Dispersive Flux Research Articles

Non-Fickian dispersion of groundwater age.

We expand the governing equation of groundwater age to account for non-Fickian dispersive fluxes using continuous random walks. Groundwater age is included as an additional (fifth) dimension on which the volumetric mass density of water is distributed and we follow the classical random walk derivation now in five dimensions. The general solution of the random walk recovers the previous conventional model of age when the low order moments of the transition density functions remain finite at their limits and describes non-Fickian age distributions when the transition densities diverge. Previously published transition densities are then used to show how the added dimension in age affects the governing differential equations. Depending on which transition densities diverge, the resulting models may be nonlocal in time, space, or age and can describe asymptotic or pre-asymptotic dispersion. A joint distribution function of time and age transitions is developed as a conditional probability and a natural result of this is that time and age must always have identical transition rate functions. This implies that a transition density defined for age can substitute for a density in time and this has implications for transport model parameter estimation. We present examples of simulated age distributions from a geologically based, heterogeneous domain that exhibit non-Fickian behavior and show that the non-Fickian model provides better descriptions of the distributions than the Fickian model.

Large‐eddy simulation of oxygen transfer to organic sediment beds

We have developed a model for dissolved oxygen (DO) transfer from water to underlying organic sediment beds. The model couples large‐eddy simulation (LES) of turbulent transport in the water column, a biogeochemical model for DO transport and consumption in the sediment, and Darcy's Law for the pore water‐driven transport. The model highlights the spatial and temporal relationship between the turbulent bursting events, the near‐wall transport of DO, and the response of the sediment layer. The numerical results—compared to data from laboratory experiments—stress the importance of analyzing instantaneous transport events (not reproducible in a Reynolds‐averaged Navier‐Stokes model) to better characterize and model the processes that lead to oxygen depletion in the sediment layer. The model's results are compared against experimental data and the sensitivity to the governing parameters has been tested. As the current velocity increases, the sediment‐oxygen demand (SOD) increases more slowly than the friction velocity at the wall, in accordance with classic heat‐and‐mass transfer laws. The overall SOD is approximately proportional to the bacterial content of the sediment layer. The predicted mean advective flux in the porous medium and across the sediment‐water interface is negligible compared to the total dispersive flux, for permeabilities typical of the sediments used in the experiments (10−7 and 10−6 cm2). Higher permeabilities (10−5 cm2) appear to yield results not consistent with experimental data. This computational tool will contribute toward the design of a process‐oriented parametrization for the SOD, currently missing in oceanographic applications, that can be easily extended to the transport of other bio‐limiting substances.

A k–ε turbulence model based on the scales of vertical shear and stem wakes valid for emergent and submerged vegetated flows

AbstractFlow and transport through aquatic vegetation is characterized by a wide range of length scales: water depth ($H$), plant height ($h$), stem diameter ($d$), the inverse of the plant frontal area per unit volume (${a}^{\ensuremath{-} 1} $) and the scale(s) over which $a$ varies. Turbulence is generated both at the scale(s) of the mean vertical shear, set in part by $a$, and at the scale(s) of the stem wakes, set by $d$. While turbulence from each of these sources is dissipated through the energy cascade, some shear-scale turbulence bypasses the lower wavenumbers as shear-scale eddies do work against the form drag of the plant stems, converting shear-scale turbulence into wake-scale turbulence. We have developed a $k$–$\varepsilon $ model that accounts for all of these energy pathways. The model is calibrated against laboratory data from beds of rigid cylinders under emergent and submerged conditions and validated against an independent data set from submerged rigid cylinders and a laboratory data set from a canopy of live vegetation. The new model outperforms existing $k$–$\varepsilon $ models, none of which include the $d$ scale, both in the emergent rigid cylinder case, where existing $k$–$\varepsilon $ models break down entirely, and in the submerged rigid cylinder and live plant cases, where existing $k$–$\varepsilon $ models fail to predict the strong dependence of turbulent kinetic energy on $d$. The new model is limited to canopies dense enough that dispersive fluxes are negligible.

