Influence of air temperature on thaw depth of a road foundation

. The expected thaw depth of the foundation soil is one of the parameters influencing the selection of technical solutions of road design in the permafrost area. The purpose of this research is a quantitative assessment of the degree of influence of air temperature on thaw depth of the road foundation soil in the permafrost area. A standard formula to calculate thaw depth of melting bodies of flat symmetry obtained through a solution of the one phase Stefan problem at boundary conditions of the first kind was used for the analysis. An algorithm using an analytical formula to assess the influence of the main initial parameters and accuracy of their determination on the resulting value, the thaw depth of the soil, is proposed. The results of the calculations are presented in a graphical form displaying the influence of average air temperature and the accuracy of its setting on the thaw depth of the road foundation soil. It was determined that the degree of change in thaw depth is non-linear and depends on both the average air temperature and on the accuracy with which it is set. For every average air temperature value there is a specific measure of required accuracy for thermal calculations so that the error of thaw depth determination does not exceed the permitted error. A 3D chart showing the expected percentage error in thaw depth forecasting depending on the average air temperature and the accuracy of its setting for an array of initial values was created.


Introduction
The expected thaw depth of the foundation soil during the warm period of the year is one of the parameters influencing the technical solutions in design of linear structures, such as automobile roads, in the permafrost area [1,2,3]. The physical and mechanical properties of the frozen soil are dependent on both the quantity and aggregate state of the pore water [4,5,6]. While in the frozen state the soil of the road foundation has sufficient strength, when thawing its load bearing capacity sharply decreases [7,8,9]. Some dispersed frozen rocks, such as sandstone, completely lose strength and become a loose, noncohesive mass [10]. Additionally, the moisture in the active layer of the soil occurs not only in pores, but also as layers or lenses of ice [11,12,13]. Thawing of such soil leads to its compression under load, which complicates the use of linear structures, damages them, and causes incidents and failures [14,15,16]. In the engineering community, significant attention is devoted to development of means and methods of protecting the linear structures in the permafrost area from adverse thermal impacts. Usually, application of various thermal insulation materials decreasing or eliminating thawing of the road foundation soils is proposed [17,18,19]. When selecting a thermal insulation material to use in road construction, the maximum thaw depth of the road foundation permissible by the technical design is set. The accuracy of determination of this parameter influences the efficiency and reliability of the thermal protection solutions used in design of roads in the permafrost area.
The aim of this research is to assess the influence of air temperature on the seasonal thaw depth of the road foundation soil in the permafrost area.

Methods
By solving a one phase Stefan problem at boundary conditions of the first kind, a formula to determine the thaw depth of the soil is obtained [1,20,21]. In [20], the formula is transformed into a form convenient for engineering calculations and analysis: Where S is the thaw depth of the soil, m, N is the number of months in the warm period of the year, units, λ is the thermal conductivity coefficient of the thawed soil, W/mK, t is the average air temperature over the thawing period, °C, ρ is the density of the soil, kg/m 3 , ω is the ice content of the soil, unitless.
For a more precise determination of the thaw depth, sometimes the concept of effective heat capacity of the rocks is used [22,23]. It is used to account for the influence of the natural temperature of the rocks on the thawing process. In that case, the denominator in the Stefan formula is, instead of a product (ρωL), written as a sum (ρωL + с p T e ). Here L is latent energy of ice thaw, J/kg, Te is the temperature of the frozen soil, °C, cp is the total heat capacity of the frozen soil, J/m 3 K. In [20], an assessment of expedience of using this improvement in forecasting of the thaw depth of the rocks in underground structures in the permafrost area. It was concluded that for most practical cases, the application of Stefan formula in its classical form does not cause an error exceeding the permitted error (usually, 10%). Thus, for the purposes of this article, it does not matter whether the natural temperature of the frozen rocks within the active layer is accounted for. The use of equation (1) is appropriate.
Following the methodology of analyzing influence of the ice content in the road foundation soil on the thaw depth provided in [22], a parameter β characterizing degree of change in thaw depth of the active layer with change in air temperature is introduced.
Where gt is the deviation of average temperature from the set temperature, °C. This equation is obtained from a relationship of two equations in the form (1) for determination of thaw depth at different air temperatures equal to t1 and t2 = t1 + gt. If the indexed notation is used and considering that the set average air temperature may be either exaggerated or underrated, gt = abs(t 2 − t 1 ). In this case, the parameter β can be determined using the equation: This approach is convenient because it allows to easily assess the possible percentage error of thaw depth. The error is found using the formula: Where the parameter β is determined using the formula (2).
In the engineering practice, the permissible calculation error is usually less than 10%. With this assumption, the equation (4) is transformed into: For example, if the error is 10%, the equation (5) is transformed into a linear equation of the parameters t and gt: From this follows a rule that the permissible inaccuracy when setting the average air temperature for the thermal calculation is higher the higher the average temperature.

