Analytical solutions to the stream-aquifer interaction problem: a critical review

The aim of the present paper is to give a systematic and critical presentation of important existing
analytical solutions for transient stream-aquifer interaction, which can be used to give answers to
simple interaction problems or to verify mathematical models. Stream-aquifer interaction is the most
common subject of papers discussing surface water-ground water interaction and a review of
analytical solutions to the problem is lacking from the literature. The analytical solutions presented in
the paper are firstly distinguished based on whether only the ground water flow equations or both the
ground water and stream flow equations are solved for their derivation and secondly based on the
type of aquifer (confined or unconfined) interacting with the stream and on the type of equations
solved. The literature review showed that there is only a small number of publications, where the
authors consider both the ground water and the stream flow equations for the development of the
analytical solutions. The majority of the available analytical solutions of stream-aquifer interaction
are derived by solving only the ground water flow equations, taking into account the stream water
level as a boundary condition.
For each analytical solution presented in the paper, its accuracy, its ease of application to simple
interaction problems and its suitability for the verification of mathematical models are discussed in
detail. Specifically for the case of predicting the water table level in unconfined aquifers interacting
with streams, an analytical solution of the non-linear Boussinesq equation is compared to two
analytical solutions of different linearized forms of the Boussinesq equation, in order to quantify the
error in estimating the water table level when using a linear solution. Among the very few analytical
solutions found in the literature, where the authors consider both the stream flow and ground water
flow equations for their development, the most comprehensive one is chosen to give an application
example, which can be used as a benchmark case for the verification of integrated stream-aquifer
mathematical models.