressions for all possible hydrate formations and dissociations as competing pseudoreactions. This involves hydrate formation on a water/carbon dioxide interface, from water solution and from carbon dioxide adsorbed on mineral surfaces, as well as all diﬀerent possible hydrate dissociation possibilities. The basic idea is that the direction of free energy minimum, under constraints of mass and heat transport, will control the progress of phase transitions in nonequilibrium systems. A new hydrate reservoir simulator based on this concept is introduced in this study. The reservoir simulator is developed from a platform previously developed for carbon dioxide storage in aquifers, RetrasoCodeBright (RCB). The main tools for generating kinetic models have been phase ﬁeld theory simulations, with thermodynamic properties derived from molecular modeling. The detailed results from these types of simulations provides information on the relative impact of mass transport, heat transport, and thermodynamics of the phase transition, which enable qualiﬁed simpliﬁcations for implementation into RCB. The primary step was to study the eﬀect of hydrate growth or dissociation with a single kinetic rate on the mechanical properties of the reservoir. Details of the simulator and numerical algorithms are discussed, and relevant examples are shown.

1. INTRODUCTION The international focus on natural gas hydrates is increasing for a number of diﬀerent reasons. Unlike conventional fossil fuels, hydrates are distributed worldwide and can be a source of energy for countries that have little or no conventional fossil fuel resources. Japan is a good example; test production at the ﬁrst oﬀshore pilot plant will take place in 2012, and full production is planned from 2018. Production of permafrost hydrates has been going on since 1970 in Russia. The worldwide amount of hydrates is uncertain, but the United States Geological Survey (USGS) estimated 10 years ago that it might amount to more than twice all known conventional fossil fuels. Hydrates in porous media are generally not stable, as will be discussed in more detail later in this paper. This implies that, basically, the demands for trapping/sealing formations to keep hydrate in place are similar to oil and gas sealings (cap rock). In the case of oﬀshore hydrates, deﬁciencies in sealing above the hydrate layers in the form of fractures and faults will lead to contact between hydrate and groundwater. The result is dissociating hydrate and methane seepage to the ocean above. Since methane is a more aggressive greenhouse gas than CO2, the global methane ﬂuxes from hydrates and free methane is a global environmental concern. Besides, in the long run, these leakage situations may lead to dement instabilities and, in the worst case, cause landslides. In some permafrost regions, 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 are described here, which are believed to be more feasible, © 2012 American Chemical Society

speciﬁcally with respect to economy. The depressurization method in which the hydrate stability condition is shifted by pressure reduction resulting in hydrate dissociation and release of methane. It is currently considered as the most feasible process considering expenses and production rate and has been investigated by many research groups through simulation studies. Thermal stimulation is another method, which is based on moving out from the stability region by temperature increase. It is considered to be costly because of the huge amount of energy waste to the surroundings. The third method is to use inhibitors such as methanol or brine to shift the equilibrium curve and dissociate hydrate, which is also costly. The fourth method is injection of CO2 into the methane hydrate reservoirs. CO2-hydrate is more stable than CH4hydrate over substantial regions of pressure and temperature. CO2 and CH4 will both form hydrate structure I, which consist of a ratio of 3 to 1 of large cavities (24 water molecules surrounding enclathrated molecule) to small cavities (20 water). Under normal conditions, CO2 will only ﬁll the large cavities and the mole fraction of CO2 is close to 0.12, while CH4 hydrate has a mole fraction of CH4 close to 0.14. A mixed hydrate in which CO2 occupies the large cavities and CH4 occupies the small cavities will be more stable than methane hydrates over all regions of temperature and pressure where either hydrate can exist. Injection of CO2 into a natural gas hydrate reservoir can release the natural gas by direct Received: February 28, 2012 Revised: May 11, 2012 Published: May 14, 2012 3564

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conversion of the in situ hydrate, which is a solid state transformation and slow.2,3 Hydrate cannot attach directly to mineral surfaces because of incompatibility with hydrogen bonding and partial charges on atoms in the mineral surfaces. Any hydrates in porous media will therefore be surrounded by ﬂuids that separate them from the mineral surfaces. Injected CO2 may therefore also form hydrates directly with free water in the pore volume. Released heat will contribute to dissociation of in situ natural gas hydrate. This heat release is substantially higher than the heat release due to solid state reformation discussed by Kvamme et al.4 The absolute smallest distance separating hydrate from the mineral surfaces is in the order of 3 nm, based on molecular dynamics simulations. The permeability would be extremely small if natural hydrates occurred with such a high pore ﬁlling. Reported ﬁlling fractions for natural gas hydrates, as well as synthetic laboratory produced hydrates in sediments, are normally substantially lower. Most reported natural gas hydrate reservoirs have below 50% of pore volume ﬁlled with hydrate. If the remaining volume is liquid water, then solid state conversion of the CH4 hydrate 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 mechanism involved in combined storage of CO2 in hydrate form and production of natural gas hydrates. The heat released from the formation of new CO2 hydrate will assist in dissociating the in situ CH4 hydrate. During more than two decades, diﬀerent research groups worldwide have applied diﬀerent theoretical modeling and simulation approaches to studies of methane production from hydrate reservoirs. The progress has been considerable, but there are still many uncertainties and unresolved challenges in the physical description of these complex systems. A limited review of past modeling eﬀorts of hydrate exploration is given in this section. Special emphasis has been on studies that have resulted in new models and/or new simulator for simulation of methane production from hydrates. Holder and colleagues5 considered a system of stratiﬁed hydrate and gas surrounded by impermeable layer from bottom and top to build a 3D model for studying hydrate dissociation eﬀect on gas production due to depressurization. Several assumptions were made to simplify the calculations. They considered the dissociation to happen only at the interface between hydrate and gas phase, and only conduction was considered to ﬁnd temperature distribution in the gas phase. They concluded that hydrate can contribute signiﬁcantly to gas production from such reservoirs.5 Burshears et al.6 developed a two-phase, 3D numerical model to study gas production from a system 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 and equilibrium conditions in the gas−hydrate interface. Water ﬂow was present in their model but only heat conduction was considered.6 Yousif et al.7 developed a one-dimensional model to simulate isothermal depressurization of hydrate in Berea sandstone samples. The model considered three phases of water, gas, and hydrate and used the kinetics model of Kim and Bishnoi8 for dissociation of hydrate. It also considered the water ﬂow due to hydrate dissociation, which according to experimental and numerical results was considerable. The variations in porosity and permeability of the gas phase were taken into account. Finally, they validated their model with experimental data.7 Xu and Ruppel9 developed an analytical

