effect of local dispersion on solute transport in randomly heterogeneous media recently an exact eulerian lagrangian theory of advective transport in space time random velocity fields was developed by one of us we present a formal extension of this theory that accounts for anisotropic local dispersion the resultant conditional mean transport equation is generally nonlocal in space time to assess the impact of local dispersion on the prediction of transport under uncertainty we adopt a first order pseudo fickian approximation for this equation we then solve it numerically by galerkin finite elements for two dimensional transport from an instantaneous square source in a uniform unconditional mean flow field subject to isotropic local dispersion we use a higher order approximation to compute explicitly the standard deviation and coefficient of variation of the predicted concentrations our theory shows in an exact manner and our numerical results demonstrate under the above closure approximations that the effect of local dispersion on first and second concentration moments varies monotonically with the magnitude of the local dispersion coefficient when this coefficient is small relative to macrodispersion its effect on the prediction of nonreactive transport under uncertainty can for all practical purposes be disregarded this is contrary to some recent assertions in the literature that local dispersion must always be taken into account no matter how small

preconditioning by fast direct methods for nonself adjoint nonseparable elliptic equations we consider the use of fast direct methods as preconditioners for iterative methods for computing the numerical solution of nonself adjoint elliptic boundary value problems we derive bounds on convergence rates that are independent of discretization mesh size for two dimensional problems on rectangular domains discretized on an n times n grid these bounds lead to asymptotic operation counts of o n 2 log n log varepsilon 1 to achieve relative error varepsilon and o n 2 log n 2 to reach truncation error

a variational multiscale method with bubble stabilization for the oseen problem based on two local gauss integrations abstract we present a variational multiscale method for the oseen equation with higher reynolds number based on the mini element discretization the stabilization term only depends on bubble functions and the best algorithmic feature of our method is using two local gauss integrations to replace projection operator without adding any extra storage a priori error estimate is proved and the effectiveness of the method is illustrated through several numerical experiments

nonasymptotic behavior of a common eulerian approximation for transport in random velocity fields lagrangian methods of analyzing solute transport in random velocity fields deal with the motion of indivisible solute particles through such fields whereas eulerian methods treat the solute concentration c as the dependent variable in a stochastic mass balance equation both approaches lead to nonlinear stochastic differential equations which are difficult to solve analytically without approximation a particular eulerian approximation which has become popular in recent years derives from the premise that in mildly fluctuating velocity fields terms involving the products of fluctuating quantities can be disregarded in comparison to terms which involve only one such quantity this leads to a non fickian low order approximation for the ensemble mean concentration c where the dispersive flux is not proportional to c but is given instead by a convolution integral the spatial moments of c based on this approximation are shown to be in conflict with those obtained from a first order lagrangian analysis we explore this apparent contradiction and conclude that the reason lies in the nonasymptotic nature of the eulerian approximation whereby terms neglected are of the same order as terms retained

subgrid stabilized projection method for 2d unsteady flows at high reynolds numbers a subgrid stabilization technique is developed for solving the two dimensional incompressible navier stokes equations at high reynolds numbers the time marching algorithm is based on a well established fractional step pressure correction projection method the advection diffusion step is enriched by an implicit subgrid stabilizing term and by an explicit dissipative shock capturing term the former is calculated by means of a hierarchical finite element setting the latter is included to avoid gibbs phenomenon in the boundary layer convergence tests on prototypical two dimensional examples are reported and the method is used to simulate the viscous incompressible flows around the airfoil naca0012 at zero incidence and reynolds numbers ranging from 105 to 106

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

a simplified two level method for the steady navier stokes equations abstract a simplified two level method for solving the steady two dimensional incompressible navier stokes equations is investigated in this article the simplified two level method involves solving one small nonlinear navier stokes problem on the coarse mesh and one linear stokes problem hence linear with positive definite symmetric part on the fine mesh the uniform optimal error estimates with respect to large for the standard galerkin method and the two level method are proven finally some numerically tests to confirm the theoretical results of the simplified two level method and the standard galerkin method are provided

new subgrid artificial viscosity galerkin methods for the navier stokes equations abstract we study subgrid artificial viscosity methods for approximating solutions to the navier stokes equations two methods are introduced that add viscous stabilization via an artificial viscosity then remove it only on a coarse mesh these methods can be considered as conforming mixed methods the first for velocity and vorticity and the second for velocity and its gradient the former incorporating a naturally arising grad div stabilization term in this paper we rigorously study the first scheme analytically showing that it is unconditionally stable and optimally convergent as well as both schemes computationally numerical experiments demonstrate the advantages of both of these methods

