Water Resources Research

Modeling and inverting reactive stream tracers undergoing two-site sorption and decay in the hyporheic zone

Authors


Corresponding author: O. A. Cirpka, Center for Applied Geoscience, University of Tübingen, Hölderlinstr. 12, 72074 Tübingen, Germany. (olaf.cirpka@uni-tuebingen.de)

Abstract

[1] Performing stream-tracer experiments is an accepted technique to assess transport characteristics of streams undergoing hyporheic exchange. Recently, combining conservative and reactive tracers, in which the latter presumably undergoes degradation exclusively within the hyporheic zone, has been suggested to study in-stream transport, hyporheic exchange, and the metabolic activity of the hyporheic zone. The combined quantitative analysis to adequately describe such tests, however, has been missing. In this paper, we present mathematical methods to jointly analyze breakthrough curves of a conservative tracer (fluorescein), a linearly degrading tracer (resazurin), and its daughter compound (resorufin), which are synchronously introduced into the stream as pulses. In-stream transport is described by the one-dimensional advection-dispersion equation, amended with a convolution term to account for transient storage within the hyporheic zone over a distribution of travel times, transformation of the reactive tracer in the hyporheic zone, and two-site sorption of the parent and daughter compounds therein. We use a shape-free approach of describing the hyporheic travel-time distribution, overcoming the difficulty of identifying the best functional parameterization for transient storage. We discuss how this model can be fitted to the breakthrough curves of all three compounds and demonstrate the method by an application to a tracer test in the third-order stream Goldersbach in Southern Germany. The entire river water passes once through the hyporheic zone over a travel distance of about 200 m with mean hyporheic residence times ranging between 16 and 23 min. We also observed a secondary peak in the transfer functions at about 1 h indicating a second hyporheic flow path. We could jointly fit the breakthrough curves of all compounds in three monitoring stations and evaluated the parameter uncertainty of the individual and joint fits by a method based on conditional realizations of the hyporheic travel-time distribution. The approach gives insight into in-stream transport, hyporheic exchange, metabolic activity, and river-bed sorption of the stream under investigation.

1. Introduction

[2] The transition zone between groundwater and surface water bodies, denoted hyporheic zone, is believed to be the key for evaluating the ecological status of the stream with respect to habitat formation, filtration of hazardous particles, biogeochemical turnover, and degradation of organic pollutants [e.g., Bencala and Walters, 1983; Findlay, 1995; Brunke and Gonser, 1997; Boulton et al., 1998; Sophocleous, 2002; Hester and Gooseff, 2010; Robertson and Wood, 2010; Krause et al., 2011]. Hyporheic exchange can take place on multiple scales, such as ripples and dunes of the river bed, riffle-pool sequences, and river meanders, leading to hyporheic travel paths of multiple lengths and with multiple travel times.

[3] In the past years, detailed modeling studies have shown that the extent of hyporheic flow cells, the strength of hyporheic exchange, and the distribution of hyporheic travel times can be predicted based on an exact morphological and hydraulic description of bedforms and bathymetry [Gooseff et al., 2006; Cardenas and Wilson, 2007; Cardenas et al., 2008; Feng and Michaelides, 2009; Boano et al., 2010a], river logs [Sawyer et al., 2011], fluvial islands [Shope et al., 2012], and meanders [Cardenas, 2008a; Revelli et al., 2008; Boano et al., 2010b, among others], to name only the most important features. The most common model used in hyporheic stream transport is the transient-storage model [Bencala and Walters, 1983], in which the hyporheic zone is conceptualized as an immobile water body exchanging solutes with the mobile stream. Solute transport within the stream is commonly described by the one-dimensional advection-dispersion equation (ADE). In the original version of the transient-storage model, a well-mixed immobile water zone is assumed, and mass transfer between mobile and immobile water is proportional to the concentration difference between the two domains. This model has been extended by accounting for multiple immobile zones, each of which having a different size and undergoing first-order mass transfer with the stream according to a different mass-transfer coefficient, in so-called multirate mass-transfer models [Haggerty and Gorelick, 1995; Carrera et al., 1998] and in multiple-storage-zone models [Marion et al., 2008]. Rather than discrete immobile zones, distributions of mass-transfer rates may also be used, in which the distributions are commonly described by functions with a few parameters [Haggerty et al., 2000]. As another variant, Deng and Jung [2009] suggested to make the mass-transfer rate of the single-transient-storage model time dependent. Conceptually similar are certain variants of the continuous-time random-walk method to describe stream transport, including a waiting-time distribution for solute particles in the immobile zone [Boano et al., 2007]. All of these extensions have been developed to improve the description of tailing behavior observed in stream-tracer tests [Drummond et al., 2012; Gooseff et al., 2003].

[4] Most of the transient-storage models listed above consider the size of the storage zone and the residence-time distribution within that zone, deriving the transport equation from mass conservation of the lumped system of the stream and the hyporheic zone. In the present study, we follow an approach similar to that of Wörman et al. [2002] that does not require the size of the transient-storage zone and depends on a volumetric exchange rate between the stream and the hyporheic zone as well as on the age distribution of the returning hyporheic water instead. These two concepts are equivalent and can be transferred into each other [e.g. Liao and Cirpka, 2011]. We denote the age distribution of the returning hyporheic as hyporheic travel-time distribution inline image, with the hyporheic travel time τ, and the discharge of exchanged water per volume of stream water as hyporheic exchange rate coefficient qhe.

[5] Several parametric models have been suggested for inline image. Wörman et al. [2002] listed: (1) the exponential distribution, which is equivalent to the single linear-storage model, (2) the Dirac delta distribution, reflecting advective transport in a single hyporheic flow path, (3) the log-normal distribution, resembling advective-dispersive transport in a single hyporheic flow path, and the uniform distribution. Power-law functions have also frequently been applied [Haggerty et al., 2002; Cardenas et al., 2008; Cardenas, 2008b]. Despite efforts to relate hyporheic travel-time distributions to geometric and other properties of single features [Sawyer et al., 2011; Cardenas et al., 2008, among others], it remains unclear which parameterization is the best in real applications affected by many features. To overcome this problem, Liao and Cirpka [2011] proposed a shape-free method to estimate the hyporheic travel-time distribution by jointly analyzing breakthrough curves of conservative and reactive tracers obtained from pulse-like tracer tests.

[6] While stream-tracer tests with conservative tracers have been used in stream characterization for several decades, Haggerty et al. [2008] only recently proposed to use resazurin (Raz) as “smart” or reactive tracer that undergoes irreversible transformation to resorufin (Rru) under mildly reducing conditions as typically present in the microbiologically active hyporheic zone [see also Haggerty et al., 2009]. In previous field experiments, Raz was continuously injected into the stream. This has the advantage that transient effects on concentrations are excluded and metabolic activity can be quantified more accurately from the Raz:Rru ratio. Conversely, it prevents using the difference between the conservative-tracer and reactive-tracer breakthrough curves for a better quantification of the strength of hyporheic exchange and the hyporheic travel-time distribution as suggested by Liao and Cirpka [2011]. Experiments with joint pulse-like injection of the conservative-tracer fluorescein (Fluo) and Raz into a stream are discussed in a companion paper [Lemke et al., 2013b] together with a simplified analysis of the observed breakthrough curves. A much closer look into the mathematical analysis of combined tracer breakthrough curves will be provided in the present paper.

