sharp l 2 error estimates and superconvergence of mixed finite element methods for non fickian flows in porous media a sharper l2 error estimate is obtained for the non fickian flow of fluid in porous media by means of a mixed ritz volterra projection instead of the mixed ritz projection used in r e ewing y lin and j wang acta math univ comenian n s 70 2001 pp 75 84 moreover local l2 superconvergence for the velocity along the gauss lines and for the pressure at the gauss points is derived for the mixed finite element method via the ritz volterra projection and global l2 superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique on the basis of the superconvergence estimates some useful a posteriori error estimators are presented for this mixed finite element method

on mass response functions field transport of reactive solute species is investigated through a class of stochastic models here termed mass response functions mrfs which incorporate simplified concepts of chemical physical nonequilibrium kinetics in the formulation of transport by travel time distributions mrfs are probability density functions pdfs associated with solute particles travel time within transport volumes the theory hinges on recent advances in modeling transport of solutes in groundwater and in basin scale transport volumes and links the approaches of surface hydrologists with recent subsurface transport models the relationship between mrfs and the theory of solute transport by continuous motions is investigated it is found that mrfs extend the basic formulation of transport of inert solutes to a particular case of sorption process the relationship between mrfs and the basic differential convection dispersion equation incorporating linear sorption is also investigated it is found here that not only are transfer functions of solutes consistent with any mechanistic three dimensional 3 d model of convection dispersion but also that they are under limit conditions the product of the travel time distribution of the carrier flow with a bounded continuous function the latter is the solution to an initial value problem which results from solving the general 3 d differential equations of convection dispersion with sorption under some simplifying assumptions and formally coincides with the resident concentration included as an assumption in the original mrf formulation travel time distributions and mrfs underlain by statistical constraints rather than by dynamical models are proposed non gaussian distributions are studied by statistical mechanical tools and are found to represent the norm rather than the exception in this formulation of transport of reactive solutes the concepts are applied to a field study and are shown to yield reliable models of solute migration in nonpoint pollution problems

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

the geostatistical characteristics of the borden aquifer a complete reexamination of sudicky s 1986 field experiment for the geostatistical characterization of hydraulic conductivity at the borden aquifer in ontario canada is performed the sampled data reveal that a number of outliers low ln k values are present in the data base these low values cause difficulties in both variogram estimation and determining population statistics the analysis shows that assuming either a normal distribution or exponential distribution for log conductivity is appropriate the classical cressie hawkins and squared median of the absolute deviations smad estimators are used to compute experimental variograms none of these estimators provides completely satisfactory variograms for the borden data with the exception of the classical estimator with outliers removed from the data set theoretical exponential variogram parameters are determined from nonlinear nl estimation differences are obtained between nl fits and those of sudicky 1986 for the classical screened estimated variogram nl fits produce an ln k variance of 0 24 nugget of 0 07 and integral scales of 5 1 m horizontal and 0 21 m vertical along a a for b b these values are 0 37 0 11 8 3 and 0 34 the fitted parameter set for b b data horizontal and vertical is statistically different than the parameter set determined for a a we also evaluate a probabilistic form of dagan s 1982 1987 equations relating geostatistical parameters to a tracer cloud s spreading moments the equations are evaluated using the parameter estimates and covariances determined from line a a as input with a velocity equal to 9 0 cm day the results are compared with actual values determined from the field test but evaluated by both freyberg 1986 and rajaram and gelhar 1988 the geostatistical parameters developed from this study produce an excellent fit to both sets of calculated plume moments when combined with dagan s stochastic theory for predicting the spread of a tracer cloud

the frisch scheme in dynamic system identification abstract the use of the frisch scheme in the identification of linear dynamic systems is investigated in order to describe the whole family of models that can explain given input output noisy sequences unlike the algebraic case it is shown that in general only a single model is compatible with the data these results are first proposed for single input single output systems and then generalized to the multivariable case

stochastic modeling of macrodispersion in heterogeneous porous media a method for the prediction of dispersion processes of inert solutes occurring in heterogeneous porous media is presented its main features are as follows 1 the velocity field is assumed to be a space random function its moments are expressed through the physical parameters of the log transmissivity and head random fields by linearizing the flow equation the random velocity being a linear function of the head and log transmissivity which are assumed to be jointly multivariate normal mvn is thus also mvn its pdf is completely defined by its first two moments 2 the statistics of the dispersion process are obtained by the particle tracking method through monte carlo simulations which are based on gaussian conditioning the method is applied to the borden natural gradient tracer test freyberg 1986 showing favorable results

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

estimation of spatial covariance structures by adjoint state maximum likelihood cross validation 1 theory this series of three papers describes a cross validation method to estimate the spatial covariance structure of intrinsic or nonintrinsic random functions from point or spatially averaged data that may be corrupted by noise any number of relevant parameters including nugget effect can be estimated the theory described in this paper is based on a maximum likelihood approach which treats the cross validation errors as gaussian various a posteriori statistical tests are used to verify this hypothesis and to show that in many cases correlation between these errors is weak the log likelihood criterion is optimized through a combination of conjugate gradient algorithms an adjoint state theory is used to efficiently calculate the gradient of the estimation criterion optimize the step size downgradient and compute a lower bound for the covariance matrix of the estimation errors issues related to the identifiability stability and uniqueness of the resulting adjoint state maximum likelihood cross validation asmlcv method are discussed this paper also describes the manner in which asmlcv allows one to use model structure identification criteria to select the best covariance model among a given set of alternatives practical aspects of asmlcv and its application to synthetic data are presented in paper 2 samper and neuman this issue a applications to real hydrogeological data transmissivities and groundwater levels have been presented elsewhere while hydrochemical and isotopic data are analyzed by asmlcv in paper 3 samper and neuman this issue b