subgrid stabilized defect correction methods for the navier stokes equations we consider the synthesis of a recent subgrid stabilization method with defect correction methods the combination is particularly efficient and combines the best algorithmic features of each we prove convergence of the method for a fixed number of corrections as the mesh size goes to zero and derive parameter scalings from the analysis we also present some numerical tests which both verify the theoretical predictions and illustrate the method s promise

eulerian lagrangian theory of transport in space time nonstationary velocity fields exact nonlocal formalism by conditional moments and weak approximation a unified eulerian lagrangian theory is presented for the transport of a conservative solute in a random velocity field that satisfies locally v x t f x t where f x t is a random function including sources and or the time derivative of head solute concentration satisfies locally the eulerian equation c x t t j x t g x t where j x t is advective solute flux and g x t is a random source independent of f x t we consider the prediction of c x t and j x t by means of their unbiased ensemble moments c x t and j x t conditioned as implied by the subscript on local hydraulic measurements through the use of the latter in obtaining a relatively smooth unbiased estimate x t of v x t these predictors satisfy c x t v t j x t g x t where j x t x t c x t q x t and q x t is a dispersive flux we show that q is given exactly by three space time convolution integrals of conditional lagrangian kernels with q with c and with c for a broad class of v x t fields including fractals this implies that q x t is generally nonlocal and non fickian rendering c x t non gaussian the direct contribution of random variations in f to q depends on c rather than on c we elucidate the nature of the above kernels discuss conditions under which the convolution of and c becomes pseudo fickian with a lagrangian dispersion tensor similar to that derived in 1921 by taylor recall a 1952 result by batchelor which yields an exact expression for c at early time use the latter to conclude that linearizations which predict that c bifurcates at early time when the probability density function of v is unimodal cannot be correct propose instead a weak approximation which leads to a nonlinear integro differential equation for c due to an instantaneous point source and which improves with the quantity and quality of hydraulic data demonstrate that the weak approximation is analogous to the direct interaction closure of turbulence theory offer non fickian and pseudo fickian weak approximations for the second conditional moment of the concentration prediction error demonstrate that it yields the so called two particle covariance as a special case conclude that the conditional variance of c does not become infinite merely as a consequence of disregarding local dispersion and discuss how to estimate explicitly the cumulative release of a contaminant across a compliance surface together with the associated estimation error

an eulerian scheme for the second order approximation of subsurface transport moments the moments of a conservative tracer cloud migrating in a mean uniform flow field are estimated using an operator approximation scheme results are presented for the second third and fourth central moments in the mean flow direction it is assumed that the spatially variable flow field and therefore the tracer migration problem itself is amenable to a probabilistic description the effects of local dispersion on cloud migration are neglected in this study variation in the flow field is assumed to be the result of spatial variation in the hydraulic conductivity spatial variation in porosity is assumed negligible the operator approximation scheme as implemented in this study is second order correct which requires a second order correct approximation of the velocity field correlation structure because estimation of the velocity correlation structure is decidedly the most difficult aspect of second order analysis an ad hoc extension of the imperfectly stratified approximation developed earlier is implemented for this purpose the first order approximation resulting from the operator expansion scheme is equivalent to small perturbation eulerian results presented earlier naff 1990 1992 the infinite order approximation resulting from this scheme is equivalent to the exponential operator results obtained by van kampen 1976

a two level subgrid stabilized oseen iterative method for the steady navier stokes equations based on two grid finite element discretization and a recent subgrid scale model a two level subgrid stabilized oseen iterative method for the convection dominated navier stokes equations is proposed and analyzed this method combines the best algorithmic features of the two grid discretization and subgrid stabilization methods it first solves a subgrid stabilized nonlinear navier stokes problem by applying m oseen iterations on a coarse grid and then solves a linear problem on a finer grid where the nonlinear convection term is fixed by the coarse grid solution stability of the method and error estimates of the discrete solution are analyzed the algorithmic parameter scalings are also derived numerical results on an example with known analytical solution the lid driven cavity flow and the backward facing step flow are given to verify the theoretical predictions and demonstrate the method s promise

probabilistic characterization of transport in heterogeneous media abstract 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 connection between subgrid scale eddy viscosity and mixed methods we consider a new mixed method related to the evss method in computational viscoelasticity for the convection dominated convection diffusion equation in which stabilization is added then removed through the extra mixed variables this consistent stabilization is equivalent to an artificial viscosity operator acting only on the fluctuations in u h by suitable choice of the mixed spaces a method of guermond is recovered exactly we show that for a different natural choice a new method results with global error estimates similar to both guermond s method and the streamline diffusion supg method

