Water age is a transient spatially distributed measure of how long water molecules are present in a hydrological system since their origin, of relevance for a variety of applications. We present a simplified analytical model for the water age distribution under arbitrary transient flow along one-dimensional hydrological pathways with mean flow approximately uniform in space and constant macrodispersivity. The proposed model is verified using Monte Carlo trajectory simulations, as well as with the analytical solution for a temporal step change of the mean velocity. The derived solution can be used for scoping calculations, for studying the coupled effects of macrodispersion and time variations on the water age distribution, as well as for benchmarking numerical simulation tools.
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 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].
 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 , and later by Niemi . 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].
 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  assumed steady conditions and focused on numerical computations of the water age first moment. Although more general than that of Goode , the approach of Varni and Carrera  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  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].
 A novel numerical method for computing the water age distribution under unsteady flow was recently presented [Cornaton, 2012] based on the theory of Ginn . 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  to combine a finite difference solution with stepwise Laplace inversion. As part of the numerical methodology verification, Cornaton  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.
 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 . 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.
2. Problem Formulation
 Consider a hydrological pathway from a recharge to a discharge area similar as was illustrated in Cvetkovic et al. . The pathway implies existence of a mean flow and is approximated as one-dimensional in a three-dimensional setting. A tracer particle originating at x = 0 at t0, is discharged at x = L, at time t = T. We wish to compute the cumulative distribution function (CDF) of the particle travel time and its age at a specified location L (e.g., discharge point of the catchment), following spatial and temporal variations of the flow velocity along the trajectories between x = 0 and x = L.
 Let the mean flow velocity be transient and approximately uniform in space, U(t). The actual three-dimensional flow velocity v(x,t) is assumed as transient and randomly varying spatially, such that . If the macroscale dispersive process is assumed Fickian with macrodispersivity λL, then the simplest one-dimensional transport equation for the water age density can be derived from the analysis of the exposure time for multicomponent mixtures as [Ginn, 1999]
where τb is the water age, or the backward travel time of a water particle entering the system at time t0 and exiting at t = T; hence . In the notation of Ginn  N = 2 and . The boundary conditions for water particle entering the system are and the initial conditions are , where δ+ is the right-handed Dirac distribution.
 The one-dimensional mass balance equation for the corresponding (forward) transport problem can be written as
where C is the tracer concentration per unit fluid volume. Note that x in both equations (1) and (2) designates a longitudinal (intrinsic) coordinate parallel to U.
 If a hydrological pathway consists of an ensemble of independent trajectories, then equation (2) provides an integrated transport model for the trajectory ensemble. The solution of equation (2) has been obtained for injection and detection in the resident concentration by Bischoff . The main objective of this work is to use the analytical solution of equation (2) and provide a simple model for computing the water age (or backward travel time) distribution at the location x = L, given λL and U(t), as an alternative to directly solving equation (1). The proposed solution will be verified against trajectory computations as well as with an analytical solution of equation (1) for a stepwise change of U(t).
3. Analytical Model
 The analytical solution of equation (2) has been derived for the tracer resident concentration C [Bischoff, 1964]. Rather than work with C, we cast the problem into a Lagrangian framework of discrete particle transport along random trajectories in the flow field v(x,t). For conservative solutes in diluted systems, , where p(x;t) is the probability density function (PDF) of particle position in the x direction parallel to U.
 If a conservative tracer is injected as a pulse at x = 0, the analytical solution for p(x;t) is [Bischoff, 1964]
and is the time dependent, mean flow velocity, with ϕ(t) being an arbitrary dimensionless function, and U0 (L/T) a constant. In other words, flow is transient in the velocity magnitude, but not in the flow direction. The solution of equation (3) is for a pulse injected in an infinite domain; if x is sufficiently large relative to macrodispersivity, the effect of the injection mode is negligible.
