Continuous time random walks for non-local radial solute transport

This study formulates and analyzes continuous time random walk (CTRW) models in radial flow geometries for the quantification of non-local solute transport induced by heterogeneous flow distributions and by mobile–immobile mass transfer processes. To this end we derive a general CTRW framework in radial coordinates starting from the random walk equations for radial particle positions and times. The particle density, or solute concentration is governed by a non-local radial advection–dispersion equation (ADE). Unlike in CTRWs for uniform flow scenarios, particle transition times here depend on the radial particle position, which renders the CTRW non-stationary. As a consequence, the memory kernel characterizing the non-local ADE, is radially dependent. Based on this general formulation, we derive radial CTRW implementations that (i) emulate non-local radial transport due to heterogeneous advection, (ii) model multirate mass transfer (MRMT) between mobile and immobile continua, and (iii) quantify both heterogeneous advection in a mobile region and mass transfer between mobile and immobile regions. The expected solute breakthrough behavior is studied using numerical random walk particle tracking simulations. This behavior is analyzed by explicit analytical expressions for the asymptotic solute breakthrough curves. We observe clear power-law tails of the solute breakthrough for broad (power-law) distributions of particle transit times (heterogeneous advection) and particle trapping times (MRMT model). The combined model displays two distinct time regimes. An intermediate regime, in which the solute breakthrough is dominated by the particle transit times in the mobile zones, and a late time regime that is governed by the distribution of particle trapping times in immobile zones. These radial CTRW formulations allow for the identification of heterogeneous advection and mobile-immobile processes as drivers of anomalous transport, under conditions relevant for field tracer tests.     

A Practical Modeling Framework for Non‐Fickian Transport and Multi‐Species Sequential First‐Order Reaction

Many studies indicate that small‐scale heterogeneity and/or mobile–immobile mass exchange produce transient non‐Fickian plume behavior that is not well captured by the use of the standard, deterministic advection‐dispersion equation (ADE). An extended ADE modeling framework is presented here that is based on continuous time random walk theory. It can be used to characterize non‐Fickian transport coupled with simultaneous sequential first‐order reactions (e.g., biodegradation or radioactive decay) for multiple degrading contaminants such as chlorinated solvents, royal demolition explosive, pesticides, and radionuclides. To demonstrate this modeling framework, new transient analytical solutions are derived and are inverted in Laplace space. Closed‐form, steady‐state, multi‐species analytical solutions are also derived for non‐Fickian transport in highly heterogeneous aquifers with linear sorption–desorption and matrix diffusion for use in spreadsheets. The solutions are general enough to allow different degradation rates for the mobile and immobile zones. The transient solutions for multi‐species transport are applied to examine the effects of source remediation on the natural attenuation of downgradient plumes of both parent and degradation products in highly heterogeneous aquifers. Results for representative settings show that the use of the standard, deterministic ADE can over‐estimate cleanup rates and under‐predict the cleanup timeframe in comparison to the extended ADE analytical model. The modeling framework and calculations introduced here are also applied for a 30 year groundwater cleanup program at a site in Palm Bay, Florida. The simulated plume concentrations using the extended ADE exhibited agreement with observed long concentration tails of trichloroethene, cis 1,2 DCE, and VC that remained above cleanup goals.     

Applicability of the Dual‐Domain Model to Nonaggregated Porous Media

More theoretical analysis is needed to investigate why a dual‐domain model often works better than the classical advection‐dispersion (AD) model in reproducing observed breakthrough curves for relatively homogeneous porous media, which do not contain distinct dual domains. Pore‐scale numerical experiments presented here reveal that hydrodynamics create preferential flow paths that occupy a small part of the domain but where most of the flow takes place. This creates a flow‐dependent configuration, where the total domain consists of a mobile and an immobile domain. Mass transfer limitations may result in nonequilibrium, or significant differences in concentration, between the apparent mobile and immobile zones. When the advection timescale is smaller than the diffusion timescale, the dual‐domain mass transfer (DDMT) model better captures the tailing in the breakthrough curve. Moreover, the model parameters (mobile porosity, mean solute velocity, dispersivity, and mass transfer coefficient) demonstrate nonlinear dependency on mean fluid velocity. The studied case also shows that when the Peclet number, Pe, is large enough, the mobile porosity approaches a constant, and the mass transfer coefficient can be approximated as proportional to mean fluid velocity. Based on detailed analysis at the pore scale, this paper provides a physical explanation why these model parameters vary in certain ways with Pe. In addition, to improve prediction in practical applications, we recommend conducting experiments for parameterization of the DDMT model at a velocity close to that of the relevant field sites, or over a range of velocities that may allow a better parameterization.     

