A finite element solution for the fractional advection–dispersion equation

The fractional advection–dispersion equation (FADE) known as its non-local dispersion, has been proven to be a promising tool to simulate anomalous solute transport in groundwater. We present an unconditionally stable finite element (FEM) approach to solve the one-dimensional FADE based on the Caputo definition of the fractional derivative with considering its singularity at the boundaries. The stability and accuracy of the FEM solution is verified against the analytical solution, and the sensitivity of the FEM solution to the fractional order α and the skewness parameter β is analyzed. We find that the proposed numerical approach converge to the numerical solution of the advection–dispersion equation (ADE) as the fractional order α equals 2. The problem caused by using the first- or third-kind boundary with an integral-order derivative at the inlet is remedied by using the third-kind boundary with a fractional-order derivative there. The problems for concentration estimation at boundaries caused by the singularity of the fractional derivative can be solved by using the concept of transition probability conservation. The FEM solution of this study has smaller numerical dispersion than that of the FD solution by Meerschaert and Tadjeran (J Comput Appl Math 2004). For a given α, the spatial distribution of concentration exhibits a symmetric non-Fickian behavior when β = 0. The spatial distribution of concentration shows a Fickian behavior on the left-hand side of the spatial domain and a notable non-Fickian behavior on the right-hand side of the spatial domain when β = 1, whereas when β = −1 the spatial distribution of concentration is the opposite of that of β = 1. Finally, the numerical approach is applied to simulate the atrazine transport in a saturated soil column and the results indicat that the FEM solution of the FADE could better simulate the atrazine transport process than that of the ADE, especially at the tail of the breakthrough curves.     

Predictive equation for longitudinal dispersion coefficient

Dye tracing field data were collected in small, steep streams in Ontario and used to calculate longitudinal dispersion coefficients for these headwater streams. A predictive equation for longitudinal dispersion coefficient is developed using combined data sets from five steeper head – water streams and 24 milder and larger rivers. The predictive equation relates the longitudinal dispersion coefficient to hydraulic and geometric parameters of the stream and has been developed using multiple regression analysis. The newly developed equation shows impressive accuracy of predictions for longitudinal dispersion coefficient (R² = 0.86, RMSE = 25, Nash–Sutcliffe coefficient E_ns = 0.86 and Index of Agreement D = 0.96) for both small, steep headwater streams as well as large, mild rivers. The Froude number has been introduced as a third key parameter to capture the effect of slope of the reach – in addition to the aspect ratio and bed material surface roughness – on the longitudinal dispersion coefficient. The pronounced improvement in the accuracy of the prediction is due to the addition of the Froude number to capture the effect of the slope of the reach on longitudinal dispersion coefficient. Copyright © 2014 John Wiley & Sons, Ltd.     

The impact of boundary on the fractional advection–dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives

The inherent heterogeneity of geological media often results in anomalous dispersion for solute transport through them, and how to model it has been an interest over the past few decades. One promising approach that has been increasingly used to simulate the anomalous transport in surface and subsurface water is the fractional advection–dispersion equation (FADE), derived as a special case of the more general continuous time random walk or the stochastic continuum model. In FADE, the dispersion is not local and the solutes have appreciable probability to move long distances, and thus reach the boundary faster than predicted by the classical advection–dispersion equation (ADE). How to deal with different boundaries associated with FADE and their consequent impact is an issue that has not been thoroughly explored. In this paper we address this by taking one-dimensional solute movement in soil columns as an example. We show that the commonly used FADE with its fractional derivatives defined by the Riemann–Liouville definition is problematic and could result in unphysical results for solute transport in bounded domains; a modified method with the fractional dispersive flux defined by the Caputo derivatives is presented to overcome this problem. A finite volume approach is given to numerically solve the modified FADE and its associated boundaries. With the numerical model, we analyse the inlet-boundary treatment in displacement experiments in soil columns, and find that, as in ADE, treating the inlet as a prescribed concentration boundary gives rise to mass-balance errors and such errors could be more significant in FADE because of its non-local dispersion. We also discuss a less-documented but important issue in hydrology: how to treat the upstream boundary in analysing the lateral movement of tracer in an aquifer when the tracer is injected as a pulse. It is shown that the use of an infinite domain, as commonly assumed in literature, leads to unphysical backward dispersion, which has a significant impact on data interpretation. To avoid this, the upstream boundary should be flux-prescribed and located at the upstream edge of the injecting point. We apply the model to simulate the movement of Cl⁻ in a tracer experiment conducted in a saturated hillslope, and analyse in details the significance of upstream-boundary treatments in parameter estimation.     

