Transport In Heterogeneous Media Research Articles

Comparison of streamtube and three-dimensional models of reactive transport in heterogeneous media

Field-scale transport of reactive solutes is influenced strongly by physical and geochemical heterogeneity over a wide range of spatial scales. Accurate numerical simulation requires fine spatial and temporal discretization to resolve the smallest scales of heterogeneity, in addition to coupling large numbers of chemical components participating in geochemical and microbiological reactions. Standard approaches based upon finite difference or finite element discretization of the governing set of advection-dispersion-reaction equations can lead to computational difficulties for large problems. To overcome these difficulties, several streamtube-based models have been presented in the literature. These models neglect transverse dispersion, and hence reactive transport occurs along independent one-dimensional streamtubes. Therefore, streamtube models are well-suited for applications where the objective is to calculate solute flux breakthrough across a control plane. We develop a reactive streamtube model by modification of the Finite Element Heat and Mass transfer code (FEHM), an existing flow and transport code. FEHM can simulate unsaturated zone conditions, heat flow, and multicomponent reactive transport, and has been used extensively by the Yucca Mountain Project. Because the same underlying numerical algorithms are used in both the FEHM and streamtube models, direct comparison of their relative performance is possible. We perform this comparison for a three-dimensional reactive transport example, to demonstrate the large potential computational advantages of the streamtube approach.

Anomalous transport in heterogeneous media demonstrated by streamline‐based simulation

We use fine‐grid streamline simulation to demonstrate that anomalous (non‐Gaussian) transport arises from purely advective movement through heterogeneous systems.The simulation model represents a channeled sandstone North Sea oil field and contains over one million grid blocks. We find the average location and spreading of the plume and the breakthrough curves. These features are consistent with anomalous transport described by an exponent β that characterizes the long‐time tail of the transit time distribution. β depends on the degree of heterogeneity and whether the tracer was injected or originally uniformly distributed across the domain. Incorporating dispersion to account for sub‐grid‐block heterogeneity does not affect the results.

Transport behavior of a passive solute in continuous time random walks and multirate mass transfer

We consider the transport behavior of a passive solute in a heterogeneous medium, modeled by continuous time random walks (CTRW) and linear multirate mass transfer (MRMT). Within the CTRW framework, we formulate a transport model which is formally equivalent to MRMT. In both approaches the total concentration is divided into mobile and immobile parts. The immobile concentration is given by the convolution in time of the mobile concentration and a memory function. The memory function is a functional of the distribution of transition and trapping times for the CTRW and the MRMT approach, respectively, and determines the transport behavior of the solute. Based on different expressions for the memory function in the two frameworks, we derive conditions for which both approaches describe the same transport behavior. We focus on anomalous transport behavior that can arise if the transition and trapping time distributions behave algebraically in a given time regime. Using an expansion of the Laplace transform of the memory function, we develop explicit expressions for the time behavior of the flux concentration as well as for the center of mass velocity and the (macro) dispersion coefficients of the solute distribution. We observe (anomalous) power law as well as (normal) Fickian transport behavior, depending on the exponents that dominate the trapping and transition time distributions, respectively. The results show that the character of the anomalous transport does not depend on the details of the transport model but only on the exponents dominating the transition or trapping time distributions. The unified transport framework presented here comprising CTRW and MRMT shows new aspects and opens new perspectives for the modeling of transport in heterogeneous media.

Stochastical analysis of surfactant-enhanced remediation of denser-than-water nonaqueous phase liquid (DNAPL)-contaminated soils.

Stochastical analysis was performed to assess the effect of soil spatial variability and heterogeneity on the recovery of denser-than-water nonaqueous phase liquids (DNAPL) during the process of surfactant-enhanced remediation. UTCHEM, a three-dimensional, multicomponent, multiphase, compositional model, was used to simulate water flow and chemical transport processes in heterogeneous soils. Soil spatial variability and heterogeneity were accounted for by considering the soil permeability as a spatial random variable and a geostatistical method was used to generate random distributions of the permeability. The randomly generated permeability fields were incorporated into UTCHEM to simulate DNAPL transport in heterogeneous media and stochastical analysis was conducted based on the simulated results. From the analysis, an exponential relationship between average DNAPL recovery and soil heterogeneity (defined as the standard deviation of log of permeability) was established with a coefficient of determination (r2) of 0.991, which indicated that DNAPL recovery decreased exponentially with increasing soil heterogeneity. Temporal and spatial distributions of relative saturations in the water phase, DNAPL, and microemulsion in heterogeneous soils were compared with those in homogeneous soils and related to soil heterogeneity. Cleanup time and uncertainty to determine DNAPL distributions in heterogeneous soils were also quantified. The study would provide useful information to design strategies for the characterization and remediation of nonaqueous phase liquid-contaminated soils with spatial variability and heterogeneity.

