nf dz

The boundary conditions are similar to those considered in the previous statement.

Formulation of the problem can be written in a more compact form, if the concept of effective heat capacity: Cef = Cfr(ifr) + 334 (d Wunf(tir))/dt is introduced. Then the equation for the frozen zone will take the more 'usual' form:

Cef(ifr>

The solution to this problem is more complex than that in the first statement and is carried out by numerical methods using fast computers.

Both the statements considered take into account the conductive heat transfer only, with exclusion of the possibility of other modes of heat transfer (notably the convective one). In addition it is assumed in these models that all the water (both free and bound) freezes at the point of its occurrence, i.e. the possibility of ground water transfer by migration or filtration is excluded from consideration. Therefore there is no way, using these statements, to calculate the process of ice lens formation, i.e. to study the principles of the freezing soils cryogenic texture and structure formation. At the same time the solution of this problem is of great importance from theoretical and practical points of view. Suffice it to say that cryogenic texture is in many ways responsible for the strength properties of frozen soils used as foundations. Ground heaving in the course of freezing and thawing etc. is also associated directly with the processes of migration. Therefore a a

Fig. 1.15. The thermal-physical conditions for the formation and change of position of a segregated ice interlayer in the soil: a - diagram of small ice layer, AB, in the freezing soil; b - the direction of heat fluxes in the vicinity of the small ice layer AB and their dependence on the trend of the temperature curve; gunf, q"r are the heat fluxes in unfrozen, frozen and freezing zones, respectively; 1-3 - soil (1 - frozen, 2 - freezing; 3 - unfrozen); 4 - the segregated ice layer.

Fig. 1.15. The thermal-physical conditions for the formation and change of position of a segregated ice interlayer in the soil: a - diagram of small ice layer, AB, in the freezing soil; b - the direction of heat fluxes in the vicinity of the small ice layer AB and their dependence on the trend of the temperature curve; gunf, q"r are the heat fluxes in unfrozen, frozen and freezing zones, respectively; 1-3 - soil (1 - frozen, 2 - freezing; 3 - unfrozen); 4 - the segregated ice layer.

investigation of the soil freezing (thawing) process in the context of the conductive problem turns out to be inadequate for the complete description of this phenomenon.

Statement of the problem of soil freezing with regard to moisture migration into the freezing zone

The process of moisture migration from the unfrozen portion of the profile into the freezing zone is often observed in soils in natural conditions. Penetrating into this zone, the bound water freezes out forming ice inclusions including those in the form of ice layers. When this process proceeds actively enough, ground heaving (surface deformation) is observed, i.e. the upper boundary does not remain stationary but moves upward. The proper allowance must be made for this in the mathematical formulation of the problem. Thus it is essential that the heat-moisture problem should be solved with regard to the possible ground heaving when considering the process of slow freezing of sandy-silty and silty-clay soils when moisture migration into the freezing zone is a possibility. This problem is very complex and even the physical model of this process has not been conclusively formulated to date, therefore each of the mathematical statements of this problem put forward by various authors has its essential disadvantages.

As a rigorous solution to the problem of soil freezing (thawing) with regard to moisture migration and the accompanying set of physico-chemical and mechanical processes is still under development, it is appropriate to consider the principles of the thermal physical aspect of the soil freezing process with regard to moisture migration and ice layer formation itself.

Water freezing in soils is possible only with the removal of the heat of crystallization i.e. with the availability of the appropriate temperature gradients in the freezing portion of soil (Fig. 1.15a, layer 0 —£fr). When high temperature gradients occur the freezing boundary (<Jfr) propagates downwards rapidly while the moisture contained in the underlying soil layer has no time to migrate into the freezing layer. As a consequence, the pore moisture in the soil is fixed by phase transition into ice in situ and massive cryogenic structure is formed. The temperature gradients in the freezing portion decrease with increasing freezing depth allowing a sufficient amount of migrational water to arrive from the unfrozen portion of the soil into the freezing zone. Freezing of this water is distributed uniformly within the freezing portion of the soil or locally in the form of segregated ice interlayers (horizon AB), and it proceeds with liberation of a great amount of crystallization heat Qph. If the heat flux q\r becomes equal to the sum of the heat flux qlfr being removed from the freezing soil and the heat being liberated in the course of phase transition, then the condition for ice lens formation is fulfilled, the movement of the isotherm AB will stop and the growth of a horizontal ice layer will be observed. In other words, the following thermal-physical condition will be fulfilled:

