Thermodynamics of freezing soil closed system saturated with gas and water

Journal Pre-proof Thermodynamics of freezing soil closed system saturated with gas and water

Vladimir A. Istomin, Evgeny M. Chuvilin, Daria V. Sergeeva, Boris A. Buhkanov, Christian Badetz, Yulia V. Stanilovskaya PII:

S0165-232X(19)30390-8

DOI:

https://doi.org/10.1016/j.coldregions.2019.102901

Reference:

COLTEC 102901

To appear in:

Cold Regions Science and Technology

Received date:

24 June 2019

Revised date:

6 September 2019

Accepted date:

19 September 2019

Please cite this article as: V.A. Istomin, E.M. Chuvilin, D.V. Sergeeva, et al., Thermodynamics of freezing soil closed system saturated with gas and water, Cold Regions Science and Technology(2018), https://doi.org/10.1016/ j.coldregions.2019.102901

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Journal Pre-proof Thermodynamics of freezing soil closed system saturated with gas and water a,b a,c, a a Vladimir A. Istomin , Evgeny M. Chuvilin *, Daria V. Sergeeva , Boris A. Buhkanov , d

Christian Badetz , Yulia V. Stanilovskaya

d

Skolkovo Institute of Science and Technology (Skoltech), Russia

b

Research Institute of Natural Gas and Gas Technologies (Gazprom VNIIGAZ JSC), Russia

d

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Department of Geology, Moscow State University (MSU), Russia Total S.A., France

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*Corresponding author: [email protected]

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Abstract

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Natural eruptions in shallow permafrost with formation of large craters reported lately from gas-producing regions of the Russian Arctic may result from pressure buildup in freezing closed

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zones of gas-saturated unfrozen soil (taliks). Under certain conditions, increasing pressure may

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lead to hydraulic fracture of the overlying permafrost, with ensuing eruption of talik material and crater formation. The conditions of pore moisture freezing in gas-saturated taliks and related pressure buildup have been modeled by thermodynamic calculations. The model uses elegant equations to provide constraints on the freezing temperature of pore fluids containing dissolved salts and gases, which depends on free gas pressure, composition and contents of dissolved pore gases (methane, carbon dioxide, and their mixtures), and salinity. Pressure increase in these conditions is limited thermodynamically by the onset of gas hydrate formation.

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Journal Pre-proof Key words: closed talik, gas-water pore fluid, freezing temperature, gas solubility, pressure buildup, gas hydrate formation, methane, carbon dioxide, nitrogen

1. Introduction A new natural phenomenon has been discovered recently in the zone of continuous permafrost in the north of Western Siberia, where gas outbursts from shallow permafrost produce

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craters as large as tens of meters in diameter and 30-60 m deep (Bogoyavlensky, 2014; Kizyakov

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et al., 2015; Olenchenko et al., 2015; Buldovicz et al., 2018). A prominent case of gas outburst

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occurred in 2014 near the Bovanenkovo gas and condensate field in the Yamal Peninsula (Fig. 1). Currently, about ten craters presumably associated with gas blasts are known from the Yamal and

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Gydan Peninsulas (Bogoyavlensky et al., 2016; Kizyakov et al., 2017). Such craters may be

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formed by explosion in freezing of unfrozen soil lenses under lakes (sub-lake taliks) saturated

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with gas and water. Freezing causes gas concentration and creates closed gas-filled porous zones exposed to increasing pressure, which leads to explosion by the hydraulic fracture mechanism

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(Buldovicz et al., 2018; Istomin et al., 2018).

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The scenario of pressure buildup in a freezing closed talik with high content of pore gas may be generally as follows (Fig. 2). First, a closed talik forms under a thermokarst lake and becomes satuarted with gas. When the lake becomes shallower, the talik begins to freeze up, first from below and from the sides and then also from above as the lake shrinking progresses. The confined freezing produces an increasingly pressurized lens saturated with gas and water and thus causes heaving of frozen soils above the talik. The arising frost mound explodes (like hydraulic fracture), provided that the volumetric content of pore gas is high enough.