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

solute transport in highly heterogeneous aquifers the paper deals with the transport of nonreactive solute in heterogeneous formations with prescribed statistical properties of the hydraulic log conductivity y ln k available solutions obtained from perturbation methods are limited to first and second order solutions valid only for small values of the log conductivity variance 2y published numerical investigations give comparative results with finite values of 2y but some discrepancies among the results generate doubts about the capability of numerical methods to capture high order effects for media with large variance values when these large 2y values are encountered in natural formations the related nonlinear effects could be significant in the velocity statistics and in the overall dispersion process the nonlinearity consequences are here investigated in two dimensional isotropic porous media by the monte carlo technique coupled with a finite element analysis the analysis includes 2y values from 0 05 to 4 enhancing the relevance of the nonlinear effects in the dispersion tensor solution to dissipate the doubts related to the numerical approach the accuracy of the solutions was defined by checking the influence of the factors which can affect the solution and by giving an estimation of the related errors the numerical results confirm the validity of the first order and second order analyses when 2y 0 the second order solution captures the nonlinear effects in a small range of log conductivity variance close to zero the nonlinear terms neglected in the first order formulation for higher 2y values give 1 late travel time longitudinal dispersion values greater than the linear solution 2 notably non gaussian distribution of the lagrangian velocity and particle displacements and 3 travel times to approach the asymptotic fickian regime longer than those obtained using the linear solution

nonergodic solute transport in physically and chemically heterogeneous porous media 1 the time dependent ensemble average of the first order longitudinal second spatial moment s11 r of nonergodic plumes of a sorbing solute in linear equilibrium and the effective longitudinal dispersivity 11 r were evaluated for the line sources of different lengths normal or parallel to the uniform mean velocity in two and three dimensional physically and chemically heterogeneous and statistically isotropic media physical and chemical heterogeneity is described with a random hydraulic conductivity field k x and a random retardation factor field r x respectively the retardation factor is defined as 1 kd x where kd x is the distribution coefficient the fields k x and kd x are assumed to be lognormally distributed and correlated three correlation models between y log k and w log kd are considered the perfectly positive correlation model a the perfectly negative correlation model b and no correlation model c the values of z11 r which is defined as s11 r minus its initial value and 11 r for all three correlation models increase as the length of a line source l increases as l z11 r or 11 r approach their respective ergodic limits x11 r or 11 r the asymptotic longitudinal dispersion due to the physical and chemical heterogeneity is fickian for a line source normal to and non fickian for a line source parallel to the positive correlation between y and w reduces z11 r and 11 r the negative correlation enhances z11 r and 11 r and no correlation slightly enhances z11 r and 11 r in comparison with nonreactive z11 and 11 the larger the value of the mean retardation factor the large its effect on z11 r and 11 r the three dimensional z11 r and 11 r are slightly larger than their respective two dimensional counterparts

higher order effects on flow and transport in randomly heterogeneous porous media advective solute transport in nonuniform geologic media is generally nonlocal and non fickian neuman 1993 cushman and ginn 1993 in statistically homogeneous log conductivity fields under uniform mean flow the transport is expected to become asymptotically local and fickian at late time during the earlier preasymptotic regime macrodispersivity a measure of the rate at which a plume spreads is expected to vary with solute residence time a first order linear in the natural log hydraulic conductivity variance 2 analysis of this variation has been performed by dagan 1984 1987 1988 he found that when local dispersion is neglected the longitudinal macrodispersivity increases monotonically from zero toward a constant asymptote however the transverse macrodispersivity first increases from zero to a peak value then decreases monotonically toward zero the first order asymptotic analyses by winter 1982 gelhar and axness 1983 and winter et al 1984 also yield zero transverse macrodispersivity when local dispersion is disregarded

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

inversion of the borden tracer experiment data investigation of stochastic moment models inversion of dagan s two and three dimensional stochastic models using freyberg s 1986 rajaram and gelhar s 1988 1991 and barry and sposito s 1990 moment data from the borden experiment is carried out to examine 1 the validity of the two dimensional dagan 1982 and three dimensional dagan 1988 models and 2 the reduction in uncertainty of the spatial moments over nonconditioned estimates a direct application of bayesian statistical inference in conjunction with monte carlo integration is used to produce posterior probability density functions for the parameters the parameter ranges from all methods show horizontal integral scales between 1 85 and 4 04 m vertical integral scales z between 0 144 and 0 459 m and log hydraulic conductivities in k between 0 120 and 0 197 these results compare well to the earlier estimates of woodbury and sudicky 1991 and robin et al 1991 we show that moment prediction uncertainty is substantially reduced when both tracer moment data and prior estimates of the in k geostatistical parameters are incorporated into theoretical formulae based on stochastic dispersion theory

stochastic subsurface hydrology from theory to applications research on stochastic analysis of subsurface flow has developed rapidly in the last decade but applications of this approach have been very limited the purpose of this paper is to illustrate how currently available techniques and results can be used to answer important questions about the large scale behavior of naturally heterogeneous aquifers perturbation based spectral theory which presumes local statistical homogeneity provides generic theoretical results for the head variance effective conductivity tensor and macrodispersivity tensor in a field these results emphasize the key role of the variance and spatial correlation scales of the log hydraulic conductivity field field information of variances and correlation scales of natural materials is summarized the validity of some of the generic stochastic results is evaluated through comparisons with monte carlo simulations and field observations a specific field application example is developed to illustrate how the stochastic results are used to estimate large scale parameters and determine the reliability of three dimensional numerical simulations using typical log conductivity covariance parameters the effective hydraulic conductivity tensor and the macrodispersivity tensor are estimated the calculated head variance based on the simulated mean hydraulic gradient is used as a measure of the adequacy of the calculation of the steady state flow model discussion emphasizes limitations and extensions of this approach and ongoing field evaluations of the results