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

arrival times and temporal moments of breakthrough curves for an imperfectly stratified aquifer using concepts developed in an earlier study a solution in laplace transform space is obtained for transport of resident concentration in an imperfectly but yet highly stratified porous medium the flow field into which an instantaneous pulse of tracer is injected is taken to be steady and mean uniform parallel to the direction of stratification from this transform space solution either temporal moments can be derived by taking derivatives with respect to the laplace parameter or the transform space solution can be inverted numerically to obtain breakthrough curves for the mean concentration when compared to an equivalent solution with a fickian dispersive flux these temporal moments indicate the extent to which transport in heterogeneous porous media deviates from classical fickian behavior the numerical inversion of the laplace transform solution gives partial breakthrough curves for the mean concentration which have the appearance of conflicting with the derived moment information a hypothesis is put forth which resolves this apparent conflict this hypothesis is verified by adding a component of local dispersion to the governing transport equation on the basis of the flux averaged concentration a form for the expected probability density function for the arrival time of a tracer particle is derived arrival time moments and an arrival time cumulative distribution function are available as a consequence arrival time moments as derived from the flux averaged concentration do not differ significantly from the resident moments as derived from the resident concentration

assessment of subgrid scale models for the incompressible navier stokes equations in this paper we assess two kinds of subgrid finite element methods for the two dimensional 2d incompressible naver stokes equations nses these methods introduce subgrid scale sgs eddy viscosity terms which do not act on the large flow structures the eddy viscous terms consist of the fluid flow fluctuation strain rate stress tensors the fluctuation tensor can be calculated by a elliptic projection or a simple l 2 projection projective filter in finite element spaces the finite element pair p 2 p 1 is adopted to numerically implement analysis and computation we give a complete error analysis based on the assumptions of some regularity conditions on the part of numerical tests the numerical computations for the stationary flows show that the numerical results agree with some benchmark solutions and theoretical analysis very well furthermore the given sgs models are applied to the non stationary fluid flows

a parallel subgrid stabilized finite element method based on fully overlapping domain decomposition for the navier stokes equations abstract based on a fully overlapping domain decomposition technique and finite element discretization a parallel subgrid stabilized method for the incompressible navier stokes equations is proposed and analyzed in this method each processor computes a local stabilized solution in its own subdomain by solving a global problem on a mesh that is fine around its own subdomain and coarse elsewhere where the stabilization term is based on an elliptic operator defined on the same mesh this method has low communication complexity it only requires the application of an existing sequential solver on the global meshes associated with each subdomain and hence can reuse the existing sequential software convergence theory of the method is developed algorithmic parameter scalings are derived numerical results are also given to verify the theoretical predictions and demonstrate the effectiveness of the method

a quasi linear theory of non fickian and fickian subsurface dispersion 1 theoretical analysis with application to isotropic media a theory is presented which accounts for nonlinearity caused by the deviation of plume particles from their mean trajectory in three dimensional statistically homogeneous but anisotropic porous media under an exponential covariance of log hydraulic conductivities existing linear theories predict that in the absence of local dispersion transverse dispersivities tend asymptotically to zero as fickian conditions are reached according to our new quasi linear theory these dispersivities ascend to peak values and then diminish gradually toward nonzero fickian asymptotes which are proportional to y4 when the log hydraulic conductivity variance y2 is much less than 1 all existing theories agree that in isotropic media the asymptotic longitudinal dispersivity is proportional to y2 when y2 1 and all are nominally restricted to mildly heterogeneous media in which this inequality is satisfied however the quasi linear theory appears to be less prone to error than linear theories when extended to strongly heterogeneous media because it deals with the above nonlinearity without formally limiting y2 it predicts that when y 1 in isotropic media both the longitudinal and transverse dispersivities ascend monotonically toward fickian asymptotes proportional to y