formulation to solve the coupled momentum, mass, and energy equations for the gas hydrate system consisting of two components (water and methane gas) and three phases (gas hydrate, free gas, water, and dissolved methane). They made several assumptions such as ignoring capillary eﬀects and ignoring kinetics of the phase transition between hydrate and aqueous phase.9 Swinkels and Drenth10 used an in-house 3D thermal reservoir simulator and modiﬁed the PVT tool to study hydrate production scenarios, as well as heat ﬂow and compaction in the reservoir and hydrate cap. They represented the reservoir ﬂuid by a gaseous phase, a hydrate phase, and an aqueous phase. Their system consisted of three components, including two hydrocarbons and a water component. Heat was considered as an extra component for all phases internally in the simulator. The Joule−Thomson cooling eﬀect was also considered automatically.10 Davie and Buﬀet11 proposed a numerical model for calculation of gas hydrate volume and distribution in marine sediments. They considered the organic material from sedimentation as the main source of carbon supply to the hydrate stable region. The rate of sedimentation, the quantity and quality of organic material, and biological productivity were considered as key parameters in this model. Hydrate formation and dissociation were controlled by thermodynamic conditions and driving force from equilibrium concentration. A ﬁrst order model with a large rate constant was used to account for the reaction rate. They compared their results with observations from practical ﬁeld. Their model was dependent on the available data of sedimentation rate and organic content.11 Goel et al.12 developed a single phase ﬂow model to predict the performance of an in situ gas hydrate. The gas withdrawal rate was assumed to be constant, and the reservoir was assumed to be inﬁnite. It used an nth order kinetics rate coupled with gas ﬂow in the porous media. The model was simple and had several limitations due to simpliﬁcations.12 Moridis13 has proposed a new module for TOUGH2 simulator named EOSHYDR2. TOUGH2 is a 3D, multicomponent, multiphase ﬂow simulator for subsurface purposes. EOSHYDR2 is designed to model hydrate behavior in both sediments and laboratory conditions. It is able to consider up to four phases of gas, liquid, ice and hydrate, and up to nine components of water, CH4-hydrate, CH4 as native and from hydrate dissociation, a second native and dissociated hydrocarbon, salt, water-soluble inhibitors, and heat as a pseudocomponent. It includes both equilibrium and kinetic models for hydrate formation and dissociation. It uses hydrate reaction model of Kim and Bishnoi8 for kinetic studies. He studied four test cases of CH4 production and concluded that both depressurization and thermal stimulation can produce a substantial amount of hydrate, and he suggested that the combination of both can be desirable. He used just the equilibrium approach because of a lack of enough suitable data necessary for the parameters of the kinetic model, while mentioning that slower processes such as depressurization follow kinetic dissociation. Finally, he discussed several uncertainties mostly due to lack of knowledge in this area of research.13 Later on, using the same module, Kowalsky and Moridis14 made a comparison study between kinetic and equilibrium approach and concluded that based on the accuracy of the available kinetic models; this approach is important for simulation of short-term and core scale systems while equilibrium approach can be used for large scale simulations of production processes. Hong et al.15,16 presented a 2D cylindrical model to study gas production from hydrate 3565

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the heat of reaction, eﬀects of water and gas permeabilities, and eﬀective porosity with possibility of variation with time. They found that the production process was sensitive with respect to temperature, pressure, and core permeability. They compared their results with experimental data as well.22 Liu et al.23 developed a 1D model to study hydrate dissociation by depressurization in porous media. They used a moving front to separate the hydrate zone from gas zone. They considered conductive and convective heat transfer and mass transfer in gas and hydrate zones and energy balance at moving front. They considered equilibrium at the front and concluded that the assumption of stationary water phase results in overprediction of dissociation front location and underprediction of gas production in the well. They also found that reservoir permeability and well pressure can highly aﬀect the production rate.23 Gamwo and Liu24 have presented a detailed theoretical description of the open source reservoir hydrate simulator HydrateResSim developed previously by Lawrence Berkeley National Laboratory (LBNL). They have also applied it to a system of three components (methane, water, and hydrate) and four phases (aqueous, gas, hydrate, and ice). Darcy’s law is used for multiphase ﬂow of mass in porous media and local thermal equilibrium is considered in the code. It considers both kinetic and equilibrium approaches, using Kim and Bishnoi8 as the kinetic model of hydrate dissociation. The studied model has been validated using the TOUGH-HYDRATE simulator. They concluded that HydrateResSim is a suitable freeware, open source code for simulating methane hydrate behavior in the reservoir. They found out that equilibrium approach usually overpredicts the hydrate dissociation compared to kinetic approach.24 From the presented literature review it is clear that the hydrate dissociation process in the reservoir is mostly treated as an equilibrium reaction and in fewer cases as kinetics. The majority of kinetic approaches are based on the kinetic model of Kim and Bishnoi,8 developed according to laboratory experiments as follows:

