N2 Mixture Injected into CH4

Article pubs.acs.org/jced

Thermodynamic Limitations of the CO2/N2 Mixture Injected into CH4 Hydrate in the Ignik Sikumi Field Trial Bjørn Kvamme* Department of Physics and Technology, University of Bergen, Allegaten 55, 5007 Bergen, Norway ABSTRACT: The huge resources of energy in the form of natural gas hydrates are widely distributed worldwide in permafrost sediments as well as in offshore sediments. A novel technology for combined production of these resources and safe long terms storage of carbon dioxide is based on the injection of carbon dioxide injection into in situ methane hydrate filled sediments. This will lead to an exchange of the in situ methane hydrate over to carbon dioxide dominated hydrate and a simultaneous release of methane gas. Recent theoretical and experimental results indicate that the conversion from natural gas hydrate to carbon dioxide hydrate and mixed carbon dioxide/methane hydrate follows two primary mechanisms. Direct solid state transformation is possible but very slow. The dominating mechanism involves formation of a new hydrate from injected carbon dioxide and associated dissociation of the in situ natural gas hydrate by the released heat. Nitrogen is frequently added in order to increase gas permeability and to reduce blocking due to new hydrate formation. In this work we examine the thermodynamic limitations of adding nitrogen. On the basis of state of the art thermodynamic analysis it is concluded that substantial amounts of nitrogen in carbon dioxide will slow down the conversion dramatically.

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INTRODUCTION Injection of carbon dioxide into sediments containing methane hydrates will lead to conversion of the in situ methane hydrate over to CO2 hydrate1−3 and mixed CO2/CH4 hydrate.4−6 This concept of simultaneous long-term CO2 storage with associated release of methane for new energy production is a win−win situation. Two primary mechanisms for this conversion have been discussed in the open literature.4,5,7,8 Various laboratories have utilized NMR9 in experiments for monitoring the hydrate exchange processes. Reported conclusions9 indicate that the conversion follows a solid state exchange mechanism. Slow mass transport through hydrate is the main kinetic limitation involved in this mechanism. Formation of a new CO2 hydrate from injected CO2 and free water in the pores4,5 is a second mechanism. The heat released from forming CO2 hydrate will then contribute to dissociation of the in situ methane hydrate. The Ignik Sikumi gas hydrate test well on the North Slope of Alaska in the Prudhoe Bay area was drilled and logged during the winter of 2010/2011, and gas hydrate production testing was carried out there during the winter of 2011/2012. The chosen area for the test is consistently sand and a representative east−west structural cross section is illustrated in Figure 1. The main reason for including this figure is to illustrate the locations and relative thickness of different sections of the different zones in the sediments of the pilot study. The figure also illustrates interpretations of connecting fracture systems which can influence the fluid flows through different sections during production of the in situ methane. E, D, and both C zones are gas hydrate-bearing sandstone. Estimated permeability10 in the hydrate sediments were 1 mDarcy or less based on both © XXXX American Chemical Society

Figure 1. A model of the test area. The cross section visualized in the bottom left figure is about 16 000 feet × 16 000 feet in plane projection. Thickness of the formation is 1045 feet from the top of the Upper F sandstone to the base of the B sandstone. Minimum depth in the model is 1136 feet and maximum depth is 3025.

TIMUR and SDR conventional methods. The TIMUR method is based on a free fluid model for estimation of permeability, Received: November 3, 2015 Accepted: January 28, 2016

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DOI: 10.1021/acs.jced.5b00930 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Figure 2. Summary of results10 from different monitoring techniques applied in the mapping of Iġnik Sikumi hydrate saturation profile. Upper section C was selected for the test. The gamma ray (Track 1) is the standard sand−shale discrimination tool, where the hydrate-bearing sand intervals are recognized by the lower GR signal. The caliper log (Track 1 HCAL), when compared to the bit-size curve (Track 1 BS) indicates a good quality borehole throughout the hydrate-bearing intervals with minimal washout. The hydrate-bearing intervals are identified by high resistivity (Track 4 AT90), low compressional slowness values (Track 5, DTCO) and separation between the conventional density and NMR porosity curves (Track 6). The deepest reading resistivity curve, AT90, was collected with the RtScanner and processed with a standard two-foot vertical resolution. A threshold value of 2 ohm-m was chosen to identify the hydrate-bearing intervals (shaded red). Lower slowness values correspond to faster velocities and indicate the presence of a porosity-reducing hydrate that also strengthens the sand. A threshold value of 140 μs/ft was used to discriminate the hydrate-bearing intervals. The bulk density measurement (RHOZ) is not affected when water is transformed into hydrate because the density of the liquid and solid are virtually the same. For this reason, the standard density log is the best option for determining the total pore volume filled with liquid and hydrate. In contrast, the fluid-sensitive NMR log does not detect the hydrate because the fast relaxation times associated with hydrate are not detectable by the conventional logging tools. The combination of the two provides a useful way to distinguish water-filled pores from hydrate-filled pores. In the figure, hydrates are identified by high RT values, low compressional slowness (DTCO), (i.e., high velocity), a subdued NPHI response (neutron porosity measurement), and the relationship between low RHOB/high DPHI (density porosity measurement) and low NMR porosity (TCMR) that results from fast T2 decay. Figure reproduced with permission.

through this study. The high resistivity regions in column 4 are complementary to the high velocity regions in column 5 and also the low NMR porosity in column 6. The resulting interpretation from these log data is an average hydrate saturation of 75% of available pore volume, 10% pore bounded water, and the remainder free water. The test was carried out using a 77.5% N2 and 22.5% CO2 by volume mixture in a “huff and puff” test using only one well. A plot of average injection rates over the injection period is given in Figure 3. More accurate data showing also the dynamic changes can be found on the NETL pages for the project (http://www.netl.doe.gov/research/oil-and-gas/methane-hydrates/co2_ch4exchange), where also the complete final report is available for download. The corresponding pressure ranges

while the SDR model is using an average of the T2 signal from the NMR response. See Zhang et al.11 and Chen et al.12 for more details on these two methods. Zones B and F in the figure are liquid water bearing sandstone with estimated permeability higher than 1 Darcy. Figure 2 below summarizes the most important log data for the actual hydrate section used in the test (upper C). The most important columns for the purpose of this study are the depth (and corresponding static pressures) in column 3 and the different complementary logging results used to quantify the estimates of hydrate saturation. The depths are needed for estimation of the range of static pressures needed for the analysis in this work while the distribution of phases are needed in follow up studies employing the thermodynamics developed B



