Water Resources Research

Water and solute transport along hydrological pathways

Authors

  • Vladimir Cvetkovic,

    Corresponding author
    1. Department of Land and Water Resources Engineering, Royal Institute of Technology,Stockholm,Sweden
      Corresponding author: V. Cvetkovic, Department of Land and Water Resources Engineering, Royal Institute of Technology, Brinellvägen 28, 10044 Stockholm, Sweden. (vdc@kth.se)
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  • Christoffer Carstens,

    1. Department of Land and Water Resources Engineering, Royal Institute of Technology,Stockholm,Sweden
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  • Jan-Olof Selroos,

    1. Department of Geoscience and Safety, Swedish Nuclear Fuel and Waste Management Co.,Stockholm,Sweden
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  • Georgia Destouni

    1. Department of Physical Geography and Quaternary Geology, Stockholm University,Stockholm,Sweden
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Corresponding author: V. Cvetkovic, Department of Land and Water Resources Engineering, Royal Institute of Technology, Brinellvägen 28, 10044 Stockholm, Sweden. (vdc@kth.se)

Abstract

[1] A Lagrangian framework for material transport along hydrological pathways is presented and consequences of statistically stationary space-time flow velocity variations on advective transport are investigated. The two specific questions addressed in this work are: How do temporal fluctuations affect forward and backward water travel time distributions when combined with spatial variability? and Can mass transfer processes be quantified using conditional probabilities in spatially and temporally variable flow? Space-time trajectories are studied for generic conditions of flow, with fully ergodic or only spatially ergodic velocity. It is shown that forward and backward distributions of advective water travel time coincide for statistically stationary space-time variations. Temporal variability alters the statistical structure of the Lagrangian velocity fluctuations. Once this is accounted for, integration of the memory function with the travel time distribution is applicable for quantifying retention. Further work is needed to better understand the statistical structure of space-time velocity variability in hydrological transport, as well as its impact on tracer retention and attenuation.

1. Introduction

[2] Material transport in catchments is important for many applications but still poses a modeling challenge [e.g., Haitjema, 1995; Kirchner et al., 2000; McGuire et al., 2005; McGuire and McDonnell, 2006; Hrachowitz et al., 2010; Godsey et al., 2010; Dunn et al., 2010; McDonnell et al., 2010]. Flow and transport through a catchment (that may be defined on a variety of scales) is three-dimensional, often taking place through complex hydrological and hydrogeological structures [Sudicky et al., 2008; Li et al., 2008]. Different source inputs and parts of a catchment system will have different impacts on the transport to the stream network and out from the catchment, depending on the timescales involved [Destouni et al., 2010]. Definition of a catchment is straightforward if it is based on surface water divides, however, some pathways may enter outside a topographically defined catchment through the subsurface. Near-coastal catchments are of particular interest with their multiple recharge and diffuse discharge zones [Destouni et al., 2008].

[3] The pathways of transport in a catchment are generally complex, although the principle components are well understood [Freeze and Harlan, 1969]. Distributed hydrological modeling poses outstanding problems of scale, subgrid representation techniques, equifinality, and model structural errors [e.g., Beven, 1989, 2001, 2009; Butts et al., 2004; Lindgren and Destouni, 2004; Refsgaard et al., 2006; Darracq and Destouni, 2007], and its very rationale has been questioned [Beven, 2001]. In spite of this, physically based numerical models of the coupled surface-subsurface system have been developed significantly over the past decade, such that relatively large-scale simulations with a fair amount of detail have improved our basic understanding of catchment hydrology [e.g., VanderKwaak and Loague, 2001; Loague et al., 2006; Sudicky et al., 2008; Kollet and Maxwell, 2008; Li et al., 2008]. Overall characterization has also improved considerably in terms of available data, better capturing the internal structure and geometry of catchments, as well as the boundary conditions at the surface. This fact has been pivotal for building confidence in simulations of large-scale space-time flow distributions based on known or hypothesized structures and boundary conditions [e.g., Loague and VanderKwaak, 2004].

[4] Although deterministic numerical simulations are a powerful and in many ways indispensable tools for understanding catchment scale flow and material transport, it is clear that some part of the heterogeneity in the structure and flow still cannot be resolved deterministically and must be treated as random. Combining numerical solutions of groundwater flow with stochastic analytical models for transport provides significant advantages in capturing macrodispersion as well as mass transfer reactions. This fact has long been recognized in engineering applications, and most recently implemented as a time domain random walk combined with numerical simulations of streamtubes, or trajectories [Kawasaki and Ahn, 2006; Painter et al., 2008], an approach closely related to Lagrangian modeling. The Lagrangian approach offers several advantages for randomizing transport along deterministic pathways, especially when focused on tracer travel times and mass flux; it has been applied to transport in streams [e.g., Rinaldo and Marani, 1987; Gupta and Cvetkovic, 2000, 2002], drainage networks [e.g., Rinaldo et al., 1991], whole catchments [e.g., Darracq et al., 2010; Persson et al., 2011], aquifers [e.g., Rainwater et al., 1987; Shapiro and Cvetkovic, 1988; Cvetkovic and Dagan, 1994; Cvetkovic et al., 1998; Ginn, 1999, 2001; Fiori et al., 2002], soil [e.g., Bresler and Dagan, 1981; Destouni and Cvetkovic, 1991], fractured rocks [e.g., Cvetkovic, 1991; Painter et al., 1998, 2002; Frampton and Cvetkovic, 2011], as well as for analytically coupling of transport through soil and groundwater [Destouni and Graham, 1995] and surface and subsurface transport in cathcments [Lindgren et al., 2004].

[5] In this work we focus on water and solute transport along the different hydrological pathways that connect a recharge with a discharge location of a catchment. With mean flow resolved, our main task is to incorporate random effects under conditions where both spatial and temporal variations are significant along the hydrological pathways through the catchment.