reservoirs. They used both analytical and numerical approaches. The numerical model considered the equations for gas and water two-phase ﬂow, conductive and convective heat ﬂow, and treated hydrate decomposition kinetically using the Kim and Bishnoi8 model. In the analytical approach, they did not consider the ﬂuid ﬂow. They studied the eﬀect of hydrate zone on the gas production from gas layer and also the importance of the intrinsic kinetics of hydrate decomposition with respect to equilibrium assumptions. They concluded that the kinetic equation was aﬀected only when it was considered 5 orders of magnitude lower than the values found by Kim and Bishnoi8 in a PVT cell.15,16 Xu17 has modeled dynamic gas hydrate systems considering two main elements. The ﬁrst element provides a model to account for dynamic phase transitions of marine gas hydrates by considering the eﬀects of ﬂuid pressure, temperature, and salinity changes on hydrate stability, solubility, and ﬂuid densities and enthalpies. This allows describing coexistence of three phases, namely hydrate, gas, and liquid solution at equilibrium. The model consists of three components (water, gas, and salt) and four phases (free gas, liquid solution, gas hydrate, and solid halite). It considers thermodynamic equilibrium for and among individual phases. The second element of the model consists of the numerical tool to solve for ﬂuid ﬂow and transport equations in porous media. It considers four components and up to ﬁve phases. The response of the model to the pressure drop and temperature rise scenarios at seaﬂoor are studied.17 Ahmadi et al.18 developed a 1D model to study methane production from hydrate dissociation in a conﬁned reservoir using depressurization method. They considered heat of dissociation, conduction, and convection in both gas and hydrate phase. They considered equilibrium conditions at the dissociation front and neglected water ﬂow in the reservoir and the Joule−Thomson cooling eﬀect.18 Sun and Mohanty19 developed a 3D simulator using the Kim and Bishnoi8 kinetic model to study formation and dissociation of hydrate 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 ice melting are considered, assuming equilibrium phase transitions. Mass transport including ﬂuid ﬂow and molecular diﬀusion plus heat transport including conduction and convection are solved implicitly.19 Phirani et al.20 upgraded the simulator to account for CO2 as a new component and CO2 hydrate as a new phase. The purpose was to study CO2 ﬂooding of methane hydrate bearing sediments. They used the model of Kim and Bishnoi8 for kinetics of dissociation and constant rates for hydrate formation.20 Uddin21 developed a general kinetic model of hydrate formation and decomposition based on Kim and Bishnoi8 model to study CO2 sequestration in methane hydrate reservoirs along with CH4 production. The model consisted of ﬁve components of water in aqueous phase, CH4 and CO2 in gas phase, and CO2-hydrate and CH4-hydrate in solid phase. The gas hydrate model was coupled with a compositional thermal reservoir simulator (CMG STARS) to study dynamics of hydrate formation and decomposition in the reservoir. It was concluded that the eﬀect of kinetic rate constant on hydrate decomposition was signiﬁcant in case it was lowered a few orders of magnitude. Reservoir permeability has also a great impact on the decomposition rate.21 Nazridoust and Ahmadi22 developed a computational hydrate module for FLUENT code. It provides the possibility to study hydrate dissociation for complex 3D geometries. They used the kinetics rate of dissociation proposed by Kim and Bishnoi,8 and they included

−

dnH = kdA s(fe − f ) dt

(1)

If the phase transition as a whole is dominated by the mass transport, the form of the equation may be adequate enough. Therefore, we have also used similar form of eq 1 as a basis for transferring results from advanced theories (see section 3) into reservoir simulator use. However, then, it is acceptable only after veriﬁcation (rate versus square root of time indicate Fick’s law) of the mass transport dominance. The use of eq 1 for ﬁtting laboratory experiments from PVT cell and then using these data for a reservoir situation might be more questionable, since it has never been proven that these data are transferable to any other experimental apparatus or real situation of hydrate phase transitions in nature. This also raises the question of how to evaluate the impact of kinetics if the rate equation is not appropriate. For pressure reduction as production method one could argue that the heat transport might have a strong impact. Some even use the heat transport rate model alone to model the kinetic rate in this case. The diﬀerent heat transport models applied are normally not documented well enough in terms of theory and/or parameters to judge whether they are appropriate or not. Heat transport to a system undergoing dynamically a phase transition that involves ﬂow is fairly complex, so it is unclear whether any reasonable model exists. Another critical question related to analysis of the impact of 3566

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kinetics is the reference case. As the system by itself cannot reach equilibrium, the P, T projections are just two dimensions of a general variable set consisting of concentration in all surrounding phases, including impact of minerals. Hydrate will dissociate toward undersaturated water, regardless of whether the system is inside stability with reference to P and T or not. Also, it will sublimate toward undersaturated gas. With hydrate saturations between 18% and 50% (which are typical ranges for reservoirs considered as “rich” enough), induced ﬂow will exchange ﬂuids surrounding the hydrate, and the impact of these ﬂuid changes on hydrate dissociation have never been evaluated. In summary, there are many uncertain issues around the kinetic models used as a basis for concluding that kinetic eﬀects are not important, as well as clearly the assumption on the reference system (read: P, T deﬁnes equilibrium regardless of concentration induced nonequilibrium). The fact that hydrates in porous media are in nonequilibrium does require kinetic models in descriptions of all relevant competing phase transitions (all phase transitions that have negative free energy changes of suﬃcient magnitude to overcome nucleation barriers). There might, however, be a time-dependent local situation where one process, or a few processes, dominates and is fast enough to qualify for some quasi-equilibrium approximation. However, those are exceptions and do not need to be in the P, T projection of the multidimensional thermodynamic dependency of the system. In this paper, a diﬀerent approach according to nonequilibrium nature of hydrate phase transitions in the reservoir will be presented. A new reservoir hydrate simulator will be introduced, which is developed on a former reactive transport reservoir simulator namely RetrasoCodeBright (RCB).25 The module is designed so that it can easily work according to the nonequilibrium thermodynamic package that is being developed in this group. The reactive nature of the code gives the possibility to distinguish between diﬀerent phase transition scenarios and treat them on a competitive basis. Therefore, it is possible to evaluate the amount of components that are used to form hydrate or produced from hydrate dissociation and include these results in the overall heat and mass balance equations. In addition, it gives the possibility to visualize the eﬀect of hydrate formation or dissociation on the solid phase properties in the form of direct changes in porosity and permeability. At this stage, kinetic models of hydrate formation and dissociation from phase ﬁeld theory simulations26,27 are used to examine the performance of the module through example cases.