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as well as dynamics of hydrate formation on gas/water interfaces are controlled by interface concentrations. Adsorption of injection gas components onto liquid water can be examined by a simplified 2D adsorption theory.16−18 For a mixture of gases μi gas =

1 1 ln(β Λi3) + ln(yi φi gasP) β β

(1)

where μi is chemical potential for component i, Λi is de Broglie wavelength for molecule i, β is the inverse of Boltzmann constant times temperature, ϕi is the fugacity coefficient for component i, P is pressure, and yi is the mole fraction of component i in the gas. The equilibrium between a gas molecule and same molecule adsorbed on the surface of liquid water may be expressed as μi gas =

1 1 1 ⎛ ∂ ln Q ⎞ ln(β Λi3) + ln(xiN ) − ⎜ ⎟ β β β ⎝ ∂Ni ⎠

T , V , Nj ≠ i

Figure 3. Average injection rates for the Iġnik Sikumi pilot test for CO2/N2 based production of CH4 from in situ hydrate. The complete report from ConocoPhillips10 contains more accurate logging data showing also the dynamic variations.

=

1 1 1 ⎛ ∂ ln Q ln(β Λi3) + ln(xiN ) − ⎜ β β β ⎝ ∂Ni

2D ⎞

⎟ ⎠T , V , N

j≠i

1 − ln Q i1D β 1 1 1 1 = ln(β Λi3) + ln(xiN ) − (βμi2D ) − ln Q i1D β β β β

during the injection period is 91 to 98 bar so in order to cover also a range of pressure for the pressure drawdown period we focus on pressures between 80 and 110 bar in this study. Corresponding temperature range in this study is limited to −10 to 10 C. Mixing N2 into the CO2 reduce the thermodynamic driving force for formation of a new hydrate from injection gas and free water in the pores. The filling of large cavities in structure I hydrate will be dominated by CO2 while some N2 will enter the small cavity. Gas permeability increases proportionally to N2/ CO2 ratio and this is one reason for adding N2. Formation of new hydrate from free pore water and injection gas reduces proportionally to increasing N2/CO2 ratio. A high N2/CO2 ratio will therefore also reduce blocking of gas flow paths through the reservoir due to the new hydrate formation. Owing to the combined first and second laws of thermodynamics the most stable hydrates will form first. One result of this is that CO2 will gradually be extracted from the gas phase and form CO2 in structure I hydrate. CO2 is superior to both CH4 and N2 in stabilization of the large cavity. Some CO2 might enter small cavities but the net stabilization of this small cavity filling is questionable. This is based on our experiences from Molecular Dynamics simulations using the approach of Kvamme and Tanaka13 on different interaction models for CO2. Movement of CO2 in the small cavity results in substantial interference with the librational movement of the water molecules in the small cavities and gives rise to a very unfavorable free energy of inclusion (eq 9). In a dynamic situation it is therefore likely that N2 will dominate small cavity filling. CO2 will also dissolve in the liquid water phase and part of this dissolved CO2 can form hydrate from the aqueous solution. Given the high N2 concentration in the pilot plant test it was decided not to evaluate this coupling in this study. It will be included naturally in our follow up study when the thermodynamics described here is implemented in our hydrate reservoir simulator. CO2 adsorbs different minerals well, including calcite,14,15 in contrast to N2 which has practically no thermodynamic benefits of mineral adsorption. Selective mineral adsorption will therefore also extract CO2 from the injection gas. The dynamics of gas solubility into liquid water,

(2)

in which the splitting into an orthonormal 2D plane assumes that the horizontal structuring of the adsorbed layer does not affect the vertical interaction with the water surface. Q in eq 2 is the configurational part of the canonical partition function. The 1D part of the partition function in the direction of the water as a function of distance z from the surface is given by Q1D ≈

∫ [exp(−β Γsmooth(z)) − 1] dz

(3)

Γsmooth(z) is the smoothed out distribution of water based on molecular dynamics simulation results19 using TIP4P for water and simplified spherical Lennard-Jones 12−16 models for CO2 and N2. Model parameters for well-depth/Boltzmann’s constant, ε/kB, for CO2 is 189.0 K, and the corresponding parameter for N2 is 101.5 K. The Lennard-Jones diameters are 3.612 and 4.486 Å for CO2 and N2, respectively. Crossinteraction parameters are calculated using the Lorentz− Berthelot mixing rules, and the Soave−Redlich−Kwong20 equation of state has been employed for the gas phase. Further details on the 2D equation of state and the adsorption theory are provided elsewhere.16−18 Applied N 2/CO2 rates in exchange experiments are frequently high. As an example for examining selective adsorption on liquid water we use a gas comprising 90 mol % N2 and 10 mol % CO2. At 30 bar and 273 K the estimated adsorption mole-fraction of CO2 is 32 mol %, that is, roughly three times the mole-fraction of CO2 in the gas phase. Given the thermodynamic benefit of CO2 in heterogeneous hydrate nucleation, the question arises as to how far CO2 concentration in the gas can be reduced before the gas is unable to create a new hydrate. Furthermore, will the in situ methane hydrate be stable in contact with this gas phase depleted of CO2 or will it dissociate? Finally, will the formed CO2-dominated hydrate remain stable in an environment where the gas-phase is dominated by CH4/N2? C



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increased salinity through exchange with surrounding water will be highly individual for each reservoir. As such the results presented here are conservative estimates intended to illustrate some important qualitative aspects of adding nitrogen to the carbon dioxide. The paper is organized as follows. The thermodynamic models are described in the next section, followed by a section for verification of the model systems through comparison with experimental data. Thermodynamic stability limits of different hydrates under influence of various surrounding fluids are discussed in five subsequent sub sections, leading to a final conclusion section.

