Analytical formulae are proposed to describe the first-order temporal evolution of the head in large groundwater systems (such as those found in North Africa or eastern Australia) that are subjected to drastic modifications of their recharge conditions (such as those in Pleistocene and Holocene times). The mathematical model is based on the hydrodynamics of a mixed-aquifer system composed of a confined aquifer connected to an unconfined one with a large storage capacity. The transient behaviour of the head following a sudden change of recharge conditions is computed with Laplace transforms for linear one-dimensional and cylindrical geometries. This transient evolution closely follows an exponential trend exp(−t/τ). The time constant τ is expressed analytically as a function of the various parameters characterizing the system. In many commonly occurring situations, τ depends on only four parameters: the width a c of the main confined aquifer, its transmissivity T c, the integrated storage situated upstream in the unconfined aquifer M = S u a u, and a curvature parameter accounting for convergence/divergence effects. This model is applied to the natural decay of large aquifer basins of the Sahara and Australia following the end of the mid-Holocene humid period. The observed persistence of the resource is discussed on the basis of the time constant estimated with the system parameters. This comparison confirms the role of the upstream water reserve, which is modelled as an unconfined aquifer, and highlights the significant increase of the time constant in case of converging flow.