Sensitivity of Simulated Hyporheic Exchange to River Bathymetry: The Steinlach River Test Site

This study determines the aspects of river bathymetry that have the greatest influence on the predictive biases when simulating hyporheic exchange. To investigate this, we build a highly parameterized HydroGeoSphere model of the Steinlach River Test Site in southwest Germany as a reference. This model is then modified with simpler bathymetries, evaluating the changes to hyporheic exchange fluxes and transit time distributions. Results indicate that simulating hyporheic exchange with a high-resolution detailed bathymetry using a three-dimensional fully coupled model leads to nested multiscale hyporheic exchange systems. A poorly resolved bathymetry will underestimate the small-scale hyporheic exchange, biasing the simulated hyporheic exchange towards larger scales, thus leading to overestimates of hyporheic exchange residence times. This can lead to gross biases in the estimation of a catchment's capacity to attenuate pollutants when extrapolated to account for all meanders along an entire river within a watershed. The detailed river slope alone is not enough to accurately simulate the locations and magnitudes of losing and gaining river reaches. Thus, local bedforms in terms of bathymetric highs and lows within the river are required. Bathymetry surveying campaigns can be more effective by prioritizing bathymetry measurements along the thalweg and gegenweg of a meandering channel. We define the gegenweg as the line that connects the shallowest points in successive cross-sections along a river opposite to the thalweg under average flow conditions. Incorporating local bedforms will likely capture the nested nature of hyporheic exchange, leading to more physically meaningful simulations of hyporheic exchange fluxes and transit times.     

Modeling surface and ground water mixing in the hyporheic zone using MODFLOW and MT3D

We used a three-dimensional MODFLOW model, paired with MT3D, to simulate hyporheic zones around debris dams and meanders along a semi-arid stream. MT3D simulates both advective transport and sink/source mixing of solutes, in contrast to particle tracking (e.g. MODPATH), which only considers advection. We delineated the hydrochemically active hyporheic zone based on a new definition, specifically as near-stream subsurface zones receiving a minimum of 10% surface water within a 10-day travel time. Modeling results indicate that movement of surface water into the hyporheic zone is predominantly an advective process. We show that debris dams are a key driver of surface water into the subsurface along the experimental reach, causing the largest flux rates of water across the streambed and creating hyporheic zones with up to twice the cross-sectional area of other hyporheic zones. Hyporheic exchange was also found in highly sinuous segments of the experimental reach, but flux rates are lower and the cross-sectional areas of these zones are generally smaller. Our modeling approach simulated surface and ground water mixing in the hyporheic zone, and thus provides numerical approximations that are more comparable to field-based observations of surface–groundwater exchange than standard particle-tracking simulations.     

Imaging hyporheic zone solute transport using electrical resistivity

Traditional characterization of hyporheic processes relies upon modelling observed in‐stream and subsurface breakthrough curves to estimate hyporheic zone size and infer exchange rates. Solute data integrate upstream behaviour and lack spatial coverage, limiting our ability to accurately quantify spatially heterogeneous exchange dynamics. Here, we demonstrate the application of near‐surface electrical resistivity imaging (ERI) methods, coupled with experiments using an electrically conductive stream tracer (dissolved NaCl), to provide in situ imaging of spatial and temporal dynamics of hyporheic exchange. Tracer‐labelled water in the stream enters the hyporheic zone, reducing electrical resistivity in the subsurface (to which subsurface ERI is sensitive). Comparison of background measurements with those recording tracer presence provides distributed characterization of hyporheic area (in this application, ∼0·5 m²). Results demonstrate the first application of ERI for two‐dimensional imaging of stream‐aquifer exchange and hyporheic extent. Future application of this technique will greatly enhance our ability to quantify processes controlling solute transport and fate in hyporheic zones, and provide data necessary to inform more complete numerical models. Copyright © 2010 John Wiley & Sons, Ltd.     

