Transport In Heterogeneous Formations Research Articles

Relevance of charge interactions for contaminant transport in heterogeneous formations: a stochastic analysis

The electrostatic properties of clay (or other charged) mineral surfaces play a significant role in the fate, transport, persistence, and remediation of subsurface contaminant plumes. This study presents a stochastic assessment of the impact and relevance of microscale electrostatic effects on macroscopic, field-scale contaminant transport in heterogeneous groundwater systems involving spatially distributed clay zones. We present Monte Carlo simulations in two-dimensional heterogeneous fields, comprising heterogeneous distributions of physical (i.e., hydraulic conductivity, porosity, tortuosity) and electrostatic (i.e., surface charge) properties, and compare scenarios with different combination and extent of physical and electrostatic processes. The simulations were performed with the multi-continua based reactive transport code, MMIT-Clay, and considering an explicit treatment of the diffuse layer processes. The results reveal that the microscopic electrostatic mechanisms within clay’s diffuse layer can significantly accelerate or retard a particular contaminant depending on its charge, leading to considerably different solute breakthroughs and mass loading/release behaviors in low permeability inclusions. Furthermore, we show that such variations in the macroscale transport behavior, solely driven by charge interactions, are statistically significant over the ensembles of Monte Carlo realizations. The simulations also demonstrate that the omission of electrostatic processes, which is still a common practice in subsurface hydrology, can lead to substantial over- or underestimation of contaminant migration.

Spatial Markov Model for the Prediction of Travel‐Time‐Based Solute Dispersion in Three‐Dimensional Heterogeneous Media

AbstractIn recent years, Spatial Markov Models have gained popularity in simulating solute transport in heterogeneous formations. They describe the transition times of particles between equidistant observation planes by statistical distributions, assuming correlation of the transit times of individual particles between subsequent steps. By this, the approach naturally captures preasymptotic solute dispersion. In this study, we analyze Spatial Markov Models assuming bivariate log‐normal distributions of the particle slowness (i.e., the inverse velocity) in subsequent transitions. The model is fully parameterized by the mean Eulerian velocity, the variance of the log‐slowness, and the correlation coefficient of log‐slowness in subsequent steps. We derive closed‐form expressions for distance‐dependent ensemble dispersion, which is defined in terms of the second‐central moments of the solute breakthrough curves. We relate the coefficients to the properties of the underlying log‐hydraulic conductivity field assuming second‐order stationarity. The results are consistent with linear stochastic theory in the limit of small log‐conductivity variances, while the approach naturally extends to high‐variance cases. We demonstrate the validity of the approach by comparison to three‐dimensional particle‐tracking simulations of advective transport in heterogeneous media with isotropic, exponential correlation structure for log‐conductivity variances up to five. This study contributes to relating solute dispersion to metrics of the porous‐medium structure in cases of strong heterogeneity.

Predicting CO2 Plume Migration in Heterogeneous Formations Using Conditional Deep Convolutional Generative Adversarial Network

AbstractNumerical simulation of flow and transport in heterogeneous formations has long been studied, especially for uncertainty quantification and risk assessment. The high computational cost associated with running large‐scale numerical simulations in a Monte Carlo sense has motivated the development of surrogate models, which aim to capture the important input‐output relations of physics‐based models but require only a fraction of the cost of full model runs. In this work, we formulate a conditional deep convolutional generative adversarial network (cDC‐GAN) surrogate model to learn the dynamic functional mappings in multiphase models. The cDC‐GAN belongs to a class of semisupervised learning methods that can be used to learn the data generation processes. Like the original GAN, a main strength of the cDC‐GAN is that it includes a self‐training scheme for improving the quality of generative modeling in a game theoretic framework, without requiring extensive statistical knowledge and assumptions on input data distributions. In particular, our cDC‐GAN model is designed to learn cross‐domain mappings between high‐dimensional input (e.g., permeability) and output (e.g., phase saturations) pairs, with the ability to incorporate conditioning information (e.g., prediction time). As a use case, we demonstrate the performance of cDC‐GAN for predicting the migration of carbon dioxide (CO2) plume in heterogeneous carbon storage reservoirs, which has both numerical and practical significance because of the safe storage requirements now mandated in many countries. Results show that cDC‐GAN achieves high accuracy in predicting the spatial and temporal evolution patterns of the injected CO2 plume, as compared to the original results obtained using a compositional reservoir simulator. The performance of cDC‐GAN models, trained using the same number of training samples, stays relatively robust when the level of spatial heterogeneity is increased. Our cDC‐GAN is pattern based and is not limited by the underlying physics. Thus, it provides a general framework for developing surrogate models, and for conducting uncertainty analyses for a wide range of physics‐based models used in both groundwater and subsurface energy exploration applications.