Results and Discussion
Calculations with varying parameter values were conducted and their results are presented as 2D and 3D charts. The figure 1 presents the dependence of the thaw depth of the soil foundation of a road on the average air temperature at different durations of the thawing process. The interval of average air temperature change typical for the permafrost area is noted in color. The calculation was done for a soil with the following characteristics: ice content is 20%, thermal conductivity coefficient in thawed state is 1.5 W/mK, the soil density in frozen state is 1500 kg/m 3 .  As the chart shows, the dependence of thaw depth on air temperature is significant for the entire period. The dependence is more pronounced the longer the thawing duration. When the temperature rises from 6°C to 12°C, all else equal, during the first month the thaw depth increases from 0.6 m to 1.0 m. The increase in thaw depth is 0.4 m. When the thawing duration is five months, the depth changes from 1.6 m to 2.3 m. The increase in depth is 0.7 m, or 1.8x. Figure 2 shows a 3D chart of change in thaw depth depending on temperature and thawing duration in months. The shape of the plane shows an increase in quantitative dependence of thaw depth on temperature in different periods. The tilt angles of the functional plane are increasing with time as temperature increases. The longer the thawing period, the greater the angle. Compare the angles at 1st (left edge) and 5th (right edge) month of soil thawing. Figure 3 shows the change in error permitted in thawing calculation when the air temperature if there is inaccuracy in setting the air temperature. The calculation was done for a thawing period of five months. The degree of inaccuracy is characterized by the parameter gt in °C.  The chart indicates that the error can reach up to 50%, which is five times over the permitted error value of 10%. At the same time, the shape of the plane on figure 3 shows that such error values are typical for a small range of parameters t and gt. It is relevant to assess the range of the parameters values for which the error will not exceed the permitted level. This dependence is presented in figure 4. The range of average temperatures of 10-12°C is green, the range of average temperatures 4-6°C is yellow. The areas differ significantly. In the range of 10-12°C, the permissible inaccuracy in setting the temperature is 2.1°C, in the range of 4-6°C it is less than 1.0°C. This confirms the assertion that the permissible level of inaccuracy is related to the absolute value of average temperature. When the permissible error increases twofold (areas indicated in blue), the indicators also increase and the contrast between them is greater. In a relative expression, the permissible inaccuracy is constant, equal to roughly 2 (2.1°C and 4.2°C at average air temperatures of 6.0°C and 12.0°C correspondingly).

Conclusion
Influence of average air temperature and accuracy of its setting on the thaw depth of road foundation soil in various climatic areas was researched. The results of numerical calculations are presented as charts displaying the influence of temperature and degree of its variation over the period of the road use on the thaw depth. It was determined that there is a significant dependence of thaw depth on air temperature over the entire period. The longer the thawing duration, the greater the dependence. For example, when the air temperature changes from 6 to 12°C, all else equal, and the thawing duration is one month, thaw depth changes from 0.6 to 1.0 m. For a thawing duration of five months, the depth changes from 1.6 to 2.3 m. The increase in depth is 1.8x. The change in thaw depth is non-linear and depends on both average air temperature and on the accuracy of setting the average air temperature. For every average air temperature value there is a specific measure of accuracy of the temperature setting at which the calculation error will be within the permissible error value. A 3D chart showing the expected percentage error when forecasting the thaw depth depending on average air temperature and the accuracy of setting the air temperature was created. The article can be useful for engineers in the field of road construction. Foremost, it has a methodological relevance because it shows an algorithm to determine the influence of parameter accuracy on the results. Further research should be directed towards researching the influence of thermal physical properties of dispersed rocks on the thaw depth of road foundations considering the ice content in the foundation soil.