Analytical and numerical adjoint solutions for cumulative streamflow depletion

Streamflow depletion is traditionally defined as the instantaneous change in the volumetric rate of aquifer–stream exchange after a finite period of continuous groundwater extraction. In the present study an alternative metric of streamflow depletion was considered: cumulative stream depletion (CSD), which described the total volumetric reduction in flow from an aquifer to a stream resulting from continuous groundwater extraction over a finite period. A novel analytical solution for the prediction of CSD was derived, based upon an existing solution that accounted for streambed conductance and partial stream penetration. Separately, a novel numerical solution for the prediction of CSD was derived, based on the derivation of an adjoint state solution. The accuracy of the two new solutions was demonstrated through benchmarking against existing analytical solutions and perturbation-based results, respectively. The derivation of the loading term used in the adjoint state solution identified three parameters of relevance to CSD prediction. First is streambed hydraulic conductivity and thickness, both of which contribute to a lumped parameterization of streambed conductance. Second is aquifer specific yield, which controls the rate at which hydraulic perturbations propagate through an aquifer. The computational advantage of the adjoint state approach was highlighted, in which a single numerical model run can be used to predict CSD resulting from any potential groundwater extraction location. The reduction in computation time achieved was proportional to the number of potential extraction well locations. Where the number of locations is large, reductions in computation times of nearly 100 % can be achieved.