Dissipation-based continuation method for multiphase flow in heterogeneous porous media

In reservoir simulation, solution of the coupled systems of nonlinear algebraic equations that are associated with fully-implicit (backward Euler) discretization is challenging. Having a robust and efficient nonlinear solver is necessary in order for reservoir simulation to serve as the primary tool for managing the recovery processes of large-scale reservoirs. However, there are several outstanding challenges that are intimately connected to the highly nonlinear nature of the problem. Given a set of sources and sinks, the variation in the total velocity can span many orders of magnitude due to extreme contrasts in the permeability field in large-scale subsurface porous formations. Moreover, multiple and complex saturation fronts must be properly resolved throughout the three-dimensional reservoir model of interest. Add to that numerical simulation studies entail making field-scale predictions over many decades, and the challenge of developing robust and efficient nonlinear solvers across a very wide parameter space becomes clear. Here, we develop a continuation method based on the use of a dissipation operator. We focus on nonlinear two-phase flow and transport in heterogeneous formations in the presence of viscous, gravitational, and capillary forces. The homotopy is constructed by adding numerical dissipation to the coupled discrete conservation equations. A continuation parameter is introduced to control the amount of dissipation. Numerical evidence of multi-dimensional models and detailed analysis of single-cell problems are used to explain how the dissipation operator improves the nonlinear convergence of the coupled system of equations. An adaptive strategy to determine the dissipation coefficient is proposed. The dissipation level is computed locally for each cell interface. We demonstrate the efficiency of the dissipation-based continuation (DBC) nonlinear solver using several examples, including 1D scalar transport and 2D heterogeneous problems with fully-coupled flow and transport. The DBC solver has better convergence properties compared with the standard damped-Newton solvers used in reservoir simulation.

Longitudinal dispersion coefficients for numerical modeling of groundwater solute transport in heterogeneous formations

Most recent research on hydrodynamic dispersion in porous media has focused on whole-domain dispersion while other research is largely on laboratory-scale dispersion. This work focuses on the contribution of a single block in a numerical model to dispersion. Variability of fluid velocity and concentration within a block is not resolved and the combined spreading effect is approximated using resolved quantities and macroscopic parameters. This applies whether the formation is modeled as homogeneous or discretized into homogeneous blocks but the emphasis here being on the latter. The process of dispersion is typically described through the Fickian model, i.e., the dispersive flux is proportional to the gradient of the resolved concentration, commonly with the Scheidegger parameterization, which is a particular way to compute the dispersion coefficients utilizing dispersivity coefficients. Although such parameterization is by far the most commonly used in solute transport applications, its validity has been questioned. Here, our goal is to investigate the effects of heterogeneity and mass transfer limitations on block-scale longitudinal dispersion and to evaluate under which conditions the Scheidegger parameterization is valid. We compute the relaxation time or memory of the system; changes in time with periods larger than the relaxation time are gradually leading to a condition of local equilibrium under which dispersion is Fickian. The method we use requires the solution of a steady-state advection-dispersion equation, and thus is computationally efficient, and applicable to any heterogeneous hydraulic conductivity K field without requiring statistical or structural assumptions. The method was validated by comparing with other approaches such as the moment analysis and the first order perturbation method. We investigate the impact of heterogeneity, both in degree and structure, on the longitudinal dispersion coefficient and then discuss the role of local dispersion and mass transfer limitations, i.e., the exchange of mass between the permeable matrix and the low permeability inclusions. We illustrate the physical meaning of the method and we show how the block longitudinal dispersivity approaches, under certain conditions, the Scheidegger limit at large Péclet numbers. Lastly, we discuss the potential and limitations of the method to accurately describe dispersion in solute transport applications in heterogeneous aquifers.