proper representations of multivariate linear differential systems abstract a proper representation of a linear differential system is a representation with no singularity at infinity it is shown that such a representation always exists it turns out that for proper representations having minimal number of rows is equivalent to having minimal total row degree one is led therefore to a natural definition of the notion of minimality what is remarkable is that a minimal proper representation is uniquely determined up to premultiplication by a unimodular polynomial matrix of special form this uniqueness result allows in particular to introduce important integer invariants

numerical assessment of 3 d macrodispersion in heterogeneous porous media 1 hydrodynamic dispersion is a key controlling factor of solute transport in heterogeneous porous media that critically depends on dimensionality the transverse macrodispersion asymptotic dispersion transverse to the mean velocity direction is known to vanish only in 2 d and not in 3 d using classical gaussian correlated permeability fields with a lognormal distribution of variance y2 we determine numerically the longitudinal and transverse dispersivities as functions of heterogeneity and dimensionality we show that the transverse macrodispersion steeply increases with y2 underlying the essential role of flow lines braiding a mechanism specific to 3 d systems that we qualitatively characterize by the increasing expansion of the flow lines transversally to the flow direction the transverse macrodispersion remains however at least two orders of magnitude smaller than the longitudinal macrodispersion which increases even more steeply with y2 at moderate to high levels of heterogeneity the transverse dispersion also converges much faster to its asymptotic regime than do the longitudinal dispersion braiding cannot be thus taken as the sole mechanism responsible for the high longitudinal macrodispersions it could be either supplemented or superseded by stronger velocity correlations in 3 d than in 2 d this assumption is supported by the much larger longitudinal macrodispersions obtained in 3 d than in 2 d up to a factor of 5 for y2 6 25

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

reduced polynomial matrices in several variables we define a higher dimensional analogue of a column reduced polynomial matrix called a reduced polynomial resolution as an exact sequence of polynomial matrices having the predictable degree property a reduced polynomial resolution is associated to any linear time invariant differential system it is uniquely determined up to isomorphism and provides a number of important integer invariants of the system

plume scale dependent dispersion in heterogeneous aquifers 1 lagrangian analysis in a stratified aquifer an analytical approach is developed for describing the ensemble average of the second moment of a solute plume in heterogeneous porous media the growth of the ensemble average second moment is related to the growth of the mean square separation of a pair of particles exact lagrangian expressions are developed for the growth of the mean square separation in a perfectly stratified aquifer these exact expressions are made possible by an exact relation between the lagrangian and eulerian velocity covariances in the perfectly stratified aquifer the ensemble average second moment is shown to depend on the initial vertical dimensions of the concentration distribution as the vertical size increases the second moment growth rate is larger the second moment expressions are contrasted with the expressions for the mean square displacement of a single particle while all scales of heterogeneity contribute to the mean square displacement of a single particle only scales of heterogeneity smaller than the plume size contribute to the second moment asymptotic large displacement expressions for the case of a gaussian covariance function describing the hydraulic conductivity variations are derived these expressions indicate that at large displacement the second moment grows as the 3 2 power of displacement however for any finite source size no matter how large the second moment growth rate approaches a different asymptote from the rate of growth of the mean square displacement of a single particle this is in conflict with the traditional notion of ergodicity which leads to the expectation that for large source sizes the dispersivity approaches the rate of growth of the mean square displacement of a single particle

temporal behavior of a solute cloud in a heterogeneous porous medium 3 numerical simulations 1 the article presents systematic numerical simulations of the temporal behavior of a passive solute in a saturated three dimensional heterogeneous medium the groundwater flow is derived from the linearized solution of the darcy equation with gauss distributed log hydraulic conductivity the transport of a passive solute is studied by a random walk method which allows for a systematic study of the temporal behavior of the effective and ensemble dispersion coefficients the numerical results are compared to the second order perturbation theory expressions given in two companion papers dentz et al 2000a 2000b and to nonperturbative results which follow from corrsin s conjecture the low order perturbation theory is intrinsically based on the assumption of small heterogeneity while corrsin s conjecture does not take into account certain contributions due to higher order terms of the perturbation series the simulations yield for the first time systematic quantitative information on the validity and the limitations of these analytic approximations for increasing heterogeneities considerable deviations from the theoretically predicted transport behavior are observed

large time behavior of macrodispersion in heterogeneous trending aquifers 1 this paper presents a stochastic analysis of the large time behavior of macrodispersion in a three dimensional heterogeneous aquifer with a linear trend in the mean log hydraulic conductivity to solve the problem analytically focus is placed on the particular case where the linear trend is aligned in the direction of mean hydraulic head gradient a spectral approach based on fourier stieltjes representations for the perturbed quantities is used to develop closed form expressions that describe variability of flow velocity the second order mean flow and asymptotic macrodispersion the impact of the mean log hydraulic conductivity gradient on these results is examined it is found that the asymptotic longitudinal and transverse macrodispersion coefficients decrease with the increasing trend gradient of mean log hydraulic conductivity in the case of finite peclet numbers this feature is a consequence of the reduction in variability of flow velocity with the trend gradient

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

stochastic system identification with noisy input output measurements using polyspectra two new classes of parametric frequency domain approaches are proposed for estimation of the parameters of scalar linear errors in variables models i e linear systems where measurements of both input and output of the system are noise contaminated the first approach consists of linear estimators where using the bispectrum or the integrated polyspectrum of the input and the cross bispectrum or the integrated cross polyspectrum the system transfer function is first estimated at a number of frequencies exceeding one half the number of unknown parameters the estimated transfer function is then used to estimate the unknown parameters using an overdetermined linear system of equations in the second class of approaches quadratic transfer function matching criteria are optimized by using the results of the linear estimators as initial guesses both classes of the parameter estimators are shown to be consistent in any measurement noise that has symmetric probability density function when the bispectral approaches are used the proposed parameter estimators are shown to be consistent in gaussian measurement noise when trispectral approaches are used