Effects of differential wall heating in street canyons on dispersion and ventilation characteristics of a passive scalar

Abstract This paper presents the first large-eddy simulation (LES) study of transfer and dispersion of a scalar released from a rough urban facet, either the street surface, the upstream-wall, or the downstream-wall, under the thermo-dynamical conditions of either the upstream-wall or the downstream-wall, plus the roof, heated by solar radiation. The boundary condition of a constant value is adopted for the scalars on the rough urban facets and a wall function is proposed for the scalars. The LES results demonstrate that dispersion inside the street canyon possesses distinctive characteristics for two conditions: the assisting condition in which the thermal-driven flow has the same direction as that of the wind-driven vortex and the opposing condition in which the thermal-driven flow has the opposite direction as that of the wind-driven vortex. For the street-released scalar under the opposing condition, the concentration fluctuations relative to the mean concentration can reach 50% and in general they are much larger than those for the assisting cases which are in the range of 25–30%. The exchange velocity of a scalar between the street canyon air and the urban boundary layer (UBL), w C B ( c ) , is one order of magnitude larger than the exchange velocity between a facet and the urban boundary layer, UBL, w 0 B ( c ) , indicating quantitatively that the resistance to the transfer of a facet-released scalar is dominated by the near-facet processes. As the temperature difference between the wall and the UBL, ΔT, increases, the total resistance to street canyon ventilation becomes more dominated by the near-facet resistance. The assisting conditions are favourable to ventilating the scalars from both walls, whereas the opposing conditions are only favourable to the ventilation of the downstream-wall released scalar. In the range of ΔT tested in this study, the exchange velocity, w C B ( c ) , linearly increases with ΔT and can be well parameterised. For the assisting cases, the advective (or dispersive) flux dominates the turbulent flux in most part of the canyon and the partition between the two alters dramatically across the roof level with turbulent processes dominating above the roof level. This study also suggests that for the street-released scalar, the resistance between the bottom canyon and the upper canyon is significant compared with the one between the upper canyon and the UBL. It is therefore necessary to adopt a two-box model in order to reasonably model such a case. The results of this study provide the guidance of improving the parameterisation schemes of transfer and dispersion for street canyons which currently do not consider the influence of wall heating.

A Phenomenological Approach to Modeling Transport in Porous Media

The objective of this article is to make use of the phenomenological approach to construct models for the transport of extensive quantities, such as mass of a fluid phase, mass of a component of a fluid phase, momentum of a phase and energy, in porous medium domains. Special attention is devoted to express the fluxes of these extensive quantities, especially the non-advective ones, as functions of their relevant driving forces, obeying the principle of minimum entropy production. It is shown that for each extensive quantity, we have a linear diffusive flux term, a non-linear diffusive term, and a dispersive flux term. The latter is shown to be proportional to the velocity squared. In each case, the number of moduli that describe fluid and porous matrix properties is determined. The momentum balance equation for a porous medium domain, which is the “motion equation,” is analyzed and simplified for special cases, leading to Darcy’s law and to Brinkman’s equation.

Spatial variation of the longitudinal dispersion coefficient in an estuary

Abstract. The effective longitudinal dispersion is a primary tool for determining property distributions in estuaries. Most previous studies have examined the longitudinal dispersion coefficient for the average tidal condition. However, information on spatial and temporal variations of this coefficient at low and high tides is scarce. Three years of hydrographic data taken at low and high tide along the main axis of the Sumjin River Estuary (SRE), Korea are used to estimate the spatial and temporal variation of the effective longitudinal dispersion coefficient. The range of the dispersion coefficient is rather broad at high water slack (HWS) and narrower at low water slack (LWS) because of the different tidal amplitudes. The spatially varying dispersion coefficient has maximal values (>300 m2 s−1) near the mouth at high water and decreases gradually upstream, with fluctuations. The temporally varying dispersion coefficient appears to be positively correlated with river discharges at both low and high tide. The dispersion varies with the square root of river discharges at HWS and LWS. The dispersive salt fluxes increases with increasing river discharges and decreases with decreasing river discharges at HWS and LWS. Estimation of the numerical values of the effective longitudinal dispersion coefficient in the SRE can be useful for better understanding of the distributions of other tracers in the SRE as well as for developing and testing hypotheses about various mixing mechanisms.