Roughness scale dependence of the relationship between tracer longitudinal dispersion and Peclet number in variable‐aperture fractures

The relationship between the longitudinal dispersion (D_L) and Peclet number (Pe) is crucial for predicting and simulating tracer through the variable‐aperture fracture. In this study, the roughness of the self‐affine fracture wall was decomposed into primary roughness (relatively large‐scale waviness) and secondary roughness (relatively small‐scale waviness) by a multiscaled wavelet analysis technique. Based on the complete dispersion mechanism (diffusion, macrodispersion, and Taylor dispersion) in the variable‐aperture fracture, three relationships (second‐order, power‐law, and linear relationships) between the D_L and Pe were investigated at large and small scales, respectively. Our results showed that the primary roughness mostly controlled the Taylor dispersion mechanism, whereas the secondary roughness was a dominant factor for the macrodispersion mechanism. Increasing the Hurst exponent and removing the secondary roughness led to the decreasing range of Pe where macrodispersion mechanism dominated the solute transport. It was found that estimating the D_L from the power‐law relationship based on Taylor dispersion theory resulted in considerable errors, even in the range of Pe where the Taylor dispersion mechanism dominated. The exponent of the power‐law relationship increased as the secondary roughness was removed. Analysing the linear relationship between the D_L and Pe revealed that the longitudinal dispersivity α_L increased linearly. However, this linear increase became weak as the Taylor dispersion mechanism dominated. In the range of Pe where the macrodispersion mechanism dominated, increasing the Hurst exponent caused the increase of α_L and the secondary roughness played a significant role in enhancing the α_L. As the Taylor dispersion mechanism dominated, the α_L was insensitive to the influence of multiscale roughness in variable‐aperture fractures.     

Estimating longitudinal dispersion in rivers using Acoustic Doppler Current Profilers

The longitudinal dispersion coefficient (D) is an important parameter needed to describe the transport of solutes in rivers and streams. The dispersion coefficient is generally estimated from tracer studies but the method can be expensive and time consuming, especially for large rivers. A number of empirical relations are available to estimate the dispersion coefficient; however, these relations are known to produce estimates within an order of magnitude of the tracer value. The focus of this paper is on using the shear-flow dispersion theory to directly estimate the dispersion coefficient from velocity measurements obtained using an Acoustic Doppler Current Profiler (ADCP). Using tracer and hydrodynamic data collected within the same river reaches, we examined conditions under which the ADCP and tracer methods produced similar results. Since dead zones / transient storage (TS) are known to influence the dispersion coefficient, we assessed the relative importance of dead zones in different stream reaches using two tracer-based approaches: (1) TS modeling which explicitly accounts for dead zones and (2) the advection–dispersion equation (ADE) which does not have separate terms for dead zones. Dispersion coefficients based on the ADE tend to be relatively high as they describe some of the effects of dead zones as well. Results based on the ADCP method were found to be in good agreement with the ADE estimates indicating that storage zones play an important role in the estimated dispersion coefficients, especially at high flows. For the river sites examined in this paper, the tracer estimates of dispersion were close to the median values of the ADCP estimates obtained from multiple datasets within a reach. The ADCP method appears to be an excellent alternative to the traditional tracer-based method if care is taken to avoid spurious data and multiple datasets are used to compute a distance-weighted average or other appropriate measure that represents reach-averaged conditions.     