 To obtain the distribution of water age from the PDF p(x;t), we first relate the travel time and position distributions using the probability (mass) balance [Shapiro and Cvetkovic, 1988; Dagan, 1989]
in which P is the CDF of particle position and F is the CDF of particle travel time between x = 0 and x; the validity condition for equation (4) is negligible reverse transport.
 Substituting equation (3) into equation (4) and carrying out the integrations, yields
which is the CDF of the tracer (forward) travel time from a specified injection time t0.
 The water age CDF can now be derived from equation (5). Given the time-dependence function ϕ(t), an inverted temporal variability function ϕb can be defined for the backward travel time (or water age) computation as , whereby the mean flow velocity for the backward computation becomes . In section 'Time Dependence and Inversion', we show that the time variability inversion is exact for a single advective trajectory, and in the following we extend this result to the mean flow. In other words, the time inversion applicable for a hydrological pathway without dispersion is now assumed applicable with dispersion.
 Inverting the mean velocity in equation (5) as , we get
 Equation (6) is the main result of this paper. It provides a simple analytical model for computing the water age CDF (or the CDF of the backward travel time t = τb) as a function of a given (constant) macrodispersivity λL, the mean flow velocity with arbitrary time variations , the observation time T at the discharge location x and the clock time at recharge point x = 0. The time sets a bounding value for the computation of the water age under given conditions, such that if looking at absolute time, water age will not be greater than . If ϕ = const., i.e., the flow is at steady state, then equations (5) (with ) and (6) yield identical distributions. In terms of boundary conditions, equations (5) and (6) are applicable for injection in the resident and detection in the flux concentration.
4. Comparison with Trajectory Simulations
 In this section, the accuracy of the model equation (6) for the water age CDF is tested against the results from Monte Carlo trajectory simulations. The effect of the temporal change on the backward transport solution is presented using a few examples.
 The trajectory simulations are described in section 'Random Trajectories'. They are based on a random lognormally distributed velocity with a log standard deviation, σY = 0.8 and log integral scale, IY = 50 (with an exponential correlation function), for a spatial domain of x = 1000 (units are arbitrary). Water age is computed from an ensemble of 200 trajectories. For the analytical model, the longitudinal dispersivity is estimated using the first-order theory [Dagan, 1989] as . Note that for the analytical calculations the mean velocity U0 is redefined as the harmonic mean to be consistent with trajectory simulations [Cvetkovic et al.2012, Appendix A].
 Consider first a linear increasing temporal trend with ϕ(t) defined by
 The function ϕ (equation (8)) and its inverted form with T = 4000 are illustrated in Figure 1a.
 The CDF of the backward travel time obtained from trajectory simulations as summarized in section 'Random Trajectories' is compared to the analytical model for the water age distribution equation (6), as blue symbols and line, respectively, in Figure 2; a close comparison is found for the water age distribution.
 Next, consider a stepwise temporal change (decrease) of the mean flow velocity as
 The function ϕ (equation (9)) and its inverted form with T = 4000 are illustrated in Figure 1b. The CDF of water age (backward travel time) computed by trajectory simulations, compares well using the model (6) (red symbols and line in Figure 2).
 As a third case, random temporal variations with ϕ(t) as illustrated in Figure 1c is considered. These variations were obtained from the normalized monthly averaged discharge measurements over 8400 days for a small catchment located in Sweden. Consistent with previous cases, trajectory simulations are done within an interval of t = 4000. The measured temporal variations have been normalized such that , with a coefficient of variation of 0.8, over the interval t = 4000; the distribution of the temporal fluctuations is approximately lognormal [Cvetkovic et al., 2012].
 Figure 2 shows the CDFs of the water age as computed by trajectory simulations (green symbols) and the analytical model (6) (green line). The model compares well with the simulations even in this case (Figure 1c). The abrupt change in temporal variations, e.g., between the interval of t = 1500 to t = 2000 (Figure 1c), results in some deviation between the simulations (symbols) and the model (line) (Figure 2).