2. Problem Formulation and Assumptions

[6] A catchment is viewed as a three-dimensional flow system (Figure 1) driven by prescribed boundary conditions. In general, the system consists of components such as rivers and streams, lakes and wetlands, and soil water and groundwater flowing through unconsolidated and consolidated (fractured) porous media. We define a “hydrological unit” (HU) as any one of the components with at least the mean properties specified. The boundary conditions for the mean flow are assumed known, and an average streamline/trajectory pattern in a catchment can be resolved, e.g., by numerical simulation based on estimated hydraulic properties of the HUs. The effect of spatial heterogeneity within HUs at, or below, a scale resolvable by simulations, can be accounted for analytically [Persson and Destouni, 2009].

Figure 1.

Sketch of source area A within a catchment with associated streamlines converging to two control discharge locations. Possibility of input from outside the topographic catchment is noted (red). We also note a discharge location (blue), e.g., due to pumping.

[7] Here the Lagrangian Stochastic Advection Reaction (LaSAR) approach [e.g., Cvetkovic and Dagan, 1994], is implemented for quantifying tracer transport when both spatial and temporal random variability of the flow velocity along a defined mean flow path, are significant. The link between water and tracer (solute) transport in this approach is the advective water travel time (WTT). The significance of WTT distributions for transport processes in catchment hydrology is widely recognized [Rinaldo and Marani, 1987; Rinaldo et al., 1989, 1991, 2011; Jobson, 1997; Simic and Destouni, 1999; Lindgren et al., 2004; McGuire and McDonnell, 2006; Kollet and Maxwell, 2008; Botter et al., 2010; Destouni et al., 2010; Godsey et al., 2010; Hrachowitz et al., 2010; McDonnell et al., 2010; Roa-Garcia and Weiler, 2010Persson et al., 2011]. Travel (or first-passage) time is also important from a more basic perspective for understanding dispersion [De Arcangelis et al., 1986; Hughes, 1995].

[8] Two key assumptions of this work are summarized as follows: First, we consider water and advective solute transport along hydrological pathways (trajectories) that are controlled by average boundary conditions; hence pathway geometry is approximately constant with velocity that may vary in space and time. Second, the solute concentrations are relatively low such that retention is linear in a generalized sense; this implies that superposition for tracer concentration or flux is applicable.

[9] If the velocity temporal fluctuations are primarily controlled by the boundary conditions, it is convenient to express inline image as separable, i.e., inline image, where inline image and inline image are approximated as independent; this approximation will be used in the following for generic illustrations. In cases where the separation does not hold, v(x, t) (inferred from measurements and/or simulations) would be used directly.

[10] Water entering a catchment at any given recharge location (Figure 1) will be partitioned between different transport pathways (e.g., red and blue trajectories in Figure 2). Temporal fluctuations will, in general, be different along the different pathways, with the red trajectory example expected to have stronger temporal fluctuations than the blue trajectory (Figure 2). Depending on prevailing boundary conditions and internal catchment structure, more or less of the water flow and solute mass will be partitioned between different pathways [Bosson et al., 2012]. The two main questions addressed are: How do the temporal fluctuations affect forward and backward water travel time distributions when combined with spatial variability? and What are the conditions for the applicability of mass transfer processes using conditional probabilities in spatially and temporally variable flow?

Figure 2.

Example of two main pathways (trajectories) emerging from a recharge (source input) location in a catchment. The thick dashed lines indicate the main average pathways over the relevant temporal scale (resolution) and the thin dashed lines indicate trajectory fluctuations pertinent to individual recharge events. Depending on boundary conditions and catchment internal structure (including prevailing average groundwater level and its fluctuations) [Bosson et al., 2012; Dahlke et al., 2012], water flow and solute flux will be partitioned differently between the main (thick, dashed) pathways. The pathways transect several hydrological units (HUs), with the blue trajectory example extending through i = 1, N HUs, from the surface into the deeper subsurface, and back to surface, finally discharging into a recipient.

[11] In sections 3 and 4 we first summarize the Lagrangian framework and then quantify water (or ideal tracer) travel time along trajectories. Retention is discussed in section 5. Finally, in section 6 our results are set in a broader context, and main conclusions drawn in section 7.

3. Lagrangian Framework

[12] Consider a catchment in three dimensions with a known structure and specified boundary conditions for flow. A typical trajectory/streamline transects N “hydrological units” (HUs) which are specified, such as a stream, lake, wetland, unsaturated subsurface flow, and shallow and deep groundwater flow in porous media as well as fracture zones, fractured rock, etc. (Figure 2). Note that here streams would be considered in a classical sense of drainage networks, i.e., as essentially known one-dimensional channels with a spatially and temporally varying velocity.

[13] Particles injected at a recharge point a with support inline image end up at a discharge location b, depending on the flow/trajectory pattern. Let inline image denote the time a water (or ideal tracer) particle takes along a trajectory from a to b in a single HU, where t0 is the starting time. In reality we typically have many particles injected at different times, at different a, which exit at different or the same discharge locations (Figure 1).

[14] A water particle is injected at inline image as inline image. The particle travels through the system with time inline image and is then discharged at b at time t = T where inline image. We refer to inline image as the water travel time, whereas T is referred to as the “arrival time.” For steady state flow, it is typical to set inline image and travel and arrival times coincide, i.e., inline image.

[15] The normalized discharge for an ideal tracer particle starting (or entering) at a at time inline image and arriving at (or exiting) the system at time inline image is obtained by the convolution,

display math

[16] Let a total amount of ideal tracer mass inline image be recharged over a given elementary area inline image at a, with a rate inline image [M/L2T]; we have

display math

[17] The ideal tracer discharge at b is

display math

where inline image is the forward WTT probability density function (pdf) conditioned on the starting time t0 at a; the corresponding expression for a nonideal tracer will be discussed in section 5.

[18] In the general case, both inline image and t0 are random, where inline image is dependent on t0. There are different pdfs that may be of interest using variables inline image, and t0. First, inline image is the pdf of t0 at a if t0 cannot be specified deterministically. Then we may consider a joint pdf inline image for instance, or inline image. We may also consider conditional pdfs such as the backward WTT pdf inline image, where travel times are conditioned on exit time T, or the forward pdf inline image where travel times are conditioned on starting time t0. This work addresses the conditional backward and forward pdfs inline image and inline image, respectively, in cases where temporal variations of the flow are significant. Note that if a catchment is conceptualized as a mixed reactor, conditional pdfs inline image and inline image can be directly related, provided that temporal variations of water storage in the catchment are known [Botter et al., 2010].

