Stagnation Points as Loci of Solute Concentration Extrema at the Evaporative Surface of a Random Porous Medium

Abstract

Evaporation of a saline solution from a porous medium often leads to the precipitation of salt at the surface of the porous medium. It is commonly observed that the crystallized salt does not form everywhere at the porous medium surface but at some specific locations. This is interpreted at the signature of spatial variations in the salt concentration at the surface of the porous medium prior to the onset of crystallization. We explore numerically the link between the ion concentration spatial variations at the surface and porous medium heterogeneities considering strongly anisotropic short-range correlated permeability Gaussian fields corresponding to a vertical layering perpendicular to the top evaporative surface for the case of the evaporation–wicking situation. It is shown that the ion concentration extrema at the surfaces correspond to stagnation points with minima corresponding to divergent stagnation points and maxima to convergent stagnation points. Counter-intuitively, the ion concentration maxima are shown to correspond to permeability minima. However, the ion concentration absolute maximum does not necessarily always correspond to the permeability absolute minimum. More generally, the study emphasizes the key role played by the impact of heterogeneities on the velocity field induced in the medium by the evaporation process. It is also shown that the number of ion mass fraction maxima at the porous medium surface is generally much lower than the naive prediction based on the number of correlation lengths spanning the medium.

Appendix

The filtration velocity distribution at the inlet of the system can be estimated as follows. Sufficiently away from the top surface the pressure only depends on z (thus is independent of x) and is the same in all media. Let us denote this pressure by P^*.

Thus, the velocity away from the evaporative surface can be expressed as

$$ V_{y} = - \frac{k}{\mu }\frac{{{\text{d}}P^{*} }}{{{\text{d}}y}} $$

(A-1)

Since the velocity is known on the top surface (V_y = V₀ = j/ρ_ℓ), expressing the flow rate conservation reads

$$ \int\limits_{0}^{{L_{x} }} {V_{y} {\text{d}}x} = - \frac{1}{\mu }\frac{{{\text{d}}P^{*} }}{{{\text{d}}y}}\int\limits_{0}^{{L_{x} }} k {\text{d}}x = L_{x} V_{0} $$

(A-2)

leading to

$$ \frac{{{\text{d}}P^{*} }}{{{\text{d}}z}} = - \frac{{\mu V_{0} }}{\left\langle k \right\rangle } $$

(A-3)

where

$$ \left\langle k \right\rangle = \frac{1}{{L_{x} }}\int\limits_{0}^{{L_{x} }} {k{\text{d}}x} $$

(A-4)

As a result, the velocity sufficiently away from the surface and thus at the inlet is given by

$$ V_{y} = \frac{k}{\left\langle k \right\rangle }V_{0} $$

(A-5)