Research Project A05

Drying and evaporation in porous media is relevant for many engineering processes and for soil cultivation. These processes occur over several scales, where the evolving liquid-gas interface is important to describe the evaporation at the pore-scale, while its effect lies on the REV-scale. Our goal is to provide a better mathematical description of REV-scale evaporation by starting with a pore-scale description of the relevant processes and using upscaling to derive an effective model at REV-scale. The effect of the evolving interface is explicitly taken into account and different approaches to describe the evolving interface at the pore scale will be investigated.

The overall goal of project A05 is to describe the effect of evolving interfaces over several scales. To achieve this, we have considered several paths; (i) how to describe evolving interfaces, (ii) how to upscale processes from a micro- to macro-scale, and (iii) how to simulate a two-scale model efficiently.

To describe the evolving liquid-gas mixture we have considered either the use of classical, averaged approaches through the saturation or using phase fields. A phase-field is an order parameter indicating which phase one is in, but with a smooth, diffuse transition zone. Phase-field models are however approximations of the "real" sharp-interface physics. However, through the asymptotic sharp-interface limit, one can show that the expected physics are indeed captured by the model. As a starting point we considered a phase-field formulation for an evolving liquid-solid interface, where a solute dissolved in the liquid can precipitate and form a mineral, or the mineral can dissolve and release solute in the liquid. The interaction between the two phases is particularly important to capture; both with respect to the change of velocity between the two phases, but also to account for the mass transfer across the diffuse interface. Through the sharp-interface limit we could show that both of these aspects were correctly captured for the considered model.

As two fluids, for example liquid and gas, flow through a porous medium, they jointly affect each other. Especially for heterogeneous porous media, where the porosity and permeability vary, the REV-scale flow of the two fluids can have a highly complicated behavior. To understand this behavior better, we considered a layered medium with flow across the strata. At the transition zones between the layers, one fluid is either trapped or can pass through depending on the capillary pressure. This leads to the fluid saturation being highly oscillatory across the layers. When we are mainly interested in the large-scale behavior of the flow, an upscaled description is more desirable. Using homogenization we derived an upscaled model, which was found to match very well with the average of the highly variable micro-scale saturation. The computational costs of the upscaled model is several orders of magnitudes less than for the original micro-scale model. The upscaled model can be solved on a much coarser grid, where small, local reconstruction problems need to be solved to calculate effective quantities.

When upscaling from pore- to REV-scale using periodic homogenization, a two-scale model is obtained. Effective parameters needed at the REV-scale are found through solving local cell problems. For the phase-field model concerning mineral precipitation and dissolution, an efficient simulation strategy was developed in collaboration with Hasselt university. In stead of recalculating the effective parameters at every REV-scale point and at every time step, we used an adaptivity criterion for when and where to update a cell problem. This way, the two-scale simulations can be much more efficient.

Formulating a phase-field model for evaporation, where the mass transfer across the liquid-gas interface is accounted for, will be of special focus. By deriving its sharp-interface limit we can analyze whether the desired sharp-interface physics are correctly captured. Having this new model available, we can upscale the model from pore- to REV-scale using homogenization. This will give a similar two-scale structure as obtained for the mineral model, and we will focus on efficient implementation of this two-scale model.

We share an interest in homogenisation techniques and implementation of two-scale models with the Computational Mathematics group at Hasselt University. In cooperation with researchers at Hasselt University, we have studied upscaling of two-phase flow models in thin strips, addressing the effect of surface tension between the fluids and dynamic contact angles at the solid wall. Together we have also developed a two-scale iterative scheme with adaptive updating of the effective properties. These concepts we wish to develop further and also apply them to evaporation processes.

We are currently working on REV-scale concepts for two-phase flow in heterogeneous porous media, where evaporation induces instabilities. In the future we wish to exted this concepts to also include pore-scale information.