In this paper we demonstrate that hyporheic exchange can be represented simply as a one-dimensional diffusion process, where the diffusivity decays exponentially with depth into the sediment bed. Based on a meta-analysis of 106 previously published laboratory measurements of hyporheic exchange (capturing a range of bed morphologies, hydraulic conditions, sediment properties, and experimental approaches) we find that the reference diffusivity and mixing length-scale are functions of the permeability Reynolds Number and Schmidt Number. These dimensionless numbers, in turn, can be estimated for a particular stream from the median grain size of the sediment bed and the stream’s depth, slope, and temperature. Application of these results to a seminal study of nitrate removal in 72 headwater streams across the United States, reveals: (1) streams draining urban and agricultural landscapes have a diminished capacity for in-stream and in-bed mixing along with smaller subsurface storage zones compared to streams draining reference landscapes; (2) under steady-state conditions nitrate uptake in the sediment bed is primarily biologically controlled; and (3) under transient conditions (e.g., spills or storms) physical transport processes are likely to influence nitrate removal rates as well. While further research is needed, the simplicity and extensibility of the framework described here should facilitate cross-disciplinary discussions and inform reach-scale studies of pollutant fate and transport and facilitate their scale-up to watersheds and beyond.

In this paper we develop and test a rigorous modeling framework, based on Duhamel’s Theorem, for the unsteady one-dimensional transport and mixing of a solute across a flat sediment-water interface (SWI) and through the benthic biolayer of a turbulent stream. The modeling framework is novel in that it allows for depth-varying diffusivity profiles, accounts for the change in porosity across the SWI and captures the two-way coupling between evolving solute concentrations in both the overlying water column and interstitial fluids of the sediment bed. We apply this new modeling framework to an extensive set of previously published laboratory measurements of turbulent mixing across a flat sediment bed, with the goal of evaluating four diffusivity profiles (constant, exponentially declining, and two hybrid models that account for molecular diffusion and enhanced turbulent mixing in the surficial portion of the bed). The exponentially declining profile is superior (based on RMSE, coefficient of determination, AICc, and model parsimony) and its reference diffusivity scales with a dimensionless measure of stream turbulence and streambed permeability called the Permeability Reynolds Number, . The diffusivity’s dependence on changes abruptly at , reflecting different modes of mixing below (dispersion) and above (turbulent diffusion) this threshold value. The depth-scale over which the diffusivity exponentially decays is about equal to the thickness of the benthic biolayer (2 to 5 cm), implying that turbulent mixing, and specifically turbulent pumping, may play an outsized role in the biogeochemical processing of nutrients and other contaminants in stream and coastal sediments.

Unsteady transit time distribution (TTD) theory is a promising new approach for merging hydrologic and water quality models at the catchment scale. A major obstacle to widespread adoption of the theory, however, has been the specification of the StorAge Selection (SAS) function, which describes how the selection of water for outflow is biased by age. In this paper we hypothesize that some unsteady hydrologic systems of practical interest can be described, to first-order, by a “shifted-uniform” SAS that falls along a continuum between plug flow sampling (for which only the oldest water in storage is sampled for outflow) and uniform sampling (for which water in storage is sampled randomly for outflow). For this choice of SAS function, explicit formulae are derived for the evolving: (1) age distribution of water in storage; (2) age distribution of water in outflow; and (3) breakthrough concentration of a conservative solute under either continuous or impulsive addition. Model predictions conform closely to chloride and deuterium breakthrough curves measured previously in a sloping lysimeter subject to periodic wetting, although refinements of the model are needed to account for the reconfiguration of flow paths at high storage levels (the so-called inverse storage effect). The analytical results derived in this paper should lower the barrier to applying TTD theory in practice, ease the computational demands associated with simulating solute transport through complex hydrologic systems, open up new opportunities for real-time control, and provide physical insights that might not be apparent from traditional numerical solutions of the governing equations.

Many water quality and ecosystem functions performed by streams occur in the benthic biolayer, the biologically active upper (~5 cm) layer of the streambed. Solute transport through the benthic biolayer is facilitated by bedform pumping, a physical process in which dynamic and static pressure variations over the surface of stationary bedforms (e.g., ripples and dunes) drive flow across the sediment-water interface. In this paper we derive two predictive modeling frameworks, one advective and the other diffusive, for solute transport through the benthic biolayer by bedform pumping. Both frameworks closely reproduce patterns and rates of bedform pumping previously measured in the laboratory, provided that the diffusion model’s dispersion coefficient declines exponentially with depth. They are also functionally equivalent, such that parameter sets inferred from the advective model can be applied to the diffusive model, and vice versa. The functional equivalence and complementary strengths of these two models expands the range of questions that can be answered, for example by adopting the advective model to study the effects of geomorphic processes (such as bedform adjustments to land use change) on flow-dependent processes, and the diffusive model to study problems where multiple transport mechanisms combine (such as bedform pumping and turbulent diffusion). By unifying advective and diffusive descriptions of bedform pumping, our analytical results provide a straightforward and computationally efficient approach for predicting, and better understanding, solute transport in the benthic biolayer of streams and coastal sediments.