Improved Box Model for Simulating Pesticide Transport in the Vadose Zone with Dispersive Flux through the Boundary Layer

AbstractA box-type transport model can be a useful tool for screening-level, long-term predictions of contamination and the relevant environmental assessment, especially when data are scarce and quick decisions are needed. In most box-type transport models, only advective flux is simulated to account for the interaction between boxes. In reality, however, significant mass transfer may also occur through the boundary between two boxes as a result of dispersion. For volatile contaminants (e.g., pesticides), the mass transfer through the boundary may involve both dissolved and vapor phases. In this study, an improved box-type analytical pesticide transport model with dispersive flux through the boundary layer (APTM-DF) was developed for simulating fate and transport of three-phase pesticides in the vadose zone, which was further divided into the surface zone, plant root zone, and deep vadose zone. The model also simulated pesticide leaching, degradation, volatilization, sorption, partition between dissolved ...

RCM-E simulation of ion acceleration during an idealized plasma sheet bubble injection

[1] In this paper, we investigate the role of plasma sheet bubbles in the ion flux variations at geosynchronous orbit during substorm injections by using the Rice Convection Model with an equilibrated magnetic field model (RCM-E). The bubble is initiated in the near-Earth plasma sheet with a localized reduction in entropy parameter PV5/3 following a substorm growth phase. In the expansion phase, characteristic features of substorm injections are reproduced; that is, there is a prominent dispersionless flux increase for energetic protons (>40 keV) and a flux decrease for lower-energy protons near midnight geosynchronous orbit while there is dispersive flux enhancement near the dusk sector. We find that the injection boundary is well coincident with the earthward boundary of the bubble, inside which the depletion of plasma content causes the magnetic field dipolarization, and in return, the magnetic field collapse energizes particles and alters the drift paths dramatically. Our results also show that a high-PV5/3 island is pushed ahead of the fast earthward propagating bubble, and a dipolarization front forms between them. Within the high-PV5/3 island, the diamagnetic effect makes the plasma pressure increase and the strength of the magnetic field decrease to a local minimum. We suggest that plasma sheet bubbles are elementary vehicles of substorm time particle injections from the main plasma sheet to the inner magnetosphere.

Modelling of block-scale macrodispersion as a random function

Numerical modelling of solute dispersion in natural heterogeneous porous media is facing several challenges. Amongst these we highlight the challenge of accounting for high-frequency variability that is filtered out by homogenization at the subgrid scale and the uncertainty in the dispersive flux for transport under non-ergodic conditions. These two effects when combined lead to inaccurate representation of the dispersive fluxes. We propose to compensate for this deficiency by defining a block-scale dispersion tensor and modelling it as a random space function ℳij. The derived dispersion tensor is a function of several length scales and time. Grid blocks will be assigned dispersion coefficients generated from the ℳijdistribution. We will show the dependence of ℳijon the spatial variability of the conductivity field, on the contaminant source size, on the travel time and on the grid-block scale. For an ergodic source, a statistically uniform conductivity field and very large grid blocks, ℳijis equal to the macrodispersion coefficients proposed by Dagan (J. Fluid Mech., vol. 145, 1984, p. 151) with zero variance. For an ergodic source and non-uniform conductivity field with a finite-size grid block, ℳijapproaches the model proposed by Rubinet al. (J. Fluid Mech., vol. 395, 1999, p. 161). In both cases, ℳijis defined by its mean value with zero variance. ℳijis subject to uncertainty when the source is non-ergodic and when the grid block is defined by a finite scale. When the grid-block scale approaches zero, which means that the spatial variability is captured completely on the numerical grid, ℳijapproaches zero with zero variance. In addition, we provide a complete statistical characterization of ℳijby invoking the concept of minimum relative entropy, thus providing upper bounds on the uncertainty associated with ℳij.

Modelling of block-scale macrodispersion as a random function

Numerical modelling of solute dispersion in natural heterogeneous porous media is facing several challenges. Amongst these we highlight the challenge of accounting for high-frequency variability that is filtered out by homogenization at the subgrid scale and the uncertainty in the dispersive flux for transport under non-ergodic conditions. These two effects when combined lead to inaccurate representation of the dispersive fluxes. We propose to compensate for this deficiency by defining a block-scale dispersion tensor and modelling it as a random space function ℳij. The derived dispersion tensor is a function of several length scales and time. Grid blocks will be assigned dispersion coefficients generated from the ℳij distribution. We will show the dependence of ℳij on the spatial variability of the conductivity field, on the contaminant source size, on the travel time and on the grid-block scale. For an ergodic source, a statistically uniform conductivity field and very large grid blocks, ℳij is equal to the macrodispersion coefficients proposed by Dagan (J. Fluid Mech., vol. 145, 1984, p. 151) with zero variance. For an ergodic source and non-uniform conductivity field with a finite-size grid block, ℳij approaches the model proposed by Rubin et al. (J. Fluid Mech., vol. 395, 1999, p. 161). In both cases, ℳij is defined by its mean value with zero variance. ℳij is subject to uncertainty when the source is non-ergodic and when the grid block is defined by a finite scale. When the grid-block scale approaches zero, which means that the spatial variability is captured completely on the numerical grid, ℳij approaches zero with zero variance. In addition, we provide a complete statistical characterization of ℳij by invoking the concept of minimum relative entropy, thus providing upper bounds on the uncertainty associated with ℳij.