[7] Our previous attempts to model transport of Raz in streams with a metabolically active hyporheic zone had two major shortcomings. Liao and Cirpka [2011] assumed that the reactive tracers do not sorb, whereas Lemke et al. [2013b] assumed linear sorption at equilibrium. Already Haggerty et al. [2008] reported nonequilibrium sorption in column experiments which is in agreement with our own column studies that we could simulate well by a two-site sorption model (data not shown here). For the rather short hyporheic residence reported by Lemke et al. [2013b], it should be clear that kinetic sorption can have an effect on the observed breakthrough curves. The general impact of sorption within the hyporheic zone on stream transport has already been discussed by Jonsson et al. [2004]. A second shortcoming of both Liao and Cirpka [2011] and Lemke et al. [2013b] is that these studies do not consider the breakthrough curves of the degradation product Rru even though the latter can be measured more easily than Raz by fluorometric methods.

[8] The main objective of the present paper is to extend the shape-free method proposed by Liao and Cirpka [2011] for the determination of hyporheic travel-time distributions from joint conservative and reactive stream-tracer tests with pulse-like injection. The extensions include (1) the consideration of two-site sorption of the reactive compounds within the hyporheic zone to be compliant with findings from column tests and (2) the analysis of the breakthrough curve of the reactive daughter compound together with the parent compound and the conservative tracer. The focus of the present study is on the derivation of the predictive models, because the inversion methodology does not significantly differ from that of Liao and Cirpka [2011]. Finally, we apply the method to the field data presented by Lemke et al. [2013b]. The proposed method yields (1) parameters of in-stream transport, namely the in-stream velocity and dispersion coefficient, (2) the characterization of hyporheic exchange by the hyporheic exchange rate and the hyporheic travel-time distribution, (3) equilibrium and kinetic sorption parameters of the hyporheic zone, and (4) a measure of metabolic activity within the hyporheic zone in the form of a first-order rate coefficient of Raz-to-Rru transformation.

2. Model Formulation

2.1. General Setup Considered

[9] The main objective of this work is to estimate transport parameters of streams from reactive tracer experiments: the velocity and in-stream dispersion coefficient, the rate of hyporheic exchange, and the distribution of times over which water parcels stay within the hyporheic zone. While breakthrough curves of tracers in observation wells directly yield the travel-time distribution of the tracer in the system by deconvoluting the input and output signals [e.g. Cirpka et al., 2007], tracer breakthrough curves in streams depend only indirectly on the travel-time distribution in the hyporheic zone, because no measurements are taken in the hyporheic zone or in the river bed itself. This leads to a coupled system of in-stream transport plus a convolution term that quantifies how the fraction of tracer, which has been transported into the hyporheic zone by infiltration, returns back into the stream.

[10] The setup of the stream-tracer application considered is shown in Figure 1. The conservative tracer (Flou) and the reactive tracer (Raz) are jointly introduced by pulse injections into the stream. Within the hyporheic zone, the reactive tracer partially undergoes reduction to its reaction product (Rru). The breakthrough curves of all three tracers are measured far downstream, to quantify hyporheic processes on the scale of a river reach, at which the effects of individual features are averaged.

Figure 1.

General setup of the model. The conservative tracer and reactive parent compound are injected as a joint pulse into the stream. No reaction takes place in the stream for all tracers, while in the hyporheic zone the reactive parent compound undergoes first-order transformation to the reaction product and both reactive tracers are subject to two-site sorption, i.e., equilibrium and kinetic sorption. The latter is shown as two different small boxes for each cell within the hyporheic zone. The strength of hyporheic exchange is quantified by qhe as discharge of exchanged water per water volume in the stream. Transport within the hyporheic zone is described as advection in travel time rather space. The hyporheic travel-time distribution inline image determines the age distribution of the water returning into the stream. Breakthrough curves of all three tracers are measured downstream.

[11] Within the model, solute transport in the stream is considered to be advective-dispersive, where a fraction of river water infiltrates into the hyporheic zone and comes back at the same location over a distribution of travel times. Both in-stream dispersion and hyporheic exchange lead to spreading of the conservative-tracer breakthrough curve, making the distinction between these two processes difficult when only conservative-tracer tests are analyzed. The reactive parent compound, however, is converted to the daughter compound on exposure to hyporheic-zone conditions. Liao and Cirpka [2011] claimed that jointly analyzing conservative and reactive tracer breakthrough curves improves our ability to distinguish between spreading by in-stream dispersion and hyporheic exchange. However, in this study, we also consider that the reactive tracers may undergo two-site sorption within the hyporheic zone. As a consequence, the reactive tracer signals are affected by the metabolic activity and sorption properties of the stream bed. Parameters describing these processes need to be inferred together with the physical transport properties in the analysis of the breakthrough curves.

2.2. Governing Equations

[12] The one-dimensional ADE is commonly used to describe solute transport in streams. As proposed by Liao and Cirpka [2011], the effect of hyporheic exchange on conservative solute transport is quantified by an exchange rate and a hyporheic travel-time distribution inline image. The governing equation for a conservative compound reads as

display math(1)

in which inline image is the tracer concentration in the river; inline image is the time since injection of the tracer into the stream; inline image is the flow velocity of the stream; inline image is the in-stream dispersion coefficient; inline image is the hyporheic exchange rate coefficient; inline image is the distribution over which water parcels stay within the hyporheic zone, and inline image is the travel time within the hyporheic zone. Equation (1) is a nonlocal-in-time, linear partial differential equation of second-order. The corresponding initial and boundary conditions are for a point injection into the flow and a semi-infinite domain:

display math(2)
display math(3)
display math(4)

in which inline image is the total mass of the tracer injected in the river; inline image is the cross-sectional area of the river channel; and inline image is the Dirac delta function with units inverse to those of the argument.

[13] We have introduced inline image as a probability density function of times over which water parcels stay within the hyporheic zone. By adopting the concept of signal processing and other fields, it may also be considered as a more generalized transfer function for any type of linear transport. A transfer function describes the response in the system output to a unit pulse input, in which the hyporheic zone is the system of interest in the current application. In this sense, inline image quantifies the concentration of a compound measured in the water returning from the hyporheic zone to the stream at time τ when the compound was introduced into the hyporheic zone as a unit pulse at time 0. The generalized hyporheic transfer function can differ from the travel-time distribution due to linear processes such as linear sorption, both kinetic and at local equilibrium, and first-order transformations. The convolution integral of equation (1), however, would not be valid if nonlinear reactions or nonlinear mass-transfer processes had to be considered [Dennery and Krzywicki, 1967, chap. 7].

