## 1. Introduction

[2] Water age is a physical measure of the time a water molecule spends in a hydrological system, from a recharge point to an arbitrary location. This quantity is of interest in hydrology, as part of catchment characterization, for instance when interpreting observations of water composition [*Wolock et al*., 1997; *McDonnell et al*., 2010]. Water age is also of interest in a broader geoscience context, such as in oceanography where for instance a general theory of the age for a constituent of seawater has been proposed [*Delhez et al*., 1999].

[3] Hydrological transport has traditionally been focused on the temporal variability. With a reservoir theory as the basis, tracer transport through a catchment has often been treated as a fully mixed system, where spatial variations are neglected and temporal variations are accounted for. If a catchment is treated as a composition of several reservoirs, some spatial variation can be included but still the overall spatial structure and dispersion are of secondary importance relative to the short-term and long-term temporal changes. The theoretical ground for quantifying tracer residence time, life expectancy and age in flow reactors with time-varying flow was first presented by *Nauman* [1969], and later by *Niemi* [1977]. Applying these theoretical results to hydrological systems has improved our understanding and expanded the modeling capabilities for relating transport and water age in hydrological systems [*Botter et al*., 2010, 2011; *Rinaldo et al*., 2011].

[4] In subsurface hydrology, the emphasis has traditionally been on the impact of spatial variability and structure on solute transport where the injection time is assumed to be known [*Dagan*, 1982, 1989; *Rubin*, 2003]. The works on tracer travel time have also considered primarily the effect of spatial variations in hydraulic properties with prescribed injection time and location [e.g., *Shapiro and Cvetkovic*, 1988; *Cvetkovic and Dagan*, 1994; *Darracq et al*., 2009]. The first analysis of groundwater age (or backward travel time) by *Goode* [1996] assumed steady conditions and focused on numerical computations of the water age first moment. Although more general than that of *Goode* [1996], the approach of *Varni and Carrera* [1998] for computing water age higher moments, still emphasizes numerical computations under steady-state flow, noting the difficulties of unsteady flow computations. The most general model for computing the groundwater age distribution of *Ginn* [1999] has been related to earlier models [*Ginn et al*., 2009], and implemented for numerical simulations in aquifers under steady-state flow [*Woolfenden and Ginn*, 2009].

[5] A novel numerical method for computing the water age distribution under unsteady flow was recently presented [*Cornaton*, 2012] based on the theory of *Ginn* [1999]. This method reduces the dimension of the governing equation from five to four (in three physical dimensions), by Laplace transform over the water age. It then uses the approach of *Sudicky* [1989] to combine a finite difference solution with stepwise Laplace inversion. As part of the numerical methodology verification, *Cornaton* [2012] presents a semianalytical solution for the water age distribution in one-dimensional uniform flow with Fickian macrodispersion and a temporal step change in the mean flow velocity. For steady -state flows, several analytical solutions of the water age distribution are available [*Ginn et al*., 2009]. One important challenge is to provide simple tools that can be used for estimating the water age distribution under arbitrary *unsteady* flow conditions.

[6] In this study, we provide an analytical model to quantify the water age distribution along one-dimensional pathways with arbitrary unsteady flow, thereby extending current analytical capability. The proposed model is verified with both trajectory simulations and the result of *Cornaton* [2012]. The main assumption of the model is that the mean flow is approximately uniform in space. The emphasis is on macrodispersion (spatial fluctuations) and temporal trends or fluctuations (long or short term). We illustrate the results for a few cases of temporal mean velocity variations, as trends, stepwise change and random fluctuations.