Multiscale Modeling of Nonlinearly Adsorbing Solute Transport

We focus our investigations on hydrologic subsurface transport processes of a solute dissolved in groundwater. Moreover, the solute adsorbs nonlinearly on the soil matrix. Starting with transport in homogeneous media, we introduce a new methodology to identify transport parameters from dissolved solute concentration data. Transport in heterogeneous media is investigated by using a stochastic modeling approach. For mildly heterogeneous media we construct a perturbation theory around the homogeneous nonlinear theory and derive explicit expressions for asymptotic and transient large-scale transport parameters. Asymptotically, we find that the large-scale parameters are exactly the same as for linearly adsorbing transport. Differences between transport with linear and nonlinear adsorption become apparent for transient times. The characteristic time or length scales to reach the asymptotic regime depend on the grade of the nonlinearity: the smaller the Freundlich exponent p, the larger the characteristic time or travel distance. The theoretical results are verified in a semianalytical approach. Ensemble and effective dispersion coefficients are obtained from integral expressions by means only of homogeneous concentration data. We found an excellent agreement between the analytical predictions and the semianalytical results.

Stochastical Analysis of Surfactant-Enhanced Remediation of Denser-than-Water Nonaqueous Phase Liquid (DNAPL)–Contaminated Soils

Stochastical analysis was performed to assess the effect of soil spatial variability and heterogeneity on the recovery of denser-than-water nonaqueous phase liquids (DNAPL) during the process of surfactant-enhanced remediation. UTCHEM, a three-dimensional, multicomponent, multiphase, compositional model, was used to simulate water flow and chemical transport processes in heterogeneous soils. Soil spatial variability and heterogeneity were accounted for by considering the soil permeability as a spatial random variable and a geostatistical method was used to generate random distributions of the permeability. The randomly generated permeability fields were incorporated into UTCHEM to simulate DNAPL transport in heterogeneous media and stochastical analysis was conducted based on the simulated results. From the analysis, an exponential relationship between average DNAPL recovery and soil heterogeneity (defined as the standard deviation of log of permeability) was established with a coefficient of determination (r2) of 0.991, which indicated that DNAPL recovery decreased exponentially with increasing soil heterogeneity. Temporal and spatial distributions of relative saturations in the water phase, DNAPL, and microemulsion in heterogeneous soils were compared with those in homogeneous soils and related to soil heterogeneity. Cleanup time and uncertainty to determine DNAPL distributions in heterogeneous soils were also quantified. The study would provide useful information to design strategies for the characterization and remediation of nonaqueous phase liquid–contaminated soils with spatial variability and heterogeneity.

Conditional moments of the breakthrough curves of kinetically sorbing solute in heterogeneous porous media using multirate mass transfer models for sorption and desorption

A methodology is presented for evaluating the temporal moments of solutes undergoing linear rate‐limited mass transfer processes based on a Lagrangian approach to solute transport in heterogeneous media. The temporal moments of sorbing solutes are written as a function of those of conservative tracers. The general continuous diffusion rate model that has recently appeared in the hydrologic literature is used to model the rate‐limited mass transfer processes. The methodology is also applied to desorption from an initially uniformly contaminated aquifer, and the concentration expected value and variance are found quasi‐analytically. The conditional temporal moments of sorbing solutes can be written as a function of the conditional moments of conservative tracers. Conditioning results in a reduction of the variance of travel time. The amount of reduction depends on the chemical model selected.

Physical pictures of transport in heterogeneous media: Advection‐dispersion, random‐walk, and fractional derivative formulations

The basic conceptual picture and theoretical basis for development of transport equations in porous media are examined. The general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations for single realizations and for ensemble averages of the domain. The application of these transport equations is focused on accounting for the appearance of non‐Fickian (anomalous) transport behavior. The general ensemble‐averaged transport equation is shown to be equivalent to a continuous time random walk (CTRW) and reduces to the conventional forms of the advection‐dispersion equation (ADE) under highly restrictive conditions. Fractional derivative formulations of the transport equations, both temporal and spatial, emerge as special cases of the CTRW. In particular, the use in this context of Lévy flights is critically examined. In order to determine chemical transport in field‐scale situations, the CTRW approach is generalized to nonstationary systems. We outline a practical numerical scheme, similar to those used with extended geological models, to account for the often important effects of unresolved heterogeneities.