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gas in a freezing talik may either be released from microbially decaying organic matter or from

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Journal Pre-proof dissociating relict gas hydrates, or it may come from greater depths through highly permable zones. As the closed talik freezes up, the pressure of gas (of whatever origin) increases due to:  cryogenic concentration or expulsion of gas inward the shrinking talik, while the volume of free gas reduces;  decrease in molar volume of water converted to ice during freezing;  increase in the content of free gas at the account of dissolved gas released during freezing

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of liquid pore water.

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In the absence of free gas, even minor volume reduction in a freezing water-saturated

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closed system leads to rapid pressure increase. In such systems would rather produce frost

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mounds (pingoes) than explosive craters (Mackay, 1998). The pore gas in sub-lake taliks consists mainly of methane, with minor carbon dioxide and nitrogen, and occasional traces of

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heavier methane homologs (Are, 1998; Kuzin, 1999; Bondarev et al., 2008). Correspondingly,

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the model below simulates a talik system with pore fluids containing methane, carbon dioxide, nitrogen, and their mixtures. Freezing of increasingly pressurized gas- and water-saturated soil

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has not widely studied before. A single relevant model we know concerns freezing of a closed

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water-gas volume (Sigunov et al., 2006). Thermodynamic calculations below address equilibrium in the system ‘gas – saline water – ice – hydrate’ for freezing gas-saturated porous materials. The problem has never been treated before, except for a few theoretical and experimental works on freezing temperature estimation. Thermodynamic calculations were applied previously to aerated water in sub-ice Lake Vostok in Antarctica (Lipenkov and Istomin 2001; Lipenkov et al., 2003) under a hydrostatic pressure of 35-40 MPa, while experiments focused on pore fluids with dissolved gases (carbon dioxide and methane) and gas hydrates (Chuvilin et al., 2003a,b; Guryeva, 2011; Mel’nikov et al., 2014). The

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Journal Pre-proof present paper continues and generalizes our recent thermodynamic studies in this line (Istomin et al., 2009, 2017 a, b) which yielded temperature curves for unfrozen and nonclathrate pore water based on its measured activity.

2. Pressure dependent freezing temperature of gas-bearing pore fluids Thermodynamic calculations are performed for the freezing temperature of pore fluids

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containing a hydrate-forming gas phase (methane, carbon dioxide, nitrogen, and their mixtures,

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as well as minor amounts of other hydrocarbon gases). The melting temperature of ice depends

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on hydrostatic pressure: pressurized ice melts at a lower temperature because ice is less dense

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than water (hence, the molar volume of frozen water exceeds that of liquid water). Thus the icewater thermodynamic equilibrium in the P-T coordinates slopes to the left. A shift of melting

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temperature from 273.15 K to 272.15 K (1 K) requires an external pressure of ~13.5 MPa. The

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slope at 273.15 K (dP/dT derivative) can be found from the Clausius-Clapeyron equation which includes the enthalpy of ice melting, as well as the ice-water molar volume difference. The

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melting temperature of hexagonal ice as a function of pressure was measured long ago

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(Bridgman, 1912) till a pressure of 210 MPa and a temperature of -22 °C. It is important in the context of our study that the three-phase equilibrium ‘ice–gas-bearing water–gas’ in the case of gas pressure (Lipenkov et al., 2001, 2003) differs markedly from that of ‘ice–water’ under hydrostatic pressure. The reason is that the phase equilibrium is controlled by gas solubility. Calculations with regard for gas solubility in water give a lower freezing temperature for gas-bearing fluids under gas pressure than for pure water at the same external pressure. The effect is prominent for highly soluble gases (CO2 and H2S) but has to be taken into account also for methane and nitrogen. The presence of dissolved components lowers the

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Journal Pre-proof freezing point of aqueous solutions at atmospheric pressure. The freezing temperature is additionally controlled by salinity of pore fluids and by the pore space structure (specific active surface area of pores): the lower the content of pore water in hydrophile soil, the lower its freezing temperature. This effect can be estimated thermodynamically using measured activity of pore water depending on its content (Istomin et al., 2009, 2017b). Thus, the freezing temperature of soil moisture depends on its salinity, contents of dissolved gases, external pressure (gas

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pressure in our case), and pore water interaction with soil skeleton.