the influence of pore scale dispersion on concentration statistical moments in transport through heterogeneous aquifers transport of an inert solute in a heterogeneous aquifer is governed by two mechanisms advection by the random velocity field v x and pore scale dispersion of coefficients ddij the velocity field is assumed to be stationary and of constant mean u and of correlation scale i much larger than the pore scale d it is assumed that ddij diju are constant the relative effect of the two mechanisms is quantified by the peclet numbers peij u ddij i dij which as a rule are much larger than unity the main aim of the study is to determine the impact of finite though high pe on c and c2 the concentration mean and variance respectively the solution derived in the past for pe is reconsidered first by assuming a normal x probability density function p d f closed form solutions are obtained for c and c2 recasting the problem in an eulerian framework leads to the same results if certain closure conditions are adopted the concentration moments for a finite pe are derived subsequently in a lagrangean framework the pore scale dispersion is viewed as a brownian motion type of displacement xd of solute subparticles of scale smaller than d added to the advective displacements x by adopting again a normal p d f for the latter explicit expressions for c and c2 are obtained in terms of quadratures over the joint p d f of advective two particles trajectories while the influence of high pe on c is generally small it has a significant impact on c2 simple results are obtained for a small v0 for which trajectories are fully correlated in particular the concentration coefficient of variation at the center tends to a constant value for large time comparison of the present solution obtained in terms of a quadrature with the monte carlo simulations of graham and mclaughlin 1989 shows a very good agreement

a nonstationary spectral method for solving stochastic groundwater problems unconditional analysis stochastic analyses of groundwater flow and transport are frequently based on partial differential equations which have random coefficients or forcing terms analytical methods for solving these equations rely on restrictive assumptions which may not hold in some practical applications numerically oriented alternatives are computationally demanding and generally not able to deal with large three dimensional problems in this paper we describe a hybrid solution approach which combines classical fourier transform concepts with numerical solution techniques our approach is based on a nonstationary generalization of the spectral representation theorem commonly used in time series analysis the generalized spectral representation is expressed in terms of an unknown transfer function which depends on space time and wave number the transfer function is found by solving a linearized deterministic partial differential equation which has the same form as the original stochastic flow or transport equation this approach can accomodate boundary conditions spatially variable mean gradients measurement conditioning and other sources of nonstationarity which cannot be included in classical spectral methods here we introduce the nonstationary spectral method and show how it can be used to derive unconditional statistics of interest in groundwater flow and transport applications

fuzzification of the miscible displacement model in heterogeneous porous media abstract typical realistic models for multiphase flow in heterogeneous formations are complex and random and must incorporate the uncertainties inherent to the mixture process these uncertainties can be modeled using differential equations coefficients such as hydrodynamic dispersivity in this work the mathematical model is expressed in terms of a nonlinear coupled system of stochastic partial differential equations a second order elliptic equation for the pressure and a hyperbolic dominated transport diffusion equation for the solvent concentration in the mixture besides the longitudinal dispersion coefficient is a fuzzy number new perspective on the quantification of uncertainty for parameter estimation problems by means of numerical simulations and membership functions is the purpose of this research in this way a fuzzification of the semiclassical solution numerical approximation is built in this regard it is proved the continuity of the function that assigns the 3 tuple comprised by longitudinal dispersion transverse dispersion and molecular diffusion to the corresponding value of the semiclassical solution at a fixed point of the domain the continuity result along with zadeh s extension principle is applied to obtain the fuzzification the relevance of this study resides in the novelty of the methodology that considers a model parameter as a fuzzy number meanwhile it is usually taken as a constant in literature other unprecedented result lays in the discovery of the link between theoretical concepts and numerical approximations to obtain a fuzzification

scale and directional dependence of macrodispersivities in colonnade networks macrodispersion in discrete colonnade network models is investigated using a series of monte carlo type numerical experiments the numerical simulations consider fluid flow and advective transport through a square flow region of a two dimensional network of hexagonal colonnades macrodispersivities are calculated as a function of the scale and orientation of the square flow region within the larger parent geometry particle tracking with flux weighted tracer injection and monitoring is used to generate experimental residence time distributions rtds the method of moments is used to characterize the longitudinal tracer spreading the simulated rtds are utilized to examine the macrodispersive behavior in colonnade networks column diameter 1 m with lognormally distributed fracture apertures b in millimeters the network models are assumed to consist of open fractures with ln b 1 945 and ln b 0 896 this translates to an equivalent continuum log conductivity variance ln k2 of 3 67 based on an ensemble average of 100 realizations the slope for spatial variance versus scale of observation ranges from 3 15 to 3 36 m for varying orientations of the hydraulic gradient for ensemble averaged data the varying orientations appear to have little effect on the macrodispersive behavior for single realization experiments the computed macrodispersivities are directionally dependent at a length scale as large as 30 times the column diameter and probably much beyond the computed asymptotic and preasymptotic macrodispersivities are compared with available stochastic solutions for two dimensional isotropic heterogeneity in the horizontal plane the ensemble based numerical data are in excellent agreement with the solutions of dagan 1984 1988 and gelhar and axness 1983 however for individual realizations nonergodic behavior is clearly apparent in the near source evolving region of transport and the numerical data are quite variable between realizations the study provides important insight on applicability of stochastic continuum theories to discrete colonnade network models having ln k2 much greater than 1

deterministic and stochastic analyses of dispersion in an unbounded stratified porous medium the dispersion of a conservative solute released instantaneously from a finite or point source in an unbounded nonrandom periodically stratified porous medium is examined theoretically by applying the moment method of r aris 1956 and p g saffman 1962 the governing moment equations are derived for a general stratified medium and then applied to study the detailed time dependent variations of various low order spatial moments of the three dimensional concentration distribution in a particular periodic sinusoidal stratified medium both for the general case when the mean flow direction is inclined to the stratification as well as for the special case of flow parallel to the stratification the present results confirm the previous results of v k gupta and k n bhattacharya 1986 regarding the asymptotic large time large scale dispersion coefficients for such a periodic medium the present results which are obtained through a deterministic analysis are compared also with the results which would be obtained by using the previous stochastic theories of g matheron and g demarsily 1980 and l w gelhar et al 1979 the comparison reveals several similarities as well as some important differences between the results of the deterministic and stochastic analyses

