Geocryology, mathematical problems in

Mathematical problems arising in the study of processes and effects taking place in frozen soils and rocks, and the geographical propagation and conditions of rock formation which are seasonally or permanently frozen (permafrost). Interesting problems in geocryology arise in studies on the interaction of temperature and humidity fields in zones with a moving boundary between separate phases. A typical feature of mathematical problems in geocryology consists in the fact that mass- and heat-exchange processes which occur during the freeze and the thaw of rocks are closely interconnected. The problem is reduced to a system of quasi-linear equations of parabolic type, since the heat-exchange and liquid-exchange characteristics of the medium substantially depend on the functions sought. Typical examples of such problems are the study of the freezing of liquid-saturated finely-dispersed rocks, which is accompanied by the migration of the liquid to the front of the freezing line and by swelling, as well as the study of the thawing of coarsely-dispersed rocks, which is accompanied by the infiltration and filtration of liquid. Of special importance in engineering geology is the solution of the multi-dimensional Stefan problem for regions with a complex configuration, in particular on the thawing density in densely-populated and industrial areas. The solution of problems in historical geocryology necessarily involves a study of the multi-front Stefan problem, with allowance for the formation of zones and their degeneration to a point. The problem of the coordination of freezing and thawing processes in the upper layers of the lithosphere with a radiation-thermal balance is of importance in this connection.

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A relevant phenomenon which can occur during ground freezing is so-called "ice lensing" , i.e. the underground formation of layers (lenses) of pure ice. Massive ice lenses have been observed in the arctic regions. Besides the mathematical models of Stefan type, other models have been proposed for the freezing of liquids in porous media, characterized by the presence of regions in which both the liquid and the solid phase are present. A possible explanation of ice lensing can be found in the thermodynamical behaviour of such zones in connection with the stress analysis of the freezing medium.

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