Table 1. Potential Phase Transition Scenarios for a System of Hydrate in Reservoir δ

1

−1

hydrate

2

−1

hydrate

3

−1

hydrate

4

+1

5

+1

6

+1

7

+1

gas + liquid water surface reformation aqueous phase adsorbed

driving force outside stability in terms of P and/ or T sublimation (gas under saturated with water) outside water under saturated with methane hydrate more stable than surrounding ﬂuids nonuniform hydrate rearranges due to mass limitations liquid super saturated with methane with reference to hydrate water and methane adsorbed on mineral surfaces

ﬁnal phase(s) gas, liquid water gas liquid water hydrate hydrate hydrate hydrate

Figure 1. Water chemical potential in empty SI (dash), empty SII (dash-dot), water as ice or liquid water (solid).

2. THEORY When a clathrate hydrate comes into contact with an aqueous solution containing its own guest molecules (CH4 or CO2), the number of the degrees of freedom available to the new system of three phases (aqueous, gas, hydrate) will be decreased compared to the two-phase system (aqueous, gas) as a result of the Gibbs phase rule. F=N−π+2

initial phase(s)

i

Figure 2. Predicted hydrate equilibrium (solid line), as estimated using the parameters from Kvamme and Tanaka1 for empty hydrate chemical potential (SI), liquid water chemical potential, and the harmonic oscillator method for evaluation of the cavity partition function. Fugacity coeﬃcients are estimated using eq 2 from Duan et al.30 Stars are measured hydrate equilibrium data from Deaton and Frost34 and Unruh and Katz.35

(2)

In this equation, F is the degree of freedom, N is the number of components, and π is the number of phases. For the twophase system comprising two components, water and gas, there will be two degrees of freedom while, in a three-phase system with hydrate as an additional phase, there will be only one degree of freedom. The Gibbs phase rule is simply conservation of mass under the constraints of equilibrium between coexisting

phases. The number of independent thermodynamic variables speciﬁed for the system must equal the number of degrees of 3567

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subsequently will dissociate in contact with groundwater from top as a result of undersaturation with CO2. As such, a kind of stationary situation of formation of new CO2 hydrate from below and dissociation from top is expected. See also Kvamme28 for examples of saturation limits of aqueous solutions toward hydrate as function of temperature and pressure. The statements about equilibrium approach in Kowalsky and Moridis14 (on page 1851 of the paper) that the system of four phases always exists at equilibrium is somewhat confusing and cannot describe a realistic hydrate reservoir, since it could happen only in a unique point consisting of a unique temperature, pressure, and composition. In hydrate systems, we are usually dealing with a range of temperatures, pressures, and compositions that can be explained only if the system is considered nonequilibrium. The typical case during the production of hydrate is three phases (gas, water, hydrate) and two components, which make the system overdetermined because temperature and pressure are deﬁned locally in every point of a real reservoir and every grid point of a reservoir model. If the system crosses the ice formation condition due to cooling, degrees of freedom will be 0 and the system even more overdetermined. Figure 1 in Kowalsky and Moridis14 has no other meaning than a possible limit to kinetics in a two thermodynamic variable projection of the nonequilibrium system, which is generally characterized by temperature, pressure, and concentrations in all coexisting phases as thermodynamic variables. Examples of two hydrate phase transitions that are thus excluded are hydrate formation from aqueous solution and hydrate dissociation toward undersaturated water. Keeping this in mind, also the comparison between equilibrium and kinetics is out of thermodynamic balance since the reference (read: equilibrium) situation is not correct. However, an additional problem is that the kinetic model of Kim and Bishnoi8 is empirical and does not contain necessary theoretical basis to be transferable outside the experimental apparatus and experimental conditions for which the model was developed. Given the applied stirring rates (and corresponding hydrodynamically induced stresses) in the experimental cell, the kinetic rates from the model with the original mode parameters are likely to be very diﬀerent from the hydrodynamic stress involved in reservoir ﬂow situations related to production. A system that cannot reach equilibrium will, however, always tend toward the minimum free energy as a consequence of the combined ﬁrst and second law and when the hydrate is inside its pressure−temperature stability region, this means that its free energy is lower than that of the aqueous solution.29 In case of hydrate formation and dissociation in the reservoir, the system will be even more complex because hydrate can form from or dissociate toward phases with diﬀerent free energies, which will produce diﬀerent phases. Consequently, the degree of freedom will decrease further and there will be no equilibrium condition and competing phase transition reactions of hydrate formation, dissociation, and reformation among diﬀerent phases will rule the system. The main factor in such a system is therefore the minimum free energy of the system and kinetics of competing reactions. Assume, for example, a system of water, methane, and hydrate in porous media. According to this system, three phases of gas, aqueous, and hydrate are evident. Methane and water are considered as the only components available in the system. The degree of freedom for such a system is 1, while it can reduce to −1 if two additional phases of adsorbed CH4 and adsorbed water are also

Figure 3. Estimated water hydrate chemical potential (solid line) for the experimental conditions of Deaton and Frost34 and Unruh and Katz35 (corresponding to the stars in Figure 2). The dashed line is for temperatures along the isobar 61 bar.36 The dash-dot line is for temperatures along the isobar 101 bar.36

Figure 4. Estimated chemical potential of water in hydrate using the extended statistical−mechanical model for hydrate due to Kvamme and Tanaka1 with chemical potential for dissolved CO2 in aqueous phase (solid line). The dashed line is pure water liquid chemical potential1 disregarding pressure changes eﬀect on liquid water chemical potential.