error correction method for navier stokes equations at high reynolds numbers in this paper we present a new iterative method for solving the stationary navier stokes equations nses at high reynolds numbers the method consists of first solving the nses by the oseen iterative scheme and then an error correction strategy is implemented to control the error arising from the linearization of the nonlinear nses the new method retains the advantage of the classical oseen scheme but it leads to a rapid rate of convergence and also enhances the capability for solving problems with higher reynolds numbers it will be shown that under the uniqueness condition the proposed method accelerates up to a factor of three in the convergence rate the stability analysis and error estimate are presented furthermore numerical simulations using the new method and other classical schemes are reported to verify the superior performance of the proposed method

an evaluation of nonlinearity in spatial second moments of ensemble mean concentration in heterogeneous porous media the integrodifferential equation for the spatial second moments x of the ensemble mean concentration in a heterogeneous aquifer is nonlinear due to statistical dependence of the particle displacement on x this nonlinear equation is either linearized or quasi linearized in previous studies to derive the linear and quasi linear theories of time dependent macrodispersion in aquifers in this study a fully nonlinear analysis is carried out by solving the integrodifferential equation for x numerically and iteratively the effects of the variance of log hydraulic conductivities y2 the local peclet number pe and the anisotropic ratio e are then investigated results show that in both statistically isotropic and anisotropic media as compared with the linear theories the effect of nonlinearity in x is to reduce the spatial longitudinal variance x11 and enhance the transverse spreading of a solute plume except in isotropic media with y2 1 where the linear theories may underestimate the longitudinal spreading of a solute it is also shown that the effect of local dispersion on x11 can be neglected when pe 10 but on the transverse macrodispersion this effect is significant for pe as large as 100 nevertheless the effect of pe on macrodispersion is secondary as compared with the effect of nonlinearity in x application of the nonlinear results shows good fits to the observed spatial variances of tracer concentration in the borden experiment and excellent agreement with the simulated variances from recent monte carlo simulations

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

on the dagan model of solute transport in groundwater foundational aspects the convection dispersion equation cde developed by dagan 1984 to predict the ensemble average concentration of a conservative solute in groundwater has been used successfully to interpret recent tracer experiments in a sand aquifer at the borden site in canada this successful application encouraged further investigation of the dagan model in respect to its physical basis and predictive characteristics it is shown in the present paper which deals with the first topic that the dagan model cde can be derived through an extension of the cumulant expansion technique applied previously by chu and sposito 1980 to develop a mean cde for solute transport in homogeneous porous media this technique also leads directly to the mean cde developed by gelhar and axness 1983 as an asymptotic large time limit and to model dispersion coefficients derived by mather on and de marsily 1980 and guven and moiz 1986 the dagan cde then is considered in detail in respect to which conceptualization of the solute concentration resident or flux the model may utilize in predictive applications general mathematical expressions relating the two conceptualizations are derived for an arbitrary solute transport problem and then are applied to the dagan model for point and prism source inputs finally the dagan model is interpreted physically using data from the recent tracer experiments of roberts and mackay 1986 at the borden site it is shown that model predictions of resident and flux concentrations are numerically indistinguishable on any time scale over which field solute concentration measurements typically are made the model prediction of dispersion coefficients on the other hand leads to dramatic differences in predicted plume behavior depending on whether finite time expressions or their asymptotic limits are used according to the dagan model the asymptotic transverse dispersion coefficient cannot replace the time dependent coefficient on any realistic time scale for solute movement at the borden site

scales of fluctuation and the propagation of uncertainty in random porous 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 heterogeneity of the medium is modeled as a superposition of scales of heterogeneity that are statistically uncorrelated the stochastic process making up a hydraulic property is thus represented as a linear combination of normalized deterministic shapes multiplied by a set of uncorrelated random variables the karhunen loeve expansion is used to formalize this representation the uncertainty contained in each of these scales is then propagated through the system using the governing differential equation as a constraint the solution process will thus consist of the original scales used to represent the porous medium as well as various interaction between these scales the objective of this paper is to develop a procedure for quantifying the uncertainty in the field variables using the scales of fluctuation of the porous medium as the measuring stick