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The phase equilibrium of the system ‘aqueous solution (water with dissolved gases and

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salts) – gas – ice’ exposed to pressure of gas (or a gas mixture) is found using quality of pore

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water (μw) and ice (μice) chemical potentials. It leads to an equation that relates temperature and pressure at a thermodynamic equilibrium of three phases: ice, water with dissolved gases and

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salts, and gas.

gases and salts is: )

(

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(

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In the general case, the chemical potential of water μ w (P, T) in a fluid containing also

)

(

)

( )

̅

(

)

(1)

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where P is the external pressure (gas pressure in our case) applied to the thermodynamic system, MPa; P0 = 0.101325 MPa is the atmospheric pressure; T is the temperature, K; T0 = 273.15 K; μw0 (P0, T) is the chemical potential of water at the atmospheric pressure P0 and the temperature T (at T < T0 it is the chemical potential of supercooled water, a metastable phase). The molar fraction of dissolved gases xg is ∑ where xi is the molar fraction of the i-th gas in a mixture; R is the universal gas constant (8.3146 J/mol K); a is the water activity in a saline pore fluid (electrolyte) at the atmospheric 5

Journal Pre-proof pressure P0; and ̅ is the partial molar volume of water in the fluid, cm3/mol. Note that equation (1) is approximate and implies joint effect of dissolved gases and salts on the chemical potential of pore water. This assumption is valid to a high accuracy at low salinity (within 30-40 g/L) and at high contents of pore moisture. The activity a of water in such fluids can be found from the approximate equation

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, where a1 and a2 are the activities of water associated with salinity and specific active

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surface area of pores, respectively. This equation stems from the assumption that the two

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variables contribute additively to the chemical potential of pore water, and their activities are

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multiplied together.

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At high water contents in a freezing talik, a2 is ~1. For hydrophilic porous materials a2 < 1 and depends on water content. The dependence of a2 on weight water content can be constrained

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in special experiments (like dew point measurements of vapor pressure). Previously we

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investigated this dependence experimentally when studied the behavior of unfrozen water (Istomin et al., 2017b) and hydrate formation (Istomin et al., 2017a) in porous materials. The

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variable xg in (1) refers to equilibrium gas solubility in water and can be found using the equation of Krichevsky-Kazarnovsky (Krichevsky et al., 1935), with empirical temperature-dependent solubility coefficients and partial molar volumes of dissolved gases. Furthermore, gas solubility can be calculated with modeling software using phase equations for multi-component mixtures. Note also that the equilibrium gas solubility depends neither on salinity nor on the pore space structure. The former effect is described approximately by the empirical equation of Sechenov (Namiot, 1991) and the latter one remains poorly constrained. Proceeding from general thermodynamic considerations, the specific active surface area of pores is expected to be lower 6

Journal Pre-proof than in bulk water for dry materials but can be neglected for wet soils, at the specified temperature and pressure. Another assumption is that the partial molar volume of water coincides with that of pure water

in the pore fluid almost

. This assumption is valid to a high accuracy for fluids that

contain dissolved gases and moderate amount of salts (within 40-50 g/L). Water, with its assumed ~1 g/cm3 density and 18.015 g/mol molecular weight has the partial molar volume Vw =

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18.015 cm3/mol. A more exact equation for the water molar volume (Bogorodsky et al., 1980)

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includes the effects of temperature and external pressure, but they can be neglected within the

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ranges of our interest (263-273 K and 3-4 MPa). The chemical potential of ice as a function of

)

(

)

(

),

(2)

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(

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temperature and pressure μice (P, T) is

where μice0(P0, T) is the ice chemical potential at atmospheric pressure P0 and temperature

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T; Vice is the molar volume of ice, cm3/mol, which is 19.65 cm3/mol, assuming an ice density of

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0.917 g/cm3. The pressure and temperature dependence of the ice partial molar volume (Bogorodsky et al., 1980) can be neglected, as in the case of water. Equation (2) does not

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include solubility of gas in ice. It can be accounted for by adding the term RTln (1 – xice), where xice is the molar content of gas in ice (found empirically as a function of gas pressure). However, gases, except for hydrogen, helium, and neon, cannot dissolve in ice because their molecules (atoms) are larger than channels in the hexagonal ice structure. The gases we consider (CH4, CO2, N2, and methane homologues) are almost insoluble in ice, and the respective term is omitted in (2). The difference between the chemical potentials of water μ w0(P0, T) and ice μice0(P0, T) at atmospheric pressure is 7

Journal Pre-proof (

)

(

)

(

).