[19] The flow field in a catchment is assumed continuous such that at an arbitrary x within the catchment, a streamline can always be found. At an arbitrary time t, one can trace inline image as the water particle age, and inline image as water particle life expectancy [Nauman, 1969]; the special case is with inline image.

4. Water Travel Time

4.1. Kinematics

[20] The basic kinematic relationship of a trajectory is well known [Dagan, 1984],

display math

[21] If the reverse flow relative to the mean velocity inline image from a to b is negligible, a trajectory can be projected onto a suitably defined mean flow direction, for instance, x of inline image. In that case, the travel (residence) time between a and b, inline image is,

display math

where inline image and a is the starting position; the boundary condition is either inline image or inline image, i.e., either the starting time t0 is known, or the arrival time inline image is known. For the separable case we write,

display math

where W(x) is a Lagrangian velocity along a trajectory parameterized with the distance [Cvetkovic et al., 1996; Gotovac et al., 2009]. If (1) is solved with inline image we have the “forward” solution that is the basis for computing inline image, and if solved with inline image then the “backward” solution is obtained as the basis for computing inline image.

[22] The conditional pdfs inline image and inline image will be site-specific for any given catchment and boundary conditions. Here we wish to investigate their basic properties under idealized flow conditions. A generic setup is defined in Appendix A (Figures A1 and A2), and is suitable for exploring the forward and backward travel time distributions. The magnitude of the temporal fluctuations are assumed uniform along the domain; in reality, these fluctuations can be expected to diminish with transport distance [Destouni, 1991; Foussereau et al., 2001]. In Appendix A, the temporal aggregation unit for the data-based temporal fluctuations is daily; however, in our illustrations we shall leave the spatial and temporal units arbitrary (unspecified), depending on the units of the mean velocity U0 which (unless specified otherwise) is set to unity.

4.2. Forward and Backward Distributions

[23] Figure 3 illustrates forward and backward WTT distributions obtained from 500 trajectory realizations. In Figure 3a, fully ergodic conditions are assumed, i.e., 500 realizations of V(x) and 500 realizations of inline image. The simulated forward and backward cumulative distribution functions (CDFs, dotted curves) are compared to the analytical form (solid curves) assuming the solution of the advection-dispersion equation (ADE), or the tempered one-sided stable (TOSS) density with inline image in (B5) (Appendix B). A steady state flow case (with 500 identical realizations of V[x]) is also shown (black). The ADE model provides a close representation for the transient case roughly after the 10th percentile; for percentiles lower than 10, deviation is obvious between the simulated (symbol) and analytical (solid) curves. Similar comparison for the illustrated steady state case is much closer over the entire range (black symbols and curves). Random temporal variations may shift the CDF relative to the steady state case, depending, for instance, on how the mean velocity is defined (see Appendix A), or whether temporal fluctuations decrease with transport distance [Destouni, 1991; Foussereau et al., 2001], etc. The forward and backward CDFs of WTT almost coincide (compare red and blue curves in Figure 3a).

Figure 3.

Forward and backward cumulative distribution functions (cdfs) for water travel time: (a) Forward (blue) and backward (red) cdfs for ergodic transient flow obtained by simulations (dotted) and analytically (solid) assuming an ADE model ((B5) with inline image), also compared to cdfs for a steady state flow case (black symbols and curve), the shift of the steady state CDF is explained at the end of Appendix A. (b) Simulated backward cdfs for different exit times (colored solid and dashed curves) compared to the forward CDF for water travel time (black dotted), assuming a single data-based realization of the temporal variability inline image as depicted in Figure A2 (Appendix A).

[24] Next, a more realistic case of the data-based inline image (top-right diagram in Figure A1 of Appendix A) is explored, where inline image is a “single realization” identical for all 500 spatially random fields V(x). The black symbol curve in Figure 3b shows the forward CDF, starting at time inline image, whereas the colored curves show the backward CDF as a function of four chosen arrival time (T) values (8000, 7500, 7000, 6500, 6000). Clearly, there are visible differences in the backward CDF for the different arrival times, and also compared to the forward CDF (symbols), since the temporal variations are now a single realization (in this case using data-based inline image). However, the differences are relatively modest indicating that even in single realizations forward and backward CDFs may be quite comparable if longer periods of time are considered.

[25] For statistically stationary temporal variations and fully ergodic conditions, the distribution of inline image is independent of t0 (forward density), and the distribution of inline image is independent of T (backward density), i.e.,

display math

where for inline image the water “life expectancy” (forward) density is defined, and for inline image the water “age” (backward) density is defined, using the terminology of Nauman [1969]. Written out in words this is, the water age and life expectancy pdfs are equivalent for the same length scale. Note that the three-dimensional vectors a, b, and x have been suitably projected onto the mean flow direction which is assumed for simplicity parallel to the x-axis, and inline image is part of the trajectory between a and b. For nonergodic conditions of temporal variations, where a single realization of inline image is considered, (2) holds approximately, provided that the fluctuations are statistically stationary and one is considering sufficiently large WTT relative to the integral scale of the temporal fluctuations.

4.3. Temporal Moments

[26] Equation (2) is further explored based on the first two moments. Specifically, water particle transport is considered for the synthetic inline image, assuming both a single realization and an ensemble of inline image (ergodic conditions). The mean this illustration is assumed to be inline image, and the backward first temporal moment is computed for different arrival (exit) times T.

[27] The mean inline image for the backward density inline image with T in the range 10,000–20,000 with intervals of 1000, is illustrated in Figure 4a; the fully ergodic case is shown with the thin red lines and the single realization case of inline image with the thin black lines. The mean inline image for the forward density inline image with inline image is also included as the thick blue line, solid for the ergodic case and dashed for the nonergodic case.

Figure 4.

First two moments of forward and backward WTT pdfs as functions of transport distance: (a) mean WTT and (b) variance of WTT. The backward moments are computed for different exit times T (10,000–20,000, in intervals of 1000).