[14] Since three tracers are actually measured and analyzed by the model, we need three governing equations, where those for the conservative tracer and the reactive parent compound look identical even though the corresponding hyporheic transfer functions differ, whereas the governing equation of the reaction product differs because it depends on the concentration of the parent compound:

display math(5)

subject to

display math(6)

in which the subscript 0 is for the conservative tracer (Flou), 1 for the reactive tracer (Raz), and 2 for the product (Rru). The governing equation for the reaction product contains two terms in the convolution integral, because one part of the tracer originates from the transformation of the reactive tracer within the hyporheic zone, whereas the other part has already been introduced into the hyporheic zone by river-water infiltration. Therefore, we use inline image to denote the hyporheic travel-time distribution, which is the hyporheic transfer function of the conservative tracer, inline image describes the hyporheic transfer function of the reactive parent compound that is not converted in the hyporheic zone, inline image denotes the hyporheic transfer function of the reaction product, already entering the hyporheic zone as reaction product, whereas inline image quantifies the response of the daughter compound concentration in the returning water caused by a unit pulse of the parent compound in the water entering the hyporheic zone. Note that the three transfer functions include the effects of all processes within the hyporheic zone, namely physical transport, the linear transformation and two-site sorption of the reactive tracers. In Appendix Solution in the Laplace Domain, we derive analytical expressions for tracer concentrations inline image in the Laplace domain, requiring the Laplace-transformed hyporheic transfer function inline image with inline image being the complex Laplace coordinate (see equation (A6)).

2.3. Hyporheic Transfer Functions for Conservative and Reactive Tracers

[15] The hyporheic transfer functions for the tracers are denoted by inline image, where τ is the travel time in the hyporheic zone starting from the time point when the water enters the river bed. All reactive processes of the solutes in the hyporheic zone, namely microbial transformations and sorption, modify inline image. Thus, while the conservative hyporheic transfer function inline image is a travel-time distribution reflecting the geometry and conductivity of the hyporheic zone and the pressure distribution at the stream bed, the reactive hyporheic transfer functions inline image of the parent compound Raz, inline image of the daughter compound Rru, and the cross-compound transfer function inline image differ in shape. In the fitting procedure, we want to estimate inline image and the reactive parameters modifying inline image to inline image, and inline image. In this paper, rather than assuming a specific functional shape of inline image, such as the exponential or log-normal distributions, we identify shape-free functions, which may exhibit nonclassical features such as broad peaks, multiple peaks, or long tails.

[16] In the following derivations, it is advantageous to express transport within the hyporheic zone as function of travel time rather than spatial coordinates, thus avoiding the explicit consideration of flow paths, velocities, and dispersion coefficients within the hyporheic zone. The latter approach is valid in linear transport with uniform reaction coefficients, in which the order of mixing water parcels of different age and linear reactions does not influence the model outcome.

2.3.1. General Relationship Between Hyporheic Travel-Time Distributions and Hyporheic Transfer Functions

[17] The general approach for calculating the transfer function of a tracer in the system for a given travel-time distribution inline image of the returning water is expressed by

display math(7)

in which inline image is the corresponding hyporheic transfer function with indices discussed above; inline image is the hyporheic travel-time distribution, indicating how the pulse of an ideal tracer, which does not undergo transformation or interphase mass-transfer processes, returns back into the stream, in which inline image is the travel time for the ideal tracer (substituting the spatial coordinate); inline image is the tracer concentration of compound i in the hyporheic zone at time τ and travel time inline image.

[18] The concept of computing inline image may be best understood by considering a pulse injection of compound i into the hyporheic zone at time inline image, as illustrated in Figure 2. As discussed below, we consider advection in travel-time coordinates, two-site sorption, and linear decay of the reactive parent compound. The solid lines of Figure 2a show the corresponding length profiles of the parent compound inline image in the travel-time domain (coordinate inline image) for different times τ as solid lines. With increasing time, the color changes from black via red to yellow. If advection was the only transport process, the length profile would be a moving Dirac pulse. At early times, such a pulse can be observed. At later times, however, kinetic sorption leads to a hump following the pulse; and at very late times the profile becomes Gaussian with respect to inline image. The dotted lines in Figure 2a show the corresponding length profiles of the reaction product with matching colors indicating identical time points. With increasing time, the reaction product starts to dominate over the parent compound, but it vanishes again due to further degradation of the reaction product. As shown in section 2.3.2, we account for an unspecified degradation of the parent compound, the transformation from parent compound to the reaction product and a further degradation of the reaction product, requiring three decay coefficients in our model.

Figure 2.

Principle of calculating transfer functions for different tracers from the hyporheic travel-time distribution inline image. (a) Length profiles of reactive-tracer concentrations with respect to travel time inline image for different times τ, indicated by changing colors (lighter colors mark later times). Solid lines: reactive parent compound; dotted lines: reactive daughter compound; black dashed line: hyporheic travel-time distribution inline image. (b) Corresponding transfer functions, reflecting the sampling of concentrations in Figure 2a according to inline image.

[19] The black dashed lines in Figures 2a and 2b mark the hyporheic travel-time distribution inline image of the water returning into the stream. As expressed in equation (7), the concentration of the compounds returning into the stream at any time τ can be computed by considering the full profile inline image at that time, weighting it with the probability density inline image that the returning water has the age inline image, and integrating over all possible values of inline image. The resulting responses on a pulse injection (that is, the transfer functions) are illustrated in Figure 2b. For an ideal tracer, the transfer function inline image equals the hyporheic travel-time distribution inline image. The transfer function of the reactive parent compound inline image rapidly drops at early times due to mass-transfer from the aqueous to the sorbing phase within the hyporheic zone. This mass is later released, leading to tailing of the transfer function inline image in comparison to the hyporheic travel-time distribution inline image. The transfer function inline image of the reaction product due to a pulse injection of the parent compound indicates to which extent the released parent compound is transformed to the daughter compound. In the profiles shown in Figure 2a, the concentration of the daughter compound significantly exceeds the concentration of the parent compound at late times. However, this happens in a range of travel-time values inline image that is hardly sampled by inline image, so that the corresponding cross-compound transfer function inline image even decreases at the decisive late times. The hyporheic transfer function inline image of the daughter compound in response to an injection already as daughter compounds, not shown in Figure 2b, is similar to that of inline image because the same process types, namely linear decay and two-site sorption, are considered.

2.3.2. Considering Decay and Two-Site Sorption

[20] The simplest case for the evaluation of the hyporheic transfer function is that of the conservative tracer, where the tracer concentration within the hyporheic zone for a pulse injection remains a pulse traveling with inline image, resulting in

display math(8)

in which the delta function inline image represents the conservative-tracer concentration within the hyporheic zone for a pulse injection.