Multiresolution wavelet coarsening and analysis of transport in heterogeneous media

Modeling transport in a disordered medium with broadly distributed and correlated heterogeneities has been hampered by the fact that a very large set of equations must be solved if the heterogeneities are distributed over several distinct length scales. We recently proposed a novel method based on the application of wavelet transformations (Phys. Rev. Lett. 79 (1997) 4385) that can be used for simulation of transport processes in such disordered media. In this paper, we extend the method so as to be able to study transport in two classes of highly heterogeneous media that are otherwise very difficult to simulate, namely, disordered anisotropic media, and those that contain more than one distinct family of transport paths (for example, natural rock with both pores and fractures). We report the results of extensive computer simulations of transport in such disordered media in which the heterogeneities are broadly distributed and contain long-range correlations. We show that, even for the type of disordered media that we consider in this paper, the wavelet-based method drastically reduces (by at least two orders of magnitude) the number of equations to be solved without neglecting any important information about the media's morphology, hence providing highly accurate estimates of their transport properties. Thus, we conclude that the wavelet-based method is applicable to any disordered medium with any type of heterogeneities.

Towards a unified framework for anomalous transport in heterogeneous media

Abstract We develop a unified framework to model the anomalous transport of tracers in highly heterogeneous media. While the framework is general, our working media in this study are geological formations. The basis of our approach takes into account the different levels of uncertainty, often associated with spatial scale, in characterizing these formations. The effects on the transport of smaller spatial scale heterogeneities are treated probabilistically with a model based on a continuous time random walk (CTRW), while the larger scale variations are included deterministically. The CTRW formulation derives from the ensemble average of a disordered system, in which the transport in each realization is described by a Master Equation. A generic example of such a system – a 3D discrete fracture network (DFN) – is treated in detail with the CTRW formalism. The key step in our approach is the derivation of a physically based ψ( s ,t) , the joint probability density for a displacement s with an event-time t. We relate the ψ( s ,t) to the velocity spectrum Φ(ξ) (|ξ|=1/v, ξ = v ) of the steady flow-field in a fluid-saturated DFN. Heterogeneous porous media are often characterized by a log-normal permeability distribution; the Φ(ξ) we use in this case is an analytic form approximating the velocity spectrum derived from this distribution. The common approximation of ψ( s ,t)∼p( s )t −1−β with a constant β, is evaluated in these cases. For the former case it is necessary to include s −t coupling while the latter case points to the presence of an effective t-dependent β. The full range of these features can be included in the CTRW solution but, as is shown, not in the fractional-time derivative equation (FDE) formulation of CTRW. Finally, the methods used for the unified framework are critically examined.

Transport in heterogeneous media: Tracer dynamics in complex flow networks

AbstractLarge‐scale inhomogeneities in natural porous media profoundly affect the dynamic response of conveyed tracers. It is proposed that tracer dynamics in heterogeneous porous media can be represented by temporal moments based on parallel and series unit models of homogeneous columns, stirred volumes, or mass‐transfer‐connected closed compartments. The rules of combining the moments for elementary volumes are based on units (1) in series, where output from the preceding units is the input to the succeeding unit, and (2) in parallel, where the splitting or joining flow paths control the response. For a network of elementary volumes in series or parallel, fate and transport of contaminants in heterogeneous media can be analyzed simply in a deterministic framework. Results for several simple series and parallel models show the wealth of responses that can be quantitatively described by this procedure.

Dispersion in Heterogeneous Geological Formations

Intensive research in stochastic subsurface hydrology over the past 2 decades has significantly improved our understanding of fieldscale transport in heterogeneous media and has significantly changed how we think about variability and uncertainty However, these studies have not translated into much impact on how predictions are made in practical groundwater modeling studies. Most models used in applications still provide only deterministic predictions that essentially ignore small‐scale heterogeneity with no accompanying information on prediction uncertainty. It is reasonable to ask why most practical models don't provide stochastic field‐scale prediction even though there is a consensus that heterogeneity has a dramatic impact on plume migration and there is a well‐established theory that deals with this topic.

A Heuristic Model of Aerobic Biodegradation of Dissolved Hydrocarbons in Aquifers

AbstractIn this paper, we present a heuristic model of mixing and aerobic degradation based on a conceptual understanding of the mixing process in heterogeneous media, rather than on a rigorous derivation from the stochastic transport equations. The mean‐behavior, heuristic approach, with a uniform velocity field, was used to reproduce organic plume behavior from a previously published simulation for random pore water velocity fields. The heuristic model presented herein requires that the mixing coefficient be calibrated to explain field data; therefore, in that sense, the model is not predictive. The numerical model uses a modified form of a particle tracking model to account for multispecies transport. Transport simulation includes a random‐walk component, while biodegradation is simulated with Monod kinetics. Numerical simulations are presented for nonreactive transport in homogeneous media, and for reactive transport in heterogeneous media. A sensitivity analysis of the model is performed to determine the effect of the mixing coefficient in the model.