 Water age is a spatiotemporally distributed quantity representing the origin of water in a flow system. If only one spatial dimension is considered following a mean flow, the simplest form of the governing equation for the water age PDF is expressed in three dimensions (equation (1)). In the general case with arbitrary variability in the mean velocity, the solution of equation (1) can be obtained only numerically. A computational method that uses Laplace transform for solving the full five-dimensional system or the three-dimensional system equation (1) was recently presented [Cornaton, 2012].
 The Lagrangian trajectory approach to travel time [Shapiro and Cvetkovic, 1988; Cvetkovic and Dagan, 1994; Fiori et al., 2002] provides an alternative formulation for computing the water age, with a transparent relationship between the forward and backward travel time (section 'Kinematical Relationships'). This formulation enabled us to take advantage of the transient advection dispersion equation (ADE) solution (3) to propose a simple model (6) suitable for studying the interplay between temporal variability of the mean velocity and macrodispersion. The analytical model (6) is limited to uniform mean flows, although the trajectory simulations are applicable to arbitrary variations of the mean velocity in space and time; the challenge of the trajectory simulation approach is to correctly represent the flow velocity trends and fluctuations. Such a representation must honor the hydrodynamics under prevailing boundary conditions, and incorporate the effect of random and deterministic structural variability; numerical simulations provide one means of computing the velocity statistics under transient flow with heterogeneous structure [Fiori and Russo, 2008].
 A comparison between the trajectory simulations and an analytical solution of equation (1) for a step temporal change of a spatially uniform mean velocity is presented in Cvetkovic et al. . The analytical solution of equation (1) for a step form of ϕ is provided in the Laplace domain by Cornaton . To establish a link with the theory of Ginn , we shall compare equation (6) with the cumulative water age distribution , where is obtained as the solution of equation (1) for a step decrease of the mean velocity for which we write ϕ(t) as
 The solution of equation (1) with equation (10) can be found in Cornaton [2012, Appendix] and Cvetkovic et al. [2012, Appendix C].
 A comparison between our analytical model and the analytical solution of equation (1) with equation (10) is to be illustrated for different values of t* and M. In Figure 3, we show a sample of trajectories for four cases, where t* and M are specified in the figure; in all cases T = 8000, x = 1000, σY = 0.8, and IY = 50, and 200 trajectories were computed. The longitudinal dispersivity is again estimated using the first-order theory as . As noted earlier, for the analytical calculations the mean velocity U0 is redefined as the harmonic mean [Cvetkovic et al., 2012].
 The CDF of water age is plotted in Figure 4a for the four cases noted in Figure 3. The two analytical models are compared with the trajectory simulations: the blue solid curve is the analytical solution of equation (1) and the dashed black curve is computed with equation (6). The analytical model (6) provides a close approximation of the trajectory simulations, although the analytical solution of equation (1) is closer to the simulated CDF (Figure 4a) for the case of a step temporal change. It is apparent from Figure 4 that temporal variability in form of a step function, can have a profound effect on the shape of the CDF and macrodispersion of water age.
 The analytical model (6) deviates from the trajectory simulations and the analytical solution of equation (1) in particular for more dispersed cases 3 and 4 (Figure 4a). This shift can be explained by the effect of the injection mode. In Figure 4b, we plot the case with uniform, constant mean velocity, i.e., ; note that the mean flow velocity is , whereby the median is around t = 1400. It is seen in Figure 4b that the slight shift persists even for the constant mean velocity case. The analytical solution of equation (1) for this case reduces to the inverse Gaussian distribution applicable for injection in the flux [Kreft and Zuber, 1978], whereas the analytical model (6) is derived from the analytical solution of Bischoff  valid for injection in the resident concentration; the trajectory simulations are consistent with the injection in the flux, hence the inverse Gaussian solution (solid line) is closer to the simulations (Figure 4). This discrepancy however has a minor effect for all the cases illustrated in Figure 2.
