Upscaling Heat Flow in Porous Media With Periodic Surface Temperature Fluctuation Using a One-Dimensional Subordinated Heat Transfer Equation

The one-dimensional (1-d), constant-parameter heat transport equation (HTE) built upon Fourier's law proposed in 1822 has been widely used to model heat transport in soil, where the core question is how to fit the observed temperature time series without detailed geomedium heterogeneity being mapped in the model. To address this question, this study extended the classical HTE by adding one parameter (the time index) using the time subordination approach to capture the impact of random operational times spent by individual heat parcels on heat dynamics. This led to a parsimonious, 1-d, Subordinated HTE (Sub-HTE), which assumes the temporally non-Fourier heat transport and contains the classical HTE as an end member, with analytical solutions aimed at upscaling local heat transport in heterogeneous porous media. To test this approach, temperature time-series were collected in unsaturated soils built in the laboratory that were subjected to periodic surface temperature fluctuations. Model applications showed that, although the HTE captures the general shape of the observed temperature time series, the Sub-HTE is a mild improvement of the HTE in characterizing the nuances of temperature amplitude attenuation (likely due to the small operational time at the intermediate/late time stages). Both Eulerian and Lagrangian frameworks were developed to interpret other dynamics in heat flow, including the early time temperature jump and phase advance. Parameter analyses also revealed that the thermal front velocity in the Sub-HTE is a function of Darcy flux, the time index, and the system's thermal properties, while the thermal diffusivity (the only parameter related to thermal properties in the Sub-HTE) can be scale independent. Therefore, the 1-d Sub-HTE can be a reliable generalization of the classical 1-d HTE in upscaling heat transport in soil.