[28] First, moment inline image maintains in all cases an approximately linear dependence on distance (Figure 4a). For the nonergodic case, inline image exhibits more dispersion for different T (black thin lines), compared to the ergodic case (red thin lines). Only 100 realizations are used for the “ergodic” case; convergence toward a more compact linear dependence of inline image with distance, relative to the single-realization case is clearly seen by comparing the red and black thin lines in Figure 4a. Note that for the pathway between a = 0 and b, the thin lines are the mean water age for different arrival times T; mean groundwater age is typically addressed for entire aquifers in steady state flows [e.g., Ginn et al., 2009].

[29] The variance inline image for the same cases is shown in Figure 4b. Here the spreading for different T is clearly more significant, in particular for the single realization of inline image; at the end of the domain, the spreading of inline image for different T is almost 50%. For full variability, the dispersion is lower: the red curves for the 10 values of T differ around 20%. Further increase in the number of trajectories would reduce this difference in the curves obtained with different T. In spite of the variations visible in Figure 4b, the overall convergence of the travel time variance for statistically stationary spatial and temporal variations, is toward a consistent linear dependence. For single realizations of inline image, (2) is one possible approximation. Note that in Figure 3b, the single realization is obtained with the data-based inline image (Figure A1, Appendix A), whereas in Figure 4 the single realization case is based on the synthetic inline image (Figure A2, Appendix A). Clearly, further work is needed to better understand the implications of (2) for different cases of temporal flow variations.

4.4. Segmented Trajectories

[30] To quantify hydrological transport through different subsystems of a whole catchment, now consider segmented trajectories representing N different HUs as illustrated schematically in Figure 2. If the HUs have different physical properties, independence of WTT between HUs is a reasonable assumption. Here we wish to address the issue of independence more fundamentally, by considering HUs as geometrical entities with identical flow properties. To convolute WTTs across N different segments, we need to identify under what conditions for spatially and temporally varying flows, the advective water travel times in each segment can be considered to be approximately independent. The answer will depend on the correlation structure of the Lagrangian velocity W(x), a result of “transforming” the fields V(x) and inline image through inline image. In the following, segmented trajectories are explored for the generic space-time velocity field (Appendix A), by computing the convolution, as well as by using the expression for macrodispersion (B5) (Appendix B).

[31] In Figure 5, forward CDFs and complementary cumulative distribution functions (CCDF = 1 – CDF) are compared with N = 4 segments, for a transient and steady state random velocity; in the transient case, the fully ergodic synthetic inline image is used. The blue curve is obtained by computing the travel time moments over each segment, then using these to compute an analytical ADE CDF/CCDF for each segment; the red curve is computed as the ADE CDF/CCDF with travel time moments obtained from simulated trajectories for the entire domain. With the four-segment example used for illustration in Figure 5, there is a slight difference between the blue and red curves, visible both for the transient and the steady state case.

Figure 5.

Cumulative and complementary cumulative distribution functions for water travel time under ergodic transient flow, obtained from simulations (dotted), by the integrated ADE model (red) and by convolution over four equal segments (blue). A comparison is made with a steady state case that is for illustration purposes defined by setting Z = 0 in (A1), thereby shifting the CDF to the right.

[32] An additional test is to use the derived macrodispersion expression (B5) for a different number of segments. In Table 1 we compare the WTT coefficient of variation inline image (equation (B5) in Appendix B) computed segment-wise, with inline image computed for the entire domain (N = 1); identical inline image values would then indicate total independence, as well as consistency with the ADE model. Note that each realization of V(x) covers 20 log-integral scales over the entire domain of inline image 1000; temporal variations tend to further increase the integral scale of inline image relative to that of V(x) (Figure A2 of Appendix A, bottom diagram).

Table 1. Comparison of WTT Coefficient of Variation inline image(B5) for a Segmented Trajectory With Different Segments N, for Transient and Steady-State Cases
Segments Nζζ Steady-State
10.3241010.257954
20.3320480.245165
40.3103850.245165
100.264950.20706

[33] Table 1 shows the increasing deviation between inline image computed over an entire domain (first row with N = 1), and inline image computed using (B5) for the same domain divided in N equal segments. For N = 2 and N = 4, the segmented case inline image agrees well with inline image calculated for the entire domain, the difference being up to a few percent. For N = 10 the deviation is more significant, ∼20%, indicating that the independence of segments is questionable. The deviation increases with further reduction of segment (or HU) size (increase of N). Note that the deviation in inline image between N = 1 and N = 10 does not differ for transient and steady state cases, although the actual value of inline image is lower for the illustrated steady state case.

5. Solute Retention

[34] All solutes (tracers) transported through catchments will be subject to some degree of retention, the minimum being molecular diffusion into stagnant water, whether in the sediment, or intragranular porosity of rocks. Transport of some tracers may be significantly delayed due to sorption combined with diffusion. The main interest for most applications is to compute the tracer residence time, or tracer discharge at specified locations b.

[35] Under steady state flow, irrespective of the time spent in the immobile regions, a tracer particle is subject to the same advective transport when mobile at any given inline image. In such a case, the methodology using conditional probabilities for relating water and tracer travel times [Villermaux, 1974; Cvetkovic and Dagan, 1994] is applicable. The main question explored in this section, is to what extent the same methodology would be applicable for both spatially and temporally variable flow.

5.1. Conceptualization

[36] Tracer particles are injected with associated (dissolving) “water particles” simultaneously and under identical conditions [Cvetkovic and Dagan, 1994; Cvetkovic and Haggerty, 2002] at a at time t0. Consider a trajectory between a and b (Figure A1, Appendix A), again projected onto the mean flow direction, here the x-axis. Let the trajectory between a and b of length x be divided into N equal segments. For simplicity, we use the same notation N for the number of segments as in section 4, where reference was made to HUs. Here N is chosen such that, given the temporal fluctuations for the sorptive (reactive) problem at hand, the selected inline image ensures that velocity temporal changes over scale inline image are relatively small.