stochastic diagrammatic analysis of groundwater flow in heterogeneous porous media the diagrammatic approach is an alternative to standard analytical methods for solving stochastic differential equations governing groundwater flow with spatially variable hydraulic conductivity this approach uses diagrams instead of abstract symbols to visualize complex multifold integrals that appear in the perturbative expansion of the stochastic flow solution and reduces the original flow problem to a closed set of equations for the mean and the covariance functions diagrammatic analysis provides an improved formulation of the flow problem over conventional first order series approximations which are based on assumptions such as constant mean hydraulic gradient infinite flow domain and neglect of cross correlation terms this formulation includes simple schemes like finite order diagrammatic perturbations that account for mean gradient trends and boundary condition effects as well as more advanced schemes like diagrammatic porous media description operators which contain infinite order correlations in other words diagrammatic analysis covers not only the cases where low order diagrams lead to good approximations of flow but also those situations where low order perturbation is insufficient and a more sophisticated analysis is needed diagrams lead to a nonlocal equation for the mean hydraulic gradient in terms of which necessary conditions are formulated for the existence of an effective hydraulic conductivity three dimensional flow in an isotropic bounded domain with dirichlet boundary conditions is considered and an integral equation for the mean hydraulic head is derived by means of diagrams this formulation provides an explicit expression for the boundary effects within the three dimensional flow domain in addition to these theoretical results the numerical performance of the diagrammatic approach is tested and useful insight is obtained by means of one dimensional flow examples where the exact stochastic solutions are available

stochastic analysis of nonstationary subsurface solute transport 1 unconditional moments this paper applies stochastic methods to the analysis and prediction of solute transport in heterogeneous saturated porous media partial differential equations for three unconditional ensemble moments the concentration mean concentration covariance and velocity concentration cross covariance are derived by applying perturbation techniques to the governing transport equation for a conservative solute concentration uncertainty is assumed to be the result of unmodeled small scale fluctuations in a steady state velocity field the moment expressions which describe how each moment evolves over time and space resemble the classic deterministic advection dispersion equation and can be solved using similar methods a solution procedure based on a galerkin finite element algorithm is illustrated with a hypothetical two dimensional example for this example the required steady state velocity statistics are obtained from an infinite domain spectral solution of the stochastic groundwater flow equation the perturbation solution is shown to reproduce the statistics obtained from a monte carlo simulation quite well for a natural log conductivity standard deviation of 0 5 and moderately well for a natural log conductivity standard deviation of 1 0 the computational effort required for a perturbation solution is significantly less than that required for a monte carlo solution of acceptable accuracy sensitivity analyses conducted with the perturbation approach provide qualitative confirmation of a number of results obtained by other investigators for more restrictive special cases

self consistent four point closure for transport in steady random flows ensemble averaging of advection dispersion equations describing transport of a passive scalar in incompressible random velocity fields requires a closure approximation commonly used approaches such as the direct interaction approximation and large eddy simulations as well as equivalent renormalization schemes employ so called two point or one loop closures these approaches have proven to be adequate for transport in zero mean unbiased time dependent random velocity fields with increasing accuracy for decreasing temporal coherence in the opposite limit of steady velocity fields with finite bias however these schemes fail to predict effective transport properties both quantitatively and qualitatively leading to an obvious inconsistency for transverse dispersion in two spatial dimensions for this case two point closures predict that macroscopic transverse dispersion increases as the square root of the disorder variance while it has been proven rigorously that there is no disorder induced contribution to macroscopic transverse dispersion for purely advective transport furthermore two point closures significantly underestimate the disorder induced contribution to longitudinal dispersion we derive a four point closure for stochastically averaged transport equations that goes beyond classical one loop schemes and demonstrate that it is exact for transverse dispersion and correctly predicts an increase of the longitudinal disorder induced dispersion coefficient with the square of the variance of the strong disorder the predicted values of asymptotic longitudinal dispersion coefficients are consistent with those obtained via monte carlo random walk simulations

universal scaling of hydraulic conductivities and dispersivities in geologic media an interpretation is offered for the observation that dispersivities increase with scale apparent longitudinal dispersivity data from a variety of hydrogeologic settings are assumed to represent a continuous hierarchy of log hydraulic conductivity fields with mutually uncorrelated increments each field having its own exponential autocovariance associated integral scale and variance that increases as a power of scale such a hierarchy is shown theoretically to form a self similar random field with homogeneous increments regardless of whether or not the underlying assumption is valid one can justify interpreting the apparent dispersivities in a manner consistent with a recent quasi linear theory of non fickian and fickian dispersion in homogeneous media which supports the notion of a self similar hierarchy a posteriori the hierarchy is revealed to possess a semivariogram s cs where c is a constant and a fractal dimension d e 0 75 where e is the topological dimension of interest this can be viewed as a universal scaling rule about which large deviations occur due to local influences including the existence of discrete natural scales at which log hydraulic conductivity is statistically homogeneous as such homogeneity is at best a local phenomenon occurring intermittently over narrow bands of the scale spectrum one must question the utility of associating medium properties with representative elementary volumes and relying on fickian models of dispersion over more than relatively narrow scale intervals porous and fractured media appear to follow the same idealized scaling rule for both flow and transport raising a question about the validity of many distinctions commonly drawn between such media finally the data suggest that conditioning transport models through calibration against hydraulic measurements has the effect of filtering out large scale modes from the hierarchy