Solitary wavepackets from oscillatory non-linear equations with damping

In this paper, an infinite family of solutions describing solitary wave packets with a finite number of nodes is presented. These structures arise from the study of damping in the framework of non-linear ordinary differential equations with oscillatory behaviour. Usually one expects to find effects of this kind in physical systems described by a set of partial differential equations. The standard argument is that the non-linear term acts against the dispersive flux and this balance explains the appearance of solitary waves. Here we show that the non-linear oscillatory behaviour can also balance the effect of damping in special cases. The theory used to discriminate among the various possibilities is plain Painleve analysis. Several physical applications are briefly discussed.

Using flushing rate to investigate spring-neap and spatial variations of gravitational circulation and tidal exchanges in an estuary

Abstract. Spring-neap and spatial variations of gravitational circulation and tidal exchanges in the Sumjin River Estuary (SRE) were investigated using the flushing rate. The flushing rate was calculated between multiple estuarine segments and the adjacent bay to examine the spatial variation of two exchanges. The strength of gravitational circulation and tidal exchanges modulated significantly between spring and neap tides, where stratification alternated between well-mixed and highly-stratified conditions over the spring-neap cycle. Tide-driven dispersive flux of salt dominated over gravitational circulation exchange near the mouth during spring tide due to the larger tidal amplitude that caused well-mixed conditions and rapid exchange. In contrast, the central and inner regimes were found to be partially stratified during spring tide due to the reduction in tidal amplitude where both gravitational circulation and tidal exchanges were important in transporting salt. The combined contributions of two fluxes were also found during neap tide along the SRE due to the significant reduction in vertical mixing that accompanied strong stratification. Gravitational circulation exchange almost entirely dominated in transporting salt at the upstream end during spring and neap tides.

Combined role of advective pumping and mechanical dispersion on time scales of bed form–induced hyporheic exchange

This study analyzes the effect of advective pumping and pore scale dispersion on bed form–induced hyporheic exchange. Advection and dispersion play a competitive role in the exchange dynamics between the porous medium and the overlying stream: Advective fluxes first lead solutes deep into the bed and then back to the stream water, whereas dispersive fluxes favor the transfer of solutes deep into the bed leading to a permanent mass retention. The combined effect of advective exchange and dispersive fluxes produces complexity in the shape of the tails of the residence time distributions (RTDs), which follow at various stages of the process either a power law or an exponential decay. The seepage velocity induced by the stream gradient and, in case of a moving bed, the celerity of the translating bed forms limit the thickness of the advective hyporheic zone, inducing the RTDs to decrease rapidly at late time. This rapid decay can be preceded by a temporal region where the probability density functions (pdf's) tend to be inversely proportional to the square of time, and is followed by a region dominated by dispersion where the pdf's tend to be inversely proportional to the 3/2 power of time. The process shows distinct temporal ranges identified here by appropriate dimensionless parameters. Because of this complex exchange dynamics, models considering pure advection in the porous medium can significantly underestimate solute transfer at long time scales, whereas purely diffusive models of hyporheic exchange appear inadequate to represent the physical processes at an intermediate stage.