More complete answers to these questions require simultaneous solutions of mass-transport, heat-transport, and associated phase transition kinetics. The present generation of hydrate simulators include only limited thermodynamic descriptions of the phase transitions that involve hydrate. In the simplest case only the temperature, pressure projection of the multidimensional dependency (which also includes concentrations of all components in all phases) is used. Yet another simplification lies in the approximation that the only hydrate formation route is through a free gas phase and available free water. The reverse situation is a corresponding dissociation of hydrate into free water and hydrate former phase. There are several ways that hydrate can be nucleated and grow as discussed in the book by Mokogan.21 Mokogan21 studied hydrate formation dynamics in systems without induced flow. Images from his experimental test cell show hydrate growing along the walls both downward into the water section as well as upward on the walls on the gas side (most experiments use methane gas). Solid, water wetting, surfaces provide excellent hydrate nucleation sites because they often permit direct adsorption of hydrate formers along with water, or secondary adsorption in low density regions of adsorbed water.22−24 Possible heterogeneous hydrate formation is also beneficial from a mass-transport point of view. Homogeneous hydrate formation from dissolved hydrate formers in solution depends on a 3D mass-transport of hydrate formers toward the growing hydrate, in contrast to a heterogeneous hydrate formation toward a solid surface or an interface, which only needs a 2D transport. While hydrate nucleation toward water wetting surfaces is beneficial, the partial charges on hydrate waters and on atoms in minerals are incompatible so that the hydrate will not attach directly but will be bridged through highly structured liquid water in the adsorbed phase toward mineral and hydrates, respectively. There are also several ways that hydrate can dissolve, including exposure to groundwater which is undersaturated with hydrate formers. There are two primary objectives of this paper. As mentioned above there is a general need for a complete thermodynamic model description of these CO2/CH4/N2 systems that can be implemented in reservoir simulation software for modeling hydrate production using CO2-based injection technologies. Evaluating transport in these multicomponent mixtures also requires an appropriate thermodynamic description. The second main goal of the paper is to shed light on some possible special features related to the addition of N2 to CO2. In particular how this approach might affect the possibilities for simultaneous CO2 storage as hydrate and CH4 production. The specific ranges of conditions (temperature, pressure, gas composition) have been chosen so as to reflect a realistic case of hydrate in nature so that the manuscript also potentially can be a supplement to other reports and publications related to analysis of data from the Ignik Sikumi test in Alaska in 2012.10 It should be emphasized that the systems are analyzed with reference to pure water. There are three reasons for this. One is that the salinity can vary substantially between different hydrate filled sediments in offshore and permafrost hydrate reservoirs. The other reason is that the addition of salinity would require a model for the impact of salinity on activity coefficient of water. It would even be possible to incorporate this by simple shifts of the water chemical potential according to the impact of salinity. And finally the third reason is that the salinity in the surrounding water will change proportionally to the consumption of water into a hydrate, but the dilution of this

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THEORY Fluid Thermodynamics. If thermodynamic equilibrium can be achieved then the temperatures, pressures, and chemical potentials of all coexisting phases have to be uniform across all phase boundaries. Minimizing free energy can be used to find phase distributions and compositions in equilibrium systems. For nonequilibrium systems a free energy analysis can be used to find the most beneficial phase distributions locally, as well thermodynamic preference for individual components to move across phase boundaries to other phases. For equilibrium systems the choice of reference state for different components in different phases is not critical as long as thermodynamic models are available. For nonequilibrium systems it will be convenient to have the same reference state for free energy of all phases. Calculation of chemical potentials of all components in the different phases based on ideal gas as the reference state is then formulated as μi (T, P, y ⃗ ) − μiideal gas (T, P, y ⃗ ) = RT ln ϕi(T, P, y ⃗ ) (4)

where ϕi is the fugacity coefficient for component i in a given phase. The ideal gas term on left-hand side also includes the ideal gas mixing term due to entropy of mixing ideal gases at constant temperature and pressure. Another reference state for the chemical potential of a liquid state component i will also be used as an intermediate step: μi (T, P, x ⃗) − μiideal liquid (T, P, x ⃗) = RT ln γi(T, P, x ⃗) (5)

lim(γi) = 1.0 when xi → 1.0, where γi is the activity coefficient for component i in the liquid mixture. The ideal mixing term is included in the chemical potential of ideal liquid on the lefthand side. Equation 3, as applied to water, can also be based on ideal gas reference state when the chemical potential of pure water liquid water is calculated from molecular interaction models using molecular simulations. More specifically, data from Kvamme and Tanaka13 are employed. The formulation in eq 5 is normally called symmetric excess. For gases with low solubility in water infinite dilution of the component in water is a more appropriate liquid reference state for those components: μi (T, P, x ⃗) − μi∞(T, P, x ⃗) = RT ln[x iγi∞(T, P, x ⃗)]

(6)

lim(γ∞ i )

= 1.0 when xi → 0, where the ∞-superscript stands for the infinite dilution. This particular convention is known as the nonsymmetric excess convention because the limit of the activity coefficient for the component i will approach unity as the mole fraction vanishes. One way to estimate values based on the ideal gas reference state for these infinite dilution

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equation of state, and the resulting chemical potential needed in eq 9 for the cavity partition functions in eq 10 can then be calculated from eq 4.

chemical potentials is through molecular dynamics simulations and the application of the Gibbs−Duhem relation.7,8 Provided that thermodynamic properties of all phases can also be specified and evaluated outside of equilibrium, the first and second laws of thermodynamics would require that the available mass of each component, and the total mass, should be distributed over all possible phases able to coexist under given local pressure and temperature conditions. This evaluation will be fairly straightforward for most of the fluid phases under consideration. The only phase requiring special attention will be the hydrate phase, which is discussed extensively in Kvamme et al.5,8 Combining thermodynamic formulations for fluids in eqs 4 to 6 with hydrate nonequilibrium formulations from Kvamme et al.5,8 makes it fairly straightforward to minimize the free energy and obtain estimates for local phase distributions obeying the first and the second law of thermodynamics. Several algorithms capable of implementing this approach are available in the open literature. Situations considered here imply very limited solubility and/ or limited concentrations. The solubility of H2O in CO2/N2 is very low. In view of this fact, the following approximation should prove sufficiently accurate for the pure liquid CO2 (or CO2 with small amounts of N2 but still in liquid) limit: μi,j(T, P, x ⃗) ≈ μi,j∞(T, P) + RT ln[x i , jγi∞ (T, P, x ⃗)] ,j

μw0,H −

∑ k = 1,2

RTvk ln(1 +

H

−Δgikinc)

i 2

(10)

The chemical potential of water in the empty hydrate structure as estimated according to Kvamme and Tanaka13 has been verified to have predictive capabilities. Empirical formulations for these chemical potentials are therefore redundant and maybe unphysical since chemical potential is a fundamental property. Throughout this study we approximate the right-hand side of (10) by pure water since there are no ions in the water and only limited amounts of dissolved gases. This approximation will imply a limited shift to the chemical potential of liquid water as corrected for dissolved CO2. For example the correction at 150 bar and 274 K will be −0.07 kJ/mol and slightly higher for 200 and 250 bar but still not dramatic for the purpose of this study. The free energy change related to a hydrate phase transition, ΔGH, can be written as

