## 1. Introduction

[2] Stream depletion is the reduction in the flow rate in a river as a result of pumping in an aquifer that is hydraulically connected to the river. Stream depletion has many negative consequences, such as reduction in water supply for municipal, agricultural, and domestic uses; failure to satisfy existing water rights; and destruction of the ecosystems that depend on streams and rivers. Quantifying stream depletion therefore is crucial for protecting water supplies, surface water rights, and environments that depend on the streams and rivers.

[3] Various analytical and semianalytical expressions have been developed to quantify stream depletion for simple systems. *Theis* [1941] and *Glover and Balmer* [1954] presented expressions for calculating stream depletion due to pumping in a two-dimensional, homogeneous, infinite aquifer for a fully penetrating, infinitely long, straight stream that is in perfect hydraulic connection with the aquifer. Later studies developed analytical expressions that relaxed many of these assumptions, including partial hydraulic connection between the river and aquifer [*Hantush*, 1965], time-varying pumping rates [*Jenkins*, 1968; *Wallace et al*., 1990], partially penetrating stream [*Hunt*, 1999; *Butler et al*., 2001], leakage across a confining unit [*Butler et al*., 2007; *Zlotnik and Tartakovsky*, 2008], and a two-layered system with confined and unconfined aquifers [*Hunt*, 2009].

[4] *Sophocleous et al*. [1995] compared the analytical solution of *Glover and Balmer* [1954] to results of numerical simulations and found that the most restrictive assumptions of Glover and Balmer's model are perfect hydraulic connection between the stream and aquifer, full penetration of the stream, and a homogeneous aquifer; thus, analytical solutions have limitations. While analytical solutions are easy to apply, they are only applicable for idealized cases, and numerical models are necessary to simulate stream depletion in more complicated systems.

[5] The standard approach for using numerical models to calculate stream depletion is to first run one groundwater flow simulation without pumping to determine the exchange of water between the river and the aquifer, and then to run an additional simulation with pumping at one location to determine the change in the flow rate of water between the river and the stream. If the location of a new well is to be chosen, many possible well locations may be under consideration. It may be necessary to choose a location that limits depletion in a nearby stream; therefore, stream depletion must be calculated for many different well locations. Assuming the aquifer is sufficiently complex to require numerical models to simulate stream depletion, the standard approach must be repeated for each potential well location, and can become computationally inefficient if many potential well locations are considered. *Neupauer and Griebling* [2012] presented an adjoint method for calculating stream depletion in a river due to pumping in an adjacent aquifer. With the adjoint method, only one simulation is needed to calculate stream depletion for a well at any location in the aquifer; thus it is more efficient than the standard approach when multiple potential well locations are considered.

[6] The goal of this paper is to present the adjoint method for calculating stream depletion for a fully coupled river and aquifer system, in which both the aquifer head and the river head are affected by pumping in the aquifer. This model is a more realistic model of surface water and groundwater interaction than has been used previously with the adjoint method. *Neupauer and Griebling* [2012] developed the adjoint model assuming that the river head was known and was independent of the head in the aquifer. That model is simplistic because stream depletion, by definition, leads to changes in the flow rate and therefore to changes in the head in the river. For that model, both the forward and adjoint equations have the same form, so the adjoint equations can be solved using a standard groundwater flow code.

[7] In the next section, we present the forward equations of groundwater and stream flow, and we provide a mathematical expression for stream depletion. With our fully coupled river and aquifer system, the forward model is nonlinear. Next, we develop the adjoint equations for this stream and aquifer system, and we discuss the approach for solving the adjoint equations using a standard groundwater flow code. Because of the nonlinearity of the forward model, the adjoint equations do not have the same form as the forward equations; thus a standard groundwater flow code must be modified to solve the adjoint equations. In our adjoint derivation, we include tributaries and evapotranspiration, which were not included in previous work on adjoint methods for calculating stream depletion. Finally, we present an example of using the adjoint method to calculate stream depletion; we use these results to investigate the importance of various system parameters on stream depletion; and we demonstrate that the adjoint method is accurate and requires substantially less computation time than the standard approach.