In this paper, we explore the use of unsteady transit time distribution (TTD) theory to model pollutant removal in biofilters, a popular form of nature-based or “green” stormwater infrastructure (GSI). TTD theory elegantly addresses many unresolved challenges associated with predicting pollutant fate and transport in these systems, including unsteadiness in the water balance (time-varying inflows, outflows, and storage), unsteadiness in pollutant loading, time-dependent reactions and scale-up to GSI networks and urban catchments. From a solution to the unsteady age conservation equation under uniform sampling, we derive an explicit expression for solute breakthrough with or without first-order decay. The solution is calibrated and validated with breakthrough data from 17 simulated storm events (+/- bromide as a conservative tracer) at a field-scale biofilter test facility in Southern California. TTD theory closely reproduces bromide breakthrough concentrations, provided that lateral exchange with the surrounding soil is accounted for. At any given time, according to theory, more than half of water in storage is from the most recent storm, while the rest is a mixture of penultimate and earlier storms. Thus, key management endpoints, such as the treatment credit attributable to GSI, are inexorably linked to the age distribution of water stored and released by these systems.

The hyporheic exchange below dune-shaped bedforms has a great impact on the stream environment. One of the most important properties of the hyporheic zone is the residence time distribution (RTD) of flow paths in the sediment domain. Here, we evaluate the influence of dimensionless sediment depths d b * = 2 π d b / λ where λ is the dune wavelength and different values of dimensionless groundwater underflow values u b * (similar to dune migration celerity), on the shape of the hyporheic exchange RTD. Empirical RTDs were generated, over a range of combinations between d b *     and u b *   values, from numerical particle tracking experiments in which 10000 particles were released over a flat domain. These empirical RTDs are represented by different distributions over the range of d b *     and u b *   . A Fréchet RTD is the best fit for deep beds ( d b *   >3.2) and negligible underglow ( u b * <0.1). A LogNormal RTD is often the best representation for u b * ≤ 0 . 8 , while a Gamma RTD performs better for larger values of u b * . In general, a LogNormal RTD provides a good representation of the empirical RTDs in all cases, as it is identified as either the best or the second-best fitting distribution according to the Anderson-Darling test. The parameters of these analytical distributions vary with d b *     and u b * , and this dependence is graphically represented in this work. These results contribute to our understanding of the physical and mixing processes underpinning hyporheic exchange in streams and paves the way for a quick evaluation of its potential impact on nutrient and contaminant processing (e.g., based on the magnitude of the Damköhler number).

The hyporheic exchange below dune-shaped bedforms has a great impact on the stream environment. One of the most important properties of the hyporheic zone is the residence time distribution (RTD) of flow paths in the sediment domain. Here we evaluate the influence of an impervious layer, at a dimensionless sediment depth of \(d_{b}^{*}=\frac{2\pi d_{b}}{\lambda}\) where \(\lambda\) is the dune wavelength, on the form of the hyporheic exchange RTD. Empirical RTDs were generated, over a range of \(d_{b}^{*\ }\ \)values, from numerical particle tracking experiments in which \(10000\) particles sinusoidally distributed over a flatbed domain were released. These empirical RTDs are best represented by the Gamma, Log-Normal and Fréchet distributions over normalized bed depth of \({0\ <=d}_{b}^{*\ }\leq 1.2\),\({1.2<d}_{b}^{*\ }\leq 3.1\), and \(d_{b}^{*\ }>3.1\), respectively. The depth dependence of the analytical distribution parameters is also presented, together with a set of regression formulae to predict these parameters based on \(d_{b}^{*\ }\)with a high degree of accuracy (\(R^{2}>99.8\%\)). These results contribute to our understanding of the physical and mixing processes underpinning hyporheic exchange in streams and allow for a quick evaluation of its likely impact on nutrient and contaminant processing (e.g., based on the magnitude of the Damköhler number).Keywords: Dunes, bedforms, residence times distribution, sediment depth effect, Hyporheic residence times, analytical representation, two parametric distributions, Damköhler Number.