Long-term model of planimetric and bathymetric evolution of a tidal lagoon

This model is based on the concept of transport concentration, defined as the time-averaged concentration in a given location of a lagoon, which determines the long-term net transport of sediments as the sum of a dispersive and an advective flux. Dispersive net flux of sediments is due to the alternate components of the tidal flow, while the advective net flux of sediments is due to the residual (Eulerian) component of the tidal, fluvial and littoral flow and possibly to the asymmetry between flow and ebb tide. The long-term deposition (or erosion) rate in a given location is expressed as proportional to the difference between the local transport concentration and the local equilibrium concentration, depending in turn on the local stirring action by waves and currents. The two-dimensional model is able to reproduce the ontogeny of a tidal lagoon as a consequence of a breaching of the littoral dune line. During the ontogenetic phase, the system tends rapidly toward a quasi-equilibrium condition apparently characterized by a stable planimetric configuration of the channel network. The structure of the channels and their watersheds is qualitatively similar to the Venice Lagoon's ones. After the relatively fast ontogenetic phase, the subsequent morphological evolution of the bottom continues at a slower pace until a stable bathymetric configuration of the lagoon is reached. The model is able to reproduce the morphological effects of subsequent perturbations of natural or anthropogenic origin.

Effective two‐phase flow in heterogeneous media under temporal pressure fluctuations

We study the combined effect of spatial heterogeneity and temporal pressure fluctuations on the dispersion of the saturation of a displacing fluid during injection into an immiscible one. In a stochastic modeling framework, we define two different dispersion quantities, one which measures the front uncertainty due to temporal pressure fluctuations and another one which quantifies the actual spreading and dispersion of the saturation front in a typical medium realization. We derive an effective large‐scale flow equation for the saturation of the displacing fluid that is characterized by a nonlocal dispersive flux term. From the latter we derive measures for the spread of the saturation front due to spatiotemporal fluctuations. Our analysis demonstrates that temporal fluctuations enhance the front uncertainty but not the average spreading of the saturation front. The analytical developments are complemented by numerical simulations of the full two‐phase flow problem. Correct assessment of the spread of the saturation front is of importance to several applications, including the assessment of the sequestration potential of a carbon dioxide storage site, for example. Spreading enhances the surface area between the fluids, which in turn enhances the dissolution and entrapment efficiency. The latter facilitates reactions and thus the sequestration of CO2 in stable forms. Our study provides a theoretical basis for the design of injection strategies to optimize dispersion and to minimize uncertainty of the saturation front.

Scale-dependent Macroscopic Balance Equations Governing Transport Through Porous Media: Theory and Observations

A theory is developed providing a rational framework for spatial scale- dependent fluid’s flow and heat transfer, and mass of a component migrating with it through porous media. Introducing the assumption of a non-Brownian type motion and referring to asymptotic expansion in powers of a small defined parameter, we develop a novel approach associated with macroscopic balance equations obtained by averaging over a Representative Elementary Volume (REV). We prove that these equations can be decomposed into a primary part that refers to the REV length scale and a secondary part valid at a length scale smaller than that of the corresponding REV length. Further to our previous development, we obtain two general forms of the primary and secondary macroscopic balance equations. One is based on the assumption that the advective flux of the extensive quantity is dominant over that of the dispersive flux, whereas the other disregards this assumption. Moreover we also introduce the primary and secondary macroscopic forms for the fluid heat- transfer equation. Considering a Newtonian fluid, the resulting primary Navier–Stokes equation can vary from a nonlinear wave equation to a drag-dominant equation at the fluid–solid interface (Darcy’s law). The secondary momentum balance equation describes a wave equation governing the concurrent propagation of the intensive momentum and the dispersive momentum flux, deviating from their corresponding average terms. The primary macroscopic fluid heat-transfer equation accounts for advective and dispersive heat fluxes and the secondary macroscopic heat-transfer equation involves the simultaneous advection of heat deviating from its corresponding intensive average quantity. The primary macroscopic solute mass balance equation accounts for advection and hydrodynamic dispersion. The secondary macroscopic component mass balance equation is in the form of pure advection governing migration of the deviation from the average component concentration. At this stage, we focus on establishing the viability of the developed theory. We do this by arguing that field observations of motion at small spatial scales are coherent with the hyperbolic characteristics of the secondary balance equations. Field observations under natural gradient flow conditions show excessive high concentration (average of 50 mg/L) of colloids under land irrigated by sewage effluents. We argue that this displacement of condensed colloidal parcels manifests the theoretical findings for the smaller spatial scale. Further evidence show the accumulation of particles moving behind the front of an emitted shockwave. We consider this as an experimental proof reinforcing the argument that colloidal migration is subject to the action of a shockwave in the fluid and pure advection transport, governed by the respective suggested hyperbolic macroscopic balance equations of fluid momentum and component mass at the smaller spatial scale.