The effects of discharge and slope on hyporheic flow in step‐pool morphologies

This paper focuses on surface–subsurface water exchange in a steep coarse‐bedded stream with a step‐pool morphology. We use both flume experiments and numerical modelling to investigate the influence of stream discharge, channel slope and sediment hydraulic conductivity on hyporheic exchange. The model step‐pool reach, whose topography is scaled from a natural river, consists of three step‐pool units with 0.1‐m step heights, discharges ranging between base and over‐bankfull flows (scaled values of 0.3–4.5 l/s) and slopes of 4% and 8%. Results indicate that the deepest hyporheic flow occurs with the steeper slope and at moderate discharges and that downwelling fluxes at the base of steps are highest at the largest stream discharges. In contrast to findings in a pool‐riffle morphology, those in this study show that steep slopes cause deeper surface–subsurface exchanges than gentle slopes. Numerical simulation results show that the portion of the hyporheic zone influenced by surface water temperature increases with sediment hydraulic conductivity. These experiments and numerical simulations emphasize the importance of topography, sediment permeability and roughness elements along the channel surface in governing the locations and magnitude of downwelling fluxes and hyporheic exchange. Our results show that hyporheic zones in these steep streams are thicker than previously expected by extending the results from streams with pool‐riffle bed forms. Copyright © 2014 John Wiley & Sons, Ltd.     

Bedform-induced hyporheic exchange with unsteady flows

The interaction between surface and subsurface water has a crucial influence on the biochemistry of stream environments. Even though the river discharge and the flow conditions can seldom be considered to be steady, the influence of this unsteadiness on the hyporheic exchange has often been neglected. In this work, a model for the study of hyporheic exchange during unsteady conditions has been developed. The model provides a sound analytical framework for the analysis of the effects of a varying stream discharge on the exchange between a stream and the hyporheic zone. The effects of the unsteadiness on the water exchange flux, the residence time of the solutes in the bed, and the stored mass are quantified. A synthetic example shows the substantial influence of a flood on the hyporheic exchange, and that the application of a steady model can lead to an underestimation of the exchanged mass, even after the flood has ended.     

Hyporheic exchange flows induced by constructed riffles and steps in lowland streams in southern Ontario,Canada

Stream–subsurface water interaction induced by natural riffles and constructed riffles/steps was examined in lowland streams in southern Ontario, Canada. The penetration of stream water into the subsurface was analysed using hydrometric data, and the zone of > 10% stream water was calculated from a chemical mixing equation using tracer injection of bromide and background chloride concentrations. The constructed riffles studied induced more extensive hyporheic exchange than the natural riffles because of their steeper longitudinal hydraulic head gradients and coarser streambed sediments. The depth of > 10% stream water zone in a small and a large constructed riffle extended to > 0·2 m and > 1·4 m depths respectively. Flux and residence time distribution of hyporheic exchange were simulated in constructed riffles using MODFLOW, a finite‐difference groundwater flow model. Hyporheic flux and residence time distribution varied along the riffles, and the exchange occurring upstream from the riffle crest was small in flux and had a long residence time. In contrast, hyporheic exchange occurring downstream from the riffle crest had a relatively short residence time and accounted for 83% and 70% of total hyporheic exchange flow in a small and large riffle respectively. Although stream restoration projects have not considered the hyporheic zone, our data indicate that constructed riffles and steps can promote vertical hydrologic exchange and increase the groundwater–surface water linkage in degraded lowland streams. Copyright © 2006 John Wiley & Sons, Ltd.     

High-Resolution Integrated Transport Model for Studying Surface Water–Groundwater Interaction

Transport processes that lead to exchange of mass between surface water and groundwater play a significant role for the ecological functioning of aquatic systems, for hydrological processes and for biogeochemical transformations. In this study, we present a novel integral modeling approach for flow and transport at the sediment–water interface. The model allows us to simultaneously simulate turbulent surface and subsurface flow and transport with the same conceptual approach. For this purpose, a conservative transport equation was implemented to an existing approach that uses an extended version of the Navier–Stokes equations. Based on previous flume studies which investigated the spreading of a dye tracer under neutral, losing and gaining flow conditions the new solver is validated. Tracer distributions of the experiments are in close agreement with the simulations. The simulated flow paths are significantly affected by in- and outflowing groundwater flow. The highest velocities within the sediment are found for losing condition, which leads to shorter residence times compared to neutral and gaining conditions. The largest extent of the hyporheic exchange flow is observed under neutral condition. The new solver can be used for further examinations of cases that are not suitable for the conventional coupled models, for example, if Reynolds numbers are larger than 10. Moreover, results gained with the integral solver provide high-resolution information on pressure and velocity distributions at the rippled streambed, which can be used to improve flow predictions. This includes the extent of hyporheic exchange under varying ambient groundwater flow conditions.     