freedom if the system should be able to reach equilibrium. As a result, when both temperature and pressure are speciﬁed, as is the case in local reservoir conditions, the system is overdetermined and will be unable to reach three-phase equilibrium. Despite this fact, it is quite common to ignore the concentration dependencies of the diﬀerent co-existing phases on phase transitions. A minimum requirement for a phase transition is negative free energy change, but in a nonequilibrium system that is not suﬃcient. All gradients in independent thermodynamic variables also have to point in the direction of lower free energy if the phase transition should go unconditionally. A practical example can be found in nature by all the seeps of hydrates that dissociates toward groundwater, which is undersaturated with methane. Injection of CO2 into aquifers which contain cold zoneswhere pressure and temperature conditions are suitable for CO 2 hydrate formationwill lead to the formation of hydrate ﬁlms, which 3568

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fraction of guest molecule k in the hydrate type i will be calculated according to eq 7.

considered. It is while temperature and pressure are provided in the reservoir and are not adjustable. All potential phase transition scenarios due to hydrate dissociation and reformation in 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. H, i H, i p ΔGi = δ[x wH, i(μwH, i − μwp ) + xCH (μCH − μCH )] 4 4

4

xki =

∑ νj ln(1 + ∑ hkji) j

k

(4)

where superscript H,0 denotes empty clathrate, νj is the fraction of cavity of type j per water molecules, and hikj is the canonical partition function for guest molecule of type k in cavity type j for hydrate of type i. Estimated chemical potentials1 for empty clathrates of structures I and II (SI and SII) are plotted in Figure 1, together with estimates for ice and liquid water chemical potential. The canonical partition functions can be expressed as hkji = exp( − β(μkji + Δgkjinclusion))

(5)

where μikj is the chemical potential of guest molecule k in cavity j for type i hydrate. The second term in the exponent is the free energy change of inclusion of the component k in cavity type j, which is independent of the speciﬁc hydrate type i. This latter term can be evaluated in the classical fashion by integrating the Boltzmann factors of the interactions between the waters in the lattice and the guest molecule over the cavity volume and normalized1 or alternatively evaluated using a harmonic oscillator approach. At equilibrium, the chemical potential for a guest in a cavity will be equal to the chemical potential for the same component in the equilibrium phase (gas, liquid, or adsorbed). In a nonequilibrium situation, these equilibrium chemical potentials can be used in a series expansion in temperature, pressure, and concentrations over to the real situation. In view of eqs 5 and 4, the chemical potentials for waters, as well as guest molecules, will be diﬀerent between diﬀerent hydrates, depending on how the hydrate was formed (read: from where the diﬀerent molecules in the hydrate come). Also, the mole fractions of water and guest molecules will be diﬀerent for each hydrate, since the ﬁlling fractions are given by θkji

=

(7)

∞ ∞ μCO (T , P , x ⃗) + RT ln γCO 2

2

⎡ ϕ (T , P ) ⎤ CO2 idealgas ⎥ = μCO (T , P) + RT ln⎢ 2 xCO2 ⎢⎣ ⎥⎦

(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 at 278.22 K from Valtz et al.31 and neglecting dissociation of dissolved CO2 into bicarbonate and carbonate ions, we obtain the following ﬁt: ∞ (278.22 K, x ⃗) μCO

∞ (278.22 K, x ⃗) + ln γCO 2 RT ∞ (278.22 K, x ⃗) μCO ∞ 2 (278.22 K, x ⃗) ≈ + ln γCO 2 RT 2

hkji 1 + ∑k hkji

i 1 + ∑j ∑k νθ j kj

For a given hydrate i to grow unconditionally, Gibbs free energy change according to eq 3 must be negative and all gradients in free energy change (temperature, pressure, and concentrations) must be negative. Practically, this implies for water, as one example, that the chemical potential of water in hydrate must be lowered from empty clathrate chemical potentials to equal or lower than the chemical potentials of ice or liquid water by means of guest stabilization through guest inclusion and chemical potentials of guests through eqs 4 and 5. The ﬁnal result is nonuniform hydrates due to all the diﬀerent hydrate phase transitions that are possible inside a porous material. The kinetics of diﬀerent phase transition is an implicit set of dynamic equations that couples the mass transport, heat transport, 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 of chemical potential for guest molecules, we apply eq 2 from Duan et al.30 in order to be able to stretch estimates beyond moderate gas condition for CO2. The results are plotted in Figures 2 and 3. Note that these estimates are pure prediction, using very simple models for the water−methane interactions from open literature, and no attempts have been made to empirically adjust the model or parameters. For the case of illustration, these estimates are considered to be fair enough, and we now apply the same model with chemical potential of CO2 from aqueous solution. Rearranging of the gas/liquid equilibrium criteria for pure CO2 in equilibrium with a CO2 ﬂuid phase gives eq 8:

(3)

In this equation, H represents hydrate phase, i represents any of the seven phase transition scenarios, p represents liquid, gas, and adsorbed phases, x represents composition, and μ represents chemical potential. δ is 1 for hydrate formation or reformation and −1 for dissociation. Keeping in mind that the phases are not in equilibrium, it might be useful to also visualize that the hydrates created along the pathways in Table 1 must also have diﬀerent ﬁllings and corresponding diﬀerent free energies, which, by deﬁnition, implies that each hydrate forming process results in a unique phase. The chemical potential for water in the hydrate of type i can be estimated using a modiﬁed version of the statistical−mechanical model1 μH,i w : μwH, i (T , P , x ⃗ H) = μwH,0 (T , P) −

i ∑j νθ j kj

2 ≈ −9.4149 + 12.67xCO2 − 5.3643xCO 2

(6)

(9)

in which we have neglected small Poynting corrections for pressure. Equation 9 can be rearranged into the total chemical potential of CO2 in aqueous solution at 278.22 K:

In eq 6, θikj is the ﬁlling fraction of guest molecule k in cavity type j for hydrate type i. Therefore, xki, which is the mole 3569

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To analyze this hydrate system, all impossible (ΔG > 0) and unlikely (|ΔG| < ε) cases must be disregarded. Taking into account the mass transport limited cases, all realistic phase transition scenarios will be determined. The purpose is to develop a hydrate reservoir simulator that has the possibility to consider the free energy changes of all phase transition scenarios and take into account the eﬀect of the realistic ones on the ﬂow and properties of the porous media through advanced kinetic models.

