Effect of microscopic heterogeneities on water transfer in frozen ground

Transcription

1 Permafrost, Phillips, Springman & Arenson (eds) 2003 Swets & Zeitlinger, Lisse, ISBN Effect of microscopic heterogeneities on water transfer in frozen ground I.A. Komarov Geological Department, Moscow State University, Vorob evy Hill, Moscow, Russia ABSTRACT: The effect of microscopic heterogeneities on the transfer of unfrozen water in dispersed frozen ground is considered. Special attention is given to the mechanisms of water transfer. The equations for the estimation of diffusion coefficients of liquid and gaseous water as well as the ions of some highly soluble salts in the pore space are presented. 1 INTRODUCTION The migration of unfrozen water and ions under the action of surface forces is described by the diffusion equations. In this case, a ground is accepted as homogeneous, which implies a similarity of conditions for the water diffusion in the different points and directions within ground. The organic and mineral matters and pore ice of natural grounds are very heterogeneous. In experiments, the structural heterogeneity is usually taken into account using the effective diffusion coefficient. The latter is determined on the basis of the comparison of experimental profile of concentration (water and ions) and the profile obtained in the solving of the diffusion equation for homogeneous conditions. A specificity of water and ions migration in the frozen ground is that the pore space is restricted and unstable, and the thickness of films of unfrozen water is comparable with the linear size of the heterogeneities, which can induce the change in the velocity, type, and direction of mass transfer. Therefore, there is a need to estimate the following: (1) bottom space and time boundaries of the applicability range of the diffusion equation, (2) the principle for averaging the effective diffusion coefficient, and (3) the time span needed for the equilibrium in the volume much larger than typical heterogeneity to be reached. These tasks were studied by Komarov (1997, 2001a). In these works, an equation was also advanced to describe the diffusion of unfrozen water (ions) in frozen grounds under action of surface forces. This equation is presented in a form of Fick s law, in which an item was added to characterise the field strength increasing or decreasing diffusion. Generally, total energetic effect on the moving water can be estimated as a gradient of chemical potential of water molecules moving under the action of the force field of ground particles. The water diffusion can also be affected by the spatial and structural heterogeneities of proper unfrozen (bonded) water, which manifests itself in the same conditions. The solving of the diffusion equation for the normal (perpendicular to the surface) transfer showed that the normal change in concentration with distance from surface in the stationary conditions is similar to the change in atmosphere density with height (Komarov, 2001a). This can be interpreted as follows: the density of pore water under action of an energetic field is modified through changing the thickness of the water film, which leads to significant differences in the water mobility within the film. The latter is supported by the NMR method (Ananyan, 1977). Thus, two categories of unfrozen water can be recognized: relatively mobile and immobile. The water content ranges corresponding to these categories are in agreement with our conception of the existence of two types of unfrozen water (Komarov, 2001b), which allows us to link the thermodynamic of equilibrium status of water and its mobility in the ground. Physically, the water transfer can be considered as diffusion, since the mobile category of unfrozen water moves the relatively immobile one. The parameters of diffusion transfer can only be attributed to mobile water (Komarov I.A., 1983). The water contents corresponding to the boundaries of the two categories of pore water are given in (Komarov, 2001b). In (Komarov, 1999), the migration of liquid water and water vapour was discussed for a number of conditions: (1) in the heterogenic temperature field and (2) under simple load and (3) in plastically deformed ground. These results will be analysed in future. In this work, other cases will be discussed. 2 IONS DIFFUSION IN PORE SOLUTION Natural pore solutions contain the numerous ions of highly soluble salts (Cl, HCO 3, CO 3, Na ). The presence of these ions can be considered as either some energetic heterogeneity of solution or the corresponding change in the surface energy of ground particles, which allows us to use a common approach for the description of diffusion (Komarov, 1997, 2001a). However, this influence will be ambiguous. According to Samoilov (1957) some ions (Fe 3, Al 3, Fe 2, Mg 2, Li, CO 2, 579