[21] For the Raz-Rru system, the expressions of inline image are somewhat more complicated. Here, we assume the following processes within the hyporheic zone: (1) linear decay of Raz to Rru, quantified by a first-order rate coefficient inline image; (2) linear decay of Raz to unidentified reaction products, quantified by inline image; (3) linear decay of Rru to unidentified reaction products, quantified by inline image; and (4) linear equilibrium sorption of both Raz and Rru, quantified by corresponding retardation factors R1 and R2; (5) linear kinetic sorption of both Raz and Rru, quantified by dimensionless distribution coefficients inline image and inline image and rate coefficients of mass transfer inline image and inline image, respectively. In general, hyporheic transport is expressed in travel-time rather than spatial coordinates.

[22] This model differs slightly from that proposed by Haggerty et al. [2009], who considered only equilibrium sorption, even though the same authors have applied multirate mass-transfer models for sorption elsewhere [Haggerty and Gorelick, 1995]. Since hyporheic travel times can range from seconds to days, it is not reasonable to assume that sorption is always at equilibrium. Also, column experiments, not shown here, indicate the need to account at least for two sorption sites. A second difference to Haggerty et al. [2009] is that in our model the transport in the travel-time domain is restricted to advection. Effects of dispersion are included in the travel-time distribution inline image appearing in equation (7): More dispersive transport would lead to a wider distribution inline image.

[23] The governing equation for the transport of the parent compound (Raz) within the hyporheic zone is

display math(9)

and that for the daughter compound (Rru)

display math(10)

in which inline image are the concentrations of the reactive parent and daughter compounds Raz and Rru in the hyporheic zone, respectively; inline image are concentrations at the kinetically sorbing sites, and inline image are dimensionless distribution coefficients. The coefficients b1 and b2 represent how much mass of the parent and daughter compounds are introduced into the hyporheic zone at time inline image. For the evaluation of transfer functions, inline image and inline image is dimensionless and can either be zero or one. Appendix Transport Equations for the Hyporheic Zone With Two-Site Sorption contains a derivation of the equations, and Appendix Solution of Governing Equation With Two-Site Sorption in the Laplace Domain the analytical solutions in the Laplace domain.

[24] Given the tracer concentrations in the hyporheic zone, the hyporheic transfer function for the reactive parent and daughter compounds can be calculated according to equation (7) as follows:

display math(11)

in which inline image is the concentration of Raz caused by injection of Raz in the system for inline image in equation (9); inline image is the concentration of Rru caused by transformation from Raz at conservative travel time inline image and time τ, stemming from the input of c1 (that is, for inline image and inline image in equation (9)), and inline image is the concentration of Rru caused by injection of the same compound into the domain, stemming from the input of c2 (that is, for inline image and inline image in equation (9)).

[25] The weighting average with respect to inline image expressed in equation (11) also holds for the Laplace-transformed hyporheic transfer functions. Here, the hyporheic concentrations inline image are replaced by their Laplace transforms inline image given in Appendix Solution of Governing Equation With Two-Site Sorption in the Laplace Domain:

display math(12)

in which we have made use of inline image according to equation (8). The inline image and inline image coefficients are derived in Appendix Solution of Governing Equation With Two-Site Sorption in the Laplace Domain. The integrals appearing in equation (12) are comparably easy to compute for a simple parametric hyporheic travel-time distribution inline image, such as the exponential function. In the shape-free approach, inline image is discretized as a piecewise linear function. The corresponding analytical expressions of the integrals in equation (12) are listed in Appendix Discretization of the Hyporheic Transfer Functions gi(τ).

2.4. Joint Estimation of the Shape-Free Hyporheic Travel-Time Distribution and Reactive Parameters

[26] The parameter-estimation scheme is essentially identical to the method of Cirpka et al. [2007], even though more data are included and more parameters are estimated. In the shape-free approach, the hyporheic travel-time distribution inline image is estimated at a set of discrete travel times inline image with regular travel-time increment inline image, chosen to be identical to the time increment inline image of the tracer measurements. The largest travel time τ must not be larger than the duration of the experiment. Therefore, the total number of parameters is in the same order as the number of measurement (hundreds to thousands), requiring regularization. Adopting the method of Cirpka et al. [2007], we assume that the hyporheic travel-time distribution inline image is a stochastic, autocorrelated travel-time variable described by a second-order intrinsic model with a linear semivariogram:

display math(13)

in which h is the travel-time difference between two values in the vector inline image. The slope inline image of the semivariogram quantifies the smoothness of inline image, where small values of θ enforce particularly smooth behavior. Although more complex semivariogram models could be used here, we have chosen the linear model because it has only one structural parameter, and the lack of information about inline image results in the simplest form of the semivariogram model. The chosen approach is identical to first-order Tikhonov regularization, in which solutions of inline image with large derivatives inline image are penalized [Kitanidis, 1997]. With the regularization, we avoid reproducing noise in the data when estimating inline image. The model is rather sensitive to the value of θ, which therefore needs to be chosen carefully in the estimation procedure. Because inline image is a probability density function, we must ensure that no element inline image is negative.

[27] Besides inline image, we estimate the following parameters: v, D, qhe, inline image and inline image, which are used to characterize stream transport and to compute the hyporheic transfer functions inline image, and inline image. By applying Bayes' theorem and assuming multi-Gaussian prior probability density functions, we obtain the objective function:

display math(14)

in which inline image is the vector of measured tracer concentrations in the stream; inline image is the simulated tracer concentration based on the set of parameters inline image, where we denote inline image for simplicity; C is the covariance matrix expressing uncertainty of the measurements; inline image is a inline image matrix, describing the discrete semivariogram values for all pairs of elements in inline image is the set of Lagrange multipliers for the non-negativity constraints on inline image; and H is a inline image selection matrix, which changes in every iteration step according to the number of constraints inline image and the position at which the non-negativity constraints are activated. The model simulation is linearized about the last estimate inline image, where J is the Jacobian matrix. Minimization of the objective function is achieved by setting the derivatives with respect to p and inline image to zero, resulting in the following system of linear equations:

display math(15)

in which inline image. This system of equations is solved iteratively with an inner iteration to identify the correct set of Lagrange multipliers and an outer iteration for the successive linearization about the last estimate. The rules to activate/deactivate the non-negativity constraints can be found in Cirpka et al. [2007]. The scheme is stabilized by a line search in the Gauss-Newton direction.

3. Application to Field Data

3.1. Experimental Setup

[28] A tracer test with the conservative and reactive fluorescent tracers Fluo and Raz was performed at River Goldersbach in southern Germany in July 2012. In the following, we give a brief overview of the field experiment. Details on the field site, the experimental setup, and the separation and correction of the tracer signals are given by Lemke et al. [2013b]. River Goldersbach is part of the Neckar catchment within the Rhine basin. The section of River Goldersbach used for the tracer is located about 30 km south of Stuttgart. The river flows along a highly meandering course through a narrow floodplain with a morphology characterized by riffle-pool sequences with abundant point bars. The river-bed material is mainly composed of coarse sand and gravels.