Stochastic Galerkin framework with locally reduced bases for nonlinear two-phase transport in heterogeneous formations

The generalized polynomial chaos method with multiwavelet basis functions is applied to the Buckley–Leverett equation. We consider a spatially homogeneous domain modeled as a random field. The problem is projected onto stochastic basis functions which yields an extended system of partial differential equations. Analysis and numerical methods leading to reduced computational cost are presented for the extended system of equations.The accurate representation of the evolution of a discontinuous stochastic solution over time requires a large number of stochastic basis functions. Adaptivity of the stochastic basis to reduce computational cost is challenging in the stochastic Galerkin setting since the change of basis affects the system matrix itself. To achieve adaptivity without adding overhead by rewriting the entire system of equations for every grid cell, we devise a basis reduction method that distinguishes between locally significant and insignificant modes without changing the actual system matrices.Results are presented for problems in one and two spatial dimensions, with varying number of stochastic dimensions. We show how to obtain stochastic velocity fields from realistic permeability fields and demonstrate the performance of the stochastic Galerkin method with local basis reduction. The system of conservation laws is discretized with a finite volume method and we demonstrate numerical convergence to the reference solution obtained through Monte Carlo sampling.

Advective Transport in Heterogeneous Formations: The Impact of Spatial Anisotropy on the Breakthrough Curve

Water flow and solute transport take place in formations of spatially variable conductivity K. The logconductivity Y = ln K is modeled as a random stationary space function, of normal univariate pdf (of mean In KG and variance \({\sigma_{Y}^{2}}\)) and of axisymmetric autocorrelation of integral scales Ih,Iv (anisotropy ratio f = Iv/Ih 1 }\) , which is of interest for many aquifers and is more difficult to solve either numerically or by approximations. We approach the three dimensional problem by modeling the structure as an ensemble of densely packed oblate spheroids of semi-major and semi-minor axis R and f R, respectively, and independent lognormal K, submerged in a matrix of uniform conductivity Kef, the effective conductivity of the ensemble. The detailed numerical simulations of transport show that the BTC is insensitive to the value of the anisotropy ratio f, i.e., μ (t, x) Ih/U depends only on \({\sigma _{Y}^{2}}\) (except for small differences in the tail). This important result implies that transport, as quantified by BTCs or spatial longitudinal mass distributions, can be modeled accurately by the much simpler solutions developed in the past for isotropic media, like e.g., the semi-analytical self-consistent approximation.

Adaptive multiscale finite-volume method for nonlinear multiphase transport in heterogeneous formations

In the previous multiscale finite-volume (MSFV) method, an efficient and accurate multiscale approach was proposed to solve the elliptic flow equation. The reconstructed fine-scale velocity field was then used to solve the nonlinear hyperbolic transport equation for the fine-scale saturations using an overlapping Schwarz scheme. A coarse-scale system for the transport equations was not derived because of the hyperbolic character of the governing equations and intricate nonlinear interactions between the saturation field and the underlying heterogeneous permeability distribution. In this paper, we describe a sequential implicit multiscale finite-volume framework for coupled flow and transport with general prolongation and restriction operations for both pressure and saturation, in which three adaptive prolongation operators for the saturation are used. In regions with rapid pressure and saturation changes, the original approach, with full reconstruction of the velocity field and overlapping Schwarz, is used to compute the saturations. In regions where the temporal changes in velocity or saturation can be represented by asymptotic linear approximations, two additional approximate prolongation operators are proposed. The efficiency and accuracy are evaluated for two-phase incompressible flow in two- and three-dimensional domains. The new adaptive algorithm is tested using various models with homogeneous and heterogeneous permeabilities. It is demonstrated that the multiscale results with the adaptive transport calculation are in excellent agreement with the fine-scale solutions. Furthermore, the adaptive multiscale scheme of flow and transport is much more computationally efficient compared with the previous MSFV method and conventional fine-scale reservoir simulation methods.