[37] The transport of water and tracer particles is conceptualized as a time domain random walk, the sketch of which is shown in Figure 6. If inline image denotes the WTT across an arbitrary inline image, and inline image denotes the retention time along the same segment for the tracer particle, then inline image is a particle residence time for segment inline image. Let inline image denote the density of inline image. Based on the mass balance for advective transport across inline image, we can write inline image in the Laplace domain as [Cvetkovic and Dagan, 1994]

display math

where ( inline image) denotes the Laplace transform. In other words, for a given segment inline image with inline image, the solute retention time inline image is drawn from a distribution dependent on inline image and a set of retention parameters denoted as a vector P. The parameters P are assumed to be known and approximately independent of time (diffusion constants, sorption coefficients, intra-aggregate porosity, etc.). The function g is referred to as the “memory function” quantifying the kinetics of mass transfer.

Figure 6.

Sketch of a segmented trajectory domain, with noted segment advective travel time transitions inline image and retention times inline image. Each advective travel time transition is associated with space-time trajectory with different starting time t0. The sum of inline image is the “virtual” WTT inline image.

[38]  inline image for the selected scale inline image will, in the general case, be correlated between segments. For transport where retention is significant, however, the retention time inline image will be independent between segments; hence, inline image may also be considered independent. In such a case, the tracer residence time for the entire trajectory is inline image and its density can be obtained by convolution,

display math

where inline image is referred to as the “virtual” water travel time between a and b. The “virtual” aspect of inline image can be explained as follows.

[39] The time domain random walk of the transport implies that at each segment we have a random water travel time inline image and a random retention time inline image (Figure 6). The retention time is the delay due to the immobilization of the tracer particle within the segment inline image. When the tracer particle is mobilized again, its retention in the next segment will be conditioned on the water particle travel time for that next segment. This means the tracer particle now follows a trajectory with another starting time t0, which in principle is obtained by backward tracking, as illustrated in Figure 6 for five trajectories. Thus, the “virtual” aspect is due to the fact that no actual water particle is transported between a and b during time inline image.

[40] A metaphor for the tracer particle under these conditions would be a passenger traveling by train between stations. At each station, the passenger disembarks and spends some time (retention due to immobilization); the passenger boards new trains at successive stations with different starting times t0 and a new space-time trajectory. Note that depending on g, t0 for each segment may be very different. For steady state flow, trajectories are independent of t0, thus irrespective of how large inline image is (i.e., how long a passenger waits at a station), the virtual (aggregated) water travel time inline image is always equal to the water travel time inline image, i.e., inline image.

[41] For approximately stationary temporal fluctuations, (2) is valid and inline image; then we write,

display math

i.e., the virtual WTT pdf is approximately equal to the actual WTT pdf, inline image.

5.2. Residence Time and Discharge

[42] Let inline image denote the pdf of tracer residence time for HU “i,” conditioned on the WTT in the same HU; in line with (3), we have,

display math

[43] In other words, for a given inline image of HU “i,” tracer residence time is drawn from a distribution dependent on a set of retention parameters denoted as a vector inline image for HU “i.”

[44] To compute an unconditional tracer residence time pdf which accounts for the variability of inline image, we ensemble average inline image to obtain by virtue of the Laplace transform definition:

display math

where inline image is the Laplace transform of the WTT pdf for the HU “i” and (4) has been used. Furthermore, h in (6) is the unconditional pdf of tracer particle residence time for pulse injection, in effect a “transfer function” for the HU “i.”

[45] The mass discharge of a nonideal, interacting tracer is obtained by,

display math

which reduces to the expression for an ideal tracer (water) with inline image. Instantaneous injection of mass inline image implies inline image. For steady state flow and inline image, the above equation reduces to the well-known expression inline image [e.g., Cvetkovic and Dagan, 1994].

6. Discussion

[46] Typically, the temporal variations are approximately stationary for longer timescales, similar to the example used in this work. When considering past events, e.g., for model verification, a single realization of the temporal variations will be available (a nonergodic case, or only a spatially ergodic case). For predictions of future transport over the decadal timescale, one will typically use a synthetic replica based on past statistics, where considering an ensemble is relevant (an ergodic case). If temporal trends are present but are relatively mild and the transport is considered over sufficiently long times, (2) and (4) may be applicable. Only in cases where abrupt changes take place over a relatively short period, such that the mean velocity exhibits a stepwise form, (2) and (4) will not be applicable. In cases where (4) is not applicable, a recently developed time domain random walk model for transient flow (J. O. Selroos, H. Cheng, S. Painter, and P. Vidstrand, Radionuclide transport during glacial cycles: Comparison of two different approaches for representing flow transients, submitted to Journal of Physics and Chemistry of the Earth, 2012) can be used for incorporating retention and possibly degradation, where each water particle is associated with an ensemble of tracer particles that are traced probabilistically, as they leave and return to the immobile zone due to mass transfer.

[47] The dispersion of a tracer in a catchment may be due to different effects. First, tracer advection will be subject to spatial and temporal flow variability. Second, even if injected over a small area, tracer particles may take different pathways, from the shallow to the deep subsurface, to surface flow; although the overall length scale of these pathways may be comparable, the hydraulic properties of the “hydrological units” are generally very different. Third, if injected over larger areas, tracer particles discharged at the same location will have different pathway lengths, which will significantly contribute to overall dispersion. All of these effects combined together may be referred to as “hydrological dispersion.” Finally, tracer particles will, in the general case, be subject to some type of mass transfer as well as possible reactive transformation.

[48] Given the complexity and variety of conditions involved, diverse conceptual approaches have been used to address hydrological dispersion. A similar approach to ours was presented for studying dispersion in a drainage network [Rinaldo et al., 1991, 2006; Rinaldo and Rodriguez-Iturbe, 1996]. In these studies, dispersion is accounted for on two levels: within an individual stream reach (hydrodynamic dispersion) and between different stream reaches within a stream network (geomorphological dispersion); the solution of the advection-dispersion equation for a given stream reach is convoluted to quantify the network effect [e.g., Rinaldo et al., 1991]. Moreover, the routing through serial stream reaches is aggregated into multiple stream pathways of varying length, which when combined, result in the dispersion on the stream network scale [Rinaldo et al., 1991, 2006].

[49] Our present work also relates to recent studies of Darracq et al. [2010], Destouni et al. [2010], and Persson et al. [2011], where subsurface flow and transport to and through the stream network is considered, with a focus on each single pathway between a recharge and discharge location. The pathways are of arbitrary length and transect a given number of “stream reaches” as well as other hydrological units, combining surface and subsurface flow.

