Various experimental and theoretical studies have shown that Fick's law, based on the assumption of a linear relation between solute dispersive mass flux and concentration gradient, is not valid when high concentration gradients are encountered in a porous medium. The value of the macrodispersivity is found to decrease as the magnitude of the concentration gradient increases. The classical, linear theory does not provide an explanation for this phenomenon. A recently developed theory suggests a nonlinear relation between concentration gradient and dispersive mass flux, introducing a new parameter in addition to the longitudinal and transversal dispersivities. Once a unique set of relevant parameters has been determined (experimentally), the nonlinear theory provides satisfactory results, matching experimental data of column tests, over a wide range of density differences between resident and invading fluids. The lower limit of the nonlinear theory, i.e. very low (tracer) density differences, recovers the linear formulation of Fick's law. The equations describing high concentration brine transport are a fluid mass balance, a salt mass balance in combination with a nonlinear dispersive mass flux equation, Darcy's law and an equation of state. We study the resulting set of nonlinear partial differential equations and derive explicit (exact) and semi-explicit solutions, under various assumptions. A comparison is made between mathematical solutions, numerical solutions and experimental data. The results indicate that the simple explicit solution can be used to simulate experiments in a wide range of density differences, given a unique set of experimentally determined parameters. The analysis shows that enhanced flow due to the compressibility effect, which is caused by local fluid density variations, is neglectable in all cases considered. The linear formulation of Fick's law appears to give an upperbound for magnitude of the compressibility effect.

Degenerate parabolic equations (msc 35K65), Heat and other parabolic equation methods (msc 58J35), Flows in porous media; filtration; seepage (msc 76S05)
Modelling, Analysis and Simulation [MAS]

Schotting, R.J, Moser, H, & Hassanizadeh, S.M. (1997). High-concentration-gradient dispersion in porous media : experiments, analysis and approximations. Modelling, Analysis and Simulation [MAS]. CWI.