6th International Conference on Gas Hydrates

DESCRIPTION OF GAS HYDRATES EQUILIBRIA IN SEDIMENTS USING EXPERIMENTAL DATA OF SOIL WATER POTENTIAL Istomin, Vladimir; Chuvilin, Evgeny; Makhonina, Natalia; Kvon, Valery; Safonov, Sergey 2008

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Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008.  DESCRIPTION OF GAS HYDRATES EQUILIBRIA IN SEDIMENTS USING EXPERIMENTAL DATA OF SOIL WATER POTENTIAL Vladimir Istomin∗ NOVATEK 8, 2-nd Brestskaya str., Moscow, 125047 Russia Evgeny Chuvilin Department of Geology Moscow State University Vorob’evy Gory, Moscow, 119899, Russia Natalia Makhonina and Valery Kvon VNIIGAZ, Moscow Region, 142717, Russia Sergey Safonov Schlumberger Moscow Research 9 Taganskaya str., Moscow, 109147 Russia ABSTRACT The purpose of the work is to show how to employ the experimental data from geocryology and soil physics for thermodynamic calculations of gas hydrate phase equilibria by taking into account pore water behavior in sediments. In fact, thermodynamic calculation is used here to determine the amount of non-clathrated pore water content in sediments in equilibrium with gas and hydrate phases. A thermodynamic model for pore water behavior in sediments is developed. Taking into account the experimental water potential data, the model calculations show good agreement with the experimentally measured unfrozen water content for different pressure and temperature conditions. The proposed thermodynamic model is applied for calculations of three-phase equilibria: multicomponent gas phase (methane, natural gas, etc.) – pore water in clay, sand, loamy sand, etc. – bulk (or pore) hydrate. As a result, correlations have been established between unfrozen and non-clathrated water content in natural sediments. Keywords: gas hydrates, unfrozen and non-clathrated water, sediment, thermodynamic simulation NOMENCLATURE c C f h M p r R T V  ∗  heat capacity Langmuir constant fugacity enthalpy molecular mass pressure pore radii universal gas constant temperature molar volume  [J/mol·K] [MPa-1] [MPa] [J/mol] [g/mol] [MPa] [m] [J/mol K] [K] [cm3/mol]  Corresponding author: Phone: +7 495 756 9188  W x γ µ ν ρ Ψ  water content gas molar fraction hydrate-water interfacial tension chemical potential gas hydrate structure parameter specific gravity water potential  E-mail: vlistomin@yandex.ru  % [H/m] [J/mol] [g/cm3] [MPa]  INTRODUCTION The first analytical relationship was developed by Makogon between hydrate dissociation pressure and vapor pressure above the pore water surface [1]. A similar method was developed by Chersky, et al., that takes into account the effective pore radii [2]. Using this approach, the decrease of the equilibrium temperature ( ∆T ) of hydrate decomposition in porous medium has been calculated and compared with that for the bulk hydrate. Depending on the hydrate structure and the hydration ratio it was found that the temperature shift can be in the range of 1.361.60 К for capillary water which fills in pores with the radius of 10 −6 ≤ r ≤ 10 −7 m. In the case of bound water which occupies pores with radii of r ≤ 10 −8 m, the expected temperature shift can reach 6.8-8.0 К. Later, Tsarev slightly improved this method [3]. Furthermore, Handa and Stupin [4] experimentally observed a shift in the thermodynamic hydrate stability curve to lower temperatures and/or higher pressures with respect to bulk conditions for both methane and propane clathrates in porous silica. Several theoretical models have been put forward [5,6] to explain this behavior. However, in spite of the good qualitative agreement between the model and the existing experiments, a strong quantitative deviation exists between them estimated to be 16 % in the pressure. Similar experiments, but with a wider range of pore sizes, were conducted by Uchida, et al. [7]. These experiments confirmed that the effect of porous media on hydrate equilibrium conditions strongly depends on the pore diameter. The pores effect on hydrate stability in natural samples was also investigated by Clennell, et al. [8], who suggested a simple model incorporating the pore size. Recent experiments have been discussed in numerous publications on the effect of narrow interconnected throats between pores on clathrate dissociation conditions in porous media [9-20]. In the majority of these studies, the GibbsThomson relationship for clathrate growth in narrow capillaries has been employed in order to develop thermodynamic models for the prediction of hydrate equilibria in porous media ∆Т/Тd bulk = -4γHW/ρHLHd, where γHW is the hydrate-water interfacial tension, d is the capillary diameter, ρH and LH are the hydrate specific gravity and enthalpy of clathrate dissociation, respectively.  