Its temperature dependence at P0 is given by the Gibbs-Helmholz equation (

(

)

)

( )

The water-ice enthalpy difference is ( )

( )

( )

(

).

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( )

The enthalpy of ice melting Δh (T0) at T = 273.15 K and P = P0 is assumed to be 6008

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J/mol (Petrenko et al., 2002); other published enthalpy values differ from one another for ~0.1

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% on average. The water-ice heat capacity difference Δc depends on temperature. This

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dependence is however negligible for the narrow temperature range around 273.15 K (from

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263.15 to 278.15 K), i.e., Δc = 36.93 J/mol at T0 (Petrenko et al., 2002). With the above assumptions, integration of the Gibbs-Helmholtz equation leads to a

)

(

)

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(

)

(

) at

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(

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temperature dependence of the chemical potential difference between water and ice

( ) (

)

(

)

:

(

(

(

))

(

))

(3)

Δh (T0) and ΔС are assumed to be 6008 J/mol and 36.93 J/mol K, respectively. The fluid (containing dissolved gases and salts in the general case) reaches thermodynamic equilibrium with ice at the gas pressure P when (

)

(

)

(4)

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Journal Pre-proof Using equations (1) and (2), we obtain from (4) that (

)

(

̅

)

(

)

(

(

)

)

or )

(

) ̅

(

)

)

(

))

)

(

̅

)

(

)

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(

), taking into account (3), we obtain

(

(

)

)

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(

(

(

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)

)

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(

Since

(

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(

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With transformed right-hand side, the latter equations becomes )

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(

(

(

(

)

( (

))

( )

)) (

(

)

),

(5)

̅ .

where

Given that T = Tfr, where Tfr is the freezing temperature (K) of gas-bearing pore fluid, equation (5) eventually becomes

(

)

(

(

))

( ) ( ) 9

Journal Pre-proof where

(

)

(

(

)

)

Equation (6) is the main result, which allows calculating the freezing temperature of a pore water fluid containing dissolved gases and salts (T = Tfr) at a specified gas pressure P ≥ P0. Thus, it is applicable to pore fluids at different gas pressures, gas compositions, and salinities. In the specific case at a = 1 and xg = 0, equation (6) describes the water-ice equilibrium

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under hydrostatic pressure. If the external pressure is produced by gas, the calculations have to

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include equilibrium solubility of gas in water xg (found as shown above). Differentiation of (6)

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along the equilibrium line gives a generalized Calusius-Clapeyron equation, which becomes the common equation for the water-ice equilibrium under hydrostatic pressure in the case of pure

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water free from gas and salts.

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Note that equation (6) describes the relation between T = Tfr and P irrespective of

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thermodynamic equilibrium between water and gas. Thus, it is also applicable to pore water fluids undersaturated or oversaturated with respect to gas, as well as to fluids equilibrated with

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gas hydrates (discussion of these issues is beyond the scope of this paper).

3. Thermodynamic calculations Thermodynamic calculations are applied below to different cases of practical interest. Equation (6) includes equilibrium gas solubilities in pressurized conditions, at temperatures near 273 K. Pressure- and temperature-dependent gas solubility in water (aqueous solutions) can be found with the existing software using equations of state for fluids which describe phase equilibrium in hydrocarbon systems, as well as some correlations (Namiot, 1991) following Henry’s law in its generalized thermodynamic formulation (Krichevsky et al., 1935). The available published data on gas solubility refer to temperatures above 25 °С, and the Henry’s law 10

Journal Pre-proof solubility constants have to be extrapolated to the range of our interest (from -10 to 0 °С), which may cause up to 5–10% error. Extrapolation may be more successful with recent empirical data on gas solubility at low pressures and at temperatures from 5 to 15 °С, i.e., within the subhydrate PT domain. The behavior of pressure-dependent solubility is modeled for a CH4+CO2 mixture at 0 °С (Figs. 3). The same calculations are performed for pressures corresponding to hydrate stability, i.e., the model includes the metastable equilibrium ‘gas-bearing water – gas – ice’ in the

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zone of possible hydrate formation. Further calculations constrain the conditions of gas hydrate

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formation for methane, carbon dioxide, nitrogen, and their mixtures that form cubic hydrates I,

