Impact of saturation on evaporation-driven density instabilities in porous media: mathematical and numerical analysis

TL;DR

This work analyzes evaporation-driven density instabilities in partially saturated porous media with dissolved salt. It develops a coupled Richards'/salt-transport model and reformulates it via a Kirchhoff potential to obtain a ground state and a quasi-steady linear stability problem, yielding a time-dependent critical Rayleigh number $R_c(\hat{t})=\min_{\hat{a}} R_E(\hat{a},\hat{t})$ that governs onset. The study shows that saturation affects onset through storage, convection, and diffusion (captured by $R_E$, $\beta$, and $\gamma$), and that partial saturation can either promote or delay instability depending on parameter balance; viscosity variations are largely stabilizing but modest. Numerical simulations with DuMuX validate linear predictions and reveal the nonlinear evolution of salt fingers, including their sensitivity to bottom pressure and perturbation structure. Overall, the results provide fast onset criteria and physical insight into how partial saturation modulates density-driven fingering in evaporating soils, with clear paths for extending the analysis to precipitation and crust formation.

Abstract

Evaporation from a porous medium partially saturated with saline water, causes the salinity (salt concentration) to increase near the top of the porous medium as water leaves while salt stays behind. As the density of the water increases with increased salt concentration, the evaporation leads to a gravitational unstable setting, where density instabilities can form. Whether density instabilities form, depends on a large range of parameters like the evaporation rate and intrinsic permeability of the porous medium, but also on the water saturation. As water saturation decreases, the storage, convection and diffusion of salt also decrease, which all influence the onset of instabilities. By performing a linear stability analysis on the governing equations, we give criteria for onset of instabilities. Numerical simulations give information about the further development of these instabilities. With this knowledge we can predict whether and when density instabilities occur, and how they will influence the further development of salt concentration in the porous medium.

Figures (14)

• Figure 1: Sketch of domain with processes governing evaporation from a partially saturated porous medium
• Figure 1: Salt concentration profile (left) and saturation profile (right) using a (non-dimensional) bottom pressure of $\hat{P}_B = 1.25$ (top row) and $\hat{P}_B=1.75$ (bottom row). Initial profile in red, while lines corresponding to dashed, dotted and dash-dotted lines are for $\beta=0.01$, $\beta=0.1$ and $\beta=1$, respectively. Black lines correspond to $R_E=10$, while increasingly brighter green correspond to $R_E=10^2,10^3,10^4,10^5$. All lines but the initial red are at non-dimensional time unit $\hat{t}=1$. Note that many of the curves lie on top of each other.
• Figure 1: Influence of either perturbing the whole domain or the top row for the random case on the onset of instabilities (red line) which is derived by $\sigma^{top}$ (black line) and on the dominant wavelength $\lambda$ (grey line).
• Figure 1: Saturation profiles for the four different bottom pressures, corresponding to four different bottom pressures.
• Figure 2: Salt concentration profile (left) and zoomed-in version (right) using a bottom pressure of $\hat{P}_B = 1.25$ (top row) and $\hat{P}_B=1.75$ (bottom row). Initial profile in red, while lines corresponding to dashed, dotted, dash-dotted and solid lines are for $\hat{t}=0.25, 0.5, 0.75, 1$, respectively. Black lines correspond to $R_E=10$, while increasingly brighter green correspond to $R_E=10^2,10^3,10^4,10^5$. Especially for the top row, the darker green lines are barely visible as the brighter green are on top of them. All lines correspond to $\beta=0.1$.

Theorems & Definitions (2)

• Remark 1
• Remark 2