Most calculations considered only the effect associated with the curvature of the interfacial surface in the pore space, or the so-called capillary pressure effect. Melnikov and Nesterov [21] suggested a model to describe the equilibrium conditions of methane and propane hydrate formation in the pores (capillary effect) and in thin wetting films (disjoining effect). It was determined that the effect of porous media on hydrate phase equilibrium becomes significant for a pore diameter smaller than 1 µm. The paper’s purpose is to improve the available thermodynamic methods for calculation of hydrate phase equilibria, taking into account the properties of pore water in natural sediments (for instance, three-phase equilibrium of “gas – pore water – gas hydrate”) in a similar way as for unfrozen water in geocryology science. A number of quantitative techniques (contact-saturation, calorimetric, dielectric, nuclear magnetic resonance, and others) are used for determination of unfrozen water content in sediments and its dependence on temperature variation. (For a detailed description of these experimental methods, please see [22,23]). The new term “non-clathrated water” is used below. This term denotes a minimal possible amount of pore water in equilibrium with both the bulk hydrate and gas phases in a soil porous sample at the given temperature and gas pressure. The unfrozen water content primarily depends on temperature and only slightly on external pressure. As for non-clathrate water content, it depends on temperature, gas pressure, and gas hydrate-former. The paper’s main aim is the application and adaptation of geocryology and soil physics methods for the thermodynamic calculation of non-clathrated water content in sediments. In fact, this paper answers the question of how to estimate the non-clathrated water content if pore water potential is known. THERMODYNAMICS OF WATER PHASE IN POROUS MEDIA For further thermodynamic description of pore water, the standard state of water phase is defined as bulk water phase at atmospheric pressure (0.101325 MPa). Thus, for temperatures below 273.15 K the presence of any liquid water phase is defined to be a metastable phase, or so-called supercooled water. According to previous experimental data [24], the supercooled water can exist as bulk phase for temperatures below  ~235 К, and its thermodynamic and thermal properties (vapor pressure, heat capacity, etc.) are precisely and reliably determined up to 245-250 K. Furthermore, the bulk hexagonal ice at pressure p0 = 0.101325 MPa and temperature below 273.15 K can be considered as the standard thermodynamic state of pore ice. Thermodynamic properties of supercooled water Now introduced are the thermodynamic characteristics of liquid (supercooled) water in respect to hexagonal ice as the functions of temperature at p0 = 0.101325 MPa and T<273.15 K and define the following parameters: ∆µ 0 w, i (T ) (J/mol) is the difference between chemical potentials of supercooled water and hexagonal ice ∆h 0 w, i (T ) (J/mol) is the difference between supercooled water and hexagonal ice enthalpies ∆c 0 w, i (T ) (J/mol·K) is the difference between supercooled water and hexagonal ice heat capacities ∆V 0 w, i (T ) (cm3/mol) is the difference between supercooled water and hexagonal ice molar volumes. Here the indexes w and i mean water and ice phases, correspondingly. The same values without upper indexes (0) are related to gas (methane or nitrogen) pressure, p, for which p>p0. The inversion of the bottom indexes causes a sign of the thermodynamic parameters to be changed, e.g., ∆µ w0 , i (T ) = −∆µi0, w . It should be mentioned that these thermodynamic characteristics can formally be defined for temperatures higher than 273.15 K where they represent the stable water phase and the metastable (superheated) ice phase. At To = 273.15 К and po = 0.101325 MPa we have: ∆µ w0 , i = 0 , meaning the equilibrium between ice and water calculated ∆hw0 ,i = ∆hw0 ,i (T0 ) = 6008.0 J/mol, using the specific heat of the “water-ice” phase transition of 333.5 kJ/mol and the water molecular mass of 18.015 g/mol ∆cw0 , i = 36.9328 J/mol K  ∆Vw0,i = −1.6421 cm3/mol.  From [24-26] the experimental values are available for heat capacities of hexagonal ice and supercooled water and their dependence on the vapor pressure and other properties of supercooled water and ice. After thermodynamic analysis of these data the following values for saturated vapor pressure of supercooled water and ice presented in Table 1 can be recommended for further use. Saturated vapor pressure, Pa×102  Temperature, К  pw pi 273.15 6.1165 6.1165 273.15 6.1121 6.1115 268.15 4.2184 4.0180 263.15 2.8656 2.5995 258.15 1.9141 1.6534 253.15 1.2558 1.0329 248.15 0.8084 0.6330 243.15 0.5099 0.3802 238.15 0.3147 0.2236 Table 1. Saturated vapor pressures of supercooled water Pw and ice Pi at temperatures below 273.15 K  An analytical approximation for the difference of o heat capacities ∆cw, i (T ) between supercooled water and hexagonal ice, obtained by processing of the experimental data, is: ∆cw0 , i = ∆cw0 ,i (T ) = 36.9328 +  + 28.3542 ⋅ 10 −6 (T − T0 ) 4 , J / mol ⋅ K at T ≤ To = 273 .15 K. The difference between supercooled water and ice chemical potentials ∆µ w0 , i = ∆µ w0 ,i (T ) as a function of temperature can be obtained from the saturated vapor pressures of supercooled water using the following relation: p (T ) ∆µ w0 , i = ∆µ w0 ,i (T ) = RT ln w + 0.101325 ⋅ ∆Vi ,0w . pi (T ) o Taking into account ∆cw, i (T ) and ∆hw, i it is also  0  possible to define ∆µ w0 , i (T ) as: ∆µ 0 (T ) RT  =  ∆µ 0 (T ) RT0  ∆h 0 (t ) dt , Rt 2 T0 T  −∫  T  ∆h(T ) = ∆h(T0 ) + ∫ ∆C p (t )dt .  (1)  T0  This study found that the experimental data for saturated vapor pressure and the thermophysical properties of supercooled water and ice are very  close to the theoretically calculated data for the following simplified form of ∆µ w0 , i (T ) : ∆ µ w0 , i = ∆ µ w0 , i (T ) = 6008 (1 − T / T 0 ) −  . (2) ⎡ ⎤ T − 38 . 2 ⋅ ⎢ T ln + (T 0 − T )⎥ T0 ⎣ ⎦ This simplified relation will be used further in these calculations. The estimated error of Eq. 2 does not exceed several percent in the temperature range of 248 – 273 K. Using the relation: T  ∆h(T ) = ∆h(T0 ) + ∫ ∆C p (T )dT , T0  the difference can be obtained between the molar enthalpies of supercooled water and ice ∆hw0 , i = ∆hw0 , i (T ) . The difference between the water and ice molar volumes can be estimated as:  where t =  between bulk and pore water molar volumes. It should be noted that ∆hw, wpor (T , W ) → 0 , when water content, W, of the sediment sample increases, but in a general case ∆hw, wpor (T , W ) ≠ 0 , contrary to the usually used in geocryology relation ∆hw, wpor (T , W ) = 0 . For  ∆cw, wpor (T , W ) in many cases it can be assumed ∆cw, wpor (T , W ) ≈ 0  ( ∆cw, wpor (T , W ) ≠ 0  only if the water content of the sample is extremely small). Therefore, in a temperature range from 260 to 300 K it is possible to accept ∆hw, wpor (T , W ) = ∆hw, wpor (W ) for the majority  T − T0 T − 273.15 = T0 273.15 .  Now consider the effect of external pressure, p, on ∆µ and ∆h. For the problems under consideration (the gas hydrate equilibrium with pore water or the influence of external pressure on the shift of unfrozen water curve) the range of gas pressure can be defined as 0.1-20 MPa. As a first approximation in this pressure range it is possible to neglect the effect of ∆Vw,i and ∆c0w,i on pressure. However, it is necessary to take into account this effect under higher pressures. Using the above assumptions, the external pressure effect on ∆µ and ∆h can be written as follows: ∆µ w, i ( p ) ≈ ∆µ w0 ,i + ∆Vw0,i ⋅ ( p − p0 ) ,  ∆hw, i ( p ) ≈ ∆hw0 ,i + ∆Vw0,i ⋅ ( p − p0 ) .  (J/mol) is the difference between chemical potentials of bulk (including supercooled) and pore water ∆h 0 w, wpor (T , W ) (J/mol) is the difference between bulk and pore water enthalpies (called differential wettability enthalpy) ∆c 0 w, wpor (T , W ) (J/mol·K) is the difference between bulk and pore water heat capacities ∆Vw0, wpor (T , W ) (cm3/mol) is the difference  that  −2  ∆Vw,i = −37.0605 ⋅ t + 164.2067 ⋅10 , 2  ∆µ 0 w, wpor (T , W )  (3)  Thermodynamic properties of pore water and pore ice in sediments Let W be the water content of the sediment sample measured in the weight percentage with respect to dry sample weight. Similar to the supercooled water, let us introduce the following thermodynamic characteristics of pore water in sediments (again at standard pressure, p0 = 0.1013 MPa):  of practical problems. However, it is necessary to keep in mind that ∆hw, wpor (W ) has a strongly nonlinear dependence on water content of the sample and ∆hw, wpor (T , W ) ≈ 0 only in the case of high water content. Using Eq. 1 the thermodynamic correlations between the introduced values under atmospheric pressure can be represented as: ∆µ w0 , wpor (T2 , W ) ∆µ w0 , wpor (T1 , W ) T ∆hw0 , wpor (t , W ) RT2  =  2  RT1  −∫  Rt 2  T1  dt  ∆hw0 , wpor (T2 , W ) = ∆hw0 , wpor (T1 ) + T2  0 0 + ∫ ∆cwpow , w (t , W )dt ≈ ∆hw , wpor (W )  .  (4)  T1  Assuming that ∆c 0 w, wpor (T , W ) ≈ 0 ,  ∆µ w0 , wpor (T2 , W ) T2  ≈  ∆µw0 , wpor (T1 , W ) T1  −  . (5) ⎛ ⎞ 1 1 − ∆hw0 , wpor ⎜⎜ − ⎟⎟ ⎝ T1 T2 ⎠ Eq. 5 allows us to recalculate the experimental data on the water chemical potential for different temperatures (i.e., from positive to negative  temperatures in Celsius). Moreover, it follows from Eq. 5 that in the case of high water content (when ∆hw, wpor (T , W ) ≈ 0 ), the temperature dependence of the difference between the pore and bulk water chemical potentials becomes ∆µ w0 , wpor (T2 , W ) ∆µ w0 , wpor (T1 , W ) . = T2 T1  Concerning the estimations for ∆Vw, wpor (T , W )  contradictory experimental data appears in the available literature. It follows from general physical and chemical reasons that the value of ∆Vw, wpor (T , W ) should be close to zero for capillary water. However, it should differ from zero for both adsorbed water (at least for three or four molecular layers adsorbed on the surface) and for interlayer water in swelling clays. The partial molar volume of interlayer water for swelling clay can be estimated from X-ray diffraction data. The partial volume of adsorbed water can be found from the geometrical analysis. Knowing ∆µ w, wpor (T , W ) it is possible to calculate thermodynamic equilibria of pore water with bulk ice or bulk gas hydrate (see below). Similar to pore water the following thermodynamic values can be introduced for the description of pore ice thermodynamics: ∆µi , ipor (T , W ) , ∆hi , ipor (T , W ) ,  ∆ci , ipor (T , W ) , ∆Vi , ipor (T , W ) .  (6)  The following question arises: what is W in this case? It should be noted that the existence of pore ice without pore water is impossible. As a result, in Eq. 6 W can be considered as the liquid pore water content as it is introduced for the pore water, but the presence of a small amount of solid phase (disperse ice) can be assumed. This statement ∆µ i , ipor (T , W ) = ∆µ w, ipor . For means that practical purposes assume that ∆ci , ipor (T , W ) ≈ 0  and ∆Vi , ipor (T , W ) ≈ 0 .  This approach does not take into account the thermodynamics of pore ice in sediments. However, it would be interesting to consider this case for the interpretation of the unfrozen water data obtained in the calorimeter study and its comparison with the direct contact-saturation method data. Similar definitions can be used for the description of pore hydrate thermodynamics in  sediments compared to bulk gas hydrate phase (at least for so-called empty hydrate lattice). Phase equilibria of pore water Pore water phase equilibrium with bulk ice and gas hydrate phases is considered below. Strictly speaking, two values of chemical potential (in other words, two values of vapor pressure for pore water) can be introduced for the same water content of the sample depending on whether the thermodynamic system moves along the adsorption or desorption curve. This hysteresis is unimportant for real soil systems, but is intended for special artificial capillary media such as porous glasses. Similar to geocryology science, only the desorption curve will be considered when dealing with the unfrozen water content. It is assumed that the internal equilibrium of pore water takes place in sediment sample (that is, all parts of the moistened sample have the same pore water chemical potential). As a first step, consider the phase equilibrium between pore water and bulk ice under atmospheric pressure and temperature T ≤ To . At such equilibrium the water and ice chemical potentials in pore space are equal to each other  µ 0 wpor (T , W ) = µ i0 (T )  or  ∆µ 0 w, wpor (T , W ) = ∆µ w0 , i (T ) .  From Eq. 2 we find: ∆ µ w0 , wpor (T , W ) = 6008 (1 − T / T 0 ) − ⎡ ⎤ T − 38 , 2 ⋅ ⎢ T ln + (T 0 − T )⎥ T0 ⎣ ⎦  .  (7)  If ∆µ 0 w, wpor (T , W ) for the sediment sample is known from the experimental data (see further discussion below), Eq. 7 represents the dependence of the equilibrium water content W of the sample on temperature T (at T<273.15 K). Thus, using Eq. 7 the unfrozen water content can be calculated as a function of temperature. The theoretically calculated data using Eq. 7 should be close to the experimental data of unfrozen water content obtained by the contact-saturation method. It is possible to define the effect of gas pressure on unfrozen water content, that is, equilibrium water content at temperature, T, and pressure, p, from the relations:  µ wpor (T , W ) + V wpor ( p − p 0 ) + RT ln (1 − x ) = = µ i0 (T ) + V i ( p − p 0 ); ∆µ w, wpor (T , W ) − RT ln(1 − x ) + ∆Vw, wpor ( p − p 0 ) =  (T ) − ∆Vi , w ; ∆µ w, wpor (T , W ) − RT ln(1 − x ) + ∆Vi , wpor ( p − p 0 ) = = ∆µ w0 , i (T ) , = ∆µ  0 w, i  where x is the molar fraction of dissolved gas in water under pressure, p , and ∆Vi , wpor (T , W ) = ∆Vi ,w (T , W ) + ∆Vw, wpor (T , W ) . Taking into account Eq. 2, the equation for the calculation of the unfrozen water content under gas pressure p can be written as: ∆µ w0 , wpor (T ,W ) − RT ln(1 − x ) + ∆Vi , wpor ( p − p0 ) = ⎡ ⎤ T = 6008(1 − T / T0 ) − 38,2 ⋅ ⎢T ln + (T0 − T )⎥ T 0 ⎣ ⎦  . (8)  The value x is the solubility of gas in water under pressure p, which can be defined by the Henry law (in its thermodynamic generalization called Krichevskii - Kazarnovskii equations), which incorporates Henry’s coefficients and partial molar volumes of gases dissolved in water. These values are well known for many gases, such as hydrocarbons, inert gases, and others, from the experimental data on gas solubility in bulk water. For the gas solubility in this case of pore water, the same Henry’s coefficients can be used approximately by introducing a small correction for the so-called “insoluble volume” if needed. Pore water mineralization can also be taken into account through an additional term defined by the Pitser method, which can be included in Eq. 8 as a changing water activity in solution. Similar relations can be written for bulk gas hydrate as for the bulk ice phase. In the case of gas hydrate, pressure is created by hydrate-forming gas. When the phase equilibrium of “pore water – gas – bulk hydrate” takes place at temperature T and pressure p, the chemical potentials of liquid water µ wpor (T , W , p ) and hydrate µ h (T , p ) become equal: µ wpor (T , w, p ) = µ h (T , p ) ,  ∆µ  (T , W , p ) = ∆µ w, h (T , p ) , ∆µ w, h (T , p ) is the difference  w, wpor  where  (9) between  chemical potentials of water in liquid and gas hydrate phases. As a result, this is obtained:  (T , W , p ) = ∆µw0 , wpor (T , W ) + . + ∆Vw0, wpor ⋅ ( p − p0 ) ∆µ  w, wpor  (10)  In major cases of practical interest ∆Vw0, wpor ≈ 0  and ∆µ w0 , wpor (T , W ) can be assumed as a known value that can be determined from the experimental data (see below). The correlations for ∆µ w, h (T , p ) can be written using the classic van-der-Waals and Barrer thermodynamic model for clathrate solution: ∆µ w, h (T , p ) = ∆µ w0 , h (T ) + ∆Vw0, h ⋅ ( p − p0 ) +  ⎡ ⎞⎤ ⎛ ⎞ ⎛ + RT ⎢ν 1 ln⎜1 + ∑ C1,k (T ) f k ⎟ + ν 2 ln⎜1 + ∑ C 2,k (T ) f k ⎟⎥ = k k ⎠⎦ ⎝ ⎠ ⎝ ⎣ 0 0 = ∆µh, w (T ) − ∆Vh, w ⋅ ( p − p0 ) + ⎡ ⎛ ⎞ ⎛ ⎞⎤ + RT ⎢ν 1 ln⎜1 + ∑C1,k (T ) f k ⎟ + ν 2 ln⎜1 + ∑C2,k (T ) f k ⎟⎥ k k ⎝ ⎠ ⎝ ⎠⎦ ⎣  ∆µ h0, w (T ) RT  =  ∆µ h0, w (T0 ) RT0  , (11)  ∆hh0, w (t ) dt , Rt 2 T0 T  −∫ T  ∆hh0, w (T ) = ∆hh0, w (T0 ) + ∫ ∆ch0,w (t , W )dt .  (12)  T0  Here ν i  is the parameters of gas hydrate  crystalline structure (ν 1 = 1 / 23, ν 2 = 3 / 23 for gas hydrate cubic structure I and ν 1 = 2 / 17, ν 2 = 1 / 17 for gas hydrate cubic structure II); f k is the fugacity of k-component of gas phase (the gas phase is considered as a multicomponent gas mixture); 0 3 ∆Vh, w = 4.6 cm /mol for gas hydrate cubic structure I, ∆Vh0, w = 5.0 cm3/mol for gas hydrate cubic structure II; and C1,k , C2 ,k are Langmuir constants for k-component of the gas phase for small and large cavities in clathrate phases (C1,k, C2,k were used for different gases from [27]). The values of ∆µ h0, w (T0 ) and ∆hh0, w (T0 ) for the thermodynamic properties of empty metastable hydrate lattice are taken from Handa and Tse parameterization [28]: ∆µ h0, w (T0 ) = 1287 J/mol K;  ∆hh0, w (T0 ) = 931 J/mol K  structure I and  ∆µ h0, w (T0 ) = 1068 J/mol K;  for  hydrate  cubic  ∆hh0, w (T0 ) = 764 J/mol K  cubic  where p wpor is the water vapor pressure (on air) in  structure II. The difference between the empty lattice and bulk water heat capacities can be found using an approximation that heat capacity of hexagonal ice is the same as the hydrate empty lattice. Using Eqs. 9-12, the equation for the calculation of three phase equilibrium of “pore water – gas – bulk gas hydrate” can be written as  the sample with water content W (under atmospheric pressure), p w is the saturated vapor pressure at the sample temperature, R is the universal gas constant, T is the Kelvin temperature of the sample, M is the molecular water mass (18.015 g/mol), and ρ is the specific gravity of pure water at the given temperature (~1.0 g/сm3). The experimental data obtained by WP 4 allows for calculation of the difference between chemical potentials of pore and bulk water ∆µ wpor , w (T , W )  for  hydrate  ⎛ ⎞ ∆µw0, wpor (T , W ) + RT ln⎜1 − ∑ xk ⎟ + ∆Vw0, wpor ⋅ ( p − p0 ) = k ⎝ ⎠ = −∆µh0, w (T ) − ∆Vh0, w ⋅ ( p − p0 ) +  ⎡ ⎛ ⎛ ⎞⎤ ⎞ + RT⎢ν1 ln⎜1 + ∑C1,k (T ) fk ⎟ +ν 2 ln⎜1 + ∑C2,k (T ) fk ⎟⎥ k k ⎝ ⎠⎦ ⎠ ⎣ ⎝  , (13)  where xk is the molar fraction of k-component in gas mixture dissolved in pore water phase (calculated from the Krichevskii - Kazarnovskii equation). Using Eq. 13 the equilibrium pressure p (at the given temperature T) can be calculated at the moment when bulk hydrate starts to form, that is, the equilibrium of “pore water - gas - bulk gas hydrate” occurs. (Generally speaking, it is easy to generalize Eq. 13 to take into account the pore hydrate phase in sediment instead of the bulk hydrate phase, which will correspond with the “metastable equilibrium” of “pore water – gas – pore hydrate”). However, in this case additional experimental data is needed (primarily the calorimetric data) for the characterization of the pore hydrate thermodynamic properties with respect to the bulk hydrate phase. THERMODYNAMIC PROPERTIES OF PORE WATER OBTAINED FROM EXPERIMENTAL DATA ON SOIL WATER POTENTIAL The experiments were conducted using Water Potential Meter WP 4T. The description of WP 4T can be found in [29]. A detailed description of the experimental result is presented in the other paper of these proceedings [30]. Using this experimental setup can determine the pore water potential ψ (measured in MPa) as a function of temperature and water content (in a temperature range of 5 ÷ 40 оC):  ψ = RT  ρ M  ln  p wpor pw  ,  (14)  for the samples with various water content as M , (15) ∆µ wpor , w (T , W ) = ψ ρ where M is the molecular water mass (18.015 g/mol), ψ is the water potential (MPa) and ρ is the specific gravity (g/cm3). Please note that ∆µ wpor , w (T , W ) = −∆µ w, wpor (T , W ) . Knowing ∆µ wpor , w (T , W ) gives us an opportunity  to calculate the “pore water – gas – bulk gas hydrate” equilibrium conditions for different pore water contents. The calculated chemical potential difference ∆µwpor, w versus temperature for kaolinite clay obtained from the experimental data on water potential ψ is presented in Fig. 1. It is obvious from Fig. 1 that ƒ  the dependence of ∆µ wpor , w on T is linear in the studied range of temperature and water content, so that ∆c 0 w, wpor (T , W ) ≈ 0 and  ∆h 0 w, wpor (T , W ) = ∆h 0 w, wpor ( W ) ƒ  ∆h 0 w, wpor ( W ) can be calculated using the slope of these linear dependences  ƒ  when W decreases the slope increases  ƒ  in the case of large W the dependence becomes non-linear meaning that ∆h 0 w, wpor ( W ) → 0 , when W increases.  -0,2 W 2,6%  -0,4  W 3,5%  ∆µwpor, w / Т  1747,2284 . W 1, 75 The smoothed values of ∆µwpor, w versus water content, W, at T=273.15 K are presented in Table 3. ∆µ wpor , w = −24,61051 −  0  -0,6  W 5%  W, %  W 5,5%  -0,8  W 6% W 9%  -1  W 9,5%  -1,2  W 12,5%  -1,4 -1,6 3,25  3,3  3,35  3,4  3,45  3,5  3,55  1/T*103  Figure 1. Kaolinite clay. Difference between chemical potentials ∆µwpor, w of pore and bulk water versus temperature (T) for samples with various water content (W, %)  The difference between pore and bulk water chemical potentials calculated from these experimental data at T=293.15 K and 273.15 K are shown in Table 2 and Fig. 2.  ∆µ wpor , w , J/mol·K  W, % (mass.)  at 273.15 K at 293.15 K 1.8 -630.4 -619.2 2.6 -397.0 -377.1 3.5 -200.9 -194.3 5.0 -131.1 -116.2 9.0 -62.4 -51.3 12.5 -31.9 -29.5 Table 2. Average values of ∆µ wpor , w (W ) at 273.15 and 293.15 К from the experimental data for kaolinite clay  The difference between pore and bulk water enthalpies ∆hwpor, w can also be calculated from the experimental results and presented in Table 4 and Fig. 3.  ∆hwpor , w , J/mol·K  W, % (mass.)  1.8 -791.6 2.6 -662.7 3.5 -385.7 5.0 -334.8 9.0 -163.0 12.5 -67.2 Table 4. Average values of ∆hwpor , w (W ) from the experiment  0 0  -200  -200  -400 ∆hwpor, w, J/mol  ∆µwpor, w, J/mol  ∆µwpor, w, J/mol·K  1.0 -170 1.5 -880 2.0 -540 2.5 -380 3.0 -280 4.0 -180 5.0 -130 6.0 -101 7.0 -83 8.0 -71 9.0 -62 10.0 -56 13.0 -30 Table 3. The chemical potential difference between pore and bulk water versus water content at T=273.15 K (calculation data)  -600 by processing of exp. data calc. data  -800  -1000  -400 -600 -800  -1000  by processing of exp. data calc. data  -1200  -1200 0  2  4  6  8  10  W, %  Figure 2. Comparison of the chemical potential difference between pore and bulk water at T=273.15 K theoretically calculated (line) and obtained from the experiment (points)  Fig. 2 uses the analytical approximation:  -1400 0,0  2,0  4,0  W, %  6,0  8,0  10,0  Figure 3. The difference between bulk and pore water enthalpies ∆hwpor, w versus water content (W) for the kaolinite clay sample  ∆hwpor , w (W )  the  following  analytical  approximation can be used: ∆h = 380,667 −  1584,303 . W 0 ,5  18  The approximated values of ∆hwpor, w are presented in Table 5. W, %  ∆hwpor, w, J/mol  1.0 -1200 1.5 -910 2.0 -740 2.5 -620 3.0 -530 4.0 -410 5.0 -330 6.0 -270 7.0 -220 8.0 -180 9.0 -160 13.0 -60 Table 5. Difference between bulk and pore water enthalpies ∆hwpor, w versus water content (W) (calculation data)  THERMODYNAMIC CALCULATIONS OF PORE WATER PHASE EQUILIBRIA 0 0 ∆hwpor Using ∆µ wpor , w (T0 , W ) , , w (T0 , W ) and 0 ∆cwpor , w (T0 , W )  to the experimental data on the non-clathrated water content in the kaolinite clay sample under methane (or carbon dioxide) gas pressure.  calculated  in  the  previous  section, the thermodynamic calculations can be performed on phase equilibrium of unfrozen or non-clathrated water content in the kaolinite clay sample under gas pressure. The following approximations were used: 0 ∆cwpor , w (T0 , W ) ≈ 0 , 0 0 ∆hwpor , w (T , W ) = ∆hwpor , w (T0 , W ) and Eq. 5.  In Fig. 4 the unfrozen water content was compared, which was experimentally measured using the contact-saturation method and presented by points and the thermodynamic calculation (lines) based on the chemical potential of pore water in kaolinite clay sample (see Tables 3, and 5 and Eq. 5). A good agreement can be seen among these data, especially in the 260 - 270 K temperature range of the intensive phase transition of “water – ice”. As a result, using the dew-point device WP 4T a new express method was formulated to measure the unfrozen water content in sediment. This method is presented here. The next step is the calculation of non-clathrated water content by considering the same data on pore water chemical potential and its comparison  16 Unfrozen water content, %  For  14  calculation data experimental data  12 10 8 6 4 2 0 258  260  262 264 266 Temperature, K  268  270  272  Figure 4. Unfrozen water content for kaolinite clay measured by the contact-saturation method and the calculation data  Let’s start with the thermodynamic calculation of “CH4 gas phase – pore water in kaolinite clay – gas hydrate of structure I” equilibrium at different temperatures with the step of 2.5 degrees from -10 up to +10 oС. To perform this calculation, first determine the pressure and temperature dependence of methane hydrate equilibrium condition with water and ice including the metastable states: supercooled water and superheated ice. For “CH4 gas phase – liquid water – gas hydrate structure I” equilibrium the following analytical dependencies can be used: 7080.1932 ln p ( MPa) = 26.86444 − T at 263.15 K < T < 273.15 K , and 7694.3008 ln p ( MPa) = 29.11273 − T at 273.15 K < T < 283.15 K For “CH4 gas phase – ice – gas hydrate structure I” equilibrium the following analytical dependence can be used:  2196.62 T at 263.15 K < T < 278.15 K ln p ( MPa) = 8.9720 −  The calculation of the equilibrium non-clathrated water content was done using the data obtained on water chemical potential in kaolinite clay as a function of the water content and temperature (see  Tables 3 and 5). The results are presented in Fig. 5. 