(7)

nH H

ΔG = δ ∑ xiH(μi H − μi p ) i=1

(11)

Superscipt H in eq 11 denotes hydrate phase property or molefraction while p denotes the same for the parent phase for the molecule i. The sum runs over all components in the hydrate phase. δ is 1 for hydrate formation and −1 for hydrate dissociation. Hydrate formation, for instance, will as minimum require that the free energy change is negative. But more rigorously it will also require that the implications of the gradients of free energy in all independent thermodynamic variables must result in negative free energy changes. For example, methane hydrate will form as long as the conditions of temperature and pressure are inside the hydrate stability zone but stability of this hydrate will also depend on the concentration of hydrate formers in the liquid water, as well water chemical potential in the hydrate former phase(s). In a pressure, volume, temperature experiment the water phase will naturally saturate in hydrate formers with reference to hydrate properties. So unless the water is replaced by undersaturated water this effect is not always seen but can be very important in real flowing situations for which the water phase may not have sufficient time to saturate with hydrate formers, due to liquid and fluid transport flux dynamics. The description of hydrate thermodynamic properties outside equilibrium due to Kvamme et al.5 can be utilized for this purpose to follow free energy gradients until the CO2/N2 phase has been mostly depleted of the most aggressive hydrate former, carbon dioxide. The analysis of eq 9 will also require the knowledge of hydrate composition. This can be found by applying statistical thermodynamic theory to the adsorption model for hydrate (left-hand side of eq 10); the composition will be given by

(8)

where subscript H denotes the hydrate phase, superscript 0 stands for the empty hydrate. vk is the fraction of cavity of type k per water molecule. For structure I hydrate, νk = 1/23 and 3/ 23 for small cavities (20 water molecules) and large cavities (24 water molecules) respectively; hik is the canonical partition function for a cavity of type k containing a “guest” molecule of type i and is given by hik = e β(μi

∑ hik)

2

∑ hik) i

RTvk ln(1 +

(T , P) + RT ln[xi ,H2Oγi ,H O(T , P , x ⃗)] = μi purewater ,H O

where subscript i refers to components; subscript j denotes the phase. In the context of this work, j is “CO2” in the case of the CO2 phase, and “H2O” for the aqueous phase, and “H” is the solid hydrate phase. For water dissolved in gas mixtures of CO2 and N2 the solubility is low enough to approximately cancel the water/water term in the attractive parameter so, at the cost of some rigor, rough estimates of liquid water drop-out can also be achieved.8 Equilibrium Thermodynamics of Hydrate. The statistical mechanical model for water in hydrate13 yields the following equation for the chemical potential of water in hydrate: μwH = μw0, H −

∑ k = 1,2

(9)

where β is the inverse of the gas constant times temperature while Δginc jk reflects the impact on hydrate water from the inclusion of the “guest” molecule i in the cavity.13 At equilibrium, chemical potential μHi has to be identical to chemical potential of molecule i in the phase from which it has been extracted. The hydrate content of all gas components can be estimated by applying eq 4 to calculate their chemical potential when dissolved in the methane phase. The typical equilibrium approximation used in most hydrate reservoir simulators is given by eq 10, within the assumption of a free hydrate former phase (gas, liquid, fluid) in which each component chemical potential is normally calculated by an

θik = E

xikH hik = vk(1 − xT ) 1 + ∑i hik

(12) DOI: 10.1021/acs.jced.5b00930 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


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where θik is the filling fraction of component in cavity type k. xHik is the mole fraction of component i in cavity type k, xT is the total mole fraction of all guests in the hydrate, and vk is, as defined above, the fraction of cavities per water of type k. Computed free energies of guest inclusion in the large cavity of structure I have been fitted to a series in inverse reduced temperature. 5

Δg inclusion =

⎡ Tc ⎤i ⎥ T⎦

∑ ki⎢⎣ i=0

consistency has been a high priority throughout this work. It was not our intention to adjust any parameters to fit experimental data. Molecular dynamics simulations reported in this work were restricted to extending the approach of Kvamme and Tanaka13 to larger hydrate systems and additional temperatures between 273.16 and 280 K, which is the temperature range in this study. Updated parameters for free energy of inclusion are given in Tables 1 to 3. Parameters for empty hydrates and ice were not significantly affected, and the parameters of Kvamme and Tanaka13 were applied. Estimates for the chemical potential of liquid water were extended from 273.15 K by means of thermodynamic relationships, and experimental data on enthalpy of dissociation and liquid water heat capacities. For more details see Kvamme and Tanaka.13

(13)

where Tc is the critical temperature of the guest molecule in question. (See Tables 1, 2 and 3 for CO2, CH4, and N2, Table 1. Coeffcient of Δginclusion in the Case of Carbon Dioxide Inclusiona k

large cavity

small cavity

0 1 2 3 4 5

14.631108552734160 −0.4714182199600883 −91.773951637448920 2.249867200986224 11.027927839813380 19.622948066966590

0 0 0 0 0 0

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RESULTS AND DISCUSSION Verification of the Model Systems. It is not the intension of this work to present a quantitative and fitted model for the CO2/N2 hydrate/fluid system. We prefer to keep the simulated (Kvamme and Tanaka13) properties of the hydrate, and a fairly simple equation of state rather than tuning the model empirically toward experimental data. Our range of interest keeps us within the gas regime for the CO2/N2 mixtures so the SRK equation of state20 is deemed accurate enough for our purpose. Results in Figures 4 and 5 below have been estimated without accounting for Poynting corrections to liquid water.

a

Critical temperature for CO2 is 304.13 K. At the present stage it is assumed that no CO2 enters the small cavity and correspondingly the coefficient is 0 for the small cavity. Units on k is kJ/mol.

Table 2. Coeffcient of Δginclusion in the Case of CH4 Inclusiona

a

k

large cavity

small cavity

0 1 2 3 4 5

15.953190676883240 −13.349289598714340 −166.526506256734500 28.068892414980430 41.945206442433370 151.1540705347040

−39.0604373928416 121.665319263939 185.436683224177 116.575797703473 −54.393598344663 −82.9143384357272

Critical temperature for CH4 is 190.56 K. Units for k is kJ/mol.