Conditional Probability Density Functions of Concentrations for Mixing-Controlled Reactive Transport in Heterogeneous Aquifers

This paper presents an approach conducive to an evaluation of the prob- ability density function (pdf) of spatio-temporal distributions of concentrations of reactive solutes (and associated reaction rates) evolving in a randomly heterogeneous aquifer. Most existing approaches to solute transport in heterogeneous media focus on providing expressions for space-time moments of concentrations. In general, only low order moments (unconditional or conditional mean and covariance) are com- puted. In some cases, this allows for obtaining a confidence interval associated with predictions of local concentrations. Common applications, such as risk assessment and vulnerability practices, require the assessment of extreme (low or high) concen- tration values. We start from the well-known approach of deconstructing the reactive transport problem into the analysis of a conservative transport process followed by speciation to (a) provide a partial differential equation (PDE) for the (conditional) pdf of conservative aqueous species, and (b) derive expressions for the pdf of re- active species and the associated reaction rate. When transport at the local scale is described by an Advection Dispersion Equation (ADE), the equation satisfied by the pdf of conservative species is non-local in space and time. It is similar to an ADE and includes an additional source term. The latter involves the contribution of dilu- tion effects that counteract dispersive fluxes. In general, the PDE we provide must be solved numerically, in a Monte Carlo framework. In some cases, an approximation can be obtained through suitable localization of the governing equation. We illustrate the methodology to depict key features of transport in randomly stratified media in

An Assembly Model for Simulation of Large‐Scale Ground Water Flow and Transport

When managing large-scale ground water contamination problems, it is often necessary to model flow and transport using finely discretized domains--for instance (1) to simulate flow and transport near a contamination source area or in the area where a remediation technology is being implemented; (2) to account for small-scale heterogeneities; (3) to represent ground water-surface water interactions; or (4) some combination of these scenarios. A model with a large domain and fine-grid resolution will need extensive computing resources. In this work, a domain decomposition-based assembly model implemented in a parallel computing environment is developed, which will allow efficient simulation of large-scale ground water flow and transport problems using domain-wide grid refinement. The method employs common ground water flow (MODFLOW) and transport (RT3D) simulators, enabling the solution of almost all commonly encountered ground water flow and transport problems. The basic approach partitions a large model domain into any number of subdomains. Parallel processors are used to solve the model equations within each subdomain. Schwarz iteration is applied to match the flow solution at the subdomain boundaries. For the transport model, an extended numerical array is implemented to permit the exchange of dispersive and advective flux information across subdomain boundaries. The model is verified using a conventional single-domain model. Model simulations demonstrate that the proposed model operated in a parallel computing environment can result in considerable savings in computer run times (between 50% and 80%) compared with conventional modeling approaches and may be used to simulate grid discretizations that were formerly intractable.

Modeling Transverse Dispersion and Variable Density Flow in Porous Media

A two-dimensional numerical model is used to study the nonlinear behavior of density gradients on transverse dispersion. Numerical simulations are conducted using d3f, a computer code for simulation of density-dependent flow in porous media. Considering a density-stratified horizontal flow in a heterogeneous porous media, a series of simulations is carried out to examine the effect of the density gradient on macro-scale transverse dispersivity. Changing salt concentration significantly affects fluid properties. This physical behavior of the fluid involves a non-linearity in modeling the interaction between salt and fresh water. It is concluded that the large-scale transport properties for high density flow deviate significantly from the tracer case due to the spatial variation of permeability, described by statistical parameters, at the local-scale. Indeed, the presence of vertical flow velocities induced by permeability variations is responsible for the reduction of the mixing zone width in the steady state in the case of a high density gradient. Uncertainties in the model simulations are studied in terms of discretization errors, boundary conditions, and convergence of ensemble averaging. With respect to the results, the gravity number appears to be the controlling parameter for dispersive flux. In addition, the applicability and limitations of the nonlinear model of Hassanizadeh (1990) and Hassanizadeh and Leijnse (1995) (Adv Water Resour 18(4):203–215, 1995) in heterogeneous porous media are investigated. We found that the main cause of the nonlinear behavior of dispersion, which is the interaction between density contrast and vertical velocity, needs to be explicitly accounted for in a macro-scale model.

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.