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., vol. 66, 1961, pp. 1185–1197). In addition to molecular diffusion ( D∗m ), the solute is redistributed by an additional spreading, in longitudinal and transverse flow directions, which is quantified by the coefficients D∗l and D∗t, 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/D∗t and Δ=D∗m/D∗t . With the aid of numerical Darcy simulations, we investigate the mixing dynamics without and with dispersion. We find that in the absence of dispersion ( Δ→∞ ) 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=104 ). A detailed analysis of the molecular and dispersive components of the mean scalar dissipation reveals a complex interplay between flow structures and solute mixing. We find that the dispersion parameters r and Δ affect the formation of fingers and their dynamics: the lower the value of Δ (or the larger the value of r ), the wider, more convoluted and iffused the fingers. We also find that for strong anisotropy, r=O(10), the role of Δ is crucial: except for the intermediate phases of the flow dynamics, dispersive flows show more efficient (or at least comparable) mixing than in non-dispersive systems. Finally, we look at the effect of the anisotropy ratio r, and we find that it produces only second-order effects, with relevant changes limited to the intermediate phase of the flow evolution, where it appears that the mixing is more efficient for small values of anisotropy. 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.