Introduction Frost Beneath and Exposed Culvert The objective of this analysis is to determine the frost penetration beneath an exposed concrete culvert that is exposed to an average air temperature of -8.5 C over a period of two months. Two cases are considered: ) no snow cover on the ground surface; and, ) mm of snow cover on the ground surface. The analysis demonstrates the insulating effect of snow, which has a lower thermal conductivity and heat capacity due to the presence of air in the void space. Numerical Model Figure presents the model geometry and stratigraphy. The foundation soil is 5.8 m thick. A mm thick region is placed at the top of the domain to model the snow pack for Case. A 5 mm thick concrete culvert is constructed near the ground surface and extends to just over 7 mm depth. The geology comprises a silty sand fill material overlying a sandy native soil material. The local water table elevation is located just below the contact between the fill and native soil. For simplicity, the native soil is assumed to be fully saturated. It should be noted that only half the domain could have been modeled in this case, as the center-line of the culvert is also a line of symmetry. Concrete 5 Elevation 3 Native Soil 3 5 7 8 Figure - Model geometry Distance TEMP/W File: Frost Beneath Exposed Culvert.gsz Page of 5
3 Material properties A full-thermal material model was used for the fill and native soil materials. Figure and Figure 3 present the thermal conductivity verses temperature function and normalized unfrozen water content function, respectively. Both functions demonstrate that the thermal properties change over a narrow temperature range below the phase change temperature. This phenomenon is a result of capillary forces that hold the water in the void space of soils. The transition from an unfrozen material property to a frozen material property occurs at a lower temperature for finer grained soils. 5 Thermal Conductivity (kj/days/m/ C) 3 9 Native 8 7 - - Figure - Thermal conductivity function Unfrozen Water Content (m³/m³).8... Native - - Figure 3 - Normalized unfrozen water content function TEMP/W File: Frost Beneath Exposed Culvert.gsz Page of 5
The water content for the fill material was assumed to be 8% by volume (~ % by weight) and the native soil was assumed to be % by volume (~ 3% by weight). The volumetric heat capacity of both soils was assumed to be 7 kj/m 3 / C for the unfrozen case and kj/m 3 / C for the frozen case. The concrete and snow were modeled using a simplified thermal model with constant thermal properties in the frozen and unfrozen states. The snow was assumed to have a thermal conductivity of 3. kj/day/m/ C (.3 W/m/K) with a heat capacity of 7 kj/m 3 / C. The concrete thermal conductivity was set at 55.5 kj/day/m/ C (.8 W/m/K) with a heat capacity of kj/m 3 / C. Initial and Boundary Conditions The initial conditions for both cases was established using a spatial function. The spatial function assumed a temperature of. C near the surface, 9 C at a depth of m, and 5 C at the base of the model (Figure ). 5 Elevation 3 3 5 7 8 Distance Figure Temperature contours for the spatial function The bottom boundary condition for both analyses was set to a constant temperature of 5 C while the top boundary is set to a temperature of -8.5 C for the duration of the analysis. In the first case, the boundary condition is applied to the top of the fill and culvert. In the second case, the snow material has been applied to the upper region, so the boundary is applied to the top of the domain. TEMP/W File: Frost Beneath Exposed Culvert.gsz Page 3 of 5
5 Results and Discussion Temperature contours and the location of the freezing front after 7 days for Case and are presented in Figure 5. The depth of the freezing front along the edges of the domain has propagated approximately. m and. m below ground for the case without snow and with snow, respectively. The reduced freezing depth for Case demonstrates the insulating effect of the snow, which has a thermal conductivity approximately % to % of the soil conductivity. There is no snow pack within the culvert; so the freezing depth is about the same in both cases at the line of symmetry. Figure 5 - Temperature contours and location of freezing front after 7 days Figure presents a plot of temperature verses time beneath the culvert ( m below ground surface) for the case that includes the snow pack. The temperature drops very rapidly as heat flow towards the freezing boundary. Eventually, the steep thermal gradient between the ground surface and the warmer TEMP/W File: Frost Beneath Exposed Culvert.gsz Page of 5
temperatures within the domain begins to diminish as the freezing front propagates downward. As a result, the upward thermal flux reduces and the cooling rate diminishes. T vs Time at m depth - - - 3 5 7 Time (days) Figure - Temperature vs time beneath the culvert for case Summary and Conclusions TEMP/W is used to model the effect of snow pack on the propagation of a freezing front over a period of two months. The analysis demonstrates the significant insulating effect of snow, which has a much lower thermal conductivity and heat capacity compared to soil. The lack of snow cover over the culvert results in little change in frost depth in this region. TEMP/W File: Frost Beneath Exposed Culvert.gsz Page 5 of 5