Laplace transforms are widely used in solving groundwater flow problems. In the groundwater literature, it has frequently been asserted or assumed that the late-time behavior is governed by the behavior of the Laplace transform for small values of the Laplace variable. Careful reading of authoritative monographs on Laplace transforms shows that this correspondence does not generally hold. In this article the proper asymptotic formula is reviewed and is applied to the problem of water influx into a slab-like matrix block in a dual-porosity medium. In doing so, we clear up a long-standing discrepancy between the fracture/matrix transfer coefficient that has been calculated by Laplace methods and the coefficient which has been found by applying Fourier methods in the time domain.