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using an thermodynamic model (Istomin et al., 1996). The curves of three-phase equilibrium

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‘gas – water – hydrate’ and ‘gas – ice – hydrate’ for methane, carbon dioxide, nitrogen, and their mixtures (Figs. 4-8) show that methane and carbon dioxide form hydrates near 273 K at

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~2.6 MPa and ~1.2 MPa, respectively. Then the freezing temperature of gas-bearing pore fluids

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is estimated using equation (6), taking into account gas solubility at T <0 °C and hydrate formation conditions. The P-T diagram (Fig. 9) covers the zone of possible hydrate formation,

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i.e., the zone of metastable equilibrium (as if no hydrate formed); the equilibrium for a four-

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phase system including hydrate is marked by dots. The presence of carbon dioxide in a gas mixture influences markedly the freezing temperature of pore fluids: at 1.0 MPa, a CO2-bearing pore water fluid freezes up at –1.4 °C. According to the available filed data, pore gas in freezing closed taliks consists mainly of methane and minor carbon dioxide, and the gas pressure does not exceed 2.0 – 2.5 MPa being limited by the onset of hydrate formation. The available published models of gas fracture show that this pressure is far enough to be responsible for the observed cryovolcanism and formation of natural craters. Joint action of gas pressure, solubility of gases (methane and carbon dioxide),

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Journal Pre-proof and pore water salinity on freezing temperature (Figs. 10 and 11) was calculated using Sechenov’s equation (Namiot, 1991), taking into account that gas solubility is lower in saline fluids. Pressure buildup in a freezing closed talik can be modeled in a simplified thermodynamic formulation (without explicit frontal freezing and geomechanics), in order to estimate the effect of a free gas phase on pressure increase upon partial water-to-ice conversion (and shrinking of the unfrozen zone).

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At the time when the talik becomes closed and exposed to confined freezing, the system

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comprises wet soil and methane (hereafter the variables that refer to this system are marked by

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subscript 1). The unfrozen zone is shrinking with time starting from the volume V1 at the onset of

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confined freezing; the pore space is occupied by equilibrated liquid water and gas phases; pore pressure is P1. Dissolved gas released during water-to-ice conversion (freezing) becomes a free

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gas phase. The current pressure P depends on the fraction of water converted to ice (δ, u.f.) in the

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volume V1. This dependence is found using a mass balance equation (Fig. 12), assuming a 0.25 MPa starting pressure of gas (methane) in the system. Gas in the unfrozen zone occupies

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different volume percentages of the pore space (from 1 % to 10%). The system in its initial state

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(V1) is assumed to contain both free and dissolved gases; dissolved gases release and become free during freezing. The pressure increase is rapid (the system is rigid) at low initial gas contents but slow at high gas contents: it reaches 1.0 MPa and higher upon considerable shrinking of the unfrozen zone.

4. Conclusions Freezing temperature of saline gas-bearing pore fluids exposed to pressure of gases or gas mixtures has been estimated by thermodynamic calculations with a new method, for different gas

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Journal Pre-proof compositions and fluid salinities. The freezing temperature of pore water fluids depends on external gas pressure and contents of dissolved gas and salts. The calculations have been performed for aqueous fluids containing methane, carbon dioxide, and their mixtures, under a large range of pressures. Gas solubility affects phase equilibrium in the system ‘ice – gas-bearing water’, especially in the presence of CO2. Pressure buildup in a freezing closed talik as a function of freezing coefficient (fraction of

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water converted to ice) is described by a simplified model, taking into account gas solubility.

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Pressure increases rapidly during freezing at low initial gas contents, while a considerable

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volume reduction of the unfrozen zone is required to reach 1.0-1.5 MPa and higher at high gas

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contents.

The maximum pressure in the system is limited by the hydrate formation conditions: it

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cannot exceed 2.0-2.5 MPa in a freezing talik system with CH4 and CO2 gases, which is

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Acknowledgements

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sufficient to cause hydraulic and gas fracture in shallow permafrost.

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This is a joint study by the Skolkovo Institute of Science and Technology (Russia) and the energy company Total (France), partly supported by the Russian Science Foundation (Project # 18-77-10063).

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Russian).

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Journal Pre-proof Figure caption. Fig. 1. A crater located 30 km from the Bovanenkovo gas field on the Yamal Peninsula. Photo credit : the Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the Russian Academy of Sciences (IPGG SB RAS).

