Thus the migration of bound water in dispersed soils can be viewed as the difference of translatory H20 molecule jumps in forward and backward directions. The rate of translatory molecule jumps, according to equation

(2.5), goes up with a rise in the temperature and a drop in the energy of bonding of water molecules to the mineral surface of particles or to the diffuse layer ions, i.e. with the increased film water mobility. Thus, with other conditions being equal, bound water will migrate from the thick films, where molecules are less bound, to the thin films, from the films of higher temperature (where molecule mobility is greater) to the films of lower temperature, from the films of lower concentration of dissolved salts to the films of higher ion concentration, etc. The intricacies of investigating (mol-ecular-kinetically) the microscopic pattern of water migration and the impossibility of making quantitative estimates on this basis to forecast the water-transfer process in dispersed soils, leads us to apply generalized thermodynamic (phenomenological) laws. By analogy with such phenomena as the transfer of heat and electricity (Fourier's and Ohm's laws), vapour diffusion and water seepage (Fick's and Darcy's laws) and others, the intensity of the bound water migratory flow Im (g(cm2 s)~~ is taken to be, under the steady-state regime, directly proportional to the gradient of thermodynamic potential (full or partial):

where A(0 is the coefficient of hydraulic conductivity (cms-1) equal to CmKco, while Kco is the coefficient of soil water characterizing soil properties relative to the lag in (time for) the development of the potential field or of the distribution of moisture content (cm2s_1). The distinctive and substantial features of Ám and Km coefficients are, in addition to their dependence on dispersivity, chemical and mineral composition, and structural and textural characteristics, their variability with soil moisture variations and clearly non-linear dependence on water content (see Fig. 2.1b).

The above equation for the steady-state migratory water flow in soils is based on the thermodynamic water potential. It can be readily rewritten in terms of moisture content (with the simple relation between /1(0 and W assumed): IC() = — K(0 grad W.

In non-steady-state conditions for the water transfer regime in soils the following differential water transfer equations apply:

2.2 Water transfer and ice formation in frozen soil

In terms of general thermodynamics, water transfer in frozen ground is due to the gradients of matrix-capillary, osmotic, temperature, electric and other potentials, which are components of the total thermodynamic potential of the unfrozen water and which result in the unfrozen water gradient (grad Wunf) in the soil system. An equation for the intensity of the steady-state unfrozen water flow in frozen ground is:

where l{(r0 is the coefficient of conductivity of water in frozen soil. Where a simple relation is observed between the water transfer potential and the unfrozen water content of soil, the equation based on water content can be applied:

where is the diffusion coefficient of unfrozen water; Wuat is the unfrozen water content of frozen soil. Note however, the unfrozen water migration is due to the action of grad /iunf, not of grad Wunf, which can only be used, as experiments have proved, in homogeneous soils.

Vapour transfer and the flow regime of vapour-gas mixtures in soil pores are mainly determined from the relation between the length of molecule free path (/ = 0.5 x 10"5 cm) and sizes of pores the soil capillaries of radius r. If r < 10"5 cm, vapour transfer would normally obey Knudsen flow theory characterized by a molecular (effusion) transfer mechanism. The effuse vapour transfer mechanism is practically ineffective since the ultrapores and narrow sections of irregular capillaries, where vapour effusion can occur, are mostly filled with bound (unfrozen) water. With r > 10"5 cm diffusion and molar vapour transfer occur, and with r > 10" 3 cm a viscous flow regime prevails, i.e. vapour transfer occurs in the molar (volume) way. The density of the vapour-gas mixture steady-state flow is determined by Poiseuille's law and proves directly proportional to the total pressure gradient and inversely proportional to gas viscosity. Molar vapour transfer is most important in large clastic rocks, in cavities and fissures of hard rocks and in karst holes. Vapour diffusion in micro- and macro-capillaries is due to molecular water migration resulting from the translatory movement of vapour molecules. Vapour diffusion (combining thermal diffusion, baric diffusion) is determined by Fick's law and caused by the gradient of saturated water vapour (grad dsat) or the partial pressure gradient of vapour-air mixture (grad P). Such a vapour transfer mechanism is applicable mostly to water-un-

saturated fine-grained soils. On the whole, the generalized equation for the steady-state vapour-flow intensity in frozen fine-grained soils can be given

where and K1? are the coefficients of vapour conductivity and vapour diffusion respectively.