an exact nonlinear correction to transverse macrodispersivity for transport in heterogeneous formations the random two dimensional and steady in time eulerian velocity field is assumed to be given it pertains to flow through heterogeneous porous media of spatially variable permeability advective transport of a passive scalar is modeled by the lagrangian approach by seeking the statistical moments of the trajectory of a tagged particle the trajectories variances and the associated dispersion coefficients are evaluated by a perturbation expansion in the velocity variances and by corrsin s conjecture first order results obtained in the past show that for flow through porous formations of log permeability two point covariance of finite integral scale the transverse dispersion coefficient tends asymptotically to zero in contrast corrsin s conjecture leads to a finite dispersion coefficient quadratic in the velocity variance to investigate this effect an exact nonlinear correction to the transverse dispersion coefficient is derived for a normal velocity field it is shown that this term tends to zero asymptotically in contrast with the result based on corrsin s conjecture furthermore an illustrative computation shows the transport related nonlinear correction to be small at any time

inertial effects in dispersion in porous media 1 in this work we develop the macroscale transport equation for dispersion of a nonreactive chemical species with a particular focus on the influence of inertial contributions at moderate reynolds numbers our starting point is the continuum level description of transport written at the subpore scale volume averaging is used to upscale these equations to develop the macroscale solute balance that applies at the darcy scale we develop a fully transient version of the ancillary closure problem that predicts the total dispersion tensor and we solve the closure using finite fourier transforms the result of this effort is a nonlocal macroscale transport equation where the nonlocal dispersion depends upon the microscale geometry of the pore space and the physical characteristics of the fluid both the longitudinal and transverse components of the total dispersion tensor are computed for a simple three dimensional unit cell the computational results indicate that a simple three dimensional periodic unit cell is able to capture the correct behavior for the longitudinal dispersion in the range 101 pep 2 5 105 although the magnitude of the longitudinal dispersion coefficient is underpredicted by up to a factor of about 4 for the transverse dispersion coefficient the simple unit cell provides much less satisfactory results when compared with experimental data the inertial effects for the longitudinal dispersion coefficient were relatively small but for the transverse dispersion coefficient inertial effects were predicted to increase the transverse dispersion coefficient up to 40 times that which would be predicted for stokes flow

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

the spatial and temporal moments of tracer concentration in disordered porous media this study develops a methodology for deriving the spatial and temporal moments of tracer concentration in disordered geological formations and suggests a geostatistical formalism for interpolation and prediction of concentration conditional to concentration measurements the required spatial and temporal moments of the concentration are obtained by a biparticle tracking scheme which yields the bivariate probability distribution function of particle displacements in terms of the input log conductivity random field and the prevailing flow conditions the structure of the moments of the concentration is discussed and we show that 1 the concentration correlation function is anisotropic nonstationary and nonsymmetrical 2 there is a larger correlation between concentrations along the mean flow direction than normal to it 3 the correlation between the concentration at any two points with fixed coordinates on a moving lagrangian coordinate system increases with travel time 4 the correlation function necessarily assumes negative values beyond a certain threshold distance and 5 the concentration variance is finite even when neglecting pore scale dispersion application of the geostatistical prediction algorithm using the concentration spatial temporal moments shows a good potential for reducing prediction uncertainty

brief paper accuracy analysis of time domain maximum likelihood method and sample maximum likelihood method for errors in variables and output error identification for identifying errors in variables models the time domain maximum likelihood tml method and the sample maximum likelihood sml method are two approaches both methods give optimal estimation accuracy but under different assumptions in the tml method an important assumption is that the noise free input signal is modelled as a stationary process with rational spectrum for sml the noise free input needs to be periodic it is interesting to know which of these assumptions contain more information to boost the estimation performance in this paper the estimation accuracy of the two methods is analyzed statistically for both errors in variables eiv and output error models oem numerical comparisons between these two estimates are also done under different signal to noise ratios snrs the results suggest that tml and sml have similar estimation accuracy at moderate or high snr for eiv for oem identification these two methods have the same accuracy at any snr

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

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

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

hydraulic conductivity velocity and the order of the fractional dispersion derivative in a highly heterogeneous system 1 a one dimensional fractional order advection dispersion equation accurately models the movement of the core of the tritium plume at the highly heterogeneous made site an a priori estimate of the parameters in that equation including the order of the fractional dispersion derivative was based on the assumption that the observed power law heavy tail of the hydraulic conductivity k field would create a similarly distributed velocity field monte carlo simulations were performed to test this hypothesis results from the monte carlo analysis show that heavy tailed k fields do give rise to heavy tailed velocity fields however the exponent of the power law the tail parameter describing these two distributions is not necessarily the same the tail parameter that characterizes a velocity distribution is not solely dependent on the tail parameter that characterizes the k distribution the k field must also have long range dependence so that water may flow through relatively continuous high k channels

eulerian lagrangian analysis of transport conditioned on hydraulic data 1 analytical numerical approach recently a unified eulerian lagrangian theory has been developed by one of us for nonreactive solute transport in space time nonstationary velocity fields we describe a combined analytical numerical method of solution based on this theory for the special case of steady state flow in a mildly fluctuating statistically homogeneous lognormal hydraulic conductivity field we take the unconditional mean velocity to be uniform but allow conditioning on measurements of log hydraulic conductivity or transmissivity and or hydraulic head this renders the velocity field nonstationary we solve the conditional transport problem analytically at early time and express it in pseudo fickian form at later time the deterministic pseudo fickian equations involve a conditional space time dependent dispersion tensor which we evaluate numerically along mean particle trajectories these equations lend themselves to accurate solution by standard galerkin finite elements on a relatively coarse grid the final step is an explicit numerical computation of lower bounds on conditional concentration prediction variance covariance and coefficient of variation travel time distribution cumulative mass release across a compliance surface the associated error and plume spatial moments our method also allows quantification of the uncertainty in the original source location of any solute particle located anywhere in the field at any time this paper describes the methodology and presents some unconditional results conditioning and more advanced computations are presented in the subsequent papers