[50] One means of incorporating varying pathway lengths is by integrating over varying distances from a finite recharge area [Destouni and Graham, 1995]. Let a total amount of solute mass M0 be released over a given recharge area, A, starting at t0. We then have,

display math

[51] In real catchments, a given discharge location typically collects water from many recharge locations or spatially distributed recharge, whereby many trajectories converge to a few discharge locations (Figure 1). Consider a finite recharge area A that is discharged into a discharge location at b. If the injected solute over area A is discharged at b, the normalized solute discharge for a noninteracting (ideal) tracer is,

display math

The variations within A, e.g., as different mean WTT due to different pathway lengths to the discharge location, enhances hydrological dispersion which was demonstrated earlier [Rinaldo and Marani, 1987; Rinaldo and Rodriguez-Iturbe, 1996; Destouni and Graham, 1995; Simic and Destouni, 1999; Lindgren et al., 2004; Persson et al., 2011].

[52] Hydrological dispersion has traditionally been analyzed using the “instantaneous unit hydrograph,” for instance, by routing instantaneous inflow through a series of linear reservoirs which leads to a gamma distribution for the entire catchment [Nash, 1959]. An alternative theory based on Boltzmann statistics was proposed by Lienhard [1964] who derived a particular form of the generalized gamma distribution for the unit hydrograph. Using a similar methodology, the result of Lienhard [1964] was extended by Rinaldo and Marani [1987] to obtain a mass response function as a generalized gamma distribution.

[53] With hydrological pathways distributed over entire catchment areas, the responses tend to exhibit exponential features characteristic of mixed reactors; hence, the exponential as well as the gamma distribution have been used in a number of works to represent catchments as one or a few mixed reservoirs [e.g., Haitjema, 1995; Rodhe et al., 1996; McGuire et al., 2005]. Note that the TOSS includes both the ADE model used, e.g., by Rinaldo et al. [1991] as well as the gamma and exponential densities [e.g., McGuire et al., 2005]; it can be shown that TOSS can reasonably well approximate even the generalized gamma density by a suitable choice of the exponent inline image in (B5).

[54] Taking advantage of analogous conceptualizations in chemical engineering [Nauman, 1969; Niemi, 1977], the mixed reservoir has recently been used to systematically address the issue of forward and backward travel time densities in catchment transport [Botter et al., 2010, 2011; Rinaldo et al., 2011]. In the present work, we also address the forward and backward travel time densities from a Lagrangian trajectory perspective, explicitly accounting for both spatial and temporal fluctuations that are illustrated for a generic setup (Appendix A). Using TOSS with the Lagrangian approach to address the forward and backward travel time densities, this work further bridges the gap between the pathway [e.g., Rinaldo et al., 1991; Destouni et al., 2010; Persson et al., 2011] and the classical mixed reactor approaches (of which Botter et al. [2011] is conceptually the most advanced) of hydrological transport.

[55] Finally, a few notes on the water age distribution. As it has been emphasized, the backward probability density function of inline image conditioned on the time of arrival T at a specified location x, inline image, is the water age density. A general theory of the water age for transient flow in a continuum advection-dispersion framework was given by Ginn [1999]. Recently, Cornaton [2012] presented an analytical solution for an abrupt change of the mean velocity using the one-dimensional formulation of Ginn [1999]. In Figure 7, our simulated inline image is compared to the corresponding cumulative distribution function obtained from the analytical solution of Cornaton [2012]; the analytical solution and the details of the illustrated case are given in Appendix C. The close comparison between the two results demonstrates consistency between a Lagrangian particle tracking approach and the continuum approach to computations of the water age distribution.

Figure 7.

Comparison of the water age cumulative distribution functions as obtained with the analytical solution (C5) of Cornaton [2012] (Appendix C), with the results of the backward particle tracking, assuming an overall setup as summarized in Appendix A. Uniform velocity changes at inline image from 1 to 0.5, and the water age is computed for different t as noted in the figure.

7. Summary and Conclusions

[56] If a tracer is applied over the entire catchment area, then given the wide range of pathway lengths, the tracer breakthrough at the exit appears consistent with a mixed reactor. Hence, catchments have frequently been conceptualized as internally fully mixed, in particular, when quantifying for instance transport of environmental isotopes.

[57] However, pathways in a catchment will typically transect different surface and subsurface hydrological units, usually with different biogeochemical properties; approximating transport as internally mixed may in such cases seriously limit the extent to which process-based descriptions of mass transfer and biogeochemistry can be applied. Furthermore, in cases where solutes (e.g., pollutants) are injected over limited areas, and where mass transfer or biogeochemical processes are of prime interest, the Lagrangian conceptualization of different hydrological pathways is most appropriate.

[58] The LaSAR methodology [Rainwater et al., 1987; Rinaldo et al., 1989; Cvetkovic and Dagan, 1994; Ginn, 2001] has proven useful in subsurface applications assuming steady state flow, for combining macrodispersion, mass transfer, and transformation (first-order loss [e.g., Cvetkovic, 2011b], or more complex chemical reactions [e.g., Rinaldo et al., 1989; Cvetkovic, 1997; Eriksson and Destouni, 1997; Ginn, 1999, 2001; Malmström et al., 2004]). The purpose of this work was to discuss conditions under which LaSAR may be applicable for hydrological transport through catchments where flow is both spatially and temporally variable. With mean properties and mean flow resolved analytically (for the subsurface see Wörman et al. [2007]; Marklund et al. [2008]) or by simulations, water and solute transport in a catchment can be analyzed in terms of flux/discharge; here the LaSAR approach is proposed for incorporating random velocity variations, as well as mass transfer.

[59] The main concluding points of this work are summarized as follows:

[60] For spatially and temporally varying velocity that are statistically stationary, (2) holds for fully ergodic conditions; if flow is only spatially ergodic, i.e., a single realization of temporal variations is considered, then (2) holds approximately, provided that the mean WTT is sufficiently large relative to the integral scale of temporal fluctuations.