Local-Global Two-Phase Upscaling of Flow and Transport in Heterogeneous Formations

Flow and transport in subsurface formations are affected by geological variability over multiple length scales. In this work, we develop a local-global two-phase upscaling approach to generate upscaled transport functions. The upscaling of multiphase flow parameters is challenging, due to their strong dependency on global flow effects. The local-global two-phase upscaling directly incorporates global coarse-scale two-phase solutions into local two-phase upscaling calculations. It effectively captures the impact of global flow, while avoiding global two-phase fine-scale simulations. The local boundary conditions are updated with time-dependent coarse-scale solutions. It therefore captures the global flow effects both spatially and temporally. The method is applied to permeability distributions with various correlation lengths and for different fluid-mobility ratios. Numerical results show that it consistently improves upon existing two-phase upscaling methods (e.g., upscaling with effective flux boundary c...

Probability density function of non-reactive solute concentration in heterogeneous porous formations

Available models of solute transport in heterogeneous formations lack in providing complete characterization of the predicted concentration. This is a serious drawback especially in risk analysis where confidence intervals and probability of exceeding threshold values are required. Our contribution to fill this gap of knowledge is a probability distribution model for the local concentration of conservative tracers migrating in heterogeneous aquifers. Our model accounts for dilution, mechanical mixing within the sampling volume and spreading due to formation heterogeneity. It is developed by modeling local concentration dynamics with an Ito Stochastic Differential Equation (SDE) that under the hypothesis of statistical stationarity leads to the Beta probability distribution function (pdf) for the solute concentration. This model shows large flexibility in capturing the smoothing effect of the sampling volume and the associated reduction of the probability of exceeding large concentrations. Furthermore, it is fully characterized by the first two moments of the solute concentration, and these are the same pieces of information required for standard geostatistical techniques employing Normal or Log-Normal distributions. Additionally, we show that in the absence of pore-scale dispersion and for point concentrations the pdf model converges to the binary distribution of [Dagan, G., 1982. Stochastic modeling of groundwater flow by unconditional and conditional probabilities, 2, The solute transport. Water Resour. Res. 18 (4), 835–848.], while it approaches the Normal distribution for sampling volumes much larger than the characteristic scale of the aquifer heterogeneity. Furthermore, we demonstrate that the same model with the spatial moments replacing the statistical moments can be applied to estimate the proportion of the plume volume where solute concentrations are above or below critical thresholds. Application of this model to point and vertically averaged bromide concentrations from the first Cape Cod tracer test and to a set of numerical simulations confirms the above findings and for the first time it shows the superiority of the Beta model to both Normal and Log-Normal models in interpreting field data. Furthermore, we show that assuming a-priori that local concentrations are normally or log-normally distributed may result in a severe underestimate of the probability of exceeding large concentrations.