Analysis of tracer tomography using temporal moments of tracer breakthrough curves

Hydraulic/partitioning tracer tomography (HPTT) was recently developed by Yeh and Zhu [Yeh T-CJ, Zhu J. Hydraulic/partitioning tracer tomography for characterization of dense nonaqueous phase liquid source zones, Water Resour Res 2007;43:W06435. doi:10.1029/2006WR004877.] for estimating spatial distribution of dense nonaqueous phase liquids (DNAPLs) in the subsurface. Since discrete tracer concentration data are directly utilized for the estimation of DNAPLs, this approach solves the hyperbolic convection–dispersion equation. Solution to the convection–dispersion equation however demands fine temporal and spatial discretization, resulting in high computational cost for an HPTT analysis. In this work, we use temporal moments of tracer breakthrough curves instead of discrete concentration data to estimate DNAPL distribution. This approach solves time independent partial differential equations of the temporal moments, and therefore avoids solving the convection–dispersion equation using a time marching scheme, resulting in a dramatic reduction of computational cost. To reduce numerical oscillations associated with convection dominated transport problems such as in inter-well tracer tests, the approach uses a finite element solver adopting the streamline upwind Petrov–Galerkin method to calculate moments and sensitivities. We test the temporal moment approach through numerical simulations. Comparing the computational costs between utilizing moments and discrete concentrations, we find that temporal moments significantly reduce the computation time. We also find that tracer moment data collected through a tomographic survey alone are able to yield reasonable estimates of hydraulic conductivity, as indicated by a correlation of 0.588 between estimated and true hydraulic conductivity fields in the synthetic case study.     

Local Equilibrium and Retardation Revisited

In modeling solute transport with mobile‐immobile mass transfer (MIMT), it is common to use an advection‐dispersion equation (ADE) with a retardation factor, or retarded ADE. This is commonly referred to as making the local equilibrium assumption (LEA). Assuming local equilibrium, Eulerian textbook treatments derive the retarded ADE, ostensibly exactly. However, other authors have presented rigorous mathematical derivations of the dispersive effect of MIMT, applicable even in the case of arbitrarily fast mass transfer. We resolve the apparent contradiction between these seemingly exact derivations by adopting a Lagrangian point of view. We show that local equilibrium constrains the expected time immobile, whereas the retarded ADE actually embeds a stronger, nonphysical, constraint: that all particles spend the same amount of every time increment immobile. Eulerian derivations of the retarded ADE thus silently commit the gambler's fallacy, leading them to ignore dispersion due to mass transfer that is correctly modeled by other approaches. We then present a particle tracking simulation illustrating how poor an approximation the retarded ADE may be, even when mobile and immobile plumes are continually near local equilibrium. We note that classic “LEA” (actually, retarded ADE validity) criteria test for insignificance of MIMT‐driven dispersion relative to hydrodynamic dispersion, rather than for local equilibrium.     

Comparison of theory and experiments for dispersion in homogeneous porous media

Modeling dispersion in homogeneous porous media with the convection–dispersion equation commonly requires computing effective transport coefficients. In this work, we investigate longitudinal and transverse dispersion coefficients arising from the method of volume averaging, for a variety of periodic, homogeneous porous media over a range of particle Péclet (Pe_p) numbers. Our objective is to validate the upscaled transverse dispersion coefficients and concentration profiles by comparison to experimental data reported in the literature, and to compare the upscaling approach to the more common approach of inverse modeling, which relies on fitting the dispersion coefficients to measured data. This work is unique in that the exact microscale geometry is available; thus, no simplifying assumptions regarding the geometry are required to predict the effective dispersion coefficients directly from theory. Transport of both an inert tracer and non-chemotactic bacteria is investigated for an experimental system that was designed to promote transverse dispersion. We highlight the occurrence of transverse dispersion coefficients that (1) depart from power-law behavior at relatively low Pe_p values and (2) are greater than their longitudinal counterparts for a specific range of Pe_p values. The upscaling theory provides values for the transverse dispersion coefficient that are within the 98% confidence interval of the values obtained from inverse modeling. The mean absolute error between experimental and upscaled concentration profiles was very similar to that between the experiments and inverse modeling. In all cases the mean absolute error did not exceed 12%. Overall, this work suggests that volume averaging can potentially be used as an alternative to inverse modeling for dispersion in homogeneous porous media.     