macrodispersivity tensor for nonreactive solute transport in isotropic and anisotropic fractal porous media analytical solutions using spectral stochastic theory macrodispersivity tensors are developed for one two and three dimensional isotropic and anisotropic fractal porous media since natural geologic aquifers are always bounded we introduce the concept of a maximum length scale lmax in the one dimensional case the asymptotic long time longitudinal macrodispersivity a is proportional to lmax2 and in the multidimensional cases a lmax in the multidimensional cases as the fractal dimension increases a decreases because a larger fractal dimension means the tortuosity of solute particles is larger leading to more mixing in the transverse direction which in turn reduces longitudinal spreading the transverse macrodispersivities are shown to be analogous to traditional spectral stochastic results in the multidimensional cases lmax is shown to be controlled by the aquifer thickness as a result ergodic conditions may be reached in relatively thin aquifers allowing the use of the asymptotic macrodispersivity expressions obtained here for single realizations

simulation of dispersion in heterogeneous porous formations statistics first order theories convergence of computations this paper discusses the results of numerical analysis of dispersion of passive solutes in two dimensional heterogeneous porous formations statistics of flow and transport variables the accuracy and the role of approximations implicit in existing first order theories and the convergence of computational results are investigated the results suggest that quite different rates of convergence with monte carlo runs hold for different spatial moments and that over 1000 realizations are required to stabilize second moments even for relatively mild heterogeneity sigma y 2 1 6 this has implications for the extent of the spatial domain for single realization numerical studies of the same type a comparison of the variance of plumes with the results of linear theories 0 05 sigma y 2 1 6 shows an unexpectedly broad validity field for the theoretical solution obtained from a suitable linearization of flow and transport reformulation of the same problem linearizing in tum the flow or the transport equations shows opposite deviations from the linear theory the interesting consequence is that the errors induced by linearizations in the flow or the transport equations have different signs and their effects on the moments of dispersing plumes are compensating thereby yielding consistent formulations unexpected features of the statistics of probability distributions of longitudinal and transverse velocities and travel times are also computed and discussed

adaptive estimation of the log fluctuating conductivity from tracer data at the cape cod site an adaptive estimation scheme is used to obtain the integral scale and variance of the log fluctuating conductivity at the cape cod site based on the fast fourier transform stochastic model of deng et al 1993 and a kalmanlike filter the filter incorporates prior estimates of the unknown parameters with tracer moment data to adaptively obtain improved estimates as the tracer evolves the results show that significant improvement in the prior estimates of the conductivity can lead to substantial improvement in the ability to predict plume movement the structure of the covariance function of the log fluctuating conductivity can be identified from the robustness of the estimation both the longitudinal and transverse spatial moment data are important to the estimation

reply to comment on an explanation of scale dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry by s w wheatcraft and s w tyler

laplace transform finite element method for non fickian flows abstract in this paper we consider numerical methods for integro differential problems based on time discretization via laplace transformation we focus our attention in models arising in the context of non fickian solute transport phenomena in porous media the mathematical models which describe the evolution of the solute concentrations are characterized by volterra equations we present and analyze an hybrid method which combines the laplace transformation with respect to the time variable with the finite element discretization in the spatial variables numerical results illustrate the performance of the method

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

parametric identification of transfer functions in the frequency domain a survey this paper gives a survey of frequency domain identification methods for rational transfer functions in the laplace s or z domain the interrelations between the different approaches are highlighted through a study of the equivalent cost functions the properties of the various estimators are discussed and illustrated by several examples

renormalization group analysis of macrodispersion in a directed random flow we apply field theoretic methods to the calculation of the effective diffusivity macrodispersion coefficient in a random flow we show how this approach can be utilized to calculate a perturbation series for the effective diffusivity of conservative tracers in an incompressible velocity field with nonvanishing mean the first order i e one loop approximation of this series coincides with classical results derived by gelhar and axness and dagan a renormalization group rng approach is utilized and the results are compared to the classical first order perturbation theory for a moderate variability of the permeability the renormalized theory predicts only small corrections to the longitudinal dispersivity however the transverse dispersivity can be larger than that predicted by the first order perturbation theory by several orders of magnitude we compare these values to the outcome of monte carlo simulations and find that the rng predictions are in much better though not perfect accordance with the results of simulations moreover the results are in good quantitative agreement with reported observations from the borden site field experiment

frequency domain system identification using non parametric noise models estimated from a small number of data sets abstract this paper discusses the problem of identifying a linear system from the frequency data when the measurements of the input and the output signals are both disturbed with noise a typical example of such a problem is the identification of a system in a feedback loop it is known that this problem can be solved using errors in varaibles methods if the covariance matrices of the disturbing noise on input and output measurements are a priori known it is shown that the exact covariance matrices can be replaced by the sample covariance matrices the system can be identified from the sample means and sample covariance matrices calculated from a small number m of independently repeated experiments it is shown that under these conditions the estimates are still strongly consistent for an increasing number of data points n in each experiment n if m 4 the loss in efficiency is quantified m 6 and the expected value of the cost function m 4 and its variance m 6 are calculated

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

an investigation of the validity of first order stochastic dispersion theories in isotropie porous media monte carlo simulations are used to 1 investigate the accuracy of approximations that are implicit in first order stochastic dispersion theories and 2 identify the accuracy limits of first order dispersion theories in isotropic porous media the fickian theory of gelhar and axness 1983 as well as the fickian and non fickian theories of dagan 1984 and neuman and zhang 1990 are investigated all monte carlo simulations are in three dimensions confidence limits of ensemble averaged monte carlo results in isotropic porous media are established for 0 1 y 1 5 these results showed that first order theoretical estimates of the eulerian velocity covariance function are quite accurate for y 1 theoretical estimates of the non fickian longitudinal dispersivity do not deviate significantly from theory for at least y 1 5 theoretical estimation of the transverse dispersivity is limited to y 1 and the fickian longitudinal dispersivity is overestimated by the theory of gelhar and axness 1983 of all first order dispersion theories the theory of dagan 1984 is most robust in estimating the dispersivity tensor