Efficient Incorporation of Global Effects in Upscaled Models of Two-Phase Flow and Transport in Heterogeneous Formations

Geological variability occurring over multiple length scales can significantly affect fluid flow in subsurface formations. In this paper we explore the impact of large scale (global) information on the accuracy of coarse scale models for two-phase flow and transport. Based on these findings, a new methodology for generating coarse models is introduced. This approach, which avoids any global fine scale calculations, entails adaptive local-global single-phase parameter upscaling coupled with subgrid models for two-phase parameters. Two related subgrid treatments for two-phase flow effects---a pseudorelative permeability model and a generalized convection-diffusion model---are investigated and applied. The upscaled single-phase parameters (transmissibilities), computed using the adaptive local-global procedure, account explicitly for global boundary conditions as they are adapted for a specific flow scenario. The two-phase coarse scale functions are computed from local simulations using effective flux boundary conditions (EFBCs), which account approximately for global effects in the local computations. The advantages of the adaptive local-global upscaled single-phase parameters, as well as the superior accuracy of EFBCs relative to standard treatments for two-phase parameters, are demonstrated for several challenging two-dimensional example cases. The overall method is also applied to highly heterogeneous systems with global boundary conditions that vary in time. Using standard upscaling methods, errors for some of these cases are very large, though the new methodology is shown to consistently provide reasonably accurate coarse models.

On the use of block‐effective macrodispersion for numerical simulations of transport in heterogeneous formations

The fundamental question we consider in this paper is how to allow flexibility in numerical grid design without discounting the dispersive action of the unmodeled variability. In doing that, we wish to preserve the interplay between all relevant length scales: those relevant to the spatial variability as well as those created by design. In this study we extend and test the concept of block‐scale macrodispersion introduced by Rubin et al. [1999] for modeling unresolved hydraulic property variations at scales smaller than the numerical grid blocks. We present closed‐form analytical results for the block‐scale macrodispersion and test them numerically. Closed‐form analytical results are presented for the large‐time aymptotic limits, and it is shown that these limits are attained very fast. The conditions of applicability are investigated, and we show that ergodicity with regard to block‐scale heterogeneity is attained surprisingly fast.

Time‐dependent transport in heterogeneous formations of bimodal structures: 2. Results

The theoretical results of part 1 [Dagan and Fiori, 2003] for modeling time‐dependent, advective transport of a conservative solute in porous formations of bimodal structure are applied to illustrate the behavior of a few trajectory statistical moments as function of time, of the permeability contrast κ, and of the inclusions volume density n. The computations are carried out for circular (2D) and spherical (3D) inclusions to represent isotropic media. Advective transport is solved by studying the distortion of a thin plume, linear (2D) or planar (3D), normal to the mean velocity U and moving through a single inclusion. The deformation of the plume is determined from the residual trajectories of solute particles that are derived numerically by a quadrature. The longitudinal macrodispersivity is defined by αL(t; n, κ) = (2U)−1dX11/dt, where X11 is the trajectories second moment in the mean flow direction. The general behavior of the time‐dependent longitudinal dispersivity αL(t; n, κ) and, in particular, its constant, large time limit are examined. The tendency of αL to the “Fickian” limit with time depends strongly on the conductivity contrast; in particular, for low permeable inclusions (κ ≪ 1) it may be extremely slow. It is shown that the first‐order approximation in the conductivity contrast κ is of limited validity (0.3 < κ < 2). The transverse moment X22 tends asymptotically to a constant value. The analysis of the trajectory high order moments shows that the probability density function (pdf) of the solute trajectories tends to normality at large time. Similar to the “Fickian” limit the normal distribution may be reached at very large time in presence of low conductivity inclusions, with the pdf characterized by significant tailing for the trailing part of the pdf.

Time‐dependent transport in heterogeneous formations of bimodal structures: 1. The model

Flow of uniform mean velocity U takes place in a heterogeneous medium made up from a matrix of conductivity K0 and inclusions of a different conductivity K. The inclusions of given shape are implanted at random and independently in the medium, without overlapping. The aim of the study is to derive simple, approximate solutions of advective transport of solutes in such heterogeneous formations for arbitrary permeability ratio κ = K/K0 and inclusions volume fraction n. Transport is characterized by the spatial moments, which in turn are equal to the one particle trajectory statistical moments for ergodic plumes. The flow and transport problems are solved for isotropic media, for circular (2D) and spherical (3D) inclusions by using the model of composite inclusions of Hashin and Shtrikman [1962]. The results tend to the dilute limit analyzed in the past by Eames and Bush [1999] for n = o(1). Asymptotic, analytical results are derived for weak heterogeneity (κ ≃ 1); they coincide with those of Rubin [1995] for a similar structure. Similarly, simple asymptotic expressions of the macrodispersivity are derived for low‐permeability inclusions, κ = o(1). The theoretical developments are applied to effectively computing trajectories moments in part 2 [Fiori and Dagan, 2003].