 The continuum formulation of Ginn  is applicable for computing the water age distribution under most general conditions, especially if advantage is taken of the recently proposed computational method [Cornaton, 2012]. In instances where hydrological pathways are relatively well defined, estimates of the age distribution using equation (6) may be sufficient, or equation (6) may be used prior to, or in addition to, numerical simulations.
6. Summary and Conclusions
 For a hydrological pathway defined in one spatial dimension along a mean flow, a simple model for the water age distribution can be obtained in the form (6). The proposed analytical model is applicable for arbitrary time variations of the mean velocity and for Fickian macrodispersion; the main simplifying assumption is that the mean flow is approximately uniform in the longitudinal direction. The derived solution is consistent with Monte Carlo trajectory simulations, as well as with the analytical solution for a temporal step change of the mean velocity presented recently [Cornaton, 2012].
 The derived solution can be used for simple calculations prior to or instead of numerical computations of the type available in the literature [Goode, 1996; Varni and Carrera, 1998; Ginn, 1999; Cornaton, 2012]. It may also be useful for studying the coupled effects of macrodispersion and time variations on the water age distribution, as well as for benchmarking numerical simulation tools for the water age distribution in particular for hydrological pathways where time fluctuations are rapid. In cases where the spatial dependence of the mean flow is important, the trajectory simulation approach can be used, once the flow velocity is adequately formulated. The method can be used for instance to assess the effect of climatic trends or hydroclimatic shifts [Destouni et al., 2013] on the water age distribution. Similarly, it can be applied in a regional linear-flood system, where the accountable heterogeneity is below the regional scale. In this context, we need to understand the trends and fluctuations of the velocity for different flow conditions and through different hydrological units, which still constitutes a significant challenge.
Appendix A: Forward and Backward Travel Time in Flows With Space-Time Variability
A1. Kinematical Relationships
 Consider a velocity field of an incompressible fluid which varies in space and time, v(x,t); our focus is on hydrological pathways with a mean flow for which in general where angular brackets denote ensemble averaging.
 The trajectory X is computed from the differential equation as [Dagan, 1982, 1989]
with the initial condition , set at the origin for simplicity.
 Assuming that is parallel to x, and reverse flow is negligible, we can write the differential equation for the travel time τ along x as
where , and for simplicity the particle originates at x = 0. Note that here x is part of a Cartesian coordinate system whereas in general it can designate an intrinsic coordinate (length) along a hydrological pathway.
 Equation (A2) can be solved for different initial conditions: If is specified, then equation (A2) yields the forward water particle travel time from x = 0 to x; if is specified, then τ quantifies water particle age, or backward travel time, from x to x = 0. In the latter case, the starting time t0 is computed as .
A2. Time Dependence and Inversion
 If the basic flow pattern of a hydrological pathway is assumed constant in space, then the following separation of velocity time dependence is useful
given here for the longitudinal component; the function ϕ(t) is specified and dimensionless. Substitution of equation (A3) into equation (A2) yields
or, projecting the velocity onto the mean flow direction,
 This is a first-order nonlinear, separable differential equation and the solution for both the forward and backward travel time are solutions of a boundary value problem. Our time inversion states that for τ = T at x = x*, we can obtain the water age at x = x0 by solving
 To prove this statement, consider two first-order separable differential equations
the solution of which can be written in an implicit form as
 We now wish to prove the following:
Theorem: Consider the two ordinary differential equations (ODEs) (A7) with solutions (A8) and (A9) in the positive domain, i.e., . Then, Y = y at x = x0 obtained by solving for y, is equal to W = w at x = x* obtained by solving for w.
Proof: Using equations (A8) and (A9), we have implicit solutions
 By changing the sign and order of integration in equation (A11), we get
 Comparison of equations (A10) and (A12) implies that the solutions of the two equations Y = y and W = w, are equivalent, i.e., Y = W, which completes the proof.