Table 3. Coeffcient of Δginclusion in the Case of Nitrogen Inclusiona

a

k

large cavity

small cavity

0 1 2 3 4 5

−45.55039046631593 18.545729464463110 −131.7881893149816 103.0832372020720 27.759086907720460 141.643579069865200

−199.1824874694727 266.23940583971190 8.448543642023667 352.2548526908923 −10.844174184315280 −60.103750139641910

Figure 4. Estimated equilibrium curves of hydrate from water and CO2 containing varying amounts of N2 (solid line). Lower curve is for pure CO2, followed by 50 mol %, 30 mol %, and 10 mol %. Corresponding experimental data are from Herri et al.26 for pure CO2 hydrate (∗) and 30 mol % (×) while experimental data for 50 mol % (○) and 10 mol % (+) are from Bouchafaa et al.27 Pressure in bars and temperature in Kelvin.

Critical temperature for N2 is 126.19 K. Units for k is kJ/mol.

And also for empty clathrate chemical potentials in eq 10 Poynting corrections are neglected. For the ranges of pressures in Figures 3 and 4, as well as for the ranges of pressures in this study the Poynting corrections to the left and right-hand side of eq 10 would both be close to unity and very similar in value. They therefore practically cancel out. We should keep in mind that the CO2 ideal gas properties are calculated from a nonspherical model.8,14,15 We are not aiming at the empirical fitting of parameters to achieve quantitative agreement in this work. Within the focus of this paper the model systems are deemed accurate enough. Estimated data in Figure 4 are limited to the range of conditions relevant to this study. It is not even recommended to estimate for regions in which pure CO2 will

respectively.) Critical temperatures for the three different guest molecule types are given in the table footnotes. There is some different experimental evidence (see for instance Kuhs et al.25) that CO2 also enters the small cavity but during our MD studies we do not find that it gives any net stabilizing effect of the hydrate. For most CO2 models we have tested the hydrate structure collapses. Whether CO2 will enter the small cavities in a highly dynamic situation of flow still remains experimentally unverified. The cavity partition function for CO2 in small cavity (eq 9) is close to zero anyway. The free energy of inclusion in eq 7 can be estimated according to Kvamme and Tanaka (1995). Thermodynamic F



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Figure 5. Estimated equilibrium curves (solid) for CO2 with different contents of CH4. From bottom: 64 mol % CO2; 52 mol % CO2; 36 mol % CO2; 11 mol % CO2. Experimental data from Herri et al.:26 (+) 64 mol % CO2, (×) 52 mol % CO2, (○) 36 mol % CO2, and (∗) 11 mol % CO2. Pressure in bars and temperature in Kelvin.

Figure 6. Estimated water chemical potential in hydrate (solid) and liquid water (dash) as a function of temperature for 80 bar and CO2 mole-fractions of 0.80, 0.6, 0.4, 0.2, 0.1, 0.05, 0.02, 0.01, with 0.80 molfraction curve lowest and 0.01 mol-fraction curve on top.

be liquid owing to limitations of SRK20 in those regions. Results for CH4 and CO2 mixtures using SRK is slightly less in agreement with experimental data than the CO2/N2 data in Figure 5 but good enough for the focus of this paper. Limits of Hydrate Stability for Mixtures of CO2 and N2. In the next section we investigate the stability of hydrates from different CO2/N2 ratios with reference to the chemical potential of water as ice or liquid water. In these calculations it is approximated that the guest molecules establish equilibrium between hydrate and gas, and that released methane arising as a result of heat generated by the new hydrate formation does not mix but escapes by buoyancy through separate pathways. Stability of mixed hydrates which also includes released methane is investigated in the subsequent section. Following that there is a section in which we examine the differences in chemical potential of methane between methane hydrate and methane diluted in CO2/N2 mixtures to shed light on the thermodynamic driving force for the dissociation of hydrate toward gas due to guest (methane) chemical potential. Hydrates dominated by CO2 as the guest species will also dissociate toward water if the chemical potential of CO2 dissolved in water is lower than the chemical potential of CO2 in the aqueous solution. Some estimates of these concentration limits are given in the last section of stability analysis. Formation of a new hydrate will increase the local salinity around the new hydrate. Experimental data from studies on this phenomena have been published elsewhere but references are included in a corresponding section below, for completeness. Hydrate Stability for Mixtures of CO2 and N2. The estimated water chemical potentials in hydrate and liquid or ice water for pressures of 80, 85, 90, 100, and 110 bar are plotted in Figures 6 to 10. For 80 bar (Figure 6) hydrate is not stable for gas mole-fraction of CO2 less than 8 mol % at 273 K and less than 26 mol % CO2 in gas at 280 K. Corresponding values for 110 bar (Figure 10) are 1% and 15%, respectively. While released heat can assist in dissociating the in situ hydrate, the estimated results in Figures 6 to 10 also makes it possible for newly formed CO2 dominated hydrate to redissociate due to low CO2 concentration in the surrounding gas and local temperature increase from the released hydrate formation heat. This is of course a positive result with respect to reducing pore blockage due to new hydrate formation but also represents an efficiency limitation



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Figure 11. Estimated water chemical potential in hydrate (solid) and liquid water (dash) as a function of temperature for 80 bar for a gas mixture containing 10 mol % CH4 and varying content of CO2 (rest N2). Mole-fractions of CO2 are 0.80 (lowest solid curve), 0.6, 0.4, 0.2, 0.1, 0.05, 0.02, 0.010 (highest solid curve).


Figure 12. Estimated water chemical potential in hydrate (solid) and liquid water (dash) as a function of temperature for 80 bar for a gas mixture containing 18 mol % CH4 and varying content of CO2 (rest N2). Mole-fractions of CO2 are 0.80 (lowest solid curve), 0.6, 0.4, 0.2, 0.1, 0.05, 0.02, 0.010 (highest solid curve).

The trend in Figures 6 to 10 is as expected as the lower limit of CO2 content for hydrate formation decreases with increasing pressure. For example, at 280 K, the minimum mole-fraction CO2 required for hydrate to be stable is 0.15 at 110 bar. Similar content limits at the same temperature at 100 bar are 0.18, at 90 bar it is 0.22, at 85 bar it is 0.25, and finally at 80 bar it is 0.30. Hydrate Stability for Mixtures of CO2, N2, and CH4. After the exchange has started the mixture of CO2 and N2 gas will be diluted with released CH4. Within the limited space of this paper we therefore investigate stability limits for two gas mixtures with 10 (Figure 11) and 18 (Figure 12) mole % CH4 respectively. At 280 K roughly 20% CO2 in the gas phase is needed to keep the mixed hydrate stable with 10% CH4 in the gas phase, while roughly 15% CO2 in the gas phase is needed to keep the CO2 dominated mixed hydrate stable with 18% CH4 in the gas phase. Gas Phase Undersaturated with CH4. On the basis of results from the previous sections it is observed that CH4 will assist in construction of a new mixed hydrate if the released CH4 mixes in with the CO2/N2 mixture. However, if the heat

generated dissociation of the in situ CH4 hydrate leads to CH4 disappearing through other pathways, which do not lead to mixing with the inflowing gas then the chemical potential difference between CH4 in the hydrate will be higher than the chemical potential of CH4 in the gas at infinite dilution. This can happen since more water than CH4 is released by dissociation, and we cannot disregard that small films of water would act as temporary barriers for mixing between the CH4 and the incoming gas to a certain extent. Some examples of driving forces between methane in hydrate and methane in the surrounding gas, at low concentration of CH4 are illustrated in Figure 13 for 1% CH4 in a CO2/N2 mixture. The minimum pressure on the CH4 equilibrium chemical potentials is 35.7 bar at 274 K, and the maximum pressure is 62.7 bar at 280 K. The values of the dashed curve should therefore have been slightly higher due to a Poynting correction H