Input data measurement-induced uncertainty in watershed modelling

Abstract

This study presents a new methodology for estimation of input data measurement-induced uncertainty in simulated dissolved oxygen (DO) and nitrate-nitrogen (NO₃-N) concentrations using the Hydrological Simulation Program–FORTRAN (HSPF) model and data from the Amite River, USA. Simulation results show that: (1) a multiplying factor of 1.3 can be used to describe the maximum error in temperature measurements; similarly, a multiplying factor of 1.9 was estimated to accommodate the maximum of ±5% error in rainfall measurements; (2) the uncertainty in simulated DO concentration due to positive temperature measurement errors can be described with a normal distribution, N(0.062, 0.567); (3) the uncertainty in simulated NO₃-N concentration due to rainfall measurement errors follows a generalized extreme value distribution; and (4) the probability density functions can be utilized to determine the measurement-induced uncertainty in simulated DO and NO₃-N concentrations according to the risk level acceptable in water quality management.

Editor D. Koutsoyiannis

Citation Patil, A. and Deng, Z.-Q., 2012. Input data measurement-induced uncertainty in watershed modelling. Hydrological Sciences Journal, 57 (1), 118–133.     

Investigation of flow and transport processes at the MADE site using ensemble Kalman filter

In this work the ensemble Kalman filter (EnKF) is applied to investigate the flow and transport processes at the macro-dispersion experiment (MADE) site in Columbus, MS. The EnKF is a sequential data assimilation approach that adjusts the unknown model parameter values based on the observed data with time. The classic advection–dispersion (AD) and the dual-domain mass transfer (DDMT) models are employed to analyze the tritium plume during the second MADE tracer experiment. The hydraulic conductivity (K), longitudinal dispersivity in the AD model, and mass transfer rate coefficient and mobile porosity ratio in the DDMT model, are estimated in this investigation. Because of its sequential feature, the EnKF allows for the temporal scaling of transport parameters during the tritium concentration analysis. Inverse simulation results indicate that for the AD model to reproduce the extensive spatial spreading of the tritium observed in the field, the K in the downgradient area needs to be increased significantly. The estimated K in the AD model becomes an order of magnitude higher than the in situ flowmeter measurements over a large portion of media. On the other hand, the DDMT model gives an estimation of K that is much more comparable with the flowmeter values. In addition, the simulated concentrations by the DDMT model show a better agreement with the observed values. The root mean square (RMS) between the observed and simulated tritium plumes is 0.77 for the AD model and 0.45 for the DDMT model at 328 days. Unlike the AD model, which gives inconsistent K estimates at different times, the DDMT model is able to invert the K values that consistently reproduce the observed tritium concentrations through all times.     

Effect of bed surface roughness on longitudinal dispersion in artificial open channels

In several empirical and modelling studies on river hydraulics, dispersion was negatively correlated to surface roughness. In this study, it was aimed to investigate the influence of surface roughness on longitudinal dispersion under controlled conditions. In artificial flow channels with a length of 104 m, tracer experiments with variations in channel bed material were performed. By use of measured tracer breakthrough curves, average flow velocity, mean longitudinal dispersion, and mean longitudinal dispersivity were calculated. Longitudinal dispersion coefficients ranged from 0·018 m² s^?1 in channels with smooth bed surface up to 0·209 m² s^?1 in channels with coarse gravel as bed material. Longitudinal dispersion was linearly related to mean flow velocity. Accordingly, longitudinal dispersivities ranged between 0·152 ± 0·017 m in channels with smooth bed surface and 0·584 ± 0·015 m in identical channels with a coarse gravel substrate. Grain size and surface roughness of the channel bed were found to correlate positively to longitudinal dispersion. This finding contradicts several existing relations between surface roughness and dispersion. Future studies should include further variation in surface roughness to derive a better‐founded empirical equation forecasting longitudinal dispersion from surface roughness. Copyright © 2011 John Wiley & Sons, Ltd.     

