Effect of Enclosed Space Configuration and Freezing Medium Volume on Refreezing Pressure

In building underground structures (bases and foundations of residential and industrial buildings, boreholes for hydrocarbon extraction, etc.) in cryolithic zones, cavities of various shapes and sizes saturated with liquid media often develop. When they refreeze, the refreezing pressure occurs; this may result in failure of the structural member continuity, and further failure of the structure as it is. The paper describes the calculation techniques for the refreezing pressure depending on cavity shapes and sizes.


Introduction
After field-geophysical research it has been stated that various enclosed spaces can develop in perennial frozen rock in-situ in structures built and being built in the Far North. As a rule, the cavities develop due to icy sands and sandy loams. The following types of cavities occur [1 -5]: -cylindrical cavities are the most common form of cavities. The diameter of these cavities usually reaches 1m, the height -over 15 m; -spherical cavities are enclosed cavities developing during freezing of the supports and bases of the idle structures in the cold season. Spherical pits and cavities not exceeding 1-1.5m in diameter develop. However, in long-term operation their sizes may increase; -slatted cavities are formed in subsidence of thawed rocks after long-term operation of structures. Freezing occurs more rapidly in the axial direction [5].
All types of cavities usually are water-saturated, therefore the phase transformation are causes the occurrence of overpressure through the ice formation, which causes the destruction of built construction.
As a result, ability of their current forecast is very relevant for substantiation of a technology that promotes the prevention of occurrence of the incident.

Analysis of calculation techniques
After the authors [6 -10], the water-ice volume balance in freezing of the enclosed volume V0 by the value dV is described as follows: , ln (5) where: λ -thermal conductivity of the frozen rocks; σ -heat of phase transition; r -current radius of the cavity; RK -initial radius of the cavity; τ -time; Tfn and Tn -temperatures of phase transition and frozen rocks correspondingly. Transforming the equation (2) and integrating the equations (3) ÷ (5), we obtain the formulas for determination of the pressure during freezing for the following cavities: -cylindrical where: R -outer diameter of the outer surface; Рmax -maximum possible freezing pressure at the given temperature.

Results
The calculation was done step by step [16 ÷ 20] in the following way: -the value ri/RK was given, Р1,τ1(τ0 = 0, τi-1/R = 1) was calculated in (Р1 + Р0)/ Рmax. Then the ratio r2/RK was given, Р2 and τ2 were calculated using the values of τ1 and (Р1 -Р0)/Рmax, etc. At each step of volume V freezing to the value Vi, the pressure Pi, was taken as constant. It was calculated as the mean pressure of the pressure obtained in the previous step Рi-1, freezing V to the value Vi-1, and the pressure at the given step Рi. Fig. 1 shows the graphs of pressure changes in freezing of cavities of various geometric shapes. The graphs show that the shape of the cavities affects only the rate of pressure increase. Thus, freezing of the cylindrical cavities develops faster than that of the slatted ones and slower than the spherical ones.
Depending on the volumetric compression of freezing media and the size of the cavities, the pressures may be lower than the maximum possible values of freezing pressure (Tab. 1). This is due to the deformation of the medium inclosing wall occurring under the influence of the refreezing pressure (Pc) [20 -22]. As a result, the volume of cavities increases and the pressure releases to the value (Рp). In this case the thermodynamic equilibrium of the system is violated: the unfrozen water is in the state of compressive suffusion. As a result, additional icing leads to pressure increase in the cavities which becomes equal to (Pc) again, and thermodynamic equilibrium returns. Significant movement of the inclosing walls can result in a situation when the whole of free water becomes icy, and the pressure never reaches its maximum possible value.
In experiments 1, 2, 4, and 7 ( 3 not freeze completely in the lower values of freezing media compressibility and the maximum pressures at the given temperatures are obtained (tests 3, 5 and 6). Thus, the refreezing pressure value depends on the ratio of the ice volume and unfrozen liquid in the cavities at the given temperature.
The results obtained make it possible to assume that if the cavity is a cylinder filled with water, with an infinitely large wall thickness and a height substantially greater than its diameter, the cavity wall (support) undergoes only the elastic deformations; according to the field data it is valid, since the heights of the cavities generated in the cryolithic zone are ten times greater than their diameter, and compression of low-temperature ice-sandy rocks is negligible.
The modulus of elasticity E (Young's modulus) and the transverse elasticity coefficient μ (Poisson ratio) are the main indicators of the elastic properties of frozen rocks. After N.A. Tsytovich [15], the Poisson ratio of frozen rocks has weak temperature-moisture dependence and can be taken as constant in calculation for rocks of various degrees of dispersion. Thus, it can be taken in the range of 0.2-0.22 for frozen sand, and for clays its mean value is approximately 0.37.
Young's modulus depends on several factors: the composition of the frozen rocks, their iciness, values of negative temperatures and the external pressure [5]. Frozen sand obtains the highest modulus of elasticity -from 8.. Let us point out the cavity volume VK with two planes perpendicular to the longitudinal axis and spaced apart by a length unit: where: Increase of the freezing cavity volume is compensated by the change of its original volume by the value ΔVК caused by the deformation of the frozen rocks, compression of ice ΔVL and the unfrozen water ΔVВ. Let us take a final value of movement Uc at the inner radius of the cavity RK from the constant internal pressure Рс which can be evaluated by the formula: (13) where: Е, μ -elastic constants of frozen rocks. Then the dependence of the pressure that occurs when water freezes in the cavity (Pp) from the amount of generated ice is as follows: (15) where: ' c (16) where: VB -volume of the suffusion water; Δt-temperature of suffusion of water.
Having taken in the equation (4) 1 c F F and solving it with regard to К, we obtain the formula showing complete freezing of the liquid phase: Thus, in the temperature range of frozen rocks from minus 2 to minus 5 ° C, corresponding to the conditions of the Far North of Western Siberia and the taken values: 1.0·103 ≤ Е ≤ 2.5·103 MPa; μ = 0.22; ε = 0.083; βl = 1.1·10-4 1/MPa, the following formulas have been obtained: -for temperature minus 2 °С: -for temperature minus 5 °С: To assess the adequacy of the obtained ratios the experiment was carried out at temperatures of minus 5o and minus 2 ° C in accordance with the technique described in the books [4,18]. The results are shown in Table. 2. The results showed that the pressure at minus 2°C and the ratio 83 , 0 / K R R did not increase, and at minus 5°C and the ratio 52 , 0 / K R R it did not exceed 20 MPa.
While at lower pressure values R/R k were close to their maximum possible ones evaluated by the formula (1) -22.9 and 57.4 MPa correspondingly; this indicated the complete phase transition of water into ice in the system simulating the cavity with the boundary conditions (18).