Adjoint-derived location and travel time probabilities for a multidimensional groundwater system

Backward location and travel time probabilities can be used to determine the former location of contamination in an aquifer. For a contaminant parcel that was detected in an aquifer the backward location probability describes its position at some time prior to sampling, and the backward travel time probability describes the amount of time required for it to travel to the sampling location from some upgradient position. These probabilities, which can provide information about the source of contamination, are related to adjoint states of resident concentration. The governing equations of the backward probabilities are adjoints of the forward governing equation, e.g., the advection-dispersion equation. We derive these backward governing equations and their boundary and final conditions for both location and travel time probabilities in a multidimensional system. Each governing equation contains the adjoint of the advection-dispersion operator and a load term that defines the particular adjoint state (probability). The load term depends on both the type of probability (location or travel time) and the sampling device (pumping well or monitoring well) with which the contamination was detected. The adjoint equation can also be used to efficiently determine forward location and travel time probabilities describing the future location of groundwater contamination, a feature most useful for delineating pumping well captures zones. We illustrate the use of the backward model for obtaining location and travel time probabilities in a hypothetical two-dimensional domain.