Short- and long-time behavior of aquifer drainage after slow and sudden recharge according to the linearized Laplace equation

During the last 25 years there has been a great interest in deriving aquifer characteristics from outflow data. This analysis has been mainly based of the drainage of a horizontal aquifer after sudden drawdown, using the Boussinesq approximation. Following the general approach of Brutsaert and Lopez Brutsaert W, Lopez, JP. Basin-scale geohydrologic drought flow features of riparian aquifers in the southern Great Plains. Water Resour Res 1998;34(2):233–40], it was determined that for this geometry the aquifer behavior could be characterized by dQ/dt ∝ Q³ for small t and by dQ/dt ∝ Q^3/2 for large t. It was remarked that dQ/dt ∝ Q for large t is often observed. In practice, it is also difficult to determine if dQ/dt ∝ Q³ for small t because this behavior can only be observed over a very short period.Here, we present a similar analysis of aquifer behavior based on the more fundamental Laplace solution for penetrated aquifers. It has been shown that also when the drain does not fully penetrate the aquifer, the solution still produces good results Szilagyi, J. Sensitivity analysis of aquifer parameter estimations based on the Laplace equation with linearized boundary conditions. Water Resour Res 2003;39(6)]. The Laplace solution quickly shows that dQ/dt ∝ Q for t → ∞ and dQ/dt ∝ Q^∞ for t → 0, after sudden drawdown. This analysis reconfirms previous findings concerning long-time behavior. More importantly, the analysis shows that the exponent B in dQ/dt ∝ Q^B does not have a fixed limited value for short times for the given geometry. Further analysis, however, shows that under certain conditions the relation dQ/dt ∝ Q³ is retained for 0 ? t < 1. Detailed examination of the Laplace solution also shows under which types of recharge dynamics a well-identifiable transition takes place between short- and long-term behavior. As long as such a clear transition exists, the aquifer characterization method proposed earlier by Brutsaert and Lopez Brutsaert W, Lopez, JP. Basin-scale geohydrologic drought flow features of riparian aquifers in the southern Great Plains. Water Resour Res 1998;34(2):233–40] can be applied. It is shown that for a sharp pulse input, the Laplace solution gives similar results as presented by Brutsaert and Lopez Brutsaert W, Lopez, JP. Basin-scale geohydrologic drought flow features of riparian aquifers in the southern Great Plains. Water Resour Res 1998;34(2):233–40]. For a smooth pulse, the transition becomes unclear. What is “smooth” and “sharp” depends on input and aquifer characteristics, whereby shallow aquifers give clearer transitions than deep aquifers for the same input. The analysis shows that when rain ceases suddenly after the aquifer has come into equilibrium with a steady rain input, a usable transition in the relation between dQ/dt and Q can be found as well. Researchers can use the present analysis to assess whether specific aquifers and recharge events can be used for the previously suggested characterization method.