Heat conduction with phase change in soils with macro-pores, snow, and cryoconite. Part I: unified model derivation and examples

TL;DR

This paper develops a comprehensive thermal conduction model that incorporates phase change in soils, snow, and cryoconite, bridging micro- and macro-scale processes and enabling coupled simulations with practical applications.

Contribution

It introduces a unified model blending soil and bulk water models, applicable to macro-pores, snow, and cryoconite, with thermodynamic consistency and invertible relationships.

Findings

Model aligns well with empirical data.

Framework supports coupled soil-snow-cryoconite simulations.

Practical models for heat conductivity are tested and validated.

Abstract

In this paper we extend models of thermal conduction with phase transition from micro- to macro-scale. Such models were previously developed for soils in permafrost regions from pore to Darcy scale, and the Darcy scale models compare well to empirical relationships. The new general model blends soil model with bulk water model and thus works well for the soils with macro-pores; it also applies to new context including modeling thermal conduction in the snow and in cryoconite, and it is consistent with rigorous thermodynamics derivations as well as with practical models from the literature. From mathematical point of view, the general model relies on carefully defined relationships between temperature, phase fraction and internal energy, which we show are invertible in a properly defined framework of multivalued graphs. We also discuss and test practical models for average heat conductivity. Our framework allows to create monolithic numerical models useful for modeling of coupled soil and snow models as well as cryoconite. We motivate and illustrate the results with practical examples and computations.