Fig. 2. Pressure buildup in a freezing gas-saturated talik. Stage I ‒ formation of a gas-

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saturated sub-lake talik; stage II ‒ onset of freezing during lake shrinking; stage III ‒ cryogenic

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gas concentration and onset of frost heaving.

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Fig. 3. Pressure-dependent solubility of gases (methane, carbon dioxide) and their mixtures in water at 0 °C. Curves from I to V correspond to different gas phase compositions: 100 % CH4

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(I), 75 % CH4 + 25 % CO2 (II), 50 % CH4 + 50 % CO2 (III), 25 % CH4 + 75 % CO2 (IV), 100 %

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CO2 (V).

Fig. 4. Three-phase equilibria ‘methane – ice – hydrate’ and ‘methane – water – hydrate’.

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Dash line is metastable equilibrium ‘methane – supercooled water – hydrate’.

Fig. 5. Three-phase equilibria ‘carbon dioxide – ice – hydrate’ and ‘carbon dioxide – water – hydrate’. Dash line is metastable equilibrium ‘carbon dioxide – supercooled water – hydrate’.

Fig. 6. Three-phase equilibria ‘nitrogen – ice – hydrate’ and ‘nitrogen – water – hydrate’. Dash line is metastable equilibrium ‘nitrogen – supercooled water – hydrate’.

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Journal Pre-proof Fig. 7. Three-phase equilibria ‘gas – water (ice) – hydrate I’ for methane, carbon dioxide and their mixtures. Curves from I to V correspond to different gas phase compositions: 100 % CH4 (I), 75 % CH4 + 25 % CO2 (II), 50 % CH4 + 50 % CO2 (III), 25 % CH4 + 75 % CO2 (IV), 100 % CO2 (V). Dots show four-phase equilibrium ‘gas – water – ice – hydrate’.

Fig. 8. Three-phase equilibrium ‘gas – water (ice) – hydrate I’ for methane, nitrogen and

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their mixtures. Curves from I to V correspond to different gas phase compositions: 100 % CH4

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(I), 75 % CH4 + 25 % N2 (II), 50 % CH4 + 50 % N2 (III), 25 % CH4 + 75 % N2 (IV), 100 % N2

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(V). Dots show four-phase equilibrium ‘gas – water – ice – hydrate’.

Fig. 9. Pressure-dependent freezing temperature of pore fluids containing dissolved gases

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(methane, carbon dioxide and their mixtures). Curves from I to VIII are labeled according to

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fluid phase composition: pure H2O free from dissolved gases (I), Н20 + 100 % CH4 (II), Н2О + 95 % CH4 + 5 % CO2 (III), H2O + 90 % CH4 + 10 % CO2 (IV), H2O + 25 % CH4 + 75 % CO2

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CO2 (VIII).

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(V), H2O + 50 % CH4 + 50 % CO2 (VI), H2O + 25 % CH4 + 75 % CO2 (VII), and H2O + 100 %

Fig. 10. Salinity-dependent freezing temperature of methane-bearing solution. Curves I to III are labeled according to gas pressure: 0.5 МPа (I), 1 MPa (II), and 2 МPа (III).

Fig. 11. Salinity-dependent freezing temperature of solution with carbon dioxide. Curves I to III are labeled according to gas pressure: 0.5 МPа (I), 1 MPa (II), and 2 МPа (III).

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Journal Pre-proof Fig. 12. The increase in pressure in a closed gas-saturated talick during its subsequent freezing. The color shows the degree of the pore space of the thawed soil (in percent) that gas (methane)

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occupied before the beginning of the closed talik freezing.

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Figures

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Fig. 1.

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Fig. 7.

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Fig. 9.

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Fig. 10.

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Fig. 11.

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Fig. 12.

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We have not conflict of interest associated with publishing this manuscript

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Highlights

1. Thermodynamic calculations of freezing of gas saturated talik zone is observed. 2. The influence of gas pressure and salinity on freezing point of pore water are investigated. 3. Gas solubility affects phase equilibrium in the system ‘ice – gas-bearing water’.

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4. Carbon dioxide have significant effect on freezing process of confined gas saturated talik

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zone.

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Figure 1

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