where £st is the rate of the ice lens (layer) growth. If a further lowering of negative temperature at the surface and the corresponding increase of the removed heat q\t takes place, the left-hand side of the equation will be greater than the right-hand side and so the isotherm AB will begin to propagate downward resulting in the ice lens growth slowing down and ceasing. And only if the condition (1.27) is fulfilled at some new level (situated below the boundary AB), will the formation and growth of a second segregated ice layer be a possibility. It was thought previously that the condition (1.27) can and must be fulfilled at the freezing boundary only, because the unfrozen water migration is practically absent in the freezing portion of the soil (above the freezing front). However it has now been proved by way of experimentation that the unfrozen water migration can be great and sufficient for the thermal physical condition (1.27) to be fulfilled, and consequently, for segregated ice layers to be formed at negative temperatures inside the freezing layer. At the same time the lower the temperature gradient in the frozen portion of the soil the closer to the freezing boundary is the isotherm of ice layer formation. Both the formation of a new segregated ice layer at quasi-stationary positions of isotherms (fulfilment of the condition (1.27)) and the transformation or even disappearance of ice layers formed before are a possibility with changes in temperature distribution within the freezing portion of the soil. As this takes place, the moisture can migrate from one layer to another providing a way for one layer to grow at the sacrifice of another.

According to B.N. Dostovalov's studies five cases of thermal conditions on the boundaries of segregated ice layers and, respectively, of dynamics of their growth and change in their position in the soil are a possibility in accordance with the temperature gradients and form of the temperature curve (see Fig. 1.15b). Thus in case A the temperature minimum falls on the layer AB. Under this condition the thermal fluxes q\r and and the migration moisture fluxes are toward the layer which can increase in thickness from the top as well as from the bottom. In case B the layer AB is situated at the temperature curve maximum. The heat fluxes q\r and as well as the moisture migration fluxes are away from the layer upward and downward resulting in the segregated ice layer decreasing in thickness. In cases C and D the fluxes q\r and q" have the same direction, with the moisture migrating in the same direction too. The segregated ice layer will respectively decrease from the top and grow from the bottom (case C) or grow from the top and decrease from the bottom (case D). In case E the temperature curve has no gradient, resulting in eliminating the possibility of heat transfer as well as of migration water transfer.

1.5 Methods for solving soil freezing (thawing) problems and approximate formulae for freezing and thawing depth calculations

The first attempts to solve the heat conduction problem taking into consideration the liberation of heat by phase transition at the moving phase boundary were undertaken in 1831 by the physicists Zh. Lyame and P. Clapeyron, members of the Russian Academy of Sciences. The problem of calculation of the depth of soil freezing had been solved for the simplest case by L.Zaal'shyuts as early as 1862. He obtained the simple formula for calculations, which has become well known as the 'Stefan formula', because the Austrian mathematician I. Stefan contributed significantly to the solution of this problem.

The exact solution of the Stefan problem was obtained for the first time by L.I. Rubenshteyn in 1947; however, this solution was not widely used because of the complexity of realization. In succeeding years V.G. Melomed's work (1957-1974) played an important role in setting up and solving the problem of soil freezing. Concurrent with this, approximate methods for solution of the Stefan problem were developed. L.S. Leybenzon, M.M. Krylov, V.S. Kovner, V.S. Luk'yanov, G.V. Porkhayev, I.A. Zolotar', G.M. Fel'dman made essential contributions in this direction. V.A. Ku-dryavtsev has made a great contribution to the solution of these problems. He put forward the approximate formulae for calculating the depth of seasonal and perennial freezing and heat storage in conditions of periodic temperature fluctuations at the ground surface which are especially valuable for the solution of many theoretical and practical geocryological problems. Let us consider some of the outlined problem solutions.

Stefan formula for determination of the depth of seasonal and perennial freezing (thawing)

The so-called Stefan formula is used for approximate calculations of the depth of ground freezing, given the thermal physical data and temperature conditions at the surface. It has been obtained using the following assumptions:

1) that consideration is being given to a homogeneous semilimited medium, the temperature of which is uniform with depth and equal to that of the phase transition, at the initial instant of time: t(z,0) = const = t{re at z > 0,

2) a constant temperature is specified for an initial instant of time on the surface and is maintained, f(0,t) = const = t0, for the case of freezing t0 < 0, and the medium is in the unfrozen state at the initial instant of time,

3) all the phase transitions proceed at temperature ffre, i.e. the case of freezing with formation of a phase boundary occurs, while ignoring the phase transition in a range of negative temperatures as well as the processes of moisture migration in the course of freezing,

4) that a far greater amount of heat is released in the course of freezing than that being released on account of soil heat capacity Cfr with decreasing temperature. This is the most important condition in this statement simplifying solution of the problem as much as possible. Formally, this assumption is written as Qph»Cfro|i0|. As a consequence of this assumption the heat capacity in the frozen zone Cfro can be neglected. The dynamics of the temperature field in the frozen zone are simplified sharply in this case; the freezing boundary £(t) moves slow enough for the temperature field in the frozen zone to tend to come into the stationary state which is characterized by the linear pattern of temperature change with depth (Fig. 1.16a):

Fig. 1.16. Diagram of the conditions for the Stefan (a) and Leybenzon (b) formulae.

It is easy to obtain the pattern of propagation of the freezing front m(q) in this case from the Stefan condition for the moving boundary:

where Qph the specific heat of phase transition. Since ¿fUnf(Z T)

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