Adjoint method for obtaining backward‐in‐time location and travel time probabilities of a conservative groundwater contaminant

Backward location and travel time probabilities can be used to determine the prior location of contamination in an aquifer. For a contaminant particle that was detected in an aquifer, the backward location probability is the probability of where the particle was located at some prior time. Backward travel time probability is the probability of when the particle was located at some position upgradient of the detection. These probabilities can be used to improve characterization of known sources of groundwater contamination, to identify previously unknown contamination sources, and to delineate capture zones. For simple model domains, backward probabilities can be obtained heuristically from a forward model of contaminant transport. For multidimensional problems and complex domain geometries, the heuristic approach is difficult to implement and verify. The adjoint method provides a formal approach for obtaining backward probabilities for all model domains and geometries. We formally show that the backward model probabilities are adjoint states of resident concentration. We provide a methodology for obtaining the governing equations and boundary and final conditions for these probabilities. The approach is illustrated using a one-dimensional, semi-infinite domain that mimics flow to a production well, and these results are compared to equivalent probabilities derived heuristically.