Research Project A05

Drying and evaporation in porous media play a crucial role in various engineering applications and soil cultivation. These processes occur across multiple scales, where the dynamics of the evolving liquid-gas interface is essential to understand the evaporation at the pore scale, while its overall impact is observed at the REV scale. Our objective is to develop a more precise mathematical model for REV-scale evaporation by first examining the underlying pore-scale mechanisms. From the pore-scale model, we apply upscaling techniques to derive an effective model at the REV scale. A key aspect of our approach is explicitly accounting for the evolving liquid-gas interface and exploring different approaches to describe its behaviour at the pore scale.

Funding Period 1

In the first funding phase, we started developing phase-field models for reactive transport and two-phase flow in collaboration with research project C02. These phase-field models introduce a diffuse transition between the phases, making them more amendable for mathematical analysis and numerical implementation. For an example, see Video 1, which illustrates a reactive transport phase-field model. Since the diffuse transition serves as a mathematical approximation of the sharp-interface physics, we therefore investigated the sharp-interface limits of these phase-field models to ensure their validity. We further started on the homogenization of models related to evaporation, specifically for two-phase flow, heat transport and reactive transport – considering each of them separately. The resulting two-scale models were numerically implemented and tested.

Funding Period 2

In the current (second) funding phase, we have built further on the findings from the first funding period. We extended the phase-field modelling to include heat transport and also in collaboration with research project C02 we developed a phase-field model for evaporation. Both models were validated by deriving their sharp-interface limits and were subsequentially implemented and tested numerically. Concerning the phase-field evaporation model we used it to investigate how a flow field around an evaporating droplet is developing. Figure 1 and Figure 2 illustrate the flow field around evaporating droplets, where no external flow field is imposed in Figure 1, while an external flow field is induced in Figure 2.

We further upscaled the phase-field models using periodic homogenisation, which resulted in a two-scale formulation. In this approach, local pore-scale cell problems provide effective parameters for the REV-scale equations. For two-phase flow we in particular investigated the behaviour of the effective parameters with respect to fluid distribution. The evaporation model was also successfully upscaled with periodic homogenisation, and we are currently implementing the resulting cell problems. Simulating the two-scale formulation is computationally beneficial as the REV-scale model equations can be solved on a coarser grid, and receive input from the local pore-scale cell problems that each can be solved on smaller domains. The two-scale formulation will eventually be implemented with the help of preCICE in collaboration with research project D02.

Concerning the vision topic, we analyzed density instabilities caused by evaporation from a porous medium in collaboration with research project A02 and associated research projects CX3 and CX5. The evaporation causes an accumulation of dissolved salts in the remining water, giving a gravitationally unstable setting. We applied linear stability analysis to investigate the onset of such density instabilities, and compared with numerical simulations. See Figure 3 for an example result of onset times found by linear stability analysis and numerical simulations.

We share an interest in homogenisation techniques with Prof. Iuliu Sorin Pop from the Computational Mathematics group at Hasselt University. Together we have studied upscaling of two-phase flow models, in particular we have addressed the effect of surface tension between the fluids and dynamic contact angles at the solid wall. We have also developed a two-scale adaptive scheme for reactive transport.

We have collaborated with Prof. C. J. van Duijn from Eindhoven University of Technology on analysis of evaporation-induced density instabilities and salt precipitation, connected to the vision topic.