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Figure 13. Estimated chemical potential for CH4 in equilibrium hydrate (−) versus chemical potential for 1 mol % CH4 mixed into a gas phase originally containing 1 mol % CO2 and remainder being N2 (---) at 80 bar pressure. Pressures on the solid curve are the estimated equilibrium pressures for pure CH4 hydrate.

Figure 14. Estimated minimum mole-fraction of CO2 in liquid water needed to keep CO2 hydrate stable. Note that since both water and CO2 are extracted into hydrate from aqueous solution the pressure sensitivity is very small and approximated to be negligible.

free energies under constraints of mass- and heat-transport, as distributed across all coexisting phases. Additional Remarks. Hydrates in porous media cannot reach equilibrium. One part of the reason is the Gibbs phase rule, which is simply conservation of mass distributed over all coexisting phases and under constraints of equilibrium. Mineral surfaces induce an active adsorbed phase for hydrate due to the favorable heterogeneous hydrate nucleation toward mineral surfaces. Hydrate water molecules, and the fairly rigid distribution of partial charges on oxygen and hydrogen, will lead to a structuring of surrounding water. On the basis of samplings from Molecular Dynamics simulations the thickness of this water structuring is roughly three water molecules. This layer of structured water might be considered as an adsorbed phase in which both density and structure is different from both liquid water and hydrate. As long as the structure and density is different it is, by definition, a separate phase. These two adsorbed phases (adsorbed on mineral and adsorbed on hydrate core) are also active in keeping a distance of hydrate from the mineral surfaces of a minimum of roughly 2.5 nm. These water molecules have low mobility and will bridge the hydrate to the mineral surface unless fluid flow results in a larger distance between mineral surfaces and hydrate surfaces. With one gas phase, liquid water, two hydrate phases (in situ CH4 hydrate and hydrate forming from the incoming CO2/N2 mixture and pore water), and two adsorbed phases, the number of active phases of importance for hydrate is 6. With four components in the system the degrees of freedom is zero, while two independent thermodynamic variables, temperature and pressure, are defined by hydrostatics, hydrodynamics, and heat exchange during phase transitions. The second factor that influences the number of phases is the combined first and second laws that will drive the system toward nonuniform hydrate phases in which the first hydrates formed from the CO2/N2 mixture and water will be dominated by CO2. As discussed in this work then the resulting gas phase depleted of some CO2 will form less and stable hydrates until the N2 content is too high for hydrate to form under the local conditions of temperature and pressure. So accounting for the

from the actual P to 100 bar, which implies very minor corrections.5,8 The effects of the subsaturation effect in filling fractions for the higher pressures are also negligible.5,8 In situ CH4 hydrate will therefore dissociate toward gas due to chemical potential gradients of CH4. Hydrate Stability toward Water Undersaturated with CO2. The mole-fraction corresponding to the solubility of CO2 from a pure CO2 gas phase into liquid water is higher than the mole-fraction CO2 in liquid water needed to keep the hydrate stable.19,28 Put in another way, hydrate can grow from dissolved CO2 in water until the stability limit of mole-fraction CO2 in water is reached. Within the focus of this paper we do not need to calculate the amount of CO2 that is being dissolved into the water, which will of course depend on the partial pressure of CO2 (or more precisely fugacity of CO2) in the gas. Before injection of CO2 it is expected that the free water in the pore spaces is CH4 saturated with reference to the hydrate CH4 chemical potential. Actual CO2 content in the water will depend on geochemistry between water and carbonate. For the simplest case of pure water with no initial dissolved CO2 the estimated amount of CO2 in water needed for coexistence between hydrate and liquid water is plotted in Figure 14. Hydrate Stability Dynamics Due to Salinity. Liquid water surrounding in situ CH4 hydrate is likely stabilized toward a limited groundwater salinity. Creation of new CO2 dominated hydrate will dynamically increase the salinity in the surrounding liquid water phase, which will also contribute to the dissociation of neighboring CH4 hydrate.29 The dynamics of these processes is very complex and sensitive to rates of dilution of the free water in the local flow fields. Within the limitations of this paper we therefore avoid speculations in favor of future simulations using a reactive transport simulator30−32 for specific hydrate reservoirs. Aspects of hydrate stability due to local salinity dynamics, as well as undersaturation of CO2 in liquid water (with reference to hydrate free energy) will of course also be sensitive to the ratio of free water to hydrate water. Any experiments outside porous media must therefore be critically examined in terms of I