μCO (278.22 K, x ⃗) 2

2 = 278.22R[− 9.4149 + 12.67xCO2 − 5.3643xCO 2

+ ln(xCO2)]

(10)

Application of eq 10 together with free energies of inclusion from Kvamme and Tanaka1 in eqs 4 and 5 gives the plot in Figure 4 for chemical potentials of water in hydrate formed from CO2 in aqueous solution. These results can be compared to the chemical potentials of free water in the same plot to ﬁnd coexistence limit for hydrate forming from solution. However, more important is the diﬀerences in the free energy of the diﬀerent hydrates. Diﬀerences in hydrate water chemical potential between hydrate forming from ﬂuid CO2 as separate phase and from solution will reﬂect back in ﬁlling fractions (eqs 6 and 7) and also in the chemical potentials that enter the free energy 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 mass transport.32,33 The local progress of the diﬀerent phase transitions and resulting phase distributions at any given time is therefore a complex function of diﬀerences in thermodynamics and diﬀerences in mass transport associated with each possible phase transition. On a reservoir scale, the enthalpy changes associated with the diﬀerent hydrate phase transitions need to be transferred to the energy ﬂuxes in the reservoir simulator. Because the chemical potential for guest molecules in hydrate is expanded from equilibrium, a direct evaluation of the temperature dependency will be more complicated than numerical estimation using the thermodynamic relationship: μ ∂⎡⎣ RTk ⎤⎦ P,N⃗

∂T

=−

H̅k RT 2

3. SIMULATION 3.1. Kinetic Model. The results from phase ﬁeld theory simulations26,32,33,37−39 have been modiﬁed to be used in the Table 2. Equations and Independent Variables equation

variable name

equilibrium of stresses balance of liquid mass balance of gas mass balance of internal energy

displacements liquid pressure gas pressure temperature

Table 3. Constitutive Equations constitutive equation

variable name

Darcy’s Law Fick’s law Fourier’s law retention curve mechanical constitutive model phase density gas law

liquid and gas advective ﬂux vapor and gas non-advective ﬂux conductive heat ﬂux liquid phase degree of saturation stress tensor liquid density gas density

Table 4. Equilibrium Restrictions (11)

for any component k in a given phase. The line above H indicates partial molar enthalpy. For each component (water and hydrate formers), eq 11 gives the relationship between partial molar enthalpy and the chemical potentials, and the chemical potentials are directly estimated outside equilibrium, so numerical diﬀerentiation of the left side of equ 11 is the easiest way to get the partial molar enthalpy for water and all hydrate formers inside the hydrate in a general nonequilibrium situation. Summing the contributions to enthalpy from the partial molar enthalpies and mole fractions of all components in each phase provides the necessary enthalpy information for the convective terms of the energy balances in the reservoir simulator. In this paper, we will limit the number of competing phase transitions to a minimum and focus mainly on describing the simulator and the speciﬁc method for implementing nonequilibrium thermodynamics in the code. More complex scenarios will be studied in the future, as the code will develop and 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 the importance of a kinetic approach in the simulation of hydrate in reservoir and to remind that there are many unattended issues in this area that leave uncertainties in comparisons between equilibrium and kinetic approach. The rest of this paper introduces the simulation approach that provides the possibility to consider, as much as possible, the kinetic theories presented in the previous sections.

equilibrium restrictions

variable name

Henry’s law psychometric law

air dissolved mass fraction vapor mass fraction

kinetic model. Phase ﬁeld simulations are based on the minimization of Gibbs’ free energy on the constraint of heat and mass transport. Extensive research has been going on in the same group on application of phase ﬁeld theory in prediction of hydrate formation and dissociation kinetics, which is still in progress.4,26−29,32,33,37−40 In this study, the simulation results for hydrate formation kinetics from such studies have been extrapolated and used as the reaction rate constant of the Kim and Bishnoi8 kinetic model in the numerical tool. Assuming that the formation process happens from gas phase and dissociation is toward aqueous phase, the diﬀusivity of CO2 in liquid and gas phase would be important. According to the available data in literature, diﬀusivity of CO2 in liquid phase is around 4 orders of magnitude lower than gas phase.41,42 Therefore, dissociation rate has been chosen accordingly in this study. 3.2. Numerical Tool. For reservoirs where hydrate may form in some regions, hydrate formation involves roughly 10% volume increase compared to water. In parallel to this, there are chemical reactions that can, for example, supply extra CO2 through dissolution of carbonates in regions of low pH, and also regions of high pH where transported ions may precipitate and even extract CO2 from water and hydrate. The formed hydrate will not be in equilibrium due to Gibbs’ phase rule, as 3570

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Table 6. Initial and Boundary Conditions boundary variable

value

pressure at the top (MPa) pressure at the bottom (MPa) temp. at the top (K) temp. at the bottom (K) initial mean stress at the top (MPa) initial mean stress at the top (MPa) CO2 injection pressure (MPa)

1.0 4.0 273.35 284.15 2.33 8.76 4.0

Table 7. Reservoir and Hydrate Properties

Figure 5. RCB solves the integrated equations sequentially in one time step.

