Density-dependent dispersion in heterogeneous porous media Part II: Comparison with nonlinear models

The results of a series of high-resolution numerical experiments are used to test and compare three nonlinear models for high-concentration-gradient dispersion. Gravity stable miscible displacement is considered. The first model, introduced by Hassanizadeh, is a modification of Fick’s law which involves a second-order term in the dispersive flux equation and an additional dispersion parameter β. The numerical experiments confirm the dependency of β on the flow rate. In addition, a dependency on travelled distance is observed. The model can successfully be applied to nearly homogeneous media (σ2 = 0.1), but additional fitting is required for more heterogeneous media. The second and third models are based on homogenization of the local scale equations describing density-dependent transport. Egorov considers media that are heterogeneous on the Darcy scale, whereas Demidov starts at the pore-scale level. Both approaches result in a macroscopic balance equation in which the dispersion coefficient is a function of the dimensionless density gradient. In addition, an expression for the concentration variance is derived. For small σ2, Egorov’s model predictions are in satisfactory agreement with the numerical experiments without the introduction of any new parameters. Demidov’s model involves an additional fitting parameter, but can be applied to more heterogeneous media as well.