## 1. Introduction

[2] A comprehensive analysis of flow processes and advective-dispersive-diffusive transport of dissolved constituents in natural fluid circulations is an important challenge in earth and environmental sciences. An approach that has become very popular consists of using water age as an ideal tracer to tag fluid masses and estimate associated time scales [*Deleersnijder et al.*, 2010]. Such time scales can lead to very helpful diagnoses that apply to interdisciplinary environmental studies, like in atmospheric and ocean circulation sciences or surface and subsurface hydrology.

[3] The characterization of water age characteristics in environmental flow systems is linked to the estimation of recharge patterns and variations, fluid flow dynamics over possibly various time scales, advective-dispersive/diffusive transport in heterogeneous circulations, and to some extent isotope geochemistry, fractionation, and interfacial reactive chemistry [*Glynn and Plummer*, 2005; *Ginn et al.*, 2009]. In particular the characterization of groundwater age distributions is of prime interest as it can act as an indicator of aquifer contamination and vulnerability, explain the modes of recharge of aquiferous reservoirs, or identify the diffusive exchanges between mobile and immobile flow regions. The literature offers a wide range of studies focusing on various aspects of water age and its conceptual representations. However, because different mathematical and conceptual foundations are used, a standardization of the results is still difficult to be achieved.

[4] Age is commonly inferred from concentrations of single isotopes/tracers (for exhaustive reviews of groundwater age dating methods, e.g., see *Clarke and Fritz* [1997] and *Kazemi et al.* [2006]), or from multiple isotope/tracer concentrations that together represent different components of age [e.g., see *Corcho Alvarado et al.*, 2007; *Troldborg et al.*, 2008; *Larocque et al.*, 2009; *Lavastre et al.*, 2010]. This concept is however very misleading because concentration-based ages are simply apparent ages that do not a priori relate to the mean age of a water sample [*Sanford*, 2011]. Moreover, the use of environmental and anthropogenic tracers to make inferences on water age goes together with the use of lumped-parameter models that assume specific age distributions regardless the complexity of the flow domain.

[5] A relatively recent representation of age and its distributions is the one that relies on the use of distributed flow models. Accounting for the relevant transport characteristics affecting age distributions however requires the use of adequate governing transport equations. Regarding this point, new advances have permitted us to consolidate the unification of the foundations on age fate by means of a physically based derivation of a transport equation for mean age [*Goode*, 1996], percentile of age and moments of age [*Varni and Carrera*, 1998], and later on, after the early works of *Campaña* [1987], of equations describing the complete age frequency distribution in the context of ocean circulations modeling [*Deleersnijder et al.*, 2001; *Delhez and Deleersnijder*, 2002] and subsurface hydrology [*Ginn*, 1999, 2000a, 2000b; *Cornaton*, 2003; *Cornaton and Perrochet*, 2006a, 2006b, 2007; *Ginn*, 2007; *Woolfenden and Ginn*, 2009].

[6] Putting aside the way age is inferred (or modeled), the transient nature of flow systems and its implication on the transient nature of age distributions have never been successfully addressed, particularly for large space and time scales and real-site applications. This aspect is however of high importance since the actual situations at which age properties are measured are a result of past-to-present perturbations [*Ginn, et al.*, 2009] at short and large time scales. *Bethke and Johnson* [2008] correctly pointed out that changing the field of groundwater age dating is a means for thinking about groundwater age in a new way. We can additionally argue that the way age is modeled is also to be refined. In particular the time dependency of age distributions play a fundamental role in the interpretation of age and tracer data. Transient groundwater age distributions are, for example, a result of natural transient hydrologic conditions and of human-induced modifications on the water cycles by artificial withdraw and recharge, but they can also originate from more specific phenomena as the temporal dependency of fluid flow to fluid density and viscosity (e.g., in thermohaline problems). When large time scales are considered, the hydrodynamic conditions of a system are changing in time, in relation to the evolutions of geomorphology, system internal structure, climate change, and thus recharge conditions. Any modification induces changes in the age distributions.

[7] This article presents the outcomes of research works on transient age distributions and their numerical solutions. We present a novel numerical formulation that is able to handle this problem in 3-D discretized domains of arbitrary complexity, and that applies to the most relevant numerical integration techniques such as the finite element, finite volume, and finite differences methods. The method is also particularly suited to fulfill the needs related to the modeling of hydrologic evolutions over very large time periods. In section 2 we summarize the mathematical foundations of transient age and constituent exposure time density equations. In section 3 we make a short review of existing numerical integration schemes and present a novel numerical formulation. The characteristics of the proposed algorithms are analyzed and discussed, and eventually validated and illustrated in section 4 using analytical and numerical solutions.