On the possibilities of the power mean as an upscaling method using the example of thermal conductivity in soil

The power mean is the generalization of the common averaging methods, such as harmonic, geometric and arithmetic mean, but also minimum and maximum. However, it also allows an infinite number of other means between these common means and can therefore be adapted very flexibly to the specific task of upscaling. This will be demonstrated in the contribution by calculating the effective thermal conductivity as the mean of the partial conductivities of soil components (typically of the solid, liquid, and gaseous phase). Soil thermal conductivity is a key factor for the soil heat balance and is widely used in many fields of science. However, it is elaborate to measure thermal conductivity of soils that have different porosities and degrees of saturation. Effective thermal conductivity of soil strongly depends on the arrangement of particles (soil structure) and on the interaction of added water to the solid phase (e.g., menisci).  To improve the prediction of soil thermal conductivity, specific information of soil structure needs to be taken into account. The relationship between the power mean exponents p and the degree of saturation is an indicator of the existing soil structure.