[29] A premixed solution of inline image mol of Fluo and inline image mol of Raz was injected into the river as a pulse, distributed uniformly over the river cross section. The injected tracers as well as the reaction product Rru were measured every 10 s using online field fluorometers [Lemke et al., 2013a]. Three monitoring stations were chosen to place the fluorometers at distances of 830, 1075, and 1210 m downstream the injection point, respectively. The experiment was started after sunset and completed before sunrise to avoid photodegradation of the tracers [Haggerty et al., 2009]. During the entire tracer test, the pH of the river was fairly stable around a value of 8. At this pH, the conservative tracer Fluo can be treated as an ideal tracer [Smith and Pretorius, 2002]. The discharge during the experimental period was also stable, which is around 60 L/s.

[30] Measured breakthrough curves are shown in the left column of Figure 3. The injected pulses were spread on transport in the stream, and reduction of Raz lead to the formation of Rru, even though this remained a small fraction. The recovery of the conservative tracer Fluo was high inline image for all three monitoring stations. The recovered number of moles of Raz and Rru at each monitoring station did not add up to the number of moles of Raz injected, which were attributed to the decay of Raz and Rru to unknown (nonfluorescent) compounds [Lemke et al., 2013b].

Figure 3.

Independent fits of breakthrough curves for each monitoring station (MS). Left column: measured and fitted breakthrough curves obtained at three MSs which are located 830, 1075, and 1210 m downstream of the injection point. Green lines: Fluo (conservative tracer), blue lines: Raz (reactive parent compound), and red lines: Rru (reaction product). Horizontal lines: quantification limits of the measurement device for each tracer. Right column: estimated transfer functions for all tracers.

3.2. Results of Data Analysis

[31] We analyze the breakthrough curves obtained at the three monitoring stations and estimate transport as well as reactive parameters along with the hyporheic travel-time distribution inline image using the shape-free approach described in section 2.2. The same data set is analyzed by Lemke et al. [2013b] who use an exponential function as the hyporheic travel-time distribution, consider only equilibrium sorption within the hyporheic zone and do not analyze the information contained in the breakthrough curves of the reaction product (Rru).

[32] The shape-free method is applied to jointly fit the breakthrough curves of the conservative tracer (Fluo), reactive parent compound (Raz), and reactive product (Rru), first at each individual monitoring station (Figure 3), followed by a joint fit of the breakthrough curves for all compounds at all monitoring stations (Figure 4). The left column of Figure 3 shows the individually fitted, semilogarithmic breakthrough curves for the time range between 1 and 12 h after tracer injection. The measured molar concentrations are shown as dots, whereas the colored lines are the simulated breakthrough curves. The three solid horizontal lines in the plots indicate the quantification limits of the fluorometers. Only measured concentrations above this line are considered reliable [Lemke et al., 2013a] and are used for parameter estimation. From top to bottom, the quantification limits are inline image for Raz, inline image for Rru, and inline image for Fluo. Because much more molar mass of Raz than that of Fluo was injected, the peak concentrations of Fluo are the smallest, but the relative accuracy of the measurements is the highest. This can be explained by the high quantum yield of Fluo in comparison to Rru, and even more so Raz, which is also indicated by the different quantification limits for the three compounds. From the breakthrough curves, we can see that the concentration of Fluo reaches its peak the earliest, Raz is somewhat retarded, which can be explained by sorption within the hyporheic zone, and Rru peaks the latest. The latter is expected since Rru needs time to form.

Figure 4.

Joint fits of all breakthrough curves and all tracers; measured and fitted breakthrough curves at the three monitoring stations (MSs); single set of hyporheic transfer functions obtained from jointly fitting the data of all three MSs. Green lines: Fluo (conservative tracer), blue lines: Raz (reactive parent compound), and red lines: Rru (reaction product). Horizontal lines: quantification limits of the measurement device for each tracer.

[33] The different tailing behavior of the three compounds is captured very well by the fit within the quantification limit. While Fluo shows an exponential tailing behavior for about 2 h, the concentration drops more slowly at late times. Such a behavior cannot be captured with a model that uses an exponential hyporheic travel-time distribution, like the one used by Lemke et al. [2013b]. The shape-free approach, by contrast, leads to near-to-perfect agreement in the tailing of the conservative tracer even at very late times. This is important because the simulated breakthrough curves of the reactive compounds directly depend on the hyporheic travel-time distribution. If we presented a good model fit of the reactive tracers together with a bad one of the conservative tracer, bias in the estimated reactive tracers was needed to cancel the error introduced by the erroneous hyporheic travel-time distribution, so that all parameters were wrong.

[34] The reactive parent compound Raz shows the steepest drop in the tail, which can be attributed to decay contributing to a loss of mass. The transformation of Raz to Rru makes the concentration of the latter drop rather slowly. The model fits agree quite well with the observations, capturing most of the tailing behavior. A little offset of the fitted Rru concentrations in comparison to the measurements can be observed at very late times and is amplified by the semilogarithmic plot, but the main pattern of the tail is still very well captured. A closer inspection of the simulated Rru breakthrough curves at all three monitoring stations reveals that the simulated concentration peaks appear a little bit later than the measured ones, and tailing is slightly overestimated. Bencala and Walters [1983] reported that sorption onto the immobile sediment in the stream becomes a major control of solute transport in low-flow periods, and we have observed similar findings in the fitting of other tracer tests with small in-stream velocity. We conjecture that the slow transport in the stream facilitates nonnegligible contact of Raz to metabolically active biofilms at the river bottom rather than within the hyporheic zone, leading to earlier transformation of Raz to Rru than in the model, in which the transformation is strictly restricted to the hyporheic zone where transport is much slower than within the stream. Altogether, however, we assess the fits of the breakthrough curves obtained by the shape-free method to be very good, including the tailing behavior. This is also corrobated by the values of the root-mean-square error (RMSE) of the model fit listed in Table 1.

