ADJOINT METHOD FOR DELINEATING A PROBABILISTIC ZONE OF CONTRIBUTION

The zone of contribution (ZOC) around a well describes the area that contributes water to the well, and any contamination within the ZOC will eventually reach the well. Because of parameter uncertainty and unsteady flow conditions, ZOC delineation is subject to uncertainty; therefore the ZOC should be defined probabilistically with the ZOC representing the probability that contamination from any location will eventually reach the well. We present an efficient method for delineating probabilistic ZOCs that account for parameter uncertainty. The probabilistic ZOC is based on travel time cumulative distribution functions (CDF) that describe the probability that contamination from each location will reach the well within a specified time. The limiting value of these CDFs, as time approaches infinity, represents the probabilistic ZOC. The travel time CDF is an adjoint state of concentration; therefore, the probabilistic ZOC is obtained by solving the adjoint of a forward contaminant transport model. In the adjoint, the flow field is reversed, and sinks of contamination in the forward model (e.g., the well) become sources of probability in the adjoint model. The well becomes a continuous source of probability, and the probability is transported away from the well to all points within the ZOC. The adjoint contains all of the same processes as the forward contaminant transport model. The dispersion coefficient in the forward model quantifies the uncertainty caused by unmodeled aquifer heterogeneity and other sources of velocity variance. The dispersion coefficient quantifies the same parameter uncertainty in the adjoint model. The adjoint model directly accounts for the effects of chemically- and biologically-reactive contaminants on the ZOC. Applications of this adjoint approach to defining the ZOC are presented for public water supply wellhead protection and aquifer remediation pump-and-treat systems.