Preferential percolation quantified by large water content sensors with artifactual macroporous envelopes

For many scientific and practical tasks, it is important to estimate the soil–water percolation fluxes. This paper builds on measurements with large horizontal time‐domain reflectometry water content sensors in a loamy Mollisol. The sensors were installed into pre‐drilled holes and the gaps between them, and the soil was filled with a slurry of local soil with water. This gave rise to envelopes around them that contained artificial macropores. The sensors reacted to intensive rains by a rapid increase of their readings, often above the native soil's porosity, followed by an almost equally rapid decrease. The paper explores the feasibility of quantifying the rapid percolation, based on these anomalous water content peaks, and demonstrates that this is possible in principle, if the processes are simulated by a suitable model. A two‐dimensional dual porosity non‐equilibrium (mobile‐immobile) model was tried. The envelope around the sensor was modelled as an annulus with higher porosity and hydraulic conductivity, which attracts preferential flow and amplifies the percolation signal. With the model at hand, the flux hydrographs can be derived from model simulations and measured precipitation. For contrast, the Durner equilibrium dual porosity model was tried but was found little suitable. However, even the mobile‐immobile model did not perform perfectly. Simulated water contents were similar to the measured ones at some depths but not in the others, and the percolation fluxes were overestimated, compared to cumulative soil–water balance. Efforts to improve model performance were not successful. Hence, the model structure needs to be improved. Copyright © 2015 John Wiley & Sons, Ltd.     

Modelling the chloride signal at Plynlimon,Wales, using a modified dynamic TOPMODEL incorporating conservative chemical mixing (with uncertainty)

The application of a modified version of dynamic TOPMODEL for two subcatchments at Plynlimon, Wales is described. Conservative chemical mixing within mobile and immobile stores has been added to the hydrological model in an attempt to simulate observed stream chloride concentrations. The model was not fully able to simulate the observed behaviour, in particular the short‐ to medium‐term dynamics. One of the primary problems highlighted by the study was the representation of dry deposition and cloud‐droplet‐deposited chloride, which formed a significant part of the long‐term chloride mass budget. Equifinality of parameter sets inhibited the ability to determine the effective catchment mixing volumes and coefficients or the most likely partition between occult mass inputs and chloride mass inputs determined by catchment immobile‐store antecedent conditions. Some success was achieved, in as much as some aspects of the dynamic behaviour of the signal were satisfactorily simulated, although spectral analysis showed that the model could not fully reproduce the 1/f power spectra of observed stream chloride concentrations with its implications of a wide distribution of residence times for water in the catchment. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical modelling-based comparison of longitudinal dispersion coefficient formulas for solute transport in rivers

River water quality models usually apply the Fischer equation to determine the longitudinal dispersion coefficient (D_x) in solving the advection–dispersion equation (ADE). Recently, more accurate formulas have been introduced to determine D_x in rivers, which could strongly affect the accuracy of the ADE results. A numerical modelling-based approach is presented to evaluate the performance of various D_x formulas using the ADE. This approach consists of a finite difference approximation of the ADE, a MATLAB code and a MS Excel interface; it was tested against the analytical ADE solution and demonstrated using eight well-known D_x formulas and tracer study data for the Chattahoochee River (USA), the Severn (UK) and the Athabasca (Canada). The results show that D_x has an important effect on tracer concentrations simulated with the ADE. Comparison between the simulated and measured concentrations confirms the appropriate performance of Zeng and Huai’s formula for D_x estimation. Use of the newly proposed equations for D_x estimation could enhance the accuracy of solving the ADE.     

Hydrodynamic dispersion during absorption of water by soil: 2. Immobile water model

The assumptions and analysis of Part 1 are extended to the three model soil-water profiles having different immobile-water contents up to 0.08 m³ m^?3. Concentration profiles in the mobile zone obtained with a constant dispersion coefficient are compared with those calculated with the same dispersion coefficient and no immobile water. The average concentration profiles were calculated from the concentration of the solution in both the mobile and immobile phases. Increasing the immobile-water content decreased and shifted the average concentration profiles further into the soil. The values of the dispersion coefficient calculated with the immobile-water fraction ignored, were found to be water content dependent and many times greater than that of the input dispersion coefficient.     