Table 1. Estimated Parameters From Breakthrough Curve of Fluo, Raz, and Rru at Three Difference Monitoring Stations (MSs) and Joint Fit of All Three Stationsa
ParameterMS1: 830 m (%)MS2: 1075 m (%)MS3: 1210 m (%)Joint Fit (%)
  1. a

    The uncertainty of the estimated parameters is given as coefficients of variation in percent. The quality of the fits is quantified by the root-mean-square error (RMSE) in the reproduction of the measured breakthrough curves by the model.

inline image0.095 ± 0.640.097 ± 0.220.098 ± 0.180.095 ± 0.61
inline image0.198 ± 7.70.118 ± 3.20.145 ± 2.80.185 ± 13.4
inline image5.10 ± 2.65.05 ± 0.74.55 ± 0.64.75 ± 2.8
inline image3.87 ± 13.02.50 ± 18.14.43 ± 19.13.73 ± 23.6
inline image6.51 ± 7.46.31 ± 5.48.42 ± 7.17.07 ± 4.0
inline image9.29 ± 37.57.81 ± 26.71.74 ± 55.17.70 ± 20.4
inline image1.21 ± 0.71.19 ± 1.21.23 ± 1.21.22 ± 1.7
inline image1.00 ± 0.61.00 ± 0.81.03 ± 4.61.00 ± 0.5
inline image0.132 ± 9.40.296 ± 2.30.515 ± 3.00.70 ± 71.0
inline image0.419 ± 21.00.716 ± 18.12.589 ± 29.50.66 ± 24.0
inline image7.75 ± 15.74.77 ± 2.83.15 ± 2.72.29 ± 76.0
inline image3.31 ± 24.61.95 ± 18.10.661 ± 36.12.74 ± 13.0
Quality of the fits    
RMSE Fluo (μmol/m3)0.20 ± 11.40.22 ± 20.60.21 ± 19.20.97 ± 1.70
RMSE Raz(μmol/m3)1.80 ± 19.02.43 ± 25.22.08 ± 23.013.60 ± 0.81
RMSE Rru (μmol/m3)1.50 ± 9.01.06 ± 9.41.50 ± 11.82.68 ± 3.80

[35] In addition, we perform a joint fit using the breakthrough curves of all compounds at all three monitoring stations to estimate a single set of parameters suitable for all stations. The results are shown in Figure 4. All breakthrough curves are fitted rather well, and the transfer functions have similar patterns as shown in the individual fits of Figure 3. In the joint fit, the mean residence time of Fluo in the hyporheic zone is 16 min, of Raz 14 min, and of Rru 23 min. As expected, the joint fit results in higher RMSE compared to the individual fits (see Table 1) but are considered still acceptable. Bencala and Walters [1983] have shown similar results. The good fit for several stations is promising for the application of our method to larger river reaches, provided that no major changes in river morphology and flow regime occur over the length of the reach under investigation.

[36] For each distance and the joint fit, we estimate one set of parameters by jointly fitting the breakthrough curves of Fluo, Raz, and Rru. Table 1 lists the estimated transport and reaction parameters. The estimation uncertainties of the parameters are expressed as coefficients of variation in percent. They are calculated by generating 1000 conditional realizations of the hyporheic travel-time distribution inline image by the method outlined by Liao and Cirpka [2011]. Each conditional realization of inline image yields a different set of the other transport and reaction parameters.

[37] The estimated in-stream velocity v of approximately inline image is very consistent across the three monitoring stations, and the value obtained by the joint fit lies in the range of individual fits. The estimated in-stream dispersion coefficients inline image show a higher variability among the monitoring stations. Here, the joint fit has the highest uncertainty. The same holds for the hyporheic exchange rate coefficient inline image. This is so because solute spreading caused by hyporheic flow paths with small hyporheic travel times can hardly be distinguished from spreading due to Fickian in-stream dispersion [see also Liao and Cirpka, 2011]. The average distance over which the stream water passes on average once through the hyporheic zone is about 195 m. The transformation from Raz to an undetected compound inline image appears to be slower than the transformation rate from Raz to Rru inline image, which is consistent with the findings of Haggerty et al. [2009]. Regarding the sorption of the reactive compounds within the hyporheic zone, Raz sorption appears to be partially at local equilibrium inline image and partially kinetically controlled inline image, whereas practically all sorption of Rru seems to be kinetically controlled according to the model fit inline image. The estimated first-order mass-transfer coefficients inline image are in the range of inline image for Raz and inline image for Rru, respectively. Overall, the estimated parameters describing chemical transformations and sorption within the hyporheic zone vary more strongly among the monitoring stations than the stream parameters and the hyporheic exchange rate coefficient. Nonetheless, the uncertainties of the joint-fit parameters are not necessarily larger than those of the individual fits. This is so, because the joint-fit parameters express a compromise solution, trying to meet the data of all measurement stations simultaneously. The range of the compromise solution is fairly small for several parameters, even though the individual fits differ to a larger extent. Joint-fit parameter values deviating too much from the compromise solution would be penalized by a severe misfit in breakthrough curves of one or two monitoring stations. The deviations between the measured and simulated breakthrough curves obtained by both the individual and joint fits that cannot be explained by parameter variability represent the conceptual model uncertainty, which is higher in the joint fit than in the individual fits, as expressed by the RMSE values listed in Table 1.

[38] The reactive parameters modify the transfer functions of Raz and Rru in comparison to the hyporheic travel-time distribution. The right column of Figure 3 shows the estimated hyporheic transfer functions for all three compounds, including the cross-compound transfer function inline image, for all three monitoring stations. The smoothness parameter θ can theoretically be inferred from the data using the expectation-maximization method requiring the repeated generation of multiple conditional realizations [Cirpka et al., 2007; Liao and Cirpka, 2011]. However, due to the associated high computational effort, we have manually chosen inline image by inspecting the model fit and the shape of the estimated hyporheic travel-time distribution, which should not fluctuate too much within very small travel-time increments.

[39] The green dotted lines in the right column of Figure 3 and the right bottom plot of Figure 4 are the estimated hyporheic transfer functions inline image for the conservative tracer Fluo, which are identical to the underlying hyporheic travel-time distributions inline image. The initial guess for inline image is an exponential function, which is equivalent to using the standard transient-storage model with a single linearly reacting storage zone. This initial guess is substantially modified in the inversion procedure. While the exponential function drops rapidly at small values of τ and more slowly at large ones, all estimated hyporheic travel-time distributions inline image estimated by the shape-free approach show first the behavior of an approximately truncated Gaussian function, which starts off comparably flat and drops off more rapidly at larger values of τ, followed by a clear secondary peak at inline image. This additional peak is needed to achieve the tailing of the observed breakthrough curves. It has independently been identified in the analysis of the breakthrough curves at all monitoring stations, and we observe the same behavior of the transfer functions in the joint fit shown in Figure 4. Our interpretation is that stream transport is affected by at least two dominant hyporheic exchange mechanisms: the exchange by microbedforms in which the stream water stays only more or less 16 min on average, and a second feature with travel times of about 1 h. Even though the ratio of two residence times is smaller than 5, which was proposed by Choi et al. [2000] as characteristic value for the separation of different regimes, we see a significant time deviation in the transfer functions. With a pulse injection and an overall duration of the experiment of 12 h, we cannot exclude that larger structures leading to considerably longer hyporheic travel times exist in the stream system. These would remain unnoticed due to quantification limits [Ward et al., 2010]. Comparing the transfer function obtained by the individual fits of the monitoring stations and by the joint fit, we observe that the joint-fit transfer function exhibits no more a clear disconnection between the two peaks.