Solute transport in heterogeneous media: The impact of anisotropy and non-ergodicity in risk assessment

Conceptual model selection is a key issue in risk assessment studies. We analyze the effect of a number of conceptual aspects related to solute transport in two-dimensional heterogeneous media. The main issues addressed are non-ergodicity, anisotropy in the correlation structure of the transmissivity field, and dispersion at the local scale. In particular, we study the development of a solute plume when mean flow is oriented at an angle with respect to the principal directions of anisotropy. The study is carried out in a Lagrangian framework using Monte Carlo analysis. Of special interest is the evolution of individual plumes. A number of aspects are analyzed, namely the location of the center of mass for each plume and the different ways to compute the angles that the main axes of the plume develop with respect to the direction of the mean flow. Stochastic theories based upon ergodicity conclude that the plume gets oriented in the mean flow direction. In our non-ergodic simulations, the mean of the offset angles, for each individual plume in each particular realization, is offset from the mean flow direction towards the direction of maximum anisotropy. If, instead, the analysis is performed on the ensemble plume (superposition of all different simulations), it is then found oriented closer to the direction of the mean flow than the average offset angle for the different plumes considered separately. This last result adds an extra word of caution to the use of ensemble averaged values in solute transport studies. Serious implications for risk assessment follow from the conceptual model adopted. First, in any single realization there will a large uncertainty in locating the plume at any given time; second, real dilution would be less than what would be expected if the macrodispersion values obtained for ergodic conditions were applied; third, the volume that is affected by a non-zero concentration is smaller than that predicted from macrodispersion concepts; fourth, the orientation of the plume does not correspond to that of the mean flow; and fifth, accounting for local dispersion helps reducing uncertainty.

Non-linear and non-local transport processes in heterogeneous media: from long-range correlated percolation to fracture and materials breakdown

We review and discuss recent progress in modelling non-linear and non-local transport processes in heterogeneous media. The non-locality that we consider is caused by long-range correlations that either exist in the morphology of the media, or are caused by the transport processes themselves. The interplay between the non-linearity and non-locality is discussed in depth with the aim of establishing that, often non-linearity and non-locality are “two sides of the same coin”, such that one may have no meaning without the presence of the other one. First, we discuss linear and scalar, but non-local transport processes and, in particular, consider those in percolation systems with long-range correlations. It appears that there are significant differences between percolative transport processes in which the long-range correlations (or the covariance function) decrease with the distance r between two points, and those in which they increase as r does. Application of this problem to flow and transport in geological formations is discussed. We then consider linear vector percolation, one type of which, the rigidity percolation, provides an example of a non-local vector transport in heterogeneous media. Applications of vector percolation to modelling elastic properties of glasses, composite solids and rock, mechanical and viscoelastic properties of polymers, and vibrations and dynamical properties of heterogeneous materials are discussed. Non-linear and non-local scalar transport processes are discussed next, including various breakdown phenomena in disordered composites, power-law transport, piecewise linear transport characterized by a threshold, and non-linear processes that arise as a result of imposing a large external potential gradient on a heterogeneous medium. Their relevance to flow of non-Newtonian fluids in porous media, to electrical currents and dielectric breakdown in composite solids and doped polycrystalline semiconductors, and several other problems is then discussed. Finally, we discuss non-linear and non-local vector transport, the most important example of which is mechanical fracture of disordered solids. We discuss exactly solvable models of fracture, quasi-static and dynamic lattice models, molecular dynamics simulations, and continuum formulation of the problem. In all cases, the predictions of the models are compared with the relevant experimental data.

Modèle d'aide à la gestion des eaux souterraines (MAGES). 2. Validation numérique du modèle de transport

MAGES is a software developed at INRS-Eau (Canada) for forecasting pollution hazards in groundwater. The transport model uses stationary truncated temporal moment equations instead of the classical time dependent advection-dispersion equation. The aim of this work is to propose a numerical validation of the method by comparison with both analytical solutions in homogeneous medium and a sophisticated lagrangian model of transport in heterogeneous medium. It is shown that the temporal moment equations perform well, while saving on computation. This gives MAGES some abilities in water management problems which compensate for the poor knowledge of the transport parameters through numerous stochastic simulations.