2 Table 1. Diffusion ion coefficients (Do 10 9 m 2 /sec) calculated for infinitely diluted solutions at different temperatures. Temperature No Ions Na 0,39 0,46 0,56 0,66 0,91 2 K 0,32 0,46 0,61 0,77 1,12 3 Ca 0,21 0,25 0,31 0,37 0,52 4 Mg 0,05 0,09 0,12 0,18 0,29 5 Cl 0,48 0,63 0,8 0,97 1,36 6 SO 4 0,08 0,15 0,23 0,32 0,52 7 NO 3 0,35 0,52 0,69 0,86 1,23 8 CO 3 0,14 0,22 0,3 0,39 0,61 9 HCO 5,36 5,46 5,57 5,68 5,91 SO 4 2, Na, and Ca 2 ) weaken the translational movement of water molecules, whereas others (Ag, K, Cl, Br, I, and Cs ) intensify it. On the other hand, multivalent ions favor coagulation, and monovalent ions induce ground dispersion. Hence, the saturation of the medium by multivalent cations (Al 3, Fe 3, Ca 2, Mg 2 ) will induce the increase in migration intensity, and the saturation of the medium by monovalent cations (Na, K, H ) results in its decrease. For infinitely diluted solutions, the diffusion coefficient Do can be defined on the basis of a variant of Nernst s equation with data on absolute ion mobility or using semi empirical relations (for example, the temperature series) as follows: Do A B/T CT, (1) where A, B, and C are the coefficients. The diffusion coefficient D of these ions in thawed ground varies from to m 2 /s, which is significantly lower than the coefficients calculated for free solutions. This result is evident, since the electrostatic surface forces and the capillary-porous structure of grounds significantly decrease the velocity of ion diffusion. However, the diffusion coefficients D of Na, K, Cl determined in the frozen ground vary from 10 9 to m 2 /s, i.e. they are significantly higher than in thawed grounds. In Ershov et al. (Ershov 1998), the value of D ranges from 0.8 to m 2 /s for Na in the kaolin clay (at t 2 C), from 0.4 to m 2 /s for Na and from 0.46 to m 2 /s for Cl in the kaolin clay, calcareous dolomite, and dolomite marl (within the temperature range 1,5 6 C). The values of D varied owing to the variation in compression load (from 1 to 18 atm) applied to the samples. It was also found that the relation between D and concentration shows extreme character for both ions of highly soluble salts and heavy elements. Our estimation for free solutions showed that the value of D monotonically decreases with increasing the solution concentration. This fact points out that freezing induces the increase of the additional factors and mechanisms of mass transfer, which intensifies diffusion. For example, these factors can be (1) the zones arising in the vicinity of the ice surface of pore ice, since an electrostatic component of surface forces of ice is absent, and (2) a high intracranial conductivity on the ice surface. 3 INFLUENCE OF PHASE AND CHEMICAL TRANSFORMATIONS The water transfer is significantly influenced by the formation and growth of ice crystals in the pore space. The migration conditions in the vicinity of ice crystals can be strongly changed, and as a result, diffusion will be induced under the effect of different tendencies: energetic reasons dictate the formation and growth of ice crystals, and normal diffusion tends to equalise the water content in ground. This can be considered as a manifestation of microscopic energetic heterogeneity. Two models for describing the evolution of ice crystals with size above a critical radius in the pore space have been considered in the work of Komarov (1999). One model is the task with mobile boundary (like Stefan s equation in the spheroid coordinates) with the assumption that ice geometry is a constant with time, and the ice centres forming in the separate pores do not interact with each other. The latter is physically based, since each ice centre is shielded by liquid films, and hence, can be considered as independent. Another model represents a diffusion equation taking into account the crystallisation kinetics (first-order equation). The surface effects of ice crystals deal with the influence of surface tension and the changing radius of Laplace s curvature. The latter plays a significant role in the initial stage of formation of the ice nucleus. As the crystal grows, its