2 - 2,5 %  28  3-3%  2  4 - 3,5 %  24 Pressure, MPa  5-4%  20  3  6-5% 7-6%  16  exp. data at T=265.65K calc. data at T=263.15K calc. data at T=265.65K calc. data at T=268.15K calc. data at T=270.65K calc. data at T=273.15K  16  1  1-2%  4  8 - bulk water  5 6 7  12 8  Pressure, MPa  32  20  12  8  4  0  8  0  4  262  264  266  268  270  272  274  276  278  280  282  284  Temperature, K  Figure 5. Equilibrium pressure of methane hydrate formation in kaolinite clay at the given water content and fixed temperatures in the range 263 – 283 K  In [30] the non-clathrated water content dependence on methane pressure at the fixed temperature of 265.65 K for the kaolinite clay sample was experimentally measured using the contact-saturation method temperature, Table 6. In Fig. 6 these measured data are compared with the calculation. A good qualitative agreement is seen between the calculation and experimental data; that is, the higher the pressure at the fixed temperature, the smaller the non-clathrated water content, which is in equilibrium with bulk methane gas hydrate. Unfortunately, a solid quantitative fit could not be obtained on the experimental data, but this will be the focus of further study. In our opinion, the experimental technique of hydrate equilibrium determination with pore water needs further improvement. Non-clathrated water Pressure, MPa content, % 1.36 8.69 1.44 7.35 1.47 6.85 1.51 5.90 1.68 4.34 2.00 3.10 1.95 3.10 2.38 2.59 3.09 1.83 3.19 1.83 3.80 0.10 Table 6. Pressure influence on non-clathrated water content in kaolinite clay at the fixed temperature 265.65 K (experimental data)  2 3 4 5 Non-clathrated water content, %  6  7  Figure 6. Experimental data obtained by the contactsaturation method at T=265.65 K (points) and the thermodynamically calculated data from water potential (lines) for kaolinite clay  Experimental data on the non-clathrated water content in kaolinite clay at the fixed average pressure of about 4.15 MPa are shown in Table 7. Fig. 7 presents the comparison between experimental and calculation data. Temperature, K P, MPa W, % 271.95 4.2 2.42…2.57 269.98 4.11 2.26…2.28…2.28 265.65 4.34 1.68…1.71…1.71 262.55 4.20 1.63…1.62 Table 7. Temperature influence on non-clathrated water content in kaolinite clay under methane pressure (experimental data) 8  Non-clathrated water content, %  0  1  6  exp. data calc. data  4 2 0 258  262  266  270  274  278  Temperature, K  Figure 7. Temperature influence on non-clathrated water content in kaolinite clay under methane pressure (P=4.1- 4.3 MPa)  The proposed method provides an opportunity to calculate the equilibrium amount of non-clathrated water for different gases and their mixtures. The pressure effect on non-clathrated water content for methane and natural mixture, that form structure II hydrates, is illustrated in fig. 8.  2 1,75  Pressure, MPa  1,5 Natural gas  1,25 1 0,75  C3H8  0,5 0,25 0 0  2  4  6  8  10  12  Water content, %  Figure 8. Pressure conditions of gas hydrate (structure II) formation in kaolinite clay versus water content at T=273.15 K (Natural gas composition: CH4 86 mol.%; C2H6 6.0 mol.%; C3H8 5.0 mol.%; i-C4H10 2.0 mol.%; n-C4H10 1.0 mol.%)  CONCLUSIONS 1. A thermodynamic model has been developed for pore water behavior in natural sediments. This model has been used for estimation of the unfrozen water content based on the experimental measurements of water potential, ψ , as a function of water content in the clay sample. The unfrozen water content data calculated from potential ψ and directly measured by the contact-saturation method are in good agreement for kaolinite clay samples. 2. A thermodynamic model has also been developed for the calculation of three-phase equilibrium: “multicomponent gas phase (methane, natural gas, etc.) – pore water in sediments (clay, sand, loamy sand, etc.) – bulk hydrate”. This method allows estimating the equilibrium non-clathrate water content both above and below the zero Celsius temperature. ACKNOWLEDGMENTS This research was supported by Grant CRDF No. RUG1-1557-MO-05 and Grand INTAS No. 03-51-5537. REFERENCES [1] Makogon YuF. Hydrates of natural gas. Moscow: Nedra, 1974 (in Russian). [2] Chersky NV, Tsarev VP, Mihailov VA. Role of hydrate formation zones in natural gas resources forming and estimation of possibility of gas hydrates deposits development. In: Exploration and estimation of gas resources