[61] If significant, temporal variability will affect the statistical structure of the Lagrangian velocity fluctuations; once this effect is accounted for, convolution is applicable provided that transition times between predefined segments are approximately independent.

[62] Macrodispersion for a single hydrological pathway can be related to dispersion in underlying independent HUs by using the derived equation (B5); this expression was shown applicable for a wide range of dispersion processes, from the ADE model to anomalous transport, using the TOSS density (B5).

[63] The memory function can be applied within a LaSAR framework for quantifying exchange kinetics, provided that the temporal fluctuations are statistically stationary such that (4) is applicable. In the case of abrupt velocity changes in time, or strong trends, a particle-based time domain random walk (TDRW) approach [Painter et al., 2008] could be suitable as an approximate computational approach, if developed further; work in this direction is currently in progress.

[64] Depending on the distribution of the temporal fluctuations, temporal moments may differ from the steady state case with identical realizations of spatially variable velocity; compared to the ADE, temporal fluctuations may emphasize more strongly early arrival (low percentiles of WTT).

[65] The backward WTT density inline image obtained by particle tracking and the water age distribution for transient flow obtained using a continuum advection-dispersion approach [Ginn, 1999; Cornaton, 2012] are consistent.

[66] A generally important task is to accurately quantify flow partitioning and mass delivery fractions through different pathways that originate from recharge locations (source inputs) of interest [Darracq et al., 2010; Destouni et al., 2010]. Flow partitioning and mass delivery fractions indicate how much of the solute mass is potentially deliverable along pathways with significant temporal variations. For pathways that carry much of the water flow and solute mass in a typical catchment, especially through the subsurface, temporal variations may be relatively small and a steady state assumption sufficient [Destouni, 1991; Foussereau et al., 2001; Russo and Fiori, 2008]. Shallower pathways capture shorter-term fluctuations [Foussereau et al., 2001; Fiori and Russo, 2008], while the full spectrum of subsurface transport pathways may reflect longer-term trends and their changes [Darracq et al., 2008]; better understanding of transport partitioning and temporal velocity variations in the deeper pathways of this spectrum are still needed. Change in the flow and transport partitioning characteristics are particularly important when studying flow and transport under climate change [Bosson et al., 2012].

[67] Another aspect to explore are the internal sources that release solute along entire pathways. The solute may then be geogenic (e.g., dissolved inorganic carbon (DIC) and dissolved organic carbon (DOC) [Lyon et al., 2010]) or anthropogenic from long-term accumulation [e.g., Eriksson and Destouni, 1997; Botter et al., 2005; Darracq et al., 2008]; these results can, in principle, be incorporated into the proposed methodology, either directly using the analytical LaSAR formulation, or using a particle-based time domain random walk. [Painter et al., 2008].

Appendix A:: Generic Example

[68] Consider an idealized space-time velocity field of the type,

display math

where Y and Z are log-normally distributed with zero mean, and variances inline image and inline image, respectively. The log-integral scales for Y and Z are IY and IZ, respectively, where the correlation structure is assumed to be a negative exponential. The mean velocity U0 is set to unity (unless specified otherwise), and will be left with arbitrary units; consequently, all temporal and spatial quantities will be left with arbitrary units. For illustration, we shall assume inline image and inline image in all cases, with the spatial domain being 20 IY = 1000.

[69] Two basic forms of inline image are to be considered. One is a normalized water discharge record over 23 yr (8400 d) from an ∼300 km2 Swedish catchment; the monthly averaged inline image is illustrated in the top right side of Figure A1; the normalization is such that inline image =1 for the data-based inline image. The distribution of inline image is approximated as lognormal (the top left side of Figure A1). In the illustrations, a synthetic inline image will be used (in addition to the “single realization” data-based inline image), for representing fully ergodic transport conditions as given by (A1); in this case inline image and inline image.

Figure A1.

Sketch of a single trajectory from a to b, where a = 0 and b = 1000, with inline image. The bottom diagram is a single realization of the velocity field with inline image and inline image. The right top diagram is the data-based discharge Q(t) [m3 m−2 d−1] over 8000 d, and the left top diagram is the CDF of the normalized Q(t), i.e., inline image (dotted) compared with a lognormal model (line).

[70] By solving (1) with the forward or backward tracking boundary condition, an example of 50 trajectories over an x = 1000 scale is illustrated in Figure A2, using the data-based inline image in the interval (0, 4000) for all 50 realizations of V(x). The backward tracking is done from inline image. A single realization of V(x) is also illustrated in Figure A2 (the thin gray line in the bottom diagram) for which the trajectory is noted with a thick blue curved line in the main diagram. For the same trajectory, the Lagrangian velocity inline image is shown in the bottom diagram as the blue thick line. The dashed line in the bottom side of Figure A2 is the Lagrangian velocity following only temporal variations, computed as inline image, where inline image is obtained with full variability. The right diagram in Figure A2 illustrates the data-based inline image (used in the calculations of all of the illustrated trajectories), as well as a single realization of the synthetic, lognormal inline image. It is clearly seen how the temporal variations alter the local trajectory slopes, consistent with fluctuations of the Lagrangian velocity shown in the bottom diagram (blue line).

Figure A2.

Forward and backward trajectories for 50 realizations of the spatially variable velocity field V(x), all subject to the data-based temporal fluctuations inline image. The right diagram shows the data-based temporal fluctuations over 4000 time units, and a single realization of the synthetic inline image. The bottom diagram shows the velocity V(x) for a single realization (gray line), the corresponding Lagrangian velocity (blue line) associated with the forward trajectory for the same realization that is (marked with a thick blue line), and the Lagrangian representation of the temporal-data–based fluctuations computed as inline image (dashed line).

[71] The mean advective WTT for a steady state flow field obtained by setting Z = 0 in (A1) (i.e., inline image) is inline image where VH is the harmonic mean velocity inline image. The mean advective WTT with the flow governed by (A1) with Y = 0 (i.e., inline image) is computed by inline image, where inline image. Finally, the mean advective WTT for a flow field defined by (A1) is well approximated by inline image, implying inline image and inline image if inline image. Note that the steady state example in Figure 3b is obtained by setting Z = 0 in (A1), hence the shift of the CDF toward later times.