Focused Groundwater Controlled Feedbacks into the Hyporheic Zone During Baseflow Recession

Groundwater surface water interaction in the hyporheic zone remains an important challenge for water resources management and ecosystem restoration. In heterogeneous stratified glacial sediments, reach‐scale environments contain an uneven distribution of focused groundwater flow occurring simultaneously with diffusely discharging groundwater. This results in a variation of stream‐aquifer interactions, where focused flow systems are able to temporally dominate exchange processes. The research presented here investigates the direct and indirect influences focused groundwater discharge exerts on the hyporheic zone during baseflow recession. Field results demonstrate that as diffuse sources of groundwater deplete during baseflow recession, focused groundwater discharge remains constant. During baseflow recession the hyporheic zone is unable to expand, while the high nitrate concentration from focused discharge changes the chemistry of the stream. The final result is a higher concentration of nitrate in the hyporheic zone as this altered surface water infiltrates into the subsurface. This indirect coupling of focused groundwater discharge and the hyporheic zone is unaccounted for in hyporheic studies at this time. Results indicate important implications for the potential reduction of agricultural degradation of water quality.     

Determining long time‐scale hyporheic zone flow paths in Antarctic streams

In the McMurdo Dry Valleys of Antarctica, glaciers are the source of meltwater during the austral summer, and the streams and adjacent hyporheic zones constitute the entire physical watershed; there are no hillslope processes in these systems. Hyporheic zones can extend several metres from each side of the stream, and are up to 70 cm deep, corresponding to a lateral cross‐section as large as 12 m², and water resides in the subsurface year around. In this study, we differentiate between the near‐stream hyporheic zone, which can be characterized with stream tracer experiments, and the extended hyporheic zone, which has a longer time‐scale of exchange. We sampled stream water from Green Creek and from the adjacent saturated alluvium for stable isotopes of D and ¹⁸O to assess the significance and extent of stream‐water exchange between the streams and extended hyporheic zones over long time‐scales (days to weeks). Our results show that water residing in the extended hyporheic zone is much more isotopically enriched (up to 11‰ D and 2·2‰ ¹⁸O) than stream water. This result suggests a long residence time within the extended hyporheic zone, during which fractionation has occurred owing to summer evaporation and winter sublimation of hyporheic water. We found less enriched water in the extended hyporheic zone later in the flow season, suggesting that stream water may be exchanged into and out of this zone, on the time‐scale of weeks to months. The transient storage model OTIS was used to characterize the exchange of stream water with the extended hyporheic zone. Model results yield exchange rates (α) generally an order magnitude lower (10^?5 s^?1) than those determined using stream‐tracer techniques on the same stream. In light of previous studies in these streams, these results suggest that the hyporheic zones in Antarctic streams have near‐stream zones of rapid stream‐water exchange, where ‘fast’ biogeochemical reactions may influence water chemistry, and extended hyporheic zones, in which slower biogeochemical reaction rates may affect stream‐water chemistry at longer time‐scales. Copyright © 2003 John Wiley & Sons, Ltd.     

How Does Subsurface Characterization Affect Simulations of Hyporheic Exchange?

We investigated the role of increasingly well‐constrained geologic structures in the subsurface (i.e., subsurface architecture) in predicting streambed flux and hyporheic residence time distribution (RTD) for a headwater stream. Five subsurface realizations with increasingly resolved lithological boundaries were simulated in which model geometries were based on increasing information about flow and transport using soil and geologic maps, surface observations, probing to depth to refusal, seismic refraction, electrical resistivity (ER) imaging of subsurface architecture, and time‐lapse ER imaging during a solute tracer study. Particle tracking was used to generate RTDs for each model run. We demonstrate how improved characterization of complex lithological boundaries and calibration of porosity and hydraulic conductivity affect model prediction of hyporheic flow and transport. Models using hydraulic conductivity calibrated using transient ER data yield estimates of streambed flux that are three orders of magnitude larger than uncalibrated models using estimated values for hydraulic conductivity based on values published for nearby hillslopes (10^?4 vs. 10^?7 m²/s, respectively). Median residence times for uncalibrated and calibrated models are 10³ and 10⁰ h, respectively. Increasingly well‐resolved subsurface architectures yield wider hyporheic RTDs, indicative of more complex hyporheic flowpath networks and potentially important to biogeochemical cycling. The use of ER imaging to monitor solute tracers informs subsurface structure not apparent from other techniques, and helps to define transport properties of the subsurface (i.e., hydraulic conductivity). Results of this study demonstrate the value of geophysical measurements to more realistically simulate flow and transport along hyporheic flowpaths.     