A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations

We extend the particle-tracking method to simulate general multi-rate mass transfer (MRMT) equations. Previous methods for single-rate equations used two-state Markov chains and found that the time a particle spends in the mobile state between waiting time epochs is random and exponentially distributed. Using Bochner’s subordination technique for Markov processes, we find that the random mobile times are still exponential for the stochastic process that corresponds to the MRMT equations. The random times in the immobile phase have a distribution that is directly related to the memory function of the MRMT equation. This connection allows us to interpret the MRMT memory function as the rate at which particles of a certain age, measured by residence time in the immobile zone, exit to become mobile once again. Because the exact distributions of mobile and immobile times are known from the MRMT equations, they can be simulated very simply and efficiently using random walks.     

Diffusive mass exchange of non‐reactive substances in dual‐porosity porous systems – column experiments under saturated conditions

Diffusive mass exchange into immobile water regions within heterogeneous porous aquifers influences the fate of solutes. The percentage of immobile water is often unidentified in natural aquifers though. Hence, the mathematical prediction of solute transport in such heterogeneous aquifers remains challenging. The objective of this study was to find a simple analytical model approach that allows quantifying properties of mobile and immobile water regions and the portion of immobile water in a porous system. Therefore, the Single Fissure Dispersion Model (SFDM), which takes into account diffusive mass exchange between mobile and immobile water zones, was applied to model transport in well‐defined saturated dual‐porosity column experiments. Direct and indirect model validation was performed by running experiments at different flow velocities and using conservative tracer with different molecular diffusion coefficients. In another column setup, immobile water regions were randomly distributed to test the model applicability and to determine the portion of immobile water. In all setups, the tracer concentration curves showed differences in normalized maximum peak concentration, tailing and mass recovery according to their diffusion coefficients. These findings were more pronounced at lower flow rates (larger flow times) indicating the dependency of diffusive mass exchange into immobile water regions on tracers' molecular diffusion coefficients. The SFDM simulated all data with high model efficiency. Successful model validation supported the physical meaning of fitted model parameters. This study showed that the SFDM, developed for fissured aquifers, is applicable in porous media and can be used to determine porosity and volume of regions with immobile water. Copyright © 2015 John Wiley & Sons, Ltd.     

On Evaluating Characteristics of the Solute Transport in the Arid Vadose Zone

The transport of bromide (Br) under matric heads of 0, ?2, ?5, and ?10 cm using undisturbed soil columns was investigated for understanding the solute transport in arid soils. Undisturbed soil cores were collected at ground surface, directly below where tension infiltrometer measurements were made in the Amargosa Desert, Nevada, United States. Laboratory experiments were conducted by introducing water containing Br tracer into a soil column maintained at steady‐state conditions. The observed data of breakthrough curves (BTC) were well fitted to an one‐region model, except for the cores at saturation, and a core at the matric head of ?5 cm, from which the observed data were better fitted to a two‐region model. Fitted pore water velocities with the one‐region model ranged from 1.2 to 56.6 cm/h, and fitted dispersion coefficients (D) ranged from 2.2 to 100 cm²/h. Results for the core analyzed with the two‐region model indicated that D ranged from 27.6 to 70.9 cm²/h at saturation, and 25.7 cm²/h at the matric head of ?5 cm; fraction of mobile water (β) ranged from 0.18 to 0.65, and mass transfer coefficient (ω) ranged from 0.006 to 0.03. In summary, the water fluxes and Br dispersion coefficients at investigated matric heads were very high due to the coarseness of the soils and possibly due to preferential flow pathways. These high water fluxes and Br dispersion coefficients would lead to a higher risk of deeper leaching accumulating nitrate nitrogen to the groundwater, and have significant effects on the desert ecosystem.     

A general approach to advective–dispersive transport with multirate mass transfer

A numerical solution that is significantly more general than other semi-analytical solutions is presented for governing equations describing advective–dispersive transport with multirate mass transfer between mobile and immobile domains. The new solution approach is general in the sense that it does not impose any restrictive assumption on the spatial or temporal variability of advective and dispersive processes in the mobile domain. A single integro-differential equation (IDE) is developed for the concentration in the mobile domain by separating the concentration in the immobile domain from the set of two partial differential equations. The solution to the IDE requires the evaluation of a temporal integral of the concentration in the mobile domain, which is a function of the Laplace transform of the distribution of the mass transfer rate coefficient. The Laplace transform is not limited to flow fields with known constant velocities. The solutions for one- and two-dimensional examples obtained using the new approach agree with those obtained by existing semi-analytical and numerical approaches.     