in gas hydrate deposits. Yakutsk: SB AS USSR, 1977. p.3-68, (in  Russian). [3] Tsarev VP. Formation peculiarities, methods of exploration and development of hydrocarbon accumulations in permafrost. Yakutsk: SB AS USSR, 1978. (in Russian). [4] Handa YP, Stupin D. Thermodynamic properties and dissociation characteristics of methane and propane hydrates in 70-Å-radius silica-gel pores. J. Phys. Chem. 1992; 96: 85998603. [5] Henry P, Thomas M, Clennell MB. Formation of natural gas hydrates in marine sediments, 2, Thermodynamic calculations of stability conditions in porous sediments. J. Geophys. Res. 1999; 104 (10): 23,005. [6] Clarke MA, Pooladi-Darvish M, Bishnoi PR. A method to predict equilibrium conditions of gas hydrate formation in porous media. Ind. Eng. Chem. Res. 1999; 38: 2485-2490. [7] Uchida T, Ebinuma T, Ishizaki T. Dissociation condition measurements of methane hydrate in confined small pores of porous glass. J. Phys. Chem. 1999; B, 103 (18): 3659-3662. [8] Clennell MB, Hovland M, Booth JS, Henry P, Winters WJ. Formation of natural gas hydrates in marine sediments, 1, Conceptual model of gas hydrate growth conditioned by host sediment properties. J. Geophys. Res. 1999; 104 (10): 22,985. [9] Uchida T, Ebinuma T, Takeya S, Nagao J, Narita H. Effects of Pore Sizes on Dissociation Temperatures and Pressures of Methane, Carbon Dioxide, and Propane Hydrates in Porous Media. J. Phys. Chem. 2002; B, 106: 820-826. [10] Uchida T, Takeya S, Chuvilin EM, Ohmura R, Nagao J, Yakushev VS, Istomin VA, Minagawa H, Ebinuma T and Narita H. Decomposition of methane hydrates in sand, sandstone, clays, and glass beads. J. Geophys. Res. 2004; 109, B05206. [11] Wilder JW, Seshradi K, Smith DH. Resolving Apparent Contradictions in Equilibrium Measurements for Clathrate Hydrates in Porous Media. J. Phys. Chem. 2001; B, 105: 9970-9972. [12] Seshadri K, Wilder JW, Smith DH. Measurements of Equilibrium Pressures and Temperatures for Propane Hydrate in Silica Gels with Different Pore-Size Distributions. J. Phys. Chem. 2001; B, 105 (13): 2627-2631. [13] Anderson R, Llamedo M, Tohidi B, Burgass RW. Characteristics of Clathrate Hydrate Equilibria in Mesopores and Interpretation of Experimental Data. J. Phys. Chem. 2003a; B, 107 (15): 3500-3506.  [14] Anderson R, Llamedo M, Tohidi B, Burgass RW. Experimental Measurement of Methane and Carbon Dioxide Clathrate Hydrate Equilibria in Mesoporous Silica. J. Phys. Chem. 2003b; B, 107 (15): 3507-3514. [15] Smith DH, Wilder JW, Seshradi K, Zhang W. Equilibrium pressures and temperatures for equilibria involving sI and sII hydrate, liquid water, and free gas in porous media. In: Proceeding of the Fourth International Conference on Gas Hydrate, Yokohama, 2002. [16] Klauda JB, Sandler SI. Modeling gas hydrate phase equilibria in laboratory nd natural porous media. Ind. Eng. Chem. Res. 2001; 40 (20): 41974208. [17] Chuvilin EM, Kozlova EV, Makhonina NA, Yakushev VS. Experimental investigations of gas hydrate and ice formation in methane-saturated sediments. In: Proceeding the 8th International Conference on Permafrost. Zurich, Switzerland, 2003. [18] Seo Y, Lee H, Uchida T. Methane and Carbon Dioxide Hydrate Phase Behavior in Small Porous Silica Gels: Three-Phase Equilibrium Determination and Thermodynamic Modeling. Langmuir, 2002; 18 (24): 9164-9170. [19] Zhang W, Wilder JW, Smith DH. Equilibrium pressures and temperatures for equilibria involving hydrate, ice, and free gas in porous media. In: Proceeding of the Fourth International Conference on Gas Hydrate, Yokohama, 2002. [20] Zhang W, Wilder JW, Smith DH. Methane Hydrate-Ice Equilibria in Porous Media. J. Phys. Chem. 2003; B, 107 (47): 13084-13089. [21] Melnikov VP, Nesterov AN. Gas hydrate formation from pore mineralized water. Earth Cryosphere 2001; 5 (1): 61-67. (in Russian). [22] Ershov ED. Physics, Chemistry and Mechanics of Frozen Ground. Moscow: MSU, 1986. (in Russian). [23] Foundations of Geocryology. V.1: Physicalchemical foundations of geocryology. ED Ershov (ed.), Moscow: MSU, 1995. (in Russian). [24] Angell CA. Supercooled water. Ann. Rev. Phys Chem. 1983; 34: 593-630. [25] Smorygin GI. Theoretical foundations of production of loose ice. Novosibirsk: Nauka, SB RAS, 1984. (in Russian). [26] Petrenko VF, Whitworth RW. Physics of Ice. Oxford Univ. Press. 1999. [27] Istomin V, Kwon V, Kolushev N, Kulkov A. Prevention of gas hydrate formation at field Conditions in Russia. In: Proceedings of Second  International Conference on Natural Gas Hydrates, Toulouse, 1996: 399-406. [28] Handa YP, Tse JS. Thermodynamic properties of empty lattice of structure I and structure II clathrate hydrates. J.Phys.Chem. 1986; 90(22): 5917-5921. [29] Decagon devices: http://www.decagon.com/geo/wp4 [30] Chuvilin EM, Gureva OM, Istomin VA, Safonov SS. Experimental Method for Determination of the Residual, Thermodynamically Equilibrium Water Content in Hydrate-Saturated Natural Sediments. In: Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008.  

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