Impact of debris dams on hyporheic interaction along a semi‐arid stream

Hyporheic exchange increases the potential for solute retention in streams by slowing downstream transport and increasing solute contact with the substrate. Hyporheic exchange may be a major mechanism to remove nutrients in semi‐arid watersheds, where livestock have damaged stream riparian zones and contributed nutrients to stream channels. Debris dams, such as beaver dams and anthropogenic log dams, may increase hyporheic interactions by slowing stream water velocity, increasing flow complexity and diverting water to the subsurface. Here, we report the results of chloride tracer injection experiments done to evaluate hyporheic interaction along a 320 m reach of Red Canyon Creek, a second order stream in the semi‐arid Wind River Range of Wyoming. The study site is part of a rangeland watershed managed by The Nature Conservancy of Wyoming, and used as a hydrologic field site by the University of Missouri Branson Geologic Field Station. The creek reach we investigated has debris dams and tight meanders that hypothetically should enhance hyporheic interaction. Breakthrough curves of chloride measured during the field experiment were modelled with OTIS‐P, a one‐dimensional, surface‐water, solute‐transport model from which we extracted the storage exchange rate α and cross‐sectional area of the storage zone A_s for hyporheic exchange. Along gaining reaches of the stream reach, short‐term hyporheic interactions associated with debris dams were comparable to those associated with severe meanders. In contrast, along the non‐gaining reach, stream water was diverted to the subsurface by debris dams and captured by large‐scale near‐stream flow paths. Overall, hyporheic exchange rates along Red Canyon Creek during snowmelt recession equal or exceed exchange rates observed during baseflow at other streams. Copyright © 2005 John Wiley & Sons, Ltd.     

Residence time of bedform-driven hyporheic exchange

Biogeochemical and ecological transformations in hyporheic zones are dependent on the timing of hyporheic exchange. We show through linked modeling of open channel turbulent flow, groundwater flow, and solute transport that the residence time distributions of solutes advected by hyporheic flow induced by current–bedform interaction follow power-laws. This tailing behavior of solutes exiting the sediments is explained by the presence of multiple path lengths coupled with very large variability in Darcy flow velocity, both occurring without heterogeneity in sediment permeability. Hyporheic exchange through bedforms will result in short-time fractal scaling of stream water chemistry.     

Effect of morphology and discharge on hyporheic exchange flows in two small streams in the Cascade Mountains of Oregon,USA

Stream‐tracer injections were used to examine the effect of channel morphology and changing stream discharge on hyporheic exchange flows. Direct observations were made from well networks to follow tracer movement through the hyporheic zone. The reach‐integrated influence of hyporheic exchange was evaluated using the transient storage model (TSM) OTIS‐P. Transient storage modelling results were compared with direct observations to evaluate the reliability of the TSM. Results from the tracer injection in the bedrock reach supported the assumption that most transient storage in headwater mountain streams results from hyporheic exchange. Direct observations from the well networks in colluvial reaches showed that subsurface flow paths tended to parallel the valley axis. Cross‐valley gradients were weak except near steps, where vertical and cross‐valley hydraulic gradients indicated a strong potential for stream water to downwell into the hyporheic zone. The TSM parameters showed that both size and residence time of transient storage were greater in reaches with a few large log‐jam‐formed steps than in reaches with more frequent, but smaller steps. Direct observations showed that residence times in the unconstrained stream were longer than in the constrained stream and that little change occurred in the location and extent of the hyporheic zone between low‐ and high‐baseflow discharges in any of the colluvial reaches. The transient storage modelling results did not agree with these observations, suggesting that the TSM was insensitive to long residence‐time exchange flows and was very sensitive to changes in discharge. Disagreements between direct observations and the transient storage modelling results highlight fundamental problems with the TSM that confound comparisons between the transient storage modelling results for tracer injections conducted under differing flow conditions. Overall, the results showed that hyporheic exchange was little affected by stream discharge (at least over the range of baseflow discharges examined in this study). The results did show that channel morphology controlled development of the hyporheic zone in these steep mountain stream channels. Copyright © 2005 John Wiley & Sons, Ltd.     

