Forward and backward temporal probability distributions of sorbing solutes in groundwater

Abstract

The travel time of a contaminant particle in an aquifer can be used to identify the release time of contamination from a known source, to delineate captures zones around wells, and for other applications. The travel time is uncertain because of spatial variability of hydrogeologic properties and fluctuations of groundwater velocity and therefore is often represented as a random variable. A forward travel time probability density function (pdf) describes the random time that a particle will reach a downgradient location, while the backward travel time pdf describes the time in the past when the particle was at an upgradient location, such as the source location. For sorbing solutes, the particle may have been in either the aqueous phase or the sorbed phase at the location of interest. Since a particle can only travel through the aquifer in the aqueous phase, the random travel time accounts only for particles in the aqueous phase. We develop new probabilistic expressions to characterize the random time in the past that a contaminant particle was sorbed at a location of interest. For a solute that exhibits linear equilibrium sorption, we develop forward and backward pdfs for the time of sorption, defined as the random time at which the particle is at a location of interest in the sorbed phase. For the linear nonequilibrium sorption case, we develop expressions for the forward and backward sorbed phase probability, defined as the discrete probability that a particle will be sorbed at a location of interest at a deterministic time. Although forward pdfs and probabilities produce the same information as backward pdfs and probabilities, backward probabilistic expressions can be calculated more efficiently when the number of observations is small relative to the number of possible source locations or release times.