 The implicit function corresponds to the backward solution for water age where the boundary value problem [x*,y*] is the current observation of arrival time (y*) at a given location (x*), and we seek the starting time (Y = y) at a given location (x0, in this case x0 < x*). We showed that this starting time can be obtained by solving another implicit equation which is the forward equation with following boundary values: the initial location (x0) and the observation time (y*); equation (A11) is then solved at the observation location (x*). The age is computed as the difference between the observation time and starting time .
 To illustrate above equivalence, consider a spatial domain x = 1000. Let the space-time variations of the velocity be defined by two dimensionless functions, κ(x) and ϕ(t) as
 We consider simple step function forms for ϕ(t) and κ(x) as
 Note that this form of ϕ(t) (with κ = 1) was used in the analytical solution of Cornaton . For illustration, T = 3000 and U0 = 0.3 is used; the function ϕ is illustrated as the black solid curve in Figure 5a.
 The backward space-time trajectory for the velocity given in equation (A13) is obtained by solving equation (A5) with τ = T at x = 1000 and is shown as the red curve in Figures 5b–5d. In Figure 5b, we first show the simplest case with only spatial velocity dependence, i.e., with ϕ = 1. The red trajectory has one inflection point at x = 500 and time 3000 − 1667 = 1333 since the age is 500/0.3 = 1667. The blue trajectory is equivalent to the red trajectory since the velocity is independent of time.
 Next, consider the case with temporal velocity change only, i.e., κ = 1. In Figure 5c, the red trajectory is obtained as the backward solution of equation (A5) with τ = 3000 at x = 1000; the inflection point is at t = 2400 when the abrupt velocity increase takes place with the trajectory slope also increasing. The blue curve in Figure 5c is obtained in the forward mode, i.e., by solving equation (A5) with τ = 0 at x = 0, and using inverted ϕ(t) as ϕ(T−t); the dashed curve in Figure 5a illustrates the inverted time dependence. In is seen in Figure 5c that the travel time of 2734 (obtained from the blue curve) is equivalent to the backward time (or age) obtained from the red curve; the starting time in this case is t0 = 3000 − 2734 = 266.
 Finally, in Figure 5d, we illustrate trajectories with both spatial and temporal variability following equation (A13). Although the red and the blue curves are different, they yield the same travel time with ϕb (blue curve) and age with ϕ (red curve) of 1400. Both curves exhibit two inflection points, one when the spatial velocity change occurs and the other when the temporal change occurs. The starting time in this case is t0 = 3000 − 1400 = 1600.
A3. Random Trajectories
 To capture the dispersion effects, we consider the same form of v(x,t) as given in equation (A13) but in this case the function κ is randomized as
where Y(x) is a normally distributed random space function N(0,σY), with a negative exponential correlation structure and IY as the integral scale. For illustration, let σY = 0.8 and IY = 50 with the domain of x = 1000. By virtue of the first-order theory [Dagan, 1989], , whereby the macroscopic Peclet number is 1000/32≈31 in this example.
 Three different cases of temporal variability are assumed for illustration as quantified by the function ϕ(t) defined in equations (8) and (9), as well as with a random ϕ; ϕ for all three cases is shown in Figure 1. Cumulative distribution functions are obtained by generating multiple realizations of the spatial dependence quantified by Y(x), and computing travel time and age by forward and backward solutions of equation (A5), respectively.
 We thank Bijan Dargahi (KTH Royal Institute of Technology, Stockholm, Sweden) for useful comments on the original version of the manuscript. We are grateful to Tim Ginn (UC Davis, USA) and Gianluca Botter (University of Padova, Italy) for providing thorough review comments that helped improve the final version of the manuscript. The support for this work was provided by the Swedish Nuclear Fuel and Waste Management Co (SKB) as a PhD research project for the first author.