Modelling Water Flow and Solute Transport in Heterogeneous Soils: A Review of Recent Approaches

A quantitative description of water flow and solute transport in the unsaturated zone of the soil is required to predict the impact of human influences on the environment. This paper starts with the basic concepts of the mathematical descriptions of transport processes in homogeneous media. However, water flow and solute transport in natural soils are significantly influenced by the occurrence of (1) macropores and structured elements (micro-heterogeneity), (2) spatial variability of soil properties (macro-heterogeneity) or (3) a combination of (1) and (2). In these cases, the classical representations of water flow and solute transport are not adequate. The paper presents an overview of some recent modelling concepts dealing with water flow and solute transport in heterogeneous media. For each model, we first introduce the underlying physical concept, and then translate the concept into a mathematical model. Each model is illustrated for a specific water flow and solute transport problem. Finally, some applications of the models are discussed. At this moment, it is difficult to specify which model should be used to solve a particular problem since no extensive validation of the models has been performed. Additional research is required to develop accurate and rapid measurement techniques for the necessary input parameters. To be useful in real environmental problems, modelling concepts for micro- and macro-heterogeneity should be coupled in one overall mathematical framework.

Probabilistic characterization of transport in heterogeneous media

The mechanics of transport and flow in a random porous medium are addressed in this paper. The hydraulic properties of the porous medium are modeled as spatial random processes. The random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented. Thus, the hydraulic processes are discretized using the spectral Karhunen — Loeve expansion. This expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes. These amplitudes form a basis in the manifold associated with the random processes. The concentrations over the whole domain are also random processes, with unknown probabilistic structure. These processes are represented using the Polynomial Chaos basis. This is a basis in the functional space described by all second order random variables. The deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator. Once the spatio-temporal variation of the concentrations has been specified in terms of the Polynomial Chaos expansion, individual realizations can be readily computed.

A Boundary Element-Finite Element Equation Solutions to Flow in Heterogeneous Porous Media

A coupled boundary element-finite element procedure, namely, the Green element method (GEM) is applied to the solution of mass transport in heterogeneous media. An equivalent integral equation of the governing differential equation is obtained by invoking the Green's second identity, and in a typical finite element fashion, the resulting equation is solved on each generic element of the problem domain. What is essentially unique about this procedure is the recognition of the particular advantages and particular features possessed by the two techniques and their effective use for the solution of engineering problems. By utilizing this approach, we observe that the range of applicability of the boundary integral methods is enhanced to cope with problems involving media heterogeneity in a straightforward and realistic manner. The method has been used to investigate problems involving various functional forms of heterogeneity, including head variations in a stream-heterogeneous aquifer interaction and in all these cases encouraging results are obtained without much difficulty.

Predicting solute transport in heterogeneous media from results obtained in homogeneous ones: an experimental approach

The aim of this paper is to show that the hydrodispersive transport parameters of a solute in a heterogeneous medium can be predicted using parameters determined by several experiments with the different homogeneous media that constitute together the heterogeneous one. Elementary rules for predicting these parameters are defined. Three types of experiments were carried out. (1) Homogeneous medium transport experiments in 1-D columns. They were fitted with the help of a new numerical particle tracking method, including advection-dispersion and exchange between mobile and immobile water with first-order kinetics. (2) Layered heterogeneous media. The experiments were simulated by the numerical model without fitting, using the parameters of each of the layers of the homogeneous media included in the heterogeneous one. The results prove that, in such layered media, the global transport problem can be considered as the convolution of elementary transports in homogeneous blocks. We will therefore conclude that the reservoir simulation methods which represent the heterogeneous reservoir as a random set of elementary homogeneous blocks can be used to represent transport if each elementary block is given its appropriate hydrodispersive parameters. (3) Homogeneous mixtures of two pure components. The experiments produced three interesting results. The immobile water volume is the direct linear combination of its equivalent in the two components, weighted by the volumetric fraction of the component in the mixture. The dispersivity of the medium also evolves linearly, but is affected by an initial sharp rise due to a strong increase in local flow path heterogeneities whenever a “foreign body” is introduced into a pure component. The exchange rate between mobile and immobile water is also perturbed by mixing. In a binary mixture of one component with dual porosity and another one without, the exchange rate decreases with the increase of the volumetric fraction of the medium without dual porosity. It seems that, even if it does not modify the volumes of immobile and mobile water, it reduces the surface/volume ratio of the solid medium and consequently the exchange rate. Mixing rules could be adopted in order to modify the values of the hydrodispersive parameters that need to be introduced for the elementary blocks of heterogeneous reservoirs.