significance decreases, and the process intensity is determined by the diffusion intensity. The crystallisation of pore water leads to the formation of stresses that change the morphology and character of the structure of the pore ice. Our estimation showed that the dominating and limiting role of normal diffusion will be realised if a parameter L 1. Its value can be determined on the basis of the combination of mechanical characteristics (Poisson s ratio and modulus of shearing) and linear sizes of ice centers and ground particles. A purely diffusion approach to the description of water transfer is valid in the late stages of ice formation, when the ice crystal becomes significantly larger than the actual critical radius of the ice centre, and stress fields created by arising disturbances have an insignificant influence on diffusion. Various chemical processes affect the diffusion of water and ions as well. In this case, the velocity of the 580

3 reaction will be controlled both by real chemical kinetics on the surface of ground particles (or in the solution) and by the intensity of transport of reagents. The kinetic first-order equations with regard to change on the surface of solution during the process can be used to estimate the dissolution of ice contacting with solution. This equation is valid when reaction parts of surface can be considered as equally accessible for water diffusion, and interaction occurs along a virtual boundary or dissolution front. We believe that the use of term dissolution front in this case (by analogy with freezing front) is allowable, since the linear size of vertical heterogeneity of concentration is smaller than plane heterogeneity of concentration. The exchanging reactions in the bulk pore space intensify water transfer due to the increase of the internal and external reaction surfaces. The concentration profile of reagents will form under the influence of local fields surrounding micro-aggregates and ultra pores of micro-aggregates, and it averages over length. A quantitative description usually uses the simplified models of natural ground. In this case, the application boundaries for diffusion or kinetic approaches are assessed on the basis of the effective diffusion coefficient D (value of D averaged over volume) and effective velocity of reaction K, which can be experimentally determined. The effective velocity of reaction averaged over volume K is related to a real constant of velocity by an effective parameter, namely, the depth of penetration of reaction into a layer. The latter characterises a distance, where the concentration of the reagent reaches equilibrium. For first-order reaction, L p can be estimated as L p ( D% / K% ) 1 2 and the summarised velocity of the process can be calculated as follows: 1 J b( C C) ( D% K% 2 ) C, (2) where C o and C are the initial and current concentrations; and D/l n transfer rate, where l n is the reduced thickness of the diffuse boundary layer, which characterises the resistance to water transfer in the vicinity of the surface. Schematising the actual ground structure such that H m is the mean micro aggregate size and d u is the equivalent diameter of ultra pores (value of d u averaged over differential curve) and in accordance with the physical pattern of transfer, four typical situations can be recognized: (1) When ( D% K% ) 1 2 b, o o the intensity of the entire process is dictated and restricted by the diffusion in the macro pore volume (intra-aggregate pore). The reagent has no time to leave the reaction space, and its concentration both at the surface, and within ultra pores is significantly higher than its concentration within macro pores, i.e. C o C; (2) When b ( D % K % ) 1 2 and H m L p d u, the restrictive factor is diffusion in ultra pores. In this case, the concentration of reagent in the vicinity of micro aggregate surface in the macro pores is close to the concentration within ultra pores, C C o, and the concentration within macro pores reduces to a minimum; (3) When b ( D % K % ) 1 2 and L p H m, the restrictive factors are macroscopic kinetics (on the micro-aggregate surface) and the real kinetics of reaction on the particle surface within ultra pores. The depth of penetration (L p ) is larger than micro-aggregate size, and whole inside surface of the micro-aggregate takes part in the reaction. The concentration of components within ultra pores is similar to the concentration close to micro-aggregate surfaces and within macro-pores (C C o ). (4) When b K (K is the real constant of kinetic velocity on the surface) and L p d u. In this case, macroscopic kinetics coincide with real ones on the micro aggregate surface. The depth of penetration (L p ) is small and comparable with ultra pore diameter, and only the external surface of micro-aggregates takes part in the reaction. 