Implications of rate-limited mass transfer for aquifer storage and recovery

Pressure to decrease reliance on surface water storage has led to increased interest in aquifer storage and recovery (ASR) systems. Recovery efficiency, which is the ratio of the volume of recovered water that meets a predefined standard to total volume of injected fluid, is a common criterion of ASR viability. Recovery efficiency can be degraded by a number of physical and geochemical processes, including rate-limited mass transfer (RLMT), which describes the exchange of solutes between mobile and immobile pore fluids. RLMT may control transport behavior that cannot be explained by advection and dispersion. We present data from a pilot-scale ASR study in Charleston, South Carolina, and develop a three-dimensional finite-difference model to evaluate the impact of RLMT processes on ASR efficiency. The modeling shows that RLMT can explain a rebound in salinity during fresh water storage in a brackish aquifer. Multicycle model results show low efficiencies over one to three ASR cycles due to RLMT degrading water quality during storage; efficiencies can evolve and improve markedly, however, over multiple cycles, even exceeding efficiencies generated by advection-dispersion only models. For an idealized ASR model where RLMT is active, our simulations show a discrete range of diffusive length scales over which the viability of ASR schemes in brackish aquifers would be hindered.     

Lattice Boltzmann Models for Flow and Transport in Saturated Karst

Flow and transport simulation in karst aquifers remains a significant challenge for the ground water modeling community. Darcy's law–based models cannot simulate the inertial flows characteristic of many karst aquifers. Eddies in these flows can strongly affect solute transport. The simple two-region conduit/matrix paradigm is inadequate for many purposes because it considers only a capacitance rather than a physical domain. Relatively new lattice Boltzmann methods (LBMs) are capable of solving inertial flows and associated solute transport in geometrically complex domains involving karst conduits and heterogeneous matrix rock. LBMs for flow and transport in heterogeneous porous media, which are needed to make the models applicable to large-scale problems, are still under development. Here we explore aspects of these future LBMs, present simple examples illustrating some of the processes that can be simulated, and compare the results with available analytical solutions. Simulations are contrived to mimic simple capacitance-based two-region models involving conduit (mobile) and matrix (immobile) regions and are compared against the analytical solution. There is a high correlation between LBM simulations and the analytical solution for two different mobile region fractions. In more realistic conduit/matrix simulation, the breakthrough curve showed classic features and the two-region model fit slightly better than the advection-dispersion equation (ADE). An LBM-based anisotropic dispersion solver is applied to simulate breakthrough curves from a heterogeneous porous medium, which fit the ADE solution. Finally, breakthrough from a karst-like system consisting of a conduit with inertial regime flow in a heterogeneous aquifer is compared with the advection-dispersion and two-region analytical solutions.     

Perspective on theories of non-Fickian transport in heterogeneous media

Subsurface fluid flow and solute transport take place in a multiscale heterogeneous environment. Neither these phenomena nor their host environment can be observed or described with certainty at all scales and locations of relevance. The resulting ambiguity has led to alternative conceptualizations of flow and transport and multiple ways of addressing their scale and space–time dependencies. We focus our attention on four approaches that give rise to nonlocal representations of advective and dispersive transport of nonreactive tracers in randomly heterogeneous porous or fractured continua. We compare these approaches theoretically on the basis of their underlying premises and the mathematical forms of the corresponding nonlocal advective–dispersive terms. One of the four approaches describes transport at some reference support scale by a classical (Fickian) advection–dispersion equation (ADE) in which velocity is a spatially (and possibly temporally) correlated random field. The randomness of the velocity, which is given by Darcy’s law, stems from random fluctuations in hydraulic conductivity (and advective porosity though this is often disregarded). Averaging the stochastic ADE over an ensemble of velocity fields results in a space–time-nonlocal representation of mean advective–dispersive flux, an approach we designate as stnADE. A closely related space–time-nonlocal representation of ensemble mean transport is obtained upon averaging the motion of solute particles through a random velocity field within a Lagrangian framework, an approach we designate stnL. The concept of continuous time random walk (CTRW) yields a representation of advective–dispersive flux that is nonlocal in time but local in space. Closely related to the latter are forms of ADE entailing fractional derivatives (fADE) which leads to representations of advective–dispersive flux that are nonlocal in space but local in time; nonlocality in time arises in the context of multirate mass transfer models, which we exclude from consideration in this paper. We describe briefly each of these four nonlocal approaches and offer a perspective on their differences, commonalities, and relative merits as analytical and predictive tools.     