Appendix B:: Dispersion Along Segmented Trajectories

[72] Let inline image denote the mean and inline image the coefficient of variation of the WTT inline image for a segment “i”; note that this segment may be a HU or any given discretized (segmented) trajectory. The WTT inline image is random due to space-time velocity variations.

[73] Let inline image denote the pdf of inline image with inline image, and let inline image denote the pdf of inline image. Assuming that inline image and inline image are given, we wish to quantify dispersion for the N number of segments.

[74] It has been shown recently that the tempered one-sided stable (TOSS) density is a general WTT distribution for hydrological transport [Cvetkovic, 2011a]. Using the TOSS density, we write for segment “i”:

display math

where s is the Laplace transform variable, and the parameters ai and ci are defined by,

display math

ai being the cut-off rate, ci the scaling coefficient, and inline image an exponent for segment “i.” Modifying the solution given, e.g., by Hughes [1995], a convergent series expansion yields

display math

[75] Assuming that inline image are independent, the WTT density for the entire system composed of N consecutive segments is obtained by convolution,

display math

Depending on inline image, ai, and ci, a wide range of dispersive phenomena may be captured. Only if the exponent inline image and ai are constant for all segments is f an infinitely divisible distribution with inline image, i.e., c scales as a sum over all segments.

[76] Using (B5), we find that the coefficient of variation inline image for WTT of the entire system is,

display math

where inline image

[77] Equation (B5) demonstrates that macrodispersion as quantified by the WTT coefficient of variation inline image, is a function of mean velocities and dispersion properties of individual segments but does not depend on the nature of transport as defined by the TOSS exponent inline image.

[78] Equation (B5) is valid if the segments are chosen such that WTT between segments can be considered approximately independent; hence, we can use (B5) as test criteria for segment independence, as summarized in Table 1 (columns two and three).

Appendix C:: Analytical Solution for Water Age Density

[79] We summarize here the analytical solution for water age density presented by Cornaton [2012] and used in Figure 7.

[80] The governing equation for the water age distribution is based on the theory presented by Ginn [1999]; in one-dimensional simplest form it writes,

display math

where inline image is the water age, or the travel time of a water particle entering the system at time t0 and exiting at t = T; hence, inline image, as noted earlier. U in (C1) is the mean velocity and inline image the dispersivity.

[81] In line with Cornaton [2012], we consider a specific system where the mean flow velocity changes abruptly at inline image, from inline image to inline image for inline image. Boundary conditions are summarized as follows:

display math
display math
display math

where inline image, and inline image is known, in this case assumed as an inverse Gaussian distribution, equivalent to the TOSS density (B4) (with inline image, inline image and inline image).

[82] The solution of the above system has been obtained in the Laplace domain by Cornaton [2012] as,

display math

where

display math
display math
display math
display math
display math
display math

with inline image defined in (C5), the cumulative distribution function of the water age that is shown in Figure 7 is computed by numerical inversion of inline image.

[83] For an illustration example of Figure 7, inline image and inline image (similar to what was used by Cornaton [2012] for illustration). In this example, the abrupt change takes place at inline image, for a domain of x = 1000 with a dispersivity of inline image. In the particle tracking simulations, we have a spatially variable velocity with a log-integral scale of 50, and a log-standard deviation of 0.6; note that the first-order result yields inline image. The overall setup is as summarized in Appendix A. The black curve in Figure 7 is the initial (steady state) inverse Gaussian distribution inline image.

Notation
a [L]

starting (entrance) location for trajectory; a specified recharge location

b [L]

end (exit) location for a trajectory; a specified discharge location

A [L2]

arbitrary area of a catchment

inline image [1/T]

forward pdf of water travel time conditioned on the starting time t0 from a specified recharge location for a hydrological pathway; corresponds to inline image in Figure 1 of Botter et al. [2011]

inline image [1/T]

backward pdf of water travel time conditioned on the arrival time T at a specified discharge location for a hydrological pathway; this is referred to as water age pdf, i.e., how old the water is since entrance/recharge; corresponds to inline image in Figure 1 of Botter et al. [2011]

inline image [1/T]

forward pdf of water travel time, from a to x, independent of the staring time, t0

inline image [1/T]

backward pdf of water travel time from x to b, independent of the exit time, T

inline image [1/T]

memory function for retention processes

inline image [1/T]

unconditional tracer residence time distribution; here we use the notion “residence time” for tracer particles which in the general case includes retention in addition to advection

inline image [M/T]

tracer discharge at a specified discharge location for a single hydrological pathway (“point source”)

inline image [M/T]

tracer discharge at a specified discharge location for a finite source area “A”

inline image [M]

tracer mass

inline image

number of segments or hydrological units

inline image

spatially dependent vector of retention parameters

s [1/T]

Laplace transform variable

t [T]

running/clock time

t0 [T]

starting time for a water or tracer particle along a trajectory

T [T]

time of arrival of a water or tracer particle at a specified discharge location; in some cases it is convenient to use t and T interchangeably

inline image [T]

water travel time obtained by solution of (1) for specified boundary conditions

inline image [L/T]

three-dimensional space-time velocity vector with components [ inline image]

inline image [L/T]

x component of the spatially variable velocity vector V; thus, V and vx are equivalent

inline image [L/T]

Lagrangian velocity in the x direction computed as inline image

x [L]

position vector with components ( inline image)

inline image [L]

equation of the trajectory inline image starting at a; assumption of a steady state flow pattern implies that X does not change in time, i.e., is independent of t0

inline image [1/T]

conditional pdf for tracer particle residence time that in general includes retention

inline image [M/TL2]

release rate at injection location

inline image [–]

normalized temporal variation function (see Figure A1).

Acknowledgments

[84] Critical comments by Andrea Rinaldo (EPFL, Switzerland) and Tim Ginn (UCD, USA) on the original version of the manuscript have significantly helped to improve this work; their effort is sincerely appreciated. The authors gratefully acknowledge the financial support by the Swedish Nuclear Fuel and Waste Management Co. (SKB), the NOVA Research and Development Center, Oskarshamn (Sweden), and the Swedish Research Council (VR, project 2009–3221).