Exact analytical solutions for contaminant transport in rivers 1. The equilibrium advection-dispersion equation

Abstract

Analytical solutions of the advection-dispersion equation and related models are indispensable for predicting or analyzing contaminant transport processes in streams and rivers, as well as in other surface water bodies. Many useful analytical solutions originated in disciplines other than surface-water hydrology, are scattered across the literature, and not always well known. In this two-part series we provide a discussion of the advection-dispersion equation and related models for predicting concentration distributions as a function of time and distance, and compile in one place a large number of analytical solutions. In the current part 1 we present a series of one-and multi-dimensional solutions of the standard equilibrium advection-dispersion equation with and without terms accounting for zero-order production and first-order decay. The solutions may prove useful for simplified analyses of contaminant transport in surface water, and for mathematical verification of more comprehensive numerical transport models. Part 2 provides solutions for advective-dispersive transport with mass exchange into dead zones, diffusion in hyporheic zones, and consecutive decay chain reactions.

Numerical simulations in the Mala Nitra stream by 1D model

“…The ADE distinguishes two transport modes: advective transport as a result of passive movement along with water, and dispersive/ /diffusive transport to account for diffusion and small-scale variations in the flow velocity as well as any other processes that contribute to solute spreading. ADE comes from the mass balance equation [Van Genuchten et al 2013], which can be formulated in a general manner by considering the accumulation or depletion of solute in a control volume over time as a result of the divergence of the flux (i.e., net inflow or outflow), possible reactions, and the injection or extraction of solute along with the fluid phase. A variety of solute source or sink terms may need to be implemented in the ADE.…”

Exact Analytical Solutions for Contaminant Transport in Rivers

“…All solutions presented in Part 1 (van Genuchten et al, 2013) hold for equilibrium contaminant transport characterized by relatively symmetrical or sigmoidal concentration distributions versus time or distance, unless modified by production and degradation processes, or special initial or boundary conditions. This ideal situation generally does not occur in streams and rivers because of the presence of relatively immobile or stagnant zones of water connected to the mean stream channel.…”

“…In part 1 (van Genuchten et al, 2013) we summarized solutions for one-and multidimensional equilibrium transport, with and without zero-order production and first-order decay. In the current part 2 we provide solutions for transport with simultaneous first-order exchange between the river and relatively immobile or stagnant water zones (transient storage models) and for situations where exchange with the hyporheic zone is modeled as a diffusion process (further referred to here as hyporheic zone diffusion models).…”

Mathematical modeling of groundwater contamination with varying velocity field

“…The second-order error was investigated through the truncated Taylor series approximation by using the explicit finite difference method to solve one-dimensional advection dispersion equation (Chaudhari, 1971). Numerical studies explored the effect of numerical dispersion (De Smedt and Wierenga, 1977;Dudley et al, 1991;van Genuchten and Gray, 1978). The root mean square was used to calculate the average error at each nodal point of the grid (Roberts and Selim, 1984).…”

“…Zamani and Bombardelli (2014) explored analytical solutions to the ADR (advection-dispersion-reaction) equation from which one can know the spatio-temporal changes in flow field and dispersivity. Van Genuchten et al (2013) presented a series of one-and multi-dimensional solutions of the standard equilibrium advection-dispersion equation with and without terms accounting for zero-order production and first-order decay, which proved useful for simplified analyses of contaminant transport in surface water, and for mathematical verification of more comprehensive numerical transport models where the isotherm concept was not used.…”