Point-source inertial particle dispersion

The dispersion of inertial particles continuously emitted from a point source is analytically investigated in the limit of small but finite inertia. Our focus is on the evolution equation of the particle joint probability density function p(x,?v,?t), x and v being the particle position and velocity, respectively. For arbitrary inertia, position and velocity variables are coupled, with the result that p(x,?v,?t) can be determined by solving a partial differential equation in a 2d-dimensional space, d being the physical-space dimensionality. For small (but nevertheless finite) inertia, (x,?v)-variables decouple and the determination of p(x,?v,?t) is reduced to solve a system of two standard forced advection–diffusion equations in the space variables x. The latter equations are derived here from first principles, i.e., from the well-known Lagrangian evolution equations for position and particle velocity.     

Modeling contaminant transport in homogeneous porous media with fractional advection-dispersion equation

The newly developed Fractional Advection-Dispersion Equation (FADE), which is FADE was extended and used in this paper for modelling adsorbing contaminant transport by adding an adsorbing term. A parameter estimation method and its corresponding FORTRAN based program named FADEMain were developed on the basis of Nonlinear Least Square Algorithm and the analytical solution for one-dimensional FADE under the conditions of step input and steady state flow. Data sets of adsorbing contaminants Cd and NH4+-N transport in short homogeneous soil columns and conservative solute NaCI transport in a long homogeneous soil column, respectively were used to estimate the transport parameters both by FADEMain and the advection-dispersion equation (ADE) based program CXTFIT2.1. Results indicated that the concentration simulated by FADE agreed well with the measured data. Compared to the ADE model, FADE can provide better simulation for the concentration in the initial lower concentration part and the late higher concentration part of the breakthrough curves for both adsorbing contaminants. The dispersion coefficients for ADE were from 0.13 to 7.06 cm2/min, while the dispersion coefficients for FADE ranged from 0.119 to 3.05 cm1.856/min for NaCI transport in the long homogeneous soil column. We found that the dispersion coefficient of FADE increased with the transport distance, and the relationship between them can be quantified with an exponential function. Less scale-dependent was also found for the dispersion coefficient of FADE with respect to ADE.     

Impact of historic land‐use change on sediment delivery to a Chesapeake Bay subestuarine delta

Historic land use in the Chesapeake Bay drainage basin induced large fluxes of fluvial sediment to subestuarine tributaries. Stratigraphic and palaeoecologic analyses of deltaic deposits may be used to infer changes on the landscape, but are not sufficient to quantify past sediment supply. When viewed as an inverse boundary‐value problem, reconstruction of the sediment supply function may be achieved by combining deltaic sedimentation chronologies with an equation governing delta progradation. We propose that the diffusion equation is appropriate for simulating delta progradation and obtaining the sediment supply function provided a suitable diffusion constant (D) can be determined. Three new methods for estimating D are presented for the case of estuarine deltas. When the inverse boundary‐value technique was applied to Otter Point Creek, a tidal freshwater delta at the head of Bush River in upper Chesapeake Bay, D values ranged from 3763 to 6199 m² a^?1. Delta growth simulations showed a 1740–1760 initial pulse, a 1760–1780 erosive/redistributive interval, a 1780–1920 growth period, and a 1920‐present erosive/redistributive era. Coupling of simulated delta elevations with an empirical plant habitat predictive equation allowed for comparison of predicted versus actual relative habitat areas. Also, the model yielded reconstructed watershed erosion rates and stream suspended sediment concentrations that could be useful for development of water quality regulations. Copyright © 2001 John Wiley & Sons, Ltd.     