4 VAPOUR WATER TRANSFER The transfer of vapour water is influenced by the heterogeneity of the structure of pore spaces (hetero-porosity) and by the discreteness of gaseous medium within pores. The condensed forms of water (free water and ice) are usually considered as continuous mediums. A gaseous medium within pores can be considered as continuous when the length of free length L fl is smaller than the capillary radius (L fl /r 1). The boundaries of application of the diffusion approach to the description of vapour transfer in various types of frozen ground were determined on the basis of experimental curves of differential pore-radius distribution and of the equations taking into account the relations between the diffusion 581

4 coefficient and capillary radius and pressure. For pressure, the lower boundary was found to range from 80 to 40 mmhg. Therefore, at atmospheric pressure and negative temperature, diffusion transfer takes place within fine- and coarse-dispersed grounds, since ultra capillaries with R 10 6 cm are occupied by unfrozen water retarding vapour transfer. The moving force of cross diffusion of water vapour and the dry air is the difference in their chemical potentials. Since vapour concentration is small (the mixture can be considered as an infinitely dilute solution), chemical potentials can be replaced by the gradient of partial pressures. In order to calculate the velocity of vapour diffusion within coarse-dispersed grounds, a number of equations based on Fick s law are available. The influence of composition and structure of ground on vapour transfer manifests itself in the decrease of the effective pore cross-sections available for diffusing molecules and the increase of their pathways owing to tortuosity. In these equations, this influence is reflected by an integrated parameter. The parameter is a ratio of vapour diffusion coefficient in the ground to that in the air. For binary gaseous mixtures in volume, the cross diffusion coefficient D 1 2 within an accepted range of pressure and temperature was calculated using the relation suggested by Hirschfelder, Kertiss and Bird (1961). In the calculation of D 1 2 for binary mixtures of water vapour with argon (Ar H 2 O), helium (He H 2 O), methane (CH 4 H 2 O) or nitrogen (N 2 H 2 O) (which are applied as models of vapour-air medium), one gas was considered as polar, and another as non-polar: D (, / P ) [ T(M M )/ 2M M ], (3) where 1 2 is the cross-section of collisions; and 1 2 is the integral of collisions, P is pressure. The values of force constants were calculated for these binary systems on the basis of the Lennard-Jones potential and modified (with consideration of hydrogen bounds for polar gas) Shtokmaier s potential (Table 2). The experimental values of D 1 2 for mixtures of He H 2 O, CH 4 H 2 O, and N 2 H 2 O were equal to 0,7 0,8 10 4, 0,18 0, , and 0,17 0, , respectively. The comparison of calculated and experimental (taken from literature) data on D 1 2 for gas-water vapour mixtures shows that they are in satisfactory agreement within temperature ranges from 273 to 240 K. The obtained coefficients of diffusion of vapour D a were used for the treatment of experimental data on vapor diffusion in the different fractions ( 0.25, , mm) of the washed and dried sand (Lyuberetsky sand). A relatively satisfactory agreement (error 15%) between experimental and calculated values was obtained when tortuosity was taken into account. In this case, we used a value of Table 2. The calculated coefficients of cross diffusion D 1 2 at different temperatures (T) and pressures (P) for various gaseous mixtures. T K No. P,at , Nitrogen vapor (N 2 H 2 O) 1 1 0,185 0,183 0,178 0,172 0, ,046 0,045 0,044 0,043 0, ,031 0,03 0,029 0,028 0, ,017 0,016 0,016 0,015 0,013 Argon water vapor (Ar H 2 O) 5 1 0,193 0,191 0,186 0,179 0,149 Helium water vapor (He H 2 O) 6 1 0,781 0,771 0,75 0,728 0,615 Methane water vapor (CH 4 H 2 O) 7 1 0,246 0,243 0,237 0,228 0,19 tortuosity equal to 0.6, which is in satisfactory agreement with data obtained by other researchers studying the diffusion of water vapor and other gases (CO 2, CS 2 ) in