[40] The blue dash-dotted lines in the right column of Figure 3 and the right bottom plot of Figure 4 are the estimated hyporheic transfer functions inline image of the reactive parent compound Raz. The most obvious difference to inline image is that the truncated Gaussian-like hump at small values of τ is lower and smaller for Raz than for Fluo. This is caused by the substantial equilibrium sorption of Raz estimated by the inversion procedure, which is also corroborated by the shift of the secondary peak. Kinetic sorption and decay have only a minor influence on the hyporheic transfer functions of Raz. The latter also becomes clear when considering the cross-compound transfer function inline image, quantifying the response of the Rru concentration in the hyporheic water returning to the stream due to a unit pulse of Raz entering the hyporheic zone. The cross-compound transfer function inline image is plotted as black solid lines in the right column of Figure 3 and the right bottom plot of Figure 4. More Rru is generated with increasing distance in stream, which is consistent with our understanding that along the river reach, Raz infiltrates more often into the hyporheic zone, where Rru is formed. On the time scales that the stream water stays within the hyporheic zone, only a small fraction of Raz is transformed to Rru. This behavior is in agreement to the finding that over the entire reach less than 20% of Raz has been transformed to Rru [Lemke et al., 2013].

[41] The red dashed line in the right column of Figure 3 and the right bottom plot of Figure 4 are the estimated hyporheic transfer functions inline image of the reaction product Rru, that is, the concentration of Rru in the returning water due to a unit pulse of Rru introduced into the hyporheic zone. Because the model fits estimate negligible equilibrium sorption of Rru, and kinetic sorption has estimated characteristic time scales of 0.8–4.2 h when most of the hyporheic water has already returned back into the stream, the hyporheic transfer functions inline image are very similar to the underlying hyporheic travel-time distributions inline image at small values of τ. Only at larger travel-time values, some modification in the shape becomes visible.

[42] Based on 1000 conditional realizations, Figure 5 shows the uncertainty of all transfer functions. The single best fit assuming a perfectly smooth hyporheic travel-time distribution lies mainly in the 95% uncertainty band of the conditional realizations, but the uncertainty itself is rather narrow for all parameters. As discussed above, the small parameter uncertainty represents the narrow parameter range for a compromise solution trying to meet all breakthrough curves as well as possible, rather than a perfect match of simulated and measured breakthrough curves.

Figure 5.

Uncertainty of the transfer functions obtained by the joint fit of all breakthrough curves at all monitoring stations. The gray band is 95% of the uncertainty for the transfer functions based on 1120 conditional realizations.

4. Discussion and Conclusions

[43] We have presented a model analyzing stream-tracer tests combining a conservative tracer, Fluo, and a linearly decaying tracer, Raz, that are jointly introduced into the stream as a single pulse, in which we also consider the concentration of the reaction product, Rru. The hyporheic travel-time distributions estimated by the shape-free approach do not resemble a simple parametric function. The independent fits for the individual monitoring stations agreed in a rather broad truncated Gaussian-like distribution of hyporheic travel times at small values, with a mean residence time of 16 min, amended by a secondary peak at about 1 h. Revealing such features is a major advantage of the shape-free approach, even though the exact details of the estimated hyporheic travel-time distributions should be handled with care. In contrast to parametric models, the shape-free method is able to reproduce the tails of the observed concentration breakthrough curves. This is only possible when accounting for relatively longer (or slower) hyporheic travel paths. Of course, we cannot make any statements about the significance of possible hyporheic travel paths with travel times beyond the duration the experiment. As discussed by Harvey et al. [1996], only the fast component of hyporheic exchange can be detected by performing stream-tracer experiments. The fits of the breakthrough curves are excellent, even though small mismatches are still unavoidable.

[44] Fits using all three tracers for each monitoring station or even across all stations is superior to only using the conservative-tracer data. Even though not shown in this paper, we have compared the results from only fitting the Fluo data and jointly fitting the breakthrough curves of all compounds. While the in-stream velocity is estimated very consistently among different monitoring stations and between fits of Fluo alone or all tracers, the in-stream dispersion coefficient tends to be overestimated and the hyporheic exchange rate coefficient underestimated, when we use only the Fluo breakthrough curves compared to the results from fitting the breakthrough curves of all compounds. This is so because both parameters explain spreading of the conservative-tracer breakthrough curves. By considering the reactive tracers in the model analysis, the distinction between in-stream mixing and spreading due to hyporheic exchange has improved, because the transformation of Raz to Rru takes place only in the hyporheic zone.

[45] Accounting for the reaction product is an important consistency check for both the experimental data and the model. Fitting only the conservative tracer and the reactive parent compound Raz, as done by Liao and Cirpka [2011] and Lemke et al. [2013b], yields a total loss coefficient of Raz, but it is not possible to quantify the fraction that is transformed to Rru. The latter information is important because the Raz-to-Rru transformation has been related to aerobic respiration of the stream bed [González-Pinzón et al., 2012; Argerich et al., 2011], whereas an unspecified loss of Raz may be caused by processes that are completely independent of metabolic activity. The mismatch between the total mass of generated Rru and the remaining Raz compared to the total mass of injected Raz may indicate long travel times in the hyporheic zone that cannot be detected within the experimental time, or unknown transformation processes.

[46] A key motivation for developing the approach was to better distinguish in-stream mixing processes from spreading due to hyporheic exchange. We cannot exclude that some mismatch still remains. The estimated in-stream dispersion coefficient has the largest value at the first monitoring station, while the hyporheic exchange coefficient decreases with travel distance. This could be an aliasing effect, because with increasing time the concentration length profiles in systems with transient storage increasingly resemble Gaussian profiles, characteristic for Fickian dispersion. The inverse model is thus allowed to trade missing hyporheic exchange for an overestimation of in-stream dispersion if only the conservative tracer is considered. The latter should be penalized by misfits of the reactive breakthrough curves because an underestimation of hyporheic exchange causes a decrease in overall simulated transformation from Raz to Rru. In the current implementation, however, such a misfit can be balanced by increasing the estimated rate coefficients, which can actually be observed in Table 1. Thus, the joint fit of breakthrough curves from several monitoring stations provides more reliable estimates of the parameters. We see that the joint-fit values of in-stream velocity v, in-stream dispersion coefficient D, and the hyporheic exchange-rate coefficient qhe all lie in the range of the individual fits. While the RMSE of the joint fit is higher than in the individual fit, we consider the joint model fit as well acceptable [see also Bencala and Walters, 1983].

[47] Our model accounts for two-site sorption of the reactive compounds within the hyporheic zone. This degree of complexity was necessary in the analysis of column tests and also when fitting other stream-tracer tests not shown in this paper. In the current application, the Raz sorption appears to be partially in local equilibrium, whereas Rru sorption is almost entirely controlled by kinetics. We cannot fully exclude that unidentified processes, such as a contribution of Raz-to-Rru transformation within the stream rather than within the hyporheic zone has introduced bias in the estimated sorption parameters. Neglecting sorption of the reactive tracers altogether, as done by Liao and Cirpka [2011], however, is clearly not valid for the Raz/Rru-system.