Solute transport in heterogeneous formations of bimodal conductivity distribution: 1. Theory

Transport of a conservative solute takes place in a formation made up from a matrix of conductivity K0 and porosity ϑ0 and inclusions of properties K, ϑ. For given inclusions shape, the system is characterized by the two parameters κ = K/K0 and the inclusions volume fraction n. In the past, approximate solutions of the flow and transport problems were obtained under the limit of low variability, i.e., κ − 1 ≪ 1, and arbitrary n [Rubin, 1995]. The present study aims at solving the problem under the opposite limit of a dilute system, i.e., n ≪ 1 and arbitrary κ. We are particularly interested in elongated inclusions (high length/thickness ratio) of high‐permeability contrast to the matrix. Such configurations are related to applications in which lenses or cracks are present in a medium of highly different conductivity (Figure 1). The basic procedure was developed by Eames and Bush [1999] for cylindrical or spherical inclusions, with no porosity contrast. They compute the macrodispersion coefficient, for advective transport past a large number of inclusions located at random. It is based on the solution for the distortion of a material surface of marked particles, moving past an individual inclusion in an unbounded domain and with uniform flow at infinity. In the present study we extend the approach to inclusions of arbitrary porosity and elliptical shape, characterized by the parameter e, the ratio between the small and large axes, with emphasis on e ≪ 1. We present the analytical solution of the flow problem and the procedure, requiring two quadratures, to calculate the macrodispersivity. Analytical solutions are obtained for two particular limits: κ ≪1 and κ ≃ 1. The latter is compared with the limit n ≪ 1 of the solution of Rubin [1995]. The theoretical results are applied to a few cases of hydrological interest [Lessoff and Dagan, this issue].

Solute transport in heterogeneous formations of bimodal conductivity distribution: 2. Applications

The theoretical results of Dagan and Lessoff [this issue] are applied to three types of media (Figure 1): horizontal lenses submerged in a homogeneous matrix, sparse cracks of random orientation in a matrix of contrasting permeability, and channels of high permeability at the surface of a homogeneous medium. These discrete features are modeled as sparse elliptical inclusions of arbitrary conductivity. The longitudinal macrodispersivity is determined by the methodology of part 1 as function of the parameters characterizing the medium: the conductivity ratio κ, the anisotropy ratio of the ellipsis e, the porosity ratio ϑ/ϑ0, and the volume fraction n ≪ 1 or the fracture number per unit volume. Unlike existing stochastic continuum solutions that are first order in the logconductivity variance, the model developed here applies for an arbitrary permeability variance. This is of great advantage in media with high conductivity contrasts between the matrix and the inclusions. Simple results are obtained for inclusions of low conductivity that lead to high macrodispersivity values that are underpredicted by the first‐order continuum approach. In contrast, the presence of thin and highly conductive cracks leads to a finite longitudinal macrodispersivity that depends mainly on their length and the number density. An attempt is made to compare the present approach with the numerical simulations of Desbarats [1990].

