Flushing factors and a sharp front solution for solute transport with multicomponent ion exchange

Abstract

Three cation exchange equations due to Vanselow, Gapon, and Gaines and Thomas are often used in transport models of ions subject to ion exchange with the solid aquifer matrix. The equations differ in the convention for the activity of adsorbed ions, and give different isotherm slopes for binary ion exchange pairs which influence calculated transport. In the binary case, the velocity of a composition is inversely related to the slope of the isotherm. The term ‘flushing factor’ is introduced as an inverse velocity that permits the easy construction of elution curves. In the multicomponent case, the flushing factors are found as eigenvalues. An analytical solution is obtained for multicomponent (heterovalent) transport with shock fronts that can be validated with the flushing factor theory. This analytical solution is applied to the freshwater injection in a brackish water aquifer presented by Valocchi and coworkers. The results indicate that the Gaines and Thomas, and Vanselow conventions can adequately describe observed data with constant exchange coefficients for the binary exchange reaction pairs. The Gapon convention gives an S-shaped isotherm for multivalent ions that can easily lead to selectivity reversals; such selectivity reversal has not been observed.