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\begin{document}

\title{Analytical and numerical adjoint solutions for cumulative streamflow
depletion}



\author[1]{Chris Turnadge}%
\author[2]{Roseanna M. Neupauer}%
\author[3]{Okke Batelaan}%
\author[1]{Russell S. Crosbie}%
\author[3]{Craig T. Simmons}%
\affil[1]{Commonwealth Scientific and Industrial Research Organisation (CSIRO)}%
\affil[2]{University of Colorado Boulder}%
\affil[3]{Flinders University}%


\vspace{-1em}



  
  \date{June 13, 2024}


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\begin{abstract}
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.%
\end{abstract}%



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\textbf{Hosted file}

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\textbf{Hosted file}

\verb`Turnadge_CSD_Supporting_Information.docx`  available at \url{https://authorea.com/users/543810/articles/788449-analytical-and-numerical-adjoint-solutions-for-cumulative-streamflow-depletion}\selectlanguage{english}
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\includegraphics[width=0.70\columnwidth]{figures/Turnadge-CSD-Figure-6/Turnadge-CSD-Figure-6}

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