A note on the influence of a constant velocity boundary condition on flow and transport in heterogeneous formations

Fluid flow and solute transport in an anisotropic, heterogeneous porous medium with mean flow normal to a constant‐flux boundary are considered. The statistical moments of flow and transport variables are determined at second order in log conductivity fluctuation, and they are expressed in terms of the log conductivity variance and integral scales, the mean flow velocity, and the distance from the boundary. The variance of the longitudinal and transverse components of velocity as well as hydraulic head variance and the longitudinal macrodispersivity are analyzed for a bounded medium with axisymmetric, Gaussian log conductivity covariance structure. In this case, all of the moments can be solved by means of a single numerical quadrature. The constant flow boundary increases the variability of head and of flow transverse to the mean flow direction and causes a reduction of the macrodispersivity in a zone adjacent to the boundary. Our results should be useful for the design and testing of numerical models and have important implications for surface infiltration of solutes.

On the Influence of Pore-Scale Dispersion in Nonergodic Transport in Heterogeneous Formations

Flow of an inert solute in an heterogeneous aquifer is usually considered as dominated by large-scale advection. As a consequence, the pore-scale dispersion, i.e. the pore scale mechanism acting at scales lower than that characteristic of the heterogeneous field, is usually neglected in the computation of global quantities like the solute plume spatial moments. Here the effect of pore-scale dispersion is taken into account in order to find its influence on the longitudinal asymptotic dispersivity D11we examine both the two-dimensional and the three-dimensional flow cases. In the calculations, we consider the finite size of the solute initial plume, i.e. we analyze both the ergodic and the nonergodic cases. With Pe the Peclat number, defined as Pe=Uλ/D, where U, λ, D are the mean fluid velocity, the heterogeneity characteristic length and the pore-scale dispersion coefficient respectively, we show that the infinite Peclat approximation is in most cases quite adequate, at least in the range of Peclat number usually encountered in practice (Pe > 102). A noteworthy exception is when the formation log-conductivity field is highly anisotropic. In this case, pore-scale may have a significant impact on D11, especially when the solute plume initial dimensions are not much larger than the heterogeneities' lengthscale. In all cases, D11 appears to be more sensitive to the pore-scale dispersive mechanisms under nonergodic conditions, i.e. for plume initial size less than about 10 log-conductivity integral scales.

Correction to “Higher‐order effects on flow and transport in randomly heterogeneous porous media”

Water Resources ResearchVolume 33, Issue 4 p. 911-912 CorrectionsFree Access Correction to “Higher-order effects on flow and transport in randomly heterogeneous porous media” Kuo-Chin Hsu, Kuo-Chin HsuSearch for more papers by this authorDongxiao Zhang, Dongxiao ZhangSearch for more papers by this authorShlomo P. Neuman, Shlomo P. NeumanSearch for more papers by this author Kuo-Chin Hsu, Kuo-Chin HsuSearch for more papers by this authorDongxiao Zhang, Dongxiao ZhangSearch for more papers by this authorShlomo P. Neuman, Shlomo P. NeumanSearch for more papers by this author First published: 01 April 1997 https://doi.org/10.1029/97WR00027AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article. References Dagan, G., An exact nonlinear correction to transverse macrodispersivity for transport in heterogeneous formations, Water Resour. Res., 3010, 2699– 2705, 1994. Volume33, Issue4April 1997Pages 911-912 ReferencesRelatedInformation

Lagrangian Analysis of Transport in Heterogeneous Formations Under Transient Flow Conditions

The coupled effects of porous media heterogeneity and flow unsteadiness on transport are investigated. We address the problem using a stochastic‐Lagrangian approach. The log condictivity is modeled as a space random function; however, the flow unsteadiness is considered as deterministic, and it affects the magnitude and direction of the mean head gradient. Using a first‐order solution in terms of the log conductivity variance, we developed a three‐dimensional solution that includes the velocity covariances, the macrodispersion coefficients, and the displacement tensor, all as a function of the media heterogeneity and the unsteady mean flow parameters. The model is applied to the case of a periodic variation in the flow direction. We show that the effects of unsteadiness on the longitudinal spread are insignificant, while it affects strongly the transverse spread, in the form of an added constant dispersivity. Results obtained in previous studies are discussed and are shown to be particular cases of the present results.