A non-linear theory of high-concentration-gradient dispersion in porous media

Abstract

The application of Fick's law to describe hydrodynamic dispersion in porous media is based on the assumption of a linear dependance of a solute dispersive mass flux on its concentration gradient. Both theoretical and experimental studies have shown that the Fickian description of dispersion is not valid when large concentration variations in the porous medium are encountered. However, an appropriate alternative is still lacking. In this work, based on a theoretical derivation of the Fickian dispersion equation, a non-linear theory of dispersion is suggested. In the non-linear theory, in addition to the longitudinal and transversal dispersivities, a new parameter is introduced. Miscible displacement experiments are carried out in order to investigate the effects of large variations in salt mass fraction and to assess the validity of the new theory. Low-concentration liquid is displaced upwards in a vertical column by a high-concentration liquid. Thus, only hydrodynamically stable flow regimes are considered. The experiments are simulated by means of both classical Fick's law and the new non-linear theory. It is found that low-concentration-gradient experiments can be simulated satisfactorily using the Fickian-type dispersion equation. However, calculated breakthrough curves for high-concentration-gradient experiments deviate substantially from the measured curves. It appears that a satisfactory fit to high-concentration-gradient data can be obtained only if the value of longitudinal dispersivity is reduced by a factor of three. Using the non-linear theory, however, it is possible to simulate both low- and high-concentration-gradient experiments with a unique set of parameter values.