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papers from the same research group. With the current extended version of our PFT with hydrodynamics and heat transport it would, however, also be possible to address flow through experiments such as those reported by Deusner et al.35 The theoretical work of Dornan et al.36 based on Molecular Dynamics simulations contains interesting aspects of methodology as well as interesting results. The estimated results are, however, not directly relevant to this work since they estimate phase transition changes for very discrete “reactions”, which again would correspond to very specific filling fractions of hydrate formers. At best their results can be regarded as possible limits in certain situations also relevant to this work. But the conclusions of their work are not directly comparable to the focus and results of this work. In a practical situation of possible real production from hydrate in Alaska under very low temperatures using CO2 or mixtures of CO2 and N2 the maximum amount of water that can be permitted in order to avoid hydrate problems becomes very small unless hydrate inhibitors are used. The minimum temperatures for alcohols do not stretch far below −20 °C. Maximum limits of water that can be tolerated have been estimated based on methods reported by Kvamme and Tanaka,13 Kvamme et al.,6 and Kvamme et al.8 Water content was estimated in pure CO2. A gas mixture containing 1/3 CO2 and 2/3 N2, and pure N2. The maximum tolerance criteria are (1) liquid water condensation (Figure 15), (2) direct hydrate formation (Figure 16) and (3) water adsorbing onto a Hematite (rust) surface (Figure 17). The SRK20 equation of state has been employed for the gas phase in these calculations, and also for water dissolved in the gas as an approximation. The justification for this approximation is that the impact of direct water−water interactions are so extremely small in the attractive parameter due to very low mole-fractions of water. Slightly better values might have been achieved through the use of Gibbs−Duhem (Kvamme et al.8) but the values are considered as sufficiently accurate for illustration of potential hydrate risk. As expected the tolerance limit for water becomes practically zero for temperatures down to 223 K. Direct hydrate formation from water dissolved in the gas is thermodynamically slightly more favorable than water condensation to liquid, but not very likely in terms of necessary amounts of water needed to even accumulate to a critical size hydrate particle. And even getting rid of the hydrate formation heat through surroundings of the heat insulating gas phase is a challenge. But the preference for water to adsorb onto Hematite (Figure 17) compared to liquid water condensation totally dominates the maximum water that can be tolerated during pipeline transport. But even if the pipeline is coated or made of nonwetting material the conclusion remains the same in view of the results in Figures 15 and 16. It should be noted that the molecular dynamics simulations involved in estimations of the parameters for free energy change due to inclusion in Tables 1 and 2 as well as for methane were limited down to 253 K. The impact of extrapolations down to temperatures below 253 K is uncertain and corresponding results should be interpreted with some caution. 220 K is also closer to the limits of validity of classical mechanics for these systems. From the studies by Kvamme and Tanaka,13 it appears that the limits for classical assumptions get increasingly questionable at temperatures below 200 K. Produced gas will also contain water even under the optimistic assumption of no free water phase following the gas flow. A similar evaluation of possible risk for hydrate formation from produced methane containing dissolved water

in situ CH4 hydrate and a range of newly formed hydrates from incoming gas free water in the pores, there are very many different phases in the system and these have all different free energies. The thermodynamic models, and results, presented here extend the thermodynamic dependency of hydrate stability beyond temperature and pressure to include the other independent variables, specifically concentrations (and corresponding chemical potentials) of all components entering the hydrate in all coexisting phases. We have so far not included hydrate instabilities associated with water phase undersaturated with nitrogen. Solubility of nitrogen in water is extremely low and chemical potential for nitrogen in water can to a fair approximation be set to infinite dilution value as either estimated according to methods described by Kvamme et al.,6,7 or by analysis of solubility data. It may even be an acceptable approximation to keep this value unaffected by the impact of lowered pH due to CO2 partial conversion into bicarbonate and carbonate. The lowered pH is expected to decrease nitrogen solubility. The effect of nitrogen in the water phase will be examined in a future extension of this work. Such an extension would naturally also include an analysis that could address available experimental results. Results and observations published by Schicks et al.33 cannot be directly analyzed and compared to the work described so far in this paper since there is no coupling to mass- and energybalances for a closed system that could tell if some component would become undersaturated in some of the coexisting phases. The result presented here and extended also to the water phase chemical potentials of all components would provide the necessary models and corresponding parameters for the thermodynamic driving forces for individual mass transport between phases (hydrate, water, fluid). For equilibrium systems this would imply incorporating the results from this work (extended with also water phase chemical potentials of carbon dioxide and nitrogen) into a solution of phase equilibrium under constraints of mass and energy. In engineering language this is called a 3-phase “flash”-calculation. For a nonequilibrium system the equivalent will be a free energy minimization under constraints of mass- and energy-conservation. As discussed in this paper there will be a selectivity in nucleation of hydrate dominated by carbon dioxide so that there will be a range of hydrate of different compositions formed as a consequence of first and second laws. This will continue until no hydrate can be formed from the supply of hydrate formers, neither from the gas side nor from hydrate formers dissolved in the water phase. There are two criteria for unconditional stability of a hydrate. All hydrate forms must have lower free energy than the surrounding fluid phases. Second all consequences of gradients in free energy lead to no positive contributions to free energy change compared to surrounding phases. Practically this raises questions such as whether the gas phase can be saturated with water or the liquid water phase can be saturated with both nitrogen and carbon dioxide. The questions above cannot be answered by thermodynamics alone and solution of the coupled dynamics of phase transition dynamics, mass-transport and heat-transport is needed. Extension of this work involves bringing the data and models over to a Phase Field Theory (PFT) analysis.1,2,5,8 In the absence of dynamic modeling along those lines, or similar theoretical approaches, it will not be possible to relate the work presented here to the experimental observations of Lee et al.,34 and to results reported in other J



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Figure 16. Estimated maximum water in gas before dropping out directly as hydrate from (a) pure CO2, (b) 1/3 mol-fraction CO2 and rest N2, and (c) pure N2. Each solid line is for a constant pressure of 50, 90, 130, 170, 210, and 250 bar, counted from top and down at highest T.

Figure 15. Estimated maximum water in gas before condensing out as liquid from (a) pure CO2, (b) 1/3 mol-fraction CO2 and rest N2, and (c) pure N2. Each solid line is for a constant pressure of 50, 90, 130, 170, 210, and 250 bar, counted from top and down at highest T. K



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results in very similar conclusions as for the incoming gases, or gas mixtures of carbon dioxide and nitrogen. For temperatures below 253 K the tolerance for water is practically zero for rusty pipelines and even for pipelines with a nonwetting surface. (See Figures 18 to 20 for evaluation of maximum water tolerance in gas mixtures of CO2, N2, and CH4.)

