Analytical Solution for Stream Depletion with Time-Varying Streambed Hydraulic Conductivity

Stream depletion is the reduction in flow rate in a stream due to pumping in an aquifer that is hydraulically connected to the stream. The amount of stream depletion is related to the pumping rate, the distance between the well and the stream, and the hydraulic properties of the aquifer and streambed. If the streambed sediment has high hydraulic conductivity, water flows easily between the aquifer and the stream, leading to high stream depletion. On the other hand, if the sediment has low hydraulic conductivity, the exchange of water between the aquifer and stream is impeded, so stream depletion is lower. In natural streambeds, streambed hydraulic conductivity varies over time due to erosion and deposition of sediment as the flow conditions in the stream change. A prior investigation used numerical simulations to predict that temporal variations in streambed hydraulic conductivity may exaggerate stream depletion. Specifically, if a well is pumping near a stream stretch that alternates between periods of high and low streambed hydraulic conductivity, the amount stream depletion during the high conductivity periods may be higher than the amount of stream depletion that would be expected if the streambed sediment permanently had high hydraulic conductivity. Similarly, the amount of stream depletion during the low hydraulic conductivity periods may be lower than the amount of stream depletion that would be expected if the stream bed sediment permanently had low hydraulic conductivity. We develop an analytical solution (in Laplace space) for stream depletion in a stream with temporally varying streambed hydraulic conductivity. We allow the streambed hydraulic conductivity to alternate between high hydraulic conductivity and low hydraulic conductivity over regular time intervals. The results show that when streambed hydraulic conductivity shifts from high to low, stream depletion drops abruptly, and when streambed hydraulic conductivity shifts from low to high, stream depletion increases abruptly. Under certain conditions, the temporal variation in streambed hydraulic conductivity leads to exaggerated stream depletion, and we use the new analytical solution to investigate the conditions under which exaggerated stream depletion occurs.