Analytical Solutions to the Linearized Boussinesq Equation for Assessing the Effects of Recharge on Aquifer Discharge

There is a need to develop a general understanding of how variations in aquifer recharge are reflected in discharge. Analytical solutions to the linearized Boussinesq equation governing flow in an unconfined aquifer provide a unified mathematical framework to quantify relationships among lag time, attenuation and distance between aquifer recharge and discharge and the effect of an up-gradient no-flow boundary. We applied this framework to three types of recharge: (1) instantaneous, (2) periodic, and (3) constant rate for a finite duration. When the temporal scale of recharge exceeds the diffusive aquifer time scale, recharge will be reflected in discharge quickly and with little attenuation. When aquifer time scale is large, most recharge events are shorter in scale than that of the aquifer, resulting in large attenuation. Attenuation is more sensitive to boundary effects than lag time, and boundary effects increase as recharge time scale increases. Boundary effects can often be ignored when the recharge source is farther than 1/3 of the domain length away from the no-flow boundary. We illustrate analytical results with application to the economically critical Eastern Snake River Plain Aquifer in Idaho. In this aquifer, detectable annual and decadal cycles in discharge can result from recharge no farther than 20 and 60 km away from the discharge point, respectively. The effects of more distant, long-term recharge can be detected only after a time lag of several decades.