Water transfer and ice formation in frozen soil due to the temperature gradient effect

The temperature gradient, created and maintained in frozen soil results in the gradients of bound water potential (grad nw), of water vapour potential (grad ¡iv), and of saturated water vapour concentration (grad dsat). These potentials result in turn in the migration of unfrozen water and vapour from regions of higher to those of lower water potentials, i.e. from regions of higher to those of lower freezing temperatures which have less unfrozen water and water vapour (18). The intensity of the total steady-state water flow in frozen soil is written in the moisture content form as follows:

JU = grad Wuni + K{J grad dM = K^ grad t + K^gradi (2.12)

where and <5^r are the thermal gradient coefficients of unfrozen water and saturated water vapour respectively with phase changes present. They are derived from curves Wunf = f(t) as the relations A Wunf/At and dsJAt. As follows from equation (2.12), the intensity of the water migration flow proves directly proportional to the gradient of freezing temperature and depends substantially on composition and structural and textural characteristics of soil and water as represented in the coefficients KK{J, <5^r.

As experiments show, the portion of vapour transfer becomes significant in the total water flow only when the soil pores are substantially empty of water and ice. For example, vapour transfer may be practically ignored in clay soil where [the coefficient of saturation] G is more than 0.5.

A gradient of temperature in freezing soil, results not only in water migration, but also in the formation of segregated ice and in complex physico-chemical and physico-mechanical processes. As water moves towards lower freezing temperatures the water flow in frozen ground diminishes, with the reduction of coefficients K£ and This leads to the freezing out of the excess amount of unfrozen water in each subsequent frozen soil cross-section since its potential proves higher than that of the ice present. This freezing out of the excess amount of migratory unfrozen water is responsible for the additional ice formation and the gradual increase of ice

0.4 0.8 1.2 a; cm

Fig. 2.3. The temperature (1), unfrozen water content (2), the total initial (3) and final (4) moisture contents along length x of frozen kaolinite clay sample without adjacent ice.

content of a frozen soil layer with a lower freezing temperature (Fig. 2.3). Evidently, the ice formation is most intensive at the section where there is a sharp decrease of density of the total water flow (\dl/dx\ = max), i.e. first of all in that range of freezing temperatures which has values of grad fVun{ (or values K£ and substantially reduced. The formation of ice, i, is calculated from the equation:

where dl/dx is the ice formation intensity.

Experiments showed that in the warmer part of the frozen soil a drop in the total initial moisture content occurred due to the steady outflow of unfrozen water. This was continually replenished in conformity with the rule of equilibrium Wunf content depending on t, by melting of pore ice. The marked loss of moisture from the warmer layer brings about intensive contraction resulting in a network of shrinkage fissures (Fig. 2.4a). Above this zone were some ice microstreaks which, gradually growing thicker and longer, and merging with each other, formed a continuous segregated ice layer (see Fig. 2.4b). The frozen sample expanded by the thickness of this ice layer, i.e. heaving deformation was recorded. Frozen samples of kaolinite clay (in comparison with samples of montmorillonite clay) showed, under similar values of grad t, a faster ice layer growth due to the presence in kaolinite clay of higher unfrozen water content gradients and higher coefficients of water diffusion which are responsible for higher density of flow.

Fig. 2.4. Ice formation in frozen polymineral clay: a-b - without moisture inflow (t = 8 days, grad f = 0.6 °C cm ') sections across and along the direction of heat and water flow; c - with moisture inflow from the ice plate (t = 14 days, grad t = 0.6°C cm ~1); 1 - ice layer; 2 - desiccated zone; 3 - ice plate.

Fig. 2.4. Ice formation in frozen polymineral clay: a-b - without moisture inflow (t = 8 days, grad f = 0.6 °C cm ') sections across and along the direction of heat and water flow; c - with moisture inflow from the ice plate (t = 14 days, grad t = 0.6°C cm ~1); 1 - ice layer; 2 - desiccated zone; 3 - ice plate.

On the whole, under 'closed' system conditions (with the sample not attached to a body of ice) water migration occurs only on account of the redistribution of moisture and this will decrease gradually. The density of water migration fVun{ is substantially greater under the effect of grad t in 'open' system conditions (with ice adfrozen to a part of the frozen ground at a higher temperature), than in the 'closed' system, This results in the formation of a thicker segregated ice layer (see Fig. 2.4c). The flow of unfrozen water to this layer is not due to the loss of water from part of the frozen sample at a warmer temperature, but to the supply of water originating in the ice adfrozen with the sample (Fig. 2.5). As to the energy used, this process is obviously favoured since the quasi-liquid water film on ice crystal surfaces

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