Can a Time Fractional‐Derivative Model Capture Scale‐Dependent Dispersion in Saturated Soils?

Time nonlocal transport models such as the time fractional advection‐dispersion equation (t‐fADE) were proposed to capture well‐documented non‐Fickian dynamics for conservative solutes transport in heterogeneous media, with the underlying assumption that the time nonlocality (which means that the current concentration change is affected by previous concentration load) embedded in the physical models can release the effective dispersion coefficient from scale dependency. This assumption, however, has never been systematically examined using real data. This study fills this historical knowledge gap by capturing non‐Fickian transport (likely due to solute retention) documented in the literature (Huang et al. 1995) and observed in our laboratory from small to intermediate spatial scale using the promising, tempered t‐fADE model. Fitting exercises show that the effective dispersion coefficient in the t‐fADE, although differing subtly from the dispersion coefficient in the standard advection‐dispersion equation, increases nonlinearly with the travel distance (varying from 0.5 to 12 m) for both heterogeneous and macroscopically homogeneous sand columns. Further analysis reveals that, while solute retention in relatively immobile zones can be efficiently captured by the time nonlocal parameters in the t‐fADE, the motion‐independent solute movement in the mobile zone is affected by the spatial evolution of local velocities in the host medium, resulting in a scale‐dependent dispersion coefficient. The same result may be found for the other standard time nonlocal transport models that separate solute retention and jumps (i.e., displacement). Therefore, the t‐fADE with a constant dispersion coefficient cannot capture scale‐dependent dispersion in saturated porous media, challenging the application for stochastic hydrogeology methods in quantifying real‐world, preasymptotic transport. Hence improvements on time nonlocal models using, for example, the novel subordination approach are necessary to incorporate the spatial evolution of local velocities without adding cumbersome parameters.     

Changes in water table level influence solute transport in uniform porous media

Changes in the water table level result in variable water saturation and variable hydrological fluxes at the interface between the unsaturated and saturated zone. This may influence the transport and fate of contaminants in the subsurface. The objective of this study was to examine the impact of a decreasing and an increasing water table on solute transport. We conducted tracer experiments at downward flow conditions in laboratory columns filled with two different uniform porous media under static and transient flow conditions either increasing or decreasing the water table. Tracer breakthrough curves were simulated using a mobile–immobile transport model. The resulting transport parameters were compared to identify dominant transport processes. Changes in the water table level affected dispersivities and mobile water fractions depending on the direction of water table movement and the grain size of the porous media. In fine glass beads, the water flow velocity was similar to the decline rate of the water table, and the mobile water fraction was decreased compared with steady‐state saturated conditions. However, immobile water was negligible. In coarse glass beads, water flow was faster because of fingered flow in the unsaturated part, and the mobile water fraction was smaller than in the fine material. Here, a rising water table led to an even smaller mobile water fraction and increased solute spreading because of diffusive interaction with immobile water. We conclude that changes of the water table need to be considered to correctly simulate transport in the subsurface at the transition of the unsaturated–saturated zone. Copyright © 2014 John Wiley & Sons, Ltd.     

An experimental study of dispersion of an interactive tracer in chemically heterogeneous medium at a column scale

This paper aims at examining the increase of phenol adsorption breakthrough curves spreading caused by the chemical heterogeneity of granular activated carbon fixed beds. The local and the thermodynamic equilibrium assumption, as well as the nonlinear adsorption obeying to Langmuir isotherm, are considered. This study particularly tempts to link the reduced variance of phenol breakthrough curves to a measurable quantifying parameter of the chemical heterogeneity. The investigated artificial heterogeneous media are prepared by alternating layers of two types of granular activated carbon, active and non-active ones, that have similar physical properties. On the one hand, the chemical heterogeneity is quantified by the active layer relative thickness of the column length, l₁/L. On the other hand, it is quantified by the mean value of the probability distribution γ. The latter also represents the mean active grains mass ratio of the total medium mass, hence the medium mean capacity. The obtained results show an increase in the reduced variance and thus the effective global dispersion with the heterogeneity; the increase is as important as the medium capacity decreases. However, the dispersion increase achieves a limit value, even when the heterogeneity increases. The results are statistically modelled using a regression equation function of the capacity variation in terms of γ and the chemical heterogeneity in terms of l₁/L. The relationship combining the medium capacity and the chemical heterogeneity is obtained. The relationship implicitly takes into account the effect of the column length.