Solute mixing in porous media with dispersion and buoyancy

Abstract

We analyse the process of convective mixing in two-dimensional, homogeneous and isotropic porous media with dispersion. We considered a Rayleigh-Taylor instability in which the presence of a solute produces density differences driving the flow. The effect of dispersion is modelled using an anisotropic Fickian dispersion tensor (Bear, J. Geophys. Res. 1961). In addition to molecular diffusion ($D_m^{\ast }$), the solute is redistributed by an additional spreading, in longitudinal and transverse flow directions, which is quantified by the coefficients $D_l^{\ast }$ and $D_t^{\ast }$, respectively, and it is produced by the presence of the pores. The flow is controlled by three dimensionless parameters: the Rayleigh-Darcy number $Ra$, defining the relative strength of convection and diffusion, and the dispersion parameters $r=D_l^{\ast }/D_t^{\ast }$ and $\Delta =D_m^{\ast }/D_t^{\ast }$. With the aid of numerical Darcy simulations, we investigate the mixing dynamics without and with dispersion. We find that in absence of dispersion ($\Delta \to \infty$) the dynamics is self-similar and independent of $Ra$, and the flow evolves following several regimes, which we analyse. Then we analyse the effect of dispersion on the flow evolution for a fixed value of the Rayleigh-Darcy number ($Ra=10^4$). A detailed analysis of the molecular and dispersive components of the mean scalar dissipation reveals a complex interplay between flow structures and solute mixing. The proposed theoretical framework, in combination with pore-scale simulations and bead packs experiments, can be used to validate and improve current dispersion models to obtain more reliable estimates of solute transport and spreading in buoyancy-driven subsurface flows.

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• Experimental and numerical study of CO${}_2$ dissolution in a heterogeneous Hele-Shaw cell