A Fractal Approach to Model Soil Structure and to Calculate Thermal Conductivity of Soils

Heat transport in soils depends on the spatial arrangement of solids, ice, air and water. In this study, we present a modified fractal approach to model the pore structure of soils and to describe its influence on the thermal conductivity. Three different fractal generators were sequentially applied to characterize a wide range of particle- and pore-size distributions. The given porosity and particle-size distribution of a clay, clay loam, silt loam and loamy sand were successfully modeled. The thermal conductivity of the fractal soil model was calculated using a network of resistors. We applied a renormalization approach to include the effects of smaller scale structures. The predictions were compared with the empirical Johansen' model (Johansen, 1975), that postulates a simple linear relationship between ice content and thermal conductivity. For high ice-saturated conditions, the calculated thermal conductivity agrees well with the empirical model. To describe partial ice saturation, we assumed that some pores were coated by ice films enclosing the air-filled center. In addition, we introduced a reduced heat exchange coefficient of the particles for unsaturated conditions. The ice-saturated and -unsaturated thermal conductivity calculated with this approach was very similar to that estimated by the empirical model. The variation of the thermal conductivities for different spatial arrangements of pores and particles in the prefractals were determined. Extreme values deviate more than 50% from the mean values.