A general framework of stream–aquifer interaction caused by variable stream stages

Stream–aquifer interface is a critical zone where surface water and groundwater exchange mass, energy, and chemicals. Previous analytical studies of stream–aquifer interaction consider stream stage fluctuations that are only functions of time. This study improves those solutions by simultaneously considering temporally and spatially variable stream stages, low-permeability streambeds, and pumping wells nearby the streams. Semi-analytical solutions of drawdown and stream–aquifer flow rate in the Laplace–Fourier domain are provided and used to obtain solutions in the real time–space domain after the inverse Laplace–Fourier transform, where the stream–aquifer flow rate refers to the rate of discharge between a stream and an adjacent aquifer. The stream–aquifer flow rate caused by the temporally variable stream stage is found to decline with time and becomes negligible when time is sufficiently long. A closed-form analytical solution for the stream–aquifer flow rate caused by a diffusive-type flood wave is obtained, and it can be used to predict the response of an aquifer upon a passing flood wave. For a diffusive-type flood wave that dies out in about 24 h, we find that the stream–aquifer flow rate drops nearly exponentially with time and dies out at about 13.4 h for a sandy aquifer and an alluvial streambed. Sensitivity analysis shows that the stream–aquifer flow rate increases with the diffusion coefficient of the flood wave (D) which controls the width and amplitude of the wave, and decreases with the average flood wave velocity (v). Low-permeability streambeds reduce stream–aquifer flow rate and also delay the response to the stream stage fluctuations. For a time-dependent cosine stream stage function with a period of 30 days, the maximal stream–aquifer flow rate with a 0.5 m thick low-permeability streambed is 24% less than that obtained without the streambed.