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CONCLUSIONS The thermodynamic stability of hydrates is a function of all independent thermodynamic variables, which include the concentrations of all components that enter the hydrates in all the coexisting phases. The primary purpose of this study was to shed more light on whether the addition of N2 to CO2 would result in a win−win situation of simultaneous CH4 production and safe long-term storage of CO2. This work focused on the changes in free energies, taking into consideration all independent thermodynamic variables rather than only pressure and temperature, which is commonly the limit of reservoir simulators that handle these systems. Results need to be evaluated in the context of mass- and heat-exchange. If the system could reach equilibrium then the appropriate suggestion would be to conduct a multiphase “flash”-calculation. This would provide equilibrium distributions of all components over all possible coexisting phases. As discussed in this work hydrates in porous media will not be able to reach equilibrium. A nonequilibrium approach instead of a “flash”-calculation will therefore involve the minimization of free energy under constraints of mass- and energy-conservation. The results of this work have not yet been completely implemented into a reservoir simulator for hydrate production but work is in progress. The fastest mechanism for exchange between in situ CH4 hydrate and injected CO2/N2 mixtures is through formation of a new hydrate. In this work we have examined hydrate formed from different mixtures of CO2, N2, and CH4 under the assumption that gas phase chemical potentials are the same in hydrate but with no equilibrium constraints on water. This makes it possible to examine differences in chemical potential for water in hydrate with liquid water chemical potential in terms of stability limits. Because of the thermodynamic nonequilibrium of hydrates in porous media, selective adsorption on liquid water and on mineral surfaces, and the combined first and second laws of thermodynamics, it is argued that CO2 will be depleted from the CO2/N2 gas mixtures under the creation of a new hydrate. For pressures as low as 80 bar this requires approximately 30 mol % CO2 in the gas mixture at 280 K. For 85 bar and the same temperature the minimum CO2 content in gas is 27 mol %. Corresponding limits for 1 °C are 6 mol % and 5 mol %, respectively, for 80 and 85 bar. This range of pressures covers the average pressure range during the injection period in the Ignik Sukomu pilot test for production of CH4 using a CO2/N2 mixture (77.5% N2 by volume). These estimates assume that released CH4 does not mix in but finds its own pathways. Also, CH4 hydrate will dissociate in chemical potential gradients of CH4 as diluted in CO2/N2 gas surroundings. Newly formed CO2-dominated hydrate will also be subject to concentrations of CO2 in the surrounding liquid water, and hydrate will redissociate if the surrounding water concentration drops below certain limits, given chemical potentials of CO2 in hydrate versus chemical potential in liquid water. Formation of new hydrate will also lead to locally increased salinity which affects stability of the newly formed hydrate as

Figure 17. Estimated maximum water in gas before adsorption onto rust (modeled as hematite) from (a) pure CO2, (b) 1/3 mol-fraction CO2 and rest N2, and (c) pure N2. Each solid line is for a constant pressure of 50, 90, 130, 170, 210, and 250 bar, counted from top and down at highest T. L



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Figure 18. Estimated maximum water in gas before condensing out as liquid from (a) pure CH4, (b) 0.5 mol-fraction CH4 and rest N2, and (c) 0.5 mol-fraction CH4, 0.1 mol-fraction CO2, 0.4 mole-fraction N2. Each solid line is for a constant pressure of 50, 90, 130, 170, 210, and 250 bar, counted from top and down at highest T.

Figure 19. Estimated maximum water in gas before direct hydrate formation from (a) pure CH4, (b) 0.5 mol-fraction CH4 and rest N2, and (c) 0.5 mol-fraction CH4, 0.1 mol-fraction CO2, 0.4 mole-fraction N2. Each solid line is for a constant pressure of 50, 90, 130, 170, 210, and 250 bar, counted from top and down at highest T. M



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well as any in situ methane hydrate close by. The dynamics of these processes are very complex and unique for the local flow situations in a specific reservoir. We propose the use of a reactive transport simulator (RCB30−32) with nonequilibrium thermodynamics for incorporation of these effects, as well as the impact of hydrate instabilities under different chemical potential gradients as discussed in this study. In summary there are several mechanisms related to the injection of CO2/N2 mixtures that can dissociate in situ CH4 hydrate. It is more uncertain whether the use of CO2/N2 mixtures will lead to safe long-terms storage of CO2. Since CO2 gets depleted from the injection gas by several mechanisms, such as dissolution into water, adsorption on minerals, and formation of new CO2-dominated hydrate, there is obviously a dependency on injection rates and subsequent fluxes of the injected gas mixture on where in the reservoir the injected gas may be unable to form new hydrate with available free water. These specific rates cannot be evaluated theoretically without implementation of the thermodynamic models into a reservoir simulator. Depending on injection rates, in situ distribution of hydrate, the completion (multiwell as in contrast to “huff and puff” as in Ignik Sikumi10), and CO2/N2 ratio, it may end up with no remaining hydrates of either the in situ CH4 hydrate or CO2-based hydrates from the injection gas. This possibility increases proportionally to the relative fraction of N2. Alaska, and other permafrost regions, are also harsh regions in terms of transport in pipelines on ground. Estimates indicate that the maximum water content that can be permitted for transport of CO2 and/or N2 in pipelines on ground is practically zero when temperatures approaches 253 K or lower. A similar concern goes for produced methane with varying content of nitrogen and carbon dioxide. Worst case scenarios occur for rusty pipelines.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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Figure 20. Estimated maximum water in gas before adsorption onto hematite from (a) pure CH4, (b) 0.5 mol-fraction CH4 and rest N2, and (c) 0.5 mol-fraction CH4, 0.1 mol-fraction CO2, 0.4 mole-fraction N2. Each solid line is for a constant pressure of 50, 90, 130, 170, 210, and 250 bar, counted from top and down at highest T. N

NOMENCLATURE C = number of components in the Gibbs phase rule EP = potential energy [kJ/mol] F = number of degrees of freedom in the Gibbs phase rule F = free energy [kJ/mol] f = free energy density [kJ/(mol m3)] f i = fugacity [Pa] g(r) = radial distribution function (RDF) G = Gibbs free energy [kJ/mol] Δginc kj = Gibbs free energy of inclusion of component k in cavity type j [kJ/mol] H = enthalpy [kJ/mol] hkj = cavity partition function of component k in cavity type j k = constants in polyonomial fitting of free energies of guest inclusion in eq 13 K = ratio of mole-fraction gas versus mole-fraction liquid of the same component (gas/liquid K-values) Ni = number of molecules N = number of phases in the Gibbs phase rule P = pressure [Pa] P0 = reference pressure [Pa] r = distance [m] DOI: 10.1021/acs.jced.5b00930 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


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R = molar gas constant [kJ/(K mol)] T = temperature [K] vj = no. of type j cavities per water molecule vm = molar volume [m3/mol] V̅ r = molar volume of rth component [m3/mol] V̅ clath = volume of clathrate [m3] x = mole fraction in liquid or adsorbed, arrow on top denote vector y = mole fraction in gas, arrow on top denote vector yw = mole fraction of water Y = residual chemical potential per Kelvin z = mole fraction α = liquid (water) phase fraction β = inverse of the gas constant times temperature μ = chemical potential [kJ/mol] μ0,H = chemical potential for water in empty hydrate w structure [kJ/mol] θkj = fractional occupancy of cavity k by comp. j γ = activity coefficient ϕ = fugacity coefficient

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N2 Mixture Injected into CH4

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