[48] In recent years, several studies have shown that transient storage occurs not only in the hyporheic zone but also within the stream itself [e.g., Harvey et al., 2005; Briggs et al., 2009, 2010; Gooseff et al., 2011]. Separating the effects of individual storage zones on stream-tracer breakthrough curves is difficult without additional information. While our stream gives no visual indication (large pools, dead zones, etc.) of intensive in-stream storage, we cannot guarantee that all transient storage identified by our method is located within the hyporheic zone. This implies that all identified parameters are effective values, describing the net behavior of the stream over the length of the reach. We do not assess this as over-critical because this effective description is what is targeted for in reach-scale solute transport anyway. The model indicates net reach-scale solute retension, metabolic activity, and sorptivity, maybe not only located in the hyporheic zone but most likely transferrable to other compounds (oxygen, sorbing contaminants) of interest.

[49] We are fully aware that hyporheic exchange is a complex three-dimensional process, depending on multiple properties of the stream, its bed, and the adjacent aquifer. Rather than resolving all influencing factors, we describe transport as a “one-and-a-half” dimensional process with one-dimensional advective-dispersive transport in the stream coupled to one-dimensional, formally purely advective transport in the hyporheic zone. The effects of the complex flow geometry on solute transport in the real system are captured by the hyporheic travel-time distribution. This approach is valid only when the reach length under consideration is much larger than the dominant flow paths of hyporheic exchange. For simplicity, we also assume stationarity of the hyporheic travel-time distribution. That is, the same distribution is considered along the entire reach length simulated. This requires that the general geomorphological and other characteristics of the stream, determining hyporheic exchange processes, do not significantly change over the distance under consideration. In its current implementation, the model does not consider gaining or losing of river water in the experiment reach. Losing river water cannot be detected by tracer tests with a single injection point, whereas gaining conditions would lead to dilution of all tracers. Toward detecting gross losses and gains, Payn et al. [2009] performed a series of conservative-tracer/dilution-gauging tests along a river reach, and Covino et al. [2011] on the scale of entire river networks. Both losing and gaining lead to discharge and thus velocity values varying over the length of the reach. These processes can be included if sufficient knowledge about the location of major gains and losses exist, detected, e.g., by differential dilution gauging. Most likely, the analytical solution of the transport equation in the Laplace domain had to be replaced by numerical simulation, whether in the time or Laplace domain.

[50] In summary, the method and model presented in this paper fits the breakthrough curves from tracer experiment in River Goldersbach very well. The shape-free method allows revealing unconventional patterns of the hyporheic travel-time distribution, yielding better fit of the breakthrough curves than obtained by parametric hyporheic travel-time distributions [Lemke et al., 2013b]. Jointly fitting high-resolution breakthrough curves of Fluo, Raz, and Rru gives insights into in-stream transport, hyporheic exchange, metabolic activity, and sorption properties of the stream bed.

Appendix A: Solution in the Laplace Domain

[51] The initial value problem equation (5) with the boundary conditions equation (6) is solved in the Laplace domain, where the convolution term becomes a multiplication, such that the partial differential equation is converted into a second-order homogeneous (for Fluo and Raz) or inhomogeneous (for Rru) linear ordinary differential equation:

display math(A1)

in which s is the complex Laplace variable and the tilde marks the corresponding Laplace transform in time. Applying the Laplace transformation on the boundary conditions, we obtain

display math(A2)

[52] For simplicity, we denote

display math(A3)

[53] Therefore, the governing equations in the Laplace domain become

display math(A4)

subject to

display math(A5)

[54] The solution in the Laplace domain reads as

display math(A6)

which is back-transformed into the time domain by the numerical method of De Hoog et al. [1982].

Appendix B: Transport Equations for the Hyporheic Zone With Two-Site Sorption

[55] We use the one-dimensional advection equation to describe solute transport in the hyporheic zone, replacing the spatial domain by the travel-time coordinate inline image. Considering first-order decay of the reactive tracer in the hyporheic zone, we arrive at

display math(B1)

subject to a zero initial condition and a pulse-like boundary condition at inline image

display math(B2)

in which inline image is the tracer concentration in the aqueous phase of the hyporheic zone; inline image is the mass of sorbing tracer per mass of solids; n is the effective porosity in the hyporheic zone; inline image is the mass density of the solids; inline image is the first-order decay coefficient; and inline image is real time, whereas inline image is travel time within the hyporheic zone, which is a substitute for space. We distinguish the sorbing-phase concentration in a contribution at equilibrium sites and another one at kinetic sites

display math(B3)

assuming linear behavior of both equilibrium and kinetic sorption

display math(B4)
display math(B5)

in which inline image is the distribution coefficient between the equilibrium sorption sites and water, inline image is the distribution coefficient between the kinetic sorption sites and water, and inline image is the rate coefficient of kinetic mass transfer.

[56] Substituting equations ((B4) and (B5)) into equation (B1), we arrive at

display math(B6)

in which we denote

display math(B7)

as the retardation factor due to equilibrium sorption.

[57] We now introduce the aqueous saturation concentration inline image with respect to the concentration at the kinetic sorption sites skin:

display math(B8)

yielding the following system of equations for equations ((B6) and (B5)):

display math(B9)

in which

display math(B10)

[58] Equation (B9) describes how the tracer concentration in the hyporheic zone changes over time in both the aqueous and sorbing phases. As seen in the governing equation (1), these values are not used directly in the model but needed for calculating inline image in equation (11).

Appendix C: Solution of Governing Equation With Two-Site Sorption in the Laplace Domain

[59] In the Laplace domain, equations ((9) and (10)) become

display math(C1)
display math(C2)

where

display math(C3)
display math(C4)
display math(C5)
display math(C6)

Appendix D: Discretization of the Hyporheic Transfer Functions inline image

[60] Using the same technique as Liao and Cirpka [2011], the hyporheic travel-time distribution inline image is not forced to follow a specific function characterized by few parameters. Instead, inline image is discretized into inline image sections with identical increment inline image, and we apply piecewise linear interpolation of discrete values inline image:

display math(D1)

in which inline image is the intersection of the linear function, and inline image is the slope.

[61] The hyporheic transfer functions inline image, and inline image depend on the hyporheic travel-time distribution inline image via equation (11). Because inline image is defined as piecewise linear function, the Laplace transforms inline image must also be computed piecewise. Substituting equation (D1) into equation (12) and performing the integration, yields

display math(D2)
display math(D3)
display math(D4)
display math(D5)

Acknowledgments

[62] The authors are thankful to the reviewers Fulvio Boano, Aaron Packman, and Roy Haggerty for their helpful comments and constructive suggestions. This work was supported by a grant from the Ministry of Science, Research and Arts of Baden-Württemberg (AZ Zu 33–721.3-2), the Helmholtz Center for Environmental Research - UFZ, Leipzig, and the German Academic Exchange Service (DAAD).