Exact analytical expressions for the piezometric profile and water exchange between stream and groundwater during and after a uniform rise of the stream level

We consider an archetypical problem relevant to a confined aquifer in contact with a stream. The model problem consists of an idealized one-dimensional region 0 ≤ x ≤ L, where the left boundary at x = 0 is held at a fixed piezometric head h₀, and the right boundary's piezometric head at x = L is increased from h_L to h₀ at a constant rate. Exact solutions for all times, all points in the aquifer, and for any possible constant rate of change of the right boundary piezometric head are presented for the piezometric head and the instantaneous flow rate. An exact expression for the exchange volume at the groundwater/stream interface for an arbitrary time is also provided. This expression shows that there is a specific critical rising rate of the stream level above which the net exchange volume is into the aquifer and below which it is out of the aquifer. The solution shows that regardless of the rise rate, a certain water volume, inversely proportional to the rise rate, enters the aquifer.