Early-Time Recession Solution from a Steady-State Initial Condition for the Horizontal Unconfined Aquifer

In this work, we present the semi-analytical solution for the early-time recession phase of the horizontal unconfined aquifer of finite length for steady-state initial conditions. This is a case where self-similarity arguments are not applicable. The solution is built as linear perturbations from the initial steady state. The solution is determined via a Sturm–Liouville eigenvalue problem, which should be solved numerically. On the other hand, the immediate response of the aquifer to the sudden switching off the recharge, i.e., in the earliest times, is obtained by deducing analytically the large eigenvalue asymptotic solutions of the problem. We find analytically that in this time regime, the outflow Q is given by Q = Q₀ − 1.4Q₀^5/3L^−4/3n^−2/3t^2/3, where Q₀ is the initial outflow rate, L is the length of the aquifer, n is the porosity of the formation and t is the time from the start of the recession. The stated result is very accurate for times t up to ~0.01 nL^3/2k^−1/2Q₀^−1/2, where k is the hydraulic conductivity of the formation. The analytical and quantitative relation of the presented solution with the classical recession phase asymptotic solutions derived in the past by Polubarinova (early-time solution) and Boussinesq (separable, late-time solution) is discussed in detail. The presented results can be used as a benchmark solution for modeling or numerical validation purposes.