Transect study on solute transport in a macroporous soil

Solute transport experiments using a non-reactive tracer were conducted on short, undisturbed, saturated columns of a sandy loam soil. All columns, 20 cm in diameter and 20 cm long, were collected along a transect of 35 m. Most of the soil columns had pre-existing macropores. The columns were leached at a steady flow-rate under ponding conditions. The resulting breakthrough curves (BTCs) showed a large heterogeneity. Several of the BTCs displayed early breakthrough and long tailing. All the data were interpreted in terms of dimensional time moments, the classical convection-dispersion equation (CDE) and the mobile-immobile transport model (MIM). Experimental time moments were found to vary significantly among the different BTCs. Analysis of the time moments also revealed that the variance of the field-scale BTC was several times larger than the average of the local-scale variance. The pore water velocity v and dispersion coefficient D were obtained by fitting the CDE to the local-scale BTCs, resulting in an average dispersivity of 7·4 cm. Frequency distributions for the CDE parameters v and D were equally well described by a normal or log-normal probability density function (pdf). When a log-normal pdf for D is considered, the variance of the log_e transformed D values (σ_{ln D}²) was found to be 2·1. For the MIM model, two additional parameters were fitted: the fraction of mobile water, θ_m/θ, and the first-order mass transfer coefficient, α. The MIM was more successful in describing the data than the CDE transport model. For the MIM model, the average dispersivity was about 2 cm. The MIM parameters v, D and θ_m/θ were best described by a log-normal pdf rather than a normal pdf. Only the parameter α was better described by a normal pdf. Mobile water fractions, θ_m/θ ranged from 0·01 to 0·98, with a mean of 0·43 (based on a log-normal pdf). When the CDE and MIM were applied to the data, the fitted pore water velocities, v, compared favourably with the effective pore water velocities, v_eff, obtained from moment analysis.     

Evaluation of temporal concentration profiles for ungauged rivers following pollution incidents

Abstract

In order to predict the impact of pollution incidents on rivers, it is necessary to predict the dispersion coefficient and the flow velocity corresponding to the discharge in the river of interest. This paper explores methods for doing this, particularly with a view to applications on ungauged rivers, i.e. those for which little hydraulic or morphometric data are available. An approach based on neural networks, trained on a wide-ranging database of optimized parameter values from tracer experiments and corresponding physical variables assembled for American and European rivers, is proposed. Tests using independent cases showed that the neural networks generally gave more reliable parameter estimates than a second-order polynomial regression approach. The quality of predictions of temporal concentration profiles was heavily influenced by the accuracy of the velocity prediction.

Citation Piotrowski, A. P., Napiorkowski, J. J., Rowinski, P. M. & Wallis, S. G. (2011) Evaluation of temporal concentration profiles for ungauged rivers following pollution incidents. Hydrol. Sci. J. 56(5), 883–894.     

Soil detachment by overland flow on steep cropland in the subtropical region of China

Accurate prediction of soil detachment capacity is fundamental to establish process-based erosion models and improve soil loss assessment. Few studies were conducted to reveal the mechanism of detachment process for yellow soil on steep cropland in the subtropical region of China using field experiments. This study was performed to determine soil detachment characteristics and explore the relationships between soil detachment capacity (D _c) and flow rate, slope gradient, mean velocity, shear stress, stream power and unit stream power. Field experiments were conducted on intact soil with flow rates ranging from 0.2 × 10⁻³ to 0.5 × 10⁻³ m⁻³ s⁻¹ and slope gradients varying from 8.8 to 42.4%. The results showed the following. (a) D _c of yellow soil was smaller than other soils because of its high clay content. (b) D _c was more susceptible to flow than to slope gradient. Power functions were derived to depict the relationship between D _c and the flow rate and slope gradient (R² = 0.91). (c) D _c was better simulated by power functions of the stream power (R² = 0.83) than functions of the shear stress or the unit stream power. (d) Considering its accuracy, simplicity and accessibility, the power function based on flow rate and slope gradient is recommended to predict D _c of yellow soil in the field. The results of this study provide useful support for revealing soil detachment mechanism and developing process-based soil erosion models for the subtropical region of China.     