Effect of streamflow stochasticity on bedform-driven hyporheic exchange

The interactions between the stream and the geomorphologic units that compose the stream channel result in an exchange of water, heat, and chemicals that is an important component of the flows of energy and nutrients in the river ecosystem. This exchange is characterized by complex spatial and temporal dynamics that depend on the characteristics of the stream flow and morphology. At present, many studies have addressed the development of spatial patterns of hyporheic exchange that are induced by many geomorphological factors at different scales. However, much less is known about the temporal evolution of the surface–subsurface exchange in response to the dynamics of the stream discharge. In order to investigate this problem, the present work analyzes the influence of streamflow variability on the hyporheic exchange induced by fluvial bedforms. A stochastic approach is employed to generate streamflow series whose statistical properties are representative of streams with different hydrological regimes. The resulting exchange fluxes and travel times are then computed, and the relationships between the streamflow regime and the dynamics of the exchange flux and travel times are investigated. The results show that the mean stream discharge can be used to estimate the average features of the temporal dynamics of hyporheic exchange. Moreover, exchange fluxes and residence times distributions exhibit significant fluctuations, which are tightly related to the coefficient of variation of the streamflow hydrograph.     

Using a fluctuating tracer to estimate hyporheic exchange in restored and unrestored reaches of the Truckee River,Nevada, USA

The goal of this research was to compare hyporheic activity in recently restored and adjacent un‐restored reaches of the Truckee River downstream from the Reno/Sparks metropolitan area. The installation of rocky riffles and raised channel bed elevations in the restored reaches may have increased the degree of surface–subsurface interaction. A fluctuating chloride concentration signal served as the tracer, induced by the variable influx of higher salinity water several miles upstream from the study reach. The solute transport model, OTIS, was used in conjunction with the hydrodynamic model, DYNHYD5, to estimate transient storage parameters under unsteady flow conditions. The model was calibrated to chloride concentrations measured over a period of three days at six in‐stream locations representing restored and un‐restored reaches. An automated parameter estimation algorithm (SCE‐UA) was used to optimize parameters for multiple reaches simultaneously and generate a distribution of parameter estimates. Results suggest that the transient storage zone cross‐sectional area (A_s) is larger in the restored reaches than in the unrestored reaches, but the exchange coefficient (α) is smaller, leading to increased hyporheic residence time and hydrologic retention in the vicinity of channel reconstructions. Scenarios were used to simulate the potential effects of increased subsurface residence time on denitrification and in‐stream NO₃‐N concentrations. Monte Carlo analysis was performed to assess uncertainty in the simulation results and show the potential for greater nutrient retention in the lower Truckee River as a result of channel restoration. Copyright © 2009 John Wiley & Sons, Ltd.     

Does streambed heterogeneity matter for hyporheic residence time distribution in sand-bedded streams?

Stream water residence times within streambed sediments are key values to quantify hyporheic processes including sediment thermal regime, solute transient storage, dilution rates and biogeochemical transformations, such as those controlling degassing nitrous oxide. Heterogeneity of the streambed sediment hydraulic properties has been shown to be potentially an important factor to characterize hyporheic processes. Here, we quantify the importance of streambed heterogeneity on residence times of dune-like bedform induced hyporheic fluxes at the bedform and reach scales. We show that heterogeneity has a net effect of compression of the hyporheic zone (HZ) toward the streambed, changing HZ volume from the homogenous case and thus inducing remarkable differences in the flow field with respect to the homogeneous case. We unravel the physical conditions for which the commonly used homogenous field assumption is applicable for quantifying hyporheic processes thus explaining why predictive measures based on a characteristic residence time, like the Damköhler number, are robust in heterogeneous sand bedded streams.     

Predictive uncertainty of a nearshore bed evolution model

Predictions of nearshore depth evolution using process-based numerical simulation models contain inherent uncertainties owing to model structural deficiencies, measurement errors, and parameter uncertainty. This paper quantifies the parameter-induced predictive uncertainty of the cross-shore depth evolution model Unibest-TC by applying the Bayesian Generalised Likelihood Uncertainty Estimation methodology to modelling depth evolution at Egmond aan Zee (Netherlands). This methodology works with multiple sets of parameter values sampled uniformly in feasible parameter space and assigns a likelihood value to each parameter set. Acceptable simulations (i.e., based on parameter sets with a nonzero likelihood) were found for a wide range of parameter values owing to parameter interdependence and insensitivity. The 95% uncertainty prediction interval of bed levels after the 33 days prediction period was largest (0.5–1 m) near the sandbar crests that characterize the Egmond depth profile, reducing to near-zero values in the sandbar troughs and the offshore area. The prediction interval built up during storms (when sediment transport rates are largest) and remained the same or even reduced slightly during less-energetic conditions. The prediction uncertainty ranges bracket the observations near the inner-bar crest, its seaward flank, and at the seaward flank of the outer bar, suggesting that elsewhere model structural errors (and, potentially, measurement errors) dominate over parameter errors. The interdependence and the non-Gaussian marginal posterior distribution functions of the free model parameters cast doubt on the ability of commonly applied multivariate normal distribution functions to estimate parameter uncertainty.     