sand. In order to eliminate the influence of transfer in the liquid phase, a number of experiments were conducted on the sand hydro phobizated by oleic acid. We could not describe the parameter using a minimum number of independent structural parameters for fine-dispersed grounds, since vapour transfer in them is always accompanied by liquid transfer. The latter is significantly dependent on the specific surface, mineral composition, dispersity, and the concentration of pore solution. Moving within capillary-pore space, water vapour interacts with water films, and the resulting intensity of vapour transfer summarises from the components of transfer over the vapour and liquid phase. For polymictic clays, D a was determined on the basis of the standard experiments conducted at negative temperatures (below 8 12 C), when migration over the liquid phase is absent. At a temperature of 12 C, D a varied from m 2 /sec for montmorillonite clay to m 2 /sec for loamy sands and within the range 0,85 1, m 2 /sec for polymictic and kaolin clays. At a temperature 1.6 C, the values of D a obtained by evaluating the total water transfer were 20% higher. 5 CONCLUSIONS 1. A specificity of migration of water and ions in the frozen ground is that the pore space is restricted and unstable, and the thickness of film of unfrozen water is comparable with linear size of heterogeneities, which can induce the change in the velocity, type, and the direction of mass transfer. 582

5 2. The presence of ions in the pore solution can be considered as a structural and energetic heterogeneity of the solution. An agreement between the calculated and measured diffusion coefficients for various ions mostly takes place at a qualitative level, and only in some cases at a quantitative level. The higher experimental values point to the fact that the ice within pores intensifies ion migration due to intra grain and surface conductivity. 3. The phase and chemical transformations (the formation and growth of ice crystals within pores, salt dissolution, and exchange with surface of ground particles) can be considered as the manifestation of structural and energetic heterogeneity. The growth of ice crystal results in the development of the surface effects and stresses changing morphology and character of final structure of pore ice. 4. The vapour transfer is effected by the discreteness of vapour gas medium within pores and by the heterogeneity of the structure of pore space (hetero porosity). The latter is in the decrease in the effective pore cross-section available for diffusing molecules and in the increase of their pathways owing to tortuosity. For coarse-dispersed grounds, the effect of their structure and composition on the migration intensity can be estimated by a parameter, which is a ratio between the diffusion coefficient of vapor in the ground and that in the air. For fine-dispersed grounds, we could not find a rational description of with the help of the minimum number of independent structural parameters, since vapour transfer is always accompanied by migration over the liquid phase. REFERENCES Ananyan A.A An estimation of average thickness films of un frozen water in thawed and frozen dispersion mountain rocks // the Connected water in dipersion systems. MSU. Hirschfelder J.O., Kertiss, and Bird H.B Molecular theory of gases and liquids. M. Komarov I.A A theory of desiccation of unconsolidated rocks in areas with negative Temperatures./ Fourth International Conference Permafrost. Fairbanks. Alaska. USA. Komarov I.A The quantitative description of mass transfert in frozen soils on a basis of using model of non-uniform environment // Results of fundamental researches of cryospere of the Earth in Arctic Region and Sub-Arctic. Novos. pp Komarov I.A Thermodynamics of freezing and frozen dispersion soils. // Dis. Sciences, pp.52. Komarov I.A. 2001a. Elements of theory of mass transfer in frozen soils, as non uniform environment Seoul. Proceeding of the 7th International Symposium on Thermal Engineering and Science for Cold Regions. Komarov I.A. 2001b. United thermodynamic model of phase adsorption and chemical equilibrium of pore moisture in frozen grounds (part 1) Seoul. Proceeding of the 7th International Symposium on Thermal Engineering and Science for Cold Regions. Samoilow O.Ya Structure of water solutions and ion aquation. M. AS of USSR. Yershov E.D General Geocryology. Cambridge University Press, pp

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