Solute transport under non-linear sorption and decay

Abstract

Contaminant transport in aquifers is usually represented by a solution to the advective–dispersive differential equation. When the contaminant is subject to non-linear degradation or decay, or it is characterized by a chemical constituent that follows a non-linear sorption isotherm, the resulting differential equation is non-linear. Using the method of decomposition, series solutions were obtained for the non-linear equation. The series were used to derive and test “simulant” solutions that arise using the concept of double decomposition. Simulant solutions are closed-form analytic expressions that approximate part of the series. These expressions are simple, stable, and flexible. They permit an accurate forecasting of contaminant propagation under non-linearity in laboratory or field investigations at early or prolonged times after the spill. In this article, the practical scenario of an instantaneous spill, and that of a constant concentration boundary condition, is studied for situations of non-linear decay, non-linear Freundlich isotherm, and non-linear Langmuir isotherm. The solutions are verified with limited well-known analytical solutions of the linear reactive and non-reactive equations with excellent agreement, and with limited finite difference solutions. Plumes undergoing non-linear decay experience a profile re-scaling with respect to that of linear decay, the degree of which is controlled by the magnitude of the non-linear parameter b. The direction of the scaling (scaling up or scaling down with respect to the linear decay plume) is controlled by the magnitude of C (whether greater or less than 1) in relation to the magnitude of b (whether greater or less than 1). When C>1, values of b<1 produce plumes that experience less decay (i.e., are scaled up) than that of the linear decay, whereas values of b>1 produce non-linear plumes that experience more decay (i.e., are scaled down) than that of the linear decay. The opposite effect is observed when concentrations are less than 1. In other words, when C<1, values of b<1 produce non-linear plumes that experience more decay (i.e., are scaled down) than that of the linear decay, whereas values of b>1, produce non-linear plumes that experience less decay (i.e., are scaled up) than that of the linear decay. A plume undergoing non-linear sorption according to a Freundlich isotherm retards the processes of advection and dispersion with respect to a plume with no sorption. Similar to the case of non-linear decay, whether this retardation is more or less pronounced than that of the linear sorption plume depends on whether the values of b and C are greater or less than 1. The solution presented here for the advective dispersive equation subject to a Freundlich sorption isotherm is restricted to concentration greater than 1. When C>1 and b<1, the decrease in mobility in the non-linear plume is not as pronounced as that of a plume modeled by a linear isotherm. Plume shape may be quite sensitive to the values of the non-linear parameters. Plumes with parameter values b<1 (and C>1) exhibit the well-known lack of symmetry with respect to their center of mass, sharp fronts, and the tailing effects observed at hazardous waste sites. As the magnitude of the non-linear parameter increases, the non-linear plume approaches the linear one. This partial non-linear “retardation” can now be observed quantitatively with the models presented herein. The models developed also simulate the case of b >1 (i.e., “unfavorable” sorption), which produce a plume even more retarded than the linear. The shape of a contaminant plume following a non-linear Langmuir isotherm is very sensitive to the magnitude of the non-linear parameter α. Approximate solutions for mild non-linearity are presented.