The effect of dispersion on the stability of density-driven flows in saturated homogeneous porous media

This work is a continuation of Musuuza et al. [37] in which a stability criterion for density-driven flow in a saturated homogeneous medium was derived. The criterion predicted the stability of a system as a function of the density and viscosity contrasts, the magnitude of the flow velocity and the concentration gradients for flow aligned orthogonal to gravity. It could not accurately predict stability transition with increasing velocity, a failure we attributed to dispersion effects that were not included. Small-scale dispersion and molecular diffusion, the main stabilising mechanisms in homogeneous media can stabilise flow if the instability wavelengths are below a certain cutoff. The width of the mixing zone is also central in controlling the range of wavelengths that persist into fingers. We propose a method of quantifying the cutoff wavelength and the width of the mixing zone, which are incorporated into the earlier criterion as constituents of the dispersive part. The earlier criterion is reformulated in terms of the Rayleigh number and with the dispersive part added, we attempt to predict the number of fingers formed which is directly related to the physical stability of the system. The inclusion of the Rayleigh number and dispersion into a single stability criterion provides new insight in the way dispersion affects vertical flow systems. Stability numbers computed with the new criterion are in reasonable agreement with numerical simulations for a range of physical variables. The numerical computations are performed with the software package d3f, which uses the cell-centred finite volume and the implicit Euler methods for the spatial and temporal discretisations, respectively. The admission of the density and dispersivities as inputs into the criterion makes it usable in practical problems.

Research highlights► Extended cirterion in Musuuza, Attinger and Radu (2009) to include dispersion. ► Reformulated criterion in terms of a Rayleigh number. ► Dispersive contribution depends on the pertubation wavelength and the mixing zone width. ► Criterion was tested for the effects of density and dispersivities. ► Dispersion is crucial in the stabilisation of vertical Elder-type systems.