iterative finite element variational multiscale method for the incompressible navier stokes equations abstract three iterative finite element variational multiscale methods are proposed and applied to the numerical simulation of the navier stokes equations the main idea of these methods is to combine the finite element variational multiscale method based on two local gauss integrations with three different iterative schemes the existence and uniqueness of approximate solutions of these iterative finite element variational multiscale methods are proved firstly and then the convergence and error estimates of them are deduced finally some numerical examples are given to support the theoretical analysis the numerical results show that the iterative finite element variational multiscale method has a wider range of reynolds numbers than standard galerkin iterative finite element method and the oseen iterative scheme is much more efficient than the other two under high reynolds numbers

stochastic modeling of groundwater flow by unconditional and conditional probabilities 1 conditional simulation and the direct problem mathematical modeling of groundwater flow in heterogeneous porous formations of large extent is investigated the formation properties hydraulic conductivity transmissivity as well as flow variables head specific discharge solute concentration are regarded as random variables subjected to uncertainty the main aim of the study is to analyze the influence of conditional probability of input variables upon the statistical structure of the dependent variables the unconditional probability density functions are supposed to be stationary multivariate normal while conditioning accounts for the measured values at a few points of the formation two problems of groundwater flow are discussed in part 1 conditional simulation and the direct problem for steady flow analytical results are obtained by using perturbation approximations average uniform head gradient flows as well as recharge and flow to wells are discussed in unconditional modeling the input variable the conductivity or transmissivity log is regarded as stationary and is represented by its constant mean and variogram the ensemble of formations on which statistical calculations are carried out represents all aquifers with same probability density distributions in conditional modeling the measured values at a few points are kept fixed and uncertainty prevails only at other points consequently statistical computations are performed for the subensemble of aquifers which preserve the measured values and as a result both input and output variables are nonstationary the main effect of conditioning is to reduce the variance i e the uncertainty of these variables this effect is particularly important in the case of flow toward wells application of the modeling method to field problems is outlined

a note on higher order corrections of the head covariances in steady aquifer flow average uniform flow takes place in a heterogeneous aquifer of infinite extent the input to the problem is the hydraulic conductivity which is regarded as a random space function that is lognormal stationary and statistically isotropic the output is the water head field which is a random function satisfying the equation of steady flow the first order approximation of the head in an asymptotic expansion for small log conductivity variance is a normal function characterized completely by the head log conductivity cross covariance and the head covariance or variogram these covariances are proportional to the log conductivity variance by using spectral methods second order corrections of the head covariances proportional to the log conductivity variance squared are derived explicitly detailed calculations are carried out for an exponential log conductivity covariance the main finding of the note is that the first order approximation is very robust and even for a log conductivity variance equal to unity the second order correction of the head variances is smaller than 10 of the first order approximation

stochastic theory of field scale fickian dispersion in anisotropic porous media a three dimensional theory is described for field scale fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities the study relies partly on earlier work by the authors the attributes of which are briefly reviewed it leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media we express the dispersion tensor d as the sum of a local component d and a field scale component the local component is assumed to be independent of velocity which is most appropriate if it represents molecular diffusion and its principal terms are taken to act parallel and normal to the mean velocity vector the field scale component is written as where is a dispersivity tensor and we show that at large peclet numbers p the dispersivity tensor reduces to a single principal component parallel to the mean velocity regardless of how is oriented this result valid for arbitrary covariance functions of log hydraulic conductivity differs from that of l w gelhar and c l axness 1983 according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value they calculate the direction of the largest principal dispersivity to be offset from the mean velocity toward the direction of least spatial correlation or away from the stratification in typical layered media we show that this principal dispersivity is offset in the opposite direction at small and intermediate peclet numbers but rotates toward the mean velocity as p increases the largest eigen value is constant and dominated by field scale velocity fluctuations at large p values the other two eigen values diminish asymptotically in proportion to p 1 and are controlled by d as well as by field scale differential convection the range of small peclet numbers has not been previously investigated under anisotropic conditions yet is of much importance for transport in low permeability rocks or soils we show that at low p values all three principal dispersivities are proportional to p and thus is proportional to 2 a phenomenon reminiscent of taylor diffusion when the mean velocity is inclined to the axes of anisotropy the eigen values of are neither parallel nor normal to however since d is dominated by d at small peclet numbers the principal dispersion coefficients are asymptotically as p 0 parallel and normal to the mean velocity just like when p is large their maximum deviation from these directions occurs at intermediate p values