Table 5. Chemical Species in the Media phase

species H2O, HCO3−, OH−, H+, CO2 (aq), CO32− CO2 (g)

value

Young’s modulus (GPa) Poisson’s ratio zero stress porosity in aquifer zero stress permeability in aquifer (m2) zero stress porosity in cap rock zero stress permeability in cap rock (m2) zero stress porosity in fracture zero stress permeability in fracture (m2) Van Genuchten’s gas entry pressure (at zero stress) (kPa) Van Genuchten’s exponent thermal conductivity of dry medium (W/m K) thermal conductivity of saturated medium (W/m K) solid phase density (kg/m3) rock speciﬁc heat (J/kg K) CO2 hydrate molecular weight (g/mol) CO2 hydrate density (kg/m3) CO2 hydrate speciﬁc heat (J/kg K) CO2 hydrate reaction enthalpy (J/mol) CO2 hydrate kinetic formation rate constant (mol/Pa m2 s) CO2 hydrate kinetic dissociation rate constant (mol/Pa m2 s)

0.5 0.25 0.3 1.0 × 10−13 0.03 1.0 × 10−17 0.5 1.0 × 10−10 196 0.457 0.5 3.1 2163 874 147.5 1100 1376 51858 1.441 × 10−12 1.441 × 10−16

fairly fast (hours to days). For this purpose, a reactive transport reservoir simulator, RetrasoCodeBright (RCB),43 is used in this study and extended with hydrate phase transitions as “pseudoreactions”. RCB is capable of realistic modeling of the reaction rates for mineral dissolution and precipitation, at least to the level of available experimental kinetic data. In contrast to some oil and gas simulators, the simulator has ﬂow description ranging 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 can be viewed as divided into two parts: spatial and temporal discretizations. The ﬁnite element method is used for the spatial discretization, while ﬁnite diﬀerences are used for the temporal discretization. The Newton−Raphson method is adopted for the iterative scheme.25,43−48 A brief overview of independent variables, constitutive equations, and equilibrium restrictions are given in Tables 2−4, respectively, while the details are presented elsewhere.43−47 3.2.1. Independent Variables. The governing equations for non-isothermal multiphase ﬂow of liquid and gas through porous 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 necessary constitutive and equilibrium laws. Tables 3 and 4 present a

Figure 6. Details of the simulation model.

aqueous gas

property

mentioned in the previous sections. Neither will it attach to the mineral surfaces due to incompatibilities of hydrogen bonds in hydrate and interactions with atomic partial charges on the mineral surfaces. For this reason, there is a need for a logistic system that can handle competing processes of formation and dissociation. A reactive transport simulator can handle that. Implicit geomechanics is needed in order to handle competing phase transitions, which are very rapid (seconds) and are dynamically coupled to geochemical reactions, which can be 3571

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summary of the constitutive laws and equilibrium restrictions that should be incorporated in the general formulation. The dependent variables that are computed using each law are also included. 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 implicit algorithm for solution of material ﬂow, heat-ﬂow, and geomechanical model equations.44−46 The Retraso extension of CodeBright involves an explicit algorithm for updating the geochemistry,25,48 as shown in the Figure 5. This new coupled tool RCB is capable of handling both saturated and unsaturated ﬂow, heat transport, and reactive transport in both liquid and gas. It is a user-friendly code for ﬂow, heat, geomechanics, and geochemistry calculations. It oﬀers the possibility to just compute the chosen unknowns of user’s interest such as hydromechanical, hydro-chemical-mechanical, hydro-thermal, hydrothermal-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 of geomechanical dynamics. According to the Figure 5, ﬂow, heat, and geomechanics are solved initially in CodeBright module through Newton−Raphson iteration, and then, the ﬂow properties are updated according to the eﬀects of reactive transport on porosity and salinity in a separate Newton− Raphson procedure but for the same time step.48 This makes it possible to study the implications of fast kinetic reactions such as hydrate formation or dissociation more realistically. Hydrate formation happens through nucleation and growth as physically well-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 and hydrodynamics can have very long induction times but our experience from hydrate formation in porous media shows little or no induction time.27 3.2.3. Modiﬁcations in RCB. The Retraso part of the code has a built-in state of the art geochemical solver and, in addition, capabilities of treating aqueous complexiation (including redox reactions) and adsorption. The density of CO2 plumes, which accumulate under traps of low permeability shale or soft clay, depends on depth and local temperature in each unique storage scenario. The diﬀerence in density and the density of the groundwater results in a buoyancy force for penetration of CO2 into the cap rock. Even if the solubility of water into CO2 is small, dissolution of water into CO2 may also lead to out-drying of clay. Mineral reactions between CO2 and shale minerals are additional eﬀects that eventually may lead to embrittlement. Linear geomechanics may not be appropriate for these eﬀects. Clay is expected to exhibit elastic nonlinear contributions to the geomechanical properties. Diﬀerent types of nonlinear models are already implemented in the CodeBright part of the code, and the structure of the code makes it easy to implement new models derived from theory and/or experiments. The current version of RCB has been extended from ideal gas into handling of CO2 according to the SRK equation of state.49 This equation of state is used for density calculations as well as the necessary calculations of fugacities of the CO2 phase, as needed in the calculation of dissolution of the CO2 into the groundwater.49−51 The Newton−Raphson method used in the original RCB has been also modiﬁed to improve the convergence of the numerical solution while increasing the range of working pressure in the system.50,51

