A porous flow approach to model thermal non-equilibrium applicable to melt migration

We develop an approach for heat exchange between a fluid and a solid phase of a porous medium where the temperatures of the fluid and matrix are not in thermal equilibrium. The formulation considers moving of the fluid within a resting or deforming porous matrix in an Eulerian coordinate system. The approach can be applied, for example, to partially molten systems or to brine transport in porous rocks. We start from an existing theory for heat exchange where the energy conservation equations for the fluid and the solid phases are separated and coupled by a heat exchange term. This term is extended to account for the full history of heat exchange. It depends on the microscopic geometry of the fluid phase. For the case of solid containing hot, fluid-filled channels, we derive an expression based on a time-dependent Fourier approach for periodic half-waves. On the macroscopic scale, the temporal evolution of the heat exchange leads to a convolution integral along the flow path of the solid, which simplifies considerably in case of a resting matrix. The evolution of the temperature in both phases with time is derived by inserting the heat exchange term into the energy equations. We explore the effects of thermal non-equilibrium between fluid and solid by considering simple cases with sudden temperature differences between fluid and solid as initial or boundary conditions, and by varying the fluid velocity with respect to the resting porous solid. Our results agree well with an analytical solution for non-moving fluid and solid. The temperature difference between solid and fluid depends on the Peclet number based on the Darcy velocity. For Peclet numbers larger than 1, the temperature difference after one diffusion time reaches 5 per cent of (T) over tilde or more ((T) over tilde is a scaling temperature, e.g. the initial temperature difference). Thus, our results imply that thermal non-equilibrium can play an important role for melt migration through partially molten systems where melt focuses into melt channels near the transition to melt ascent by dykes. Our method is based on solving the convolution integration for the heat exchange over the full flow history, which is numerically expensive. We tested to replace the heat exchange term by an instantaneous, approximate term. We found considerable errors on the short timescale, but a good agreement on the long timescale if appropriate parameters for the approximate terms are used. We derived these parameters which may be implemented in fully dynamical two-phase flow formulations of melt migration in the Earth.