Piezometric level response to periodic forcings, like solid earth tide strain, barometric loading or oceanic tide loading, is a useful alternative way to characterize aquifer properties. It was first spread as a tool for aquifer characterization 34 years ago with the model of Hsieh et al. (1987). It was promising in the sense it only needs classical monitoring data (hourly water level) and no expensive field work like a pumping test. Yet its usage is still far from its potential as expressed in the review of McMillan et al. (2019). On the one hand, the pumping test literature, relatively old, covers a wide variety of aquifer geometries and boundary conditions, including, without comprehensiveness, models for confined radial aquifer (Theis, 1935), leaky aquifers (Hantush & Jacob, 1955), double porosity aquifers (Warren & Root, 1963), unconfined aquifer (Neuman, 1972, spatially heterogeneous aquifers with the general radial flow model (Barker, 1988). Research is still ongoing namely on numerical modeling to complexify the possible responses. On the other hand, literature on periodic responses only recently represented a decent variety of aquifer geometry, starting from Roeloffs (1996) and Rojstaczer (1988) for unconfined aquifers, to Wang et al. (2018) for leaky aquifers. Research on this topic now faces the challenge of proposing a comprehensive view and practical guidelines, like did in its time the useful tutorial of Doan et al. (2008). Now the two major and somewhat opposing pitfalls scientists face are the following. The first one is that the existing models are based on strong hypotheses (perfect confinement for Hsieh et al. (1987), negligible storage in the aquitard in Wang et al. (2018),) which of course are not in general met. There is still room for a more general derivation of analytical models, like what Odling et al. (2015) did in the case of the model of Rojstaczer (1988), precising the low frequency behavior which was improperly predicted. The second pitfall, opposing to the first, is that we need to remain as simple as possible, and keep in mind the mathematical limitations of modeling inversion: from n observed variables, we can invert at best n parameters. In that sense, periodic response problems are nearly always over-parametrized, since observations are often limited to phase and amplitude response at one or two frequencies, while hydrodynamic and poroelastic parameters (hydraulic conductivity, storativity, etc.), grow numerous with the complexity of the models. In this paper, we do not claim to give the comprehensive view we