The following modiﬁcations are additional in this study. To account for nonequilibrium thermodynamics of hydrate and to determine the kinetic rates of diﬀerent competing scenarios in each node and each time step according to the temperature, pressure, and composition of the system, CO2 and CH4 hydrates are added into the simulator as pseudomineral components with kinetic approach for hydrate formation and dissociation. Hydrate formation and dissociation can directly be observed through porosity changes in the speciﬁc areas of the porous media. Porosity reduction indicates hydrate formation and porosity increase indicates hydrate dissociation. The consequences on the intrinsic permeability are implemented through Kozeny’s model, as has been used in the code originally.43 The code also supports several models for calculating relative permeability, which are described in detail elsewhere.43 The surface area of hydrate is calculated in a similar approach to ordinary minerals assuming spherical particles.43 Temperature, pressure, and concentrations are three factors that inﬂuence hydrate formation or dissociation. The kinetic rate used in this study is calculated from extrapolated results of phase ﬁeld theory simulations.26,27,29,32,33,37−40 In the next stage, it will be replaced by a thermodynamic code, which is already in the ﬁnal stages, to account for all diﬀerent competing reactions in a nonequilibrium approach. Hydrate reaction is considered based on the eq 12, which is good enough for illustration purposes. For more rigorous analysis the numbers in this equation should contain empty cavities, but there are many model parameters which are even more uncertain. So, for a model case, these numbers are used. The energy balance for the gas phase is modiﬁed from ideal gas to real gas according to the eq 13 using SRK equation of state to calculate fugacity coeﬃcient and derivatives. The energy balance for solid phase is also modiﬁed according to hydrate reaction enthalpy of Table 7. 8(CH4 /CO2 ) + 46H 2O ↔ Hydrate

H = H (id.gas) − RT 2

d(ln ϕ) dT

(12)

(13)

3.3. Model Description. A system of CO2 injection into the reservoir presented in Figure 6 is selected to show the performance of the simulator. It consists of a 2D model of 300 m × 1000 m located at the depth of 100 m. The pressure gradient in the reservoir is 1 MPa/100 m, and the temperature gradient is 3.6 °C/100 m. The model is discretized into 1500 elements with dimensions of 10 m × 20 m. The cap rock starts at the depth of 270 m and goes down to 310 m. There is a fracture in the middle of the cap rock. CO2 is injected at constant pressure of 4 MPa at the speciﬁed location on Figure 6. Equation 14 is used to describe CO2 hydrate equilibrium conditions in the model. This is based on the model developed by Kvamme and Tanaka,1 while the SRK equation of state is used to calculate the fugacity of the liquid phase. P = 9.968156693851430 × 10−7T 6−1.721355993747740 × 10−3T 5+1.237931459591990T 4−4.745780290305340 × 102T 3+1.022898518566810 × 105T 2 −1.175309918126070 × 107T +5.624214942384240 × 108

(14)

In this equation, P is calculated in MPa and T is in Kelvin. 3572

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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.

3573

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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).

in regions of signiﬁcant exchange of aqueous phase due to ﬂow. In regions of lower ﬂow rates, more stationary situations of local saturations may result in hydrate formation and dissociation, which maintains very low permeability and induces constraints of directions of ﬂow. Relevant aquifer storage reservoirs with hydrate zones are available oﬀshore Norway. An example is the Snøhvit injection project oﬀshore north Norway. The basic data of location, seaﬂoor depth, and seaﬂoor temperature indicates that the upper few hundred meters are inside formation region for CO2 hydrate. Very limited detailed

Tables 5−7 present the information regarding available species in diﬀerent phases, initial and boundary conditions, and material properties.

4. RESULTS AND DISCUSSION Storage of carbon dioxide in reservoirs that contain zones of carbon dioxide hydrate stability with respect to temperature and pressure results in special ﬂow paths, since the hydrates formed are not in unconditional equilibrium. This implies that the hydrate may dissociate again toward under saturated water 3574

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data are available in the open literature on this reservoir, and an example is used to illustrate the new hydrate reservoir simulator. The simulation results processed by RCB are visualized using GiD visual window. 52 Figure 7 presents the suitable thermodynamic conditions for CO 2 hydrate formation according 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. Pressure increase can be observed below the cap rock, in Figure 8, which is parallel to gas phase ﬂux pattern in Figure 10. Gas phase ﬂow upward is limited due to cap rock low porosity and permeability and stretches toward the fracture. Figure 11 shows porosity change after 615 days. It is presented in three diﬀerent scales for better illustration. Porosity change due to hydrate formation process becomes visible after around 550 days which is an indication that gas ﬂow has reached the upper aquifer where thermodynamic conditions are suitable for hydrate formation. Formation of hydrate results in increase of hydrate concentration as a pseudomineral component in the system, which will be added to total concentration of solid phase leading to change of porosity. It should be mentioned that porosity here refers to the 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 panel of the ﬁgure, hydrate formation results in porosity change, while in the bottom panel, hydrate formation is excluded from simulation. The more limited expansion of gas phase above the fracture in the top panel compared to the bottom one can be explained by presence of hydrate.

REFERENCES

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5. CONCLUSION The nonequilibrium nature of hydrate in porous media has been investigated in this paper. On the basis of the statistical− mechanical model of Kvamme and Tanaka,1 diﬀerences in properties of hydrates formed from diﬀerent phases is illustrated and discussed. All possible phase transitions involving hydrate are formulated and corresponding simpliﬁed kinetic models are derived on the basis of phase ﬁeld theory simulations. The resulting set of phase transition kinetic models is implemented into a reservoir simulator as pseudoreactions. The platform used to develop this new hydrate simulator is the RetrasoCodeBright reactive transport simulator. The implementation is illustrated by a carbon dioxide injection example into a formation containing hydrate stability regions.

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AUTHOR INFORMATION

Corresponding Author

* Phone: (+47) 55580000/(+47) 55583310. Fax: (+47) 55583380. E-mail: [email protected] Notes

The authors declare no competing ﬁnancial interest.

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ACKNOWLEDGMENTS We acknowledge the grant and support from Research Council of Norway through the following projects: SSC-Ramore, “Subsurface storage of CO2Risk assessment, monitoring and remediation”, Project No. 178008/I30. FME-SUCCESS, Project No. 804831. PETROMAKS, “CO2 injection for extra production”, Research Council of Norway, Project No. 801445. 3575

dx.doi.org/10.1021/ef300348r | Energy Fuels 2012, 26, 3564−3576

Energy & Fuels

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