Successional changes in soil and hyporheic nitrogen fertility on an alluvial flood plain: implications for riparian vegetation

In floodplain primary succession, vegetation colonizes nitrogen-poor alluvial deposits and fertility improves as soil nitrogen accumulates over time. It is generally assumed that vegetation assimilates the vast majority of its nitrogen from the soil; however, recent studies have suggested that the hyporheic zone also may be an important nitrogen source. We investigated the potential relative importance of hyporheic nitrogen by comparing fertility indices, specifically total (TN), dissolved inorganic (DIN), potentially mineralizable (PMN) and ion exchange resin nitrogen (IERN) in both soils and the hyporheic zone at early, mid and late succession stands on an expansive river flood plain. We also constructed mesocosms to assess growth of cottonwood cuttings with access to soil and/or hyporheic water. We found TN and PMN increased from early to mid succession in both the soil (to 10 cm) and hyporheic zone (in a 10 cm layer). While TN, DIN and PMN were an order of magnitude higher in the soil than in the hyporheic zone, IERN was higher in the hyporheic zone, indicating that subsurface flow through the flood plain may be important in delivering nitrogen to the root zone. However, even when flux was added to the hyporheic PMN pool, nitrogen availability in the hyporheic zone (in a 10 cm layer) was vastly lower than soil PMN (to 10 cm). Further, the instantaneous standing stock of DIN in the surface soil alone was about equal to the sum of the DIN pool, the mean subsurface flux and the PMN pool in a 10 cm layer of hyporheic zone. In the mesocosm experiment, cottonwood cuttings with access to both soil and hyporheic water grew fastest; however, they also had the lowest foliar nitrogen concentrations, indicating that this was not due to greater nitrogen availability. In the field, nitrogen content of cottonwood foliage increased along with soil (but potentially hyporheic as well) nitrogen accumulation during succession, suggesting the vegetation responded to increasing nitrogen fertility. We conclude that at least on a per unit-volume basis, the hyporheic zone probably provides little nitrogen relative to the surface soil, except on new alluvial bars that characteristically are nitrogen poor. Therefore, the hyporheic zone is probably a much smaller nitrogen source for mature forests relative to the surface soil unless the vegetation exploits a much larger volume of the hyporheic zone than surface soil.     

Integrating tracer experiments with modeling to assess runoff processes and water transit times

Representing runoff process complexity in a simple model structure remains a challenge in hydrology. We present an integrated approach to investigate runoff processes using a hillslope tracer experiment and modeling exercise to explore model parameterization, process representation, and transit times. A spatially-explicit model constrained by soil hydrologic properties, runoff, and applied tracer data was used to identify the dominant processes necessary to explain both water and solute flux from a steep hillslope. The tracer data allowed for the rejection of model parameter sets based on the calibration to runoff data alone, thus reducing model uncertainty. The additional calibration to tracer data, improved parameter identifiability and provided further insight to process controls on hillslope-scale water and solute flux. Transit time distributions developed using the model provided further insight to model structure such as subsurface volume, mixing assumptions, and the water table dynamics. Combining field experiments with the modeling exercise may lead to a more comprehensive assessment of runoff process representation in models.     

Interpretation and nonuniqueness of CTRW transition distributions: Insights from an alternative solute transport formulation

The continuous time random walk (CTRW) has both an elegant mathematical theory and a successful record at modeling solute transport in the subsurface. However, there are some interpretation ambiguities relating to the relationship between the discrete CTRW transition distributions and the underlying continuous movement of solute that have not been addressed in existing literature. These include the exact definition of “transition”, and the extent to which transition probability distributions are unique/quantifiable from data. Here, we present some theoretical results which address these uncertainties in systems with an advective bias. Simultaneously, we present an alternative, reduced parameter CTRW formulation for general advective transport in heterogeneous porous media, which models early- and late-time transport by use of random transition times between sparse, imaginary planes normal to flow. We show that even in the context of this reduced-parameter formulation there is nonuniqueness in the definitions of both transition lengths and waiting time distributions, and that neither may be uniquely determined from experimental data. For practical use of this formulation, we suggest Pareto transition time distributions, leading to a two-degree-of-freedom modeling approach. We then demonstrate the power of this approach in fitting two sets of existing experimental data. While the primary focus is the presentation of new results, the discussion is designed to be pedagogical and to provide a good entry point into practical modeling of solute transport with the CTRW.