Dispersions for the pushover‐based risk assessment of reinforced concrete frames and cantilever walls

The paper presents the results of an investigation into the dispersion values, expressed in terms of limit‐state spectral accelerations, which could be used for the pushover‐based risk assessment of low‐height to mid‐height reinforced concrete frames and cantilever walls. The results of an extensive parametric study of a portfolio of test structures indicated that the dispersion values due to record‐to‐record variability and modelling uncertainty (β_LS,RU) are within the range from 0.3 to 0.55 for the near collapse limit state, and between 0.35 and 0.60 for the collapse limit state. The dispersions β_LS,RU proposed for the code‐conforming and the majority of old (non code‐conforming) frames are in between these values. On the other hand, the dispersions proposed for the old frames with a soft storey and an invariant plastic mechanism, and for the code‐conforming cantilever walls, are at the lower and upper bounds of the presented values, respectively. The structural parameters that influence these dispersions were identified, and the influence of different ground motion sets, and of the models used for the calculation of the rotation capacities of the columns, on the calculated fragility parameters was examined and quantified. The proposed dispersion values were employed in a practice‐oriented pushover‐based method for the estimation of failure probability for eight selected examples. The pushover‐based risk assessment method, although extremely simple and economical when compared with more rigorous probabilistic methods, was able to predict seismic risk with reasonable accuracy, thus showing it to be a practical tool for engineers. Copyright © 2016 John Wiley & Sons, Ltd.     

Hydraulic characteristics of a small Coastal Plain stream of the southeastern United States: effects of hydrology and season

Various physical and biological properties affect solute transport patterns in streams. We measured hydraulic characteristics of Payne Creek, a low‐gradient upper Coastal Plain stream, using tracer experiments and parameter estimation with OTIS‐P (one‐dimensional transport with inflow and storage with parameter optimization). The primary objective of this study was to estimate the effects of varying discharge, season, and litter accumulation on hydraulic parameters. Channel area A ranged from 0·081 to 0·371 m² and transient storage area A_s ranged from 0·027 to 0·111 m². Dispersion D ranged from 1·5 to 11·1 m² min⁻¹ and exchange coefficient α ranged from 0·009 to 0·038 min⁻¹. Channel area and dispersion were positively correlated to discharge Q, whereas storage area and exchange coefficient were not. Relative storage size A_s/A ranged from 0·17 to 0·59, and was higher during fall than other seasons under a similar Q. The fraction of median travel time due to transient storage ranged from 8·8 to 34·5% and was significantly correlated with Q through a negative power function. Both metrics indicated that transient storage was a significant component affecting solute transport in Payne Creek, especially during the fall. Comparison between the measured channel area A_c and A suggested that surface storage was dominant in Payne Creek. During fall, accumulation of leaf litter resulted in larger A and A_s and lower velocity and D than during other seasons with similar discharge. Seasonal changes in discharge and organic matter accumulation, and dynamic channel morphology affected the magnitude of transient storage and overall hydraulic characteristics of Payne Creek. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite analytic method for solving the 2-D time-dependent advection–diffusion equation with time-invariant coefficients

Difficulty in solving the transient advection–diffusion equation (ADE) stems from the relationship between the advection derivatives and the time derivative. For a solution method to be viable, it must account for this relationship by being accurate in both space and time. This research presents a unique method for solving the time-dependent ADE that does not discretize the derivative terms but rather solves the equation analytically in the space–time domain. The method is computationally efficient and numerically accurate and addresses the common limitations of numerical dispersion and spurious oscillations that can be prevalent in other solution methods. The method is based on the improved finite analytic (IFA) solution method [Lowry TS, Li S-G. A characteristic based finite analytic method for solving the two-dimensional steady-state advection–diffusion equation. Water Resour Res 38 (7), 10.1029/2001WR000518] in space coupled with a Laplace transformation in time. In this way, the method has no Courant condition and maintains accuracy in space and time, performing well even at high Peclet numbers. The method is compared to a hybrid method of characteristics, a random walk particle tracking method, and an Eulerian–Lagrangian Localized Adjoint Method using various degrees of flow-field heterogeneity across multiple Peclet numbers. Results show the IFALT method to be computationally more efficient while producing similar or better accuracy than the other methods.