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

a monte carlo assessment of eulerian flow and transport perturbation models monte carlo studies of flow and transport in two dimensional synthetic conductivity fields are employed to evaluate first order flow and eulerian transport theories hydraulic conductivity is assumed to obey fractional brownian motion fbm statistics with infinite integral scale or to have an exponential covariance structure with finite integral scale the flow problem is solved via a block centered finite difference scheme and a random walk approach is employed to solve the transport equation for a conservative tracer the model is tested for mass conservation and convergence of computed statistics and found to yield accurate results it is then used to address several issues in the context of flow and transport the validity of the first order relation between the fluctuating velocity covariance and the fluctuating log conductivity is examined the simulations show that for exponential covariance this approximation is justified in the mean flow direction for log conductivity variance f2 of the order of unity however as f2 increases the relation for the transverse velocity component deviates from the fully nonlinear monte carlo results eulerian transport models neglect triplet correlation functions that appear in the nonlocal macroscopic flux the relative importance of the triplet correlation term for conservative chemicals is examined this term appears to be small relative to the convolution flux term in mildly heterogeneous media as f2 increases or the integral scale grows the triplet correlation becomes significant in purely convective transport the triplet correlation term is significant if the heterogeneity is evolving the exact nonlocal macroscale flux for the purely convective case significantly differs from that of the convective dispersive transport this is in agreement with recent theoretical analysis and numerical studies and it suggests that neglecting local scale dispersion may lead to large errors localization errors in the flux term are evaluated using monte carlo simulations the nonlocal in time model significantly differs from the fully nonlocal model for small variance and integral scale there is a slight difference between the fully localized flux and the fully nonlocal convolution flux this is also in agreement with recent theories that suggest that moments through the second for the two models are nearly identical for conservative tracers the fully localized model does not perform well in the purely convective cases

eulerian lagrangian approach for modeling of flow and transport in heterogeneous geological formations this paper presents a new eulerian lagrangian method for modeling flow and transport of passive solutes in heterogeneous porous formations the physically plausible random velocity fields are generated by a geostatistical based model the ability of the method to correctly reproduce the velocity field statistics and to satisfy mass balance is tested and demonstrated for the case of two dimensional flow the method is found particularly useful in dealing with large fields where numerical fixed grid methods such as finite difference or finite elements become very demanding from a computational point of view the transport problem is solved in a lagrangian framework by the particle tracking approach which uses a suitable grid refinement to correctly handle the velocity fluctuation at any a priori defined resolution the new method is employed to compute the probability distribution functions of the concentration and travel times and to investigate the limitations of existing methods for predicting concentrations

stochastic analysis of solute transport in heterogeneous aquifers subject to spatiotemporal random recharge this paper extends the work of li and graham 1998 to deal with the unconditional moments of head velocity and concentration under transient flow conditions which are assumed to be caused by a spatiotemporally random recharge semianalytical solutions are derived for the unconditional covariances for transient velocity with a constant mean recharge using a fourier transform approach results demonstrate that the velocity covariance derived for the steady state random recharge field is a limiting case of the spatiotemporally variable velocity covariance with an infinite temporal correlation scale another limiting case indicates that introduction of temporally random but spatially uniform recharge has no effect on the velocity covariances over that induced by the mean recharge on the mean head gradient thus for this limiting case there is no increased effect on the ensemble mean concentration plume spreading or the concentration prediction uncertainty following deng et al 1993 the equations for mean concentration and macrodispersive flux under zero mean transient recharge are decoupled in the laplace fourier domain and solved using a fast fourier transform algorithm which significantly reduces the computational demand the first order concentration variance is solved using three different approximate techniques an approximate fast fourier transform technique a finite element method and a direct numerical integration the simulation results show that introduction of a spatiotemporally random recharge enhances both longitudinal and lateral mean concentration plume spreading compared to the no recharge case however transient recharge produces less spreading and less concentration prediction uncertainty than the steady state spatially random recharge case

a meshless method on non fickian flows with mixing length growth in porous media based on radial basis functions a comparative study the present study aims to introduce a solution for parabolic integro differential equations arising in heat conduction in materials with memory which naturally occur in many applications two radial basis functions rbfs collocation schemes are employed for solving this equation the first method tested is an unsymmetric method and the second one which appears to be more efficient is a symmetric one the convergence of these two schemes is accelerated as we use the cartesian nodes as the center nodes

estimation of spatial covariance structures by adjoint state maximum likelihood cross validation 2 synthetic experiments paper 2 of this three part series uses synthetic data to investigate the properties of the adjoint state maximum likelihood cross validation asmlcv method presented in paper 1 samper and neuman this issue a more than 40 synthetic experiments are performed to compare various conjugate gradient algorithms investigate the manner in which computer time varies with asmlcv parameters study the effect of sample size and choice of kriging points on asmlcv estimates evaluate the ability of various model structure identification criteria to help select the most appropriate semivariogram model among given alternatives study the conditions required for parameter identifiability uniqueness and stability quantify the statistics of cross validation errors test hypotheses concerning the distribution and autocorrelation of these errors and illustrate the computation of approximate quality indicators for asmlcv parameter estimates

stochastic uncertainty analysis for solute transport in randomly heterogeneous media using a karhunen loeve based moment equation approach 1 a new approach has been developed for solving solute transport problems in randomly heterogeneous media using the karhunen loeve based moment equation klme technique proposed by zhang and lu 2004 the klme approach combines the karhunen loeve decomposition of the underlying random conductivity field and the perturbative and polynomial expansions of dependent variables including the hydraulic head flow velocity dispersion coefficient and solute concentration the equations obtained in this approach are sequential and their structure is formulated in the same form as the original governing equations such that any existing simulator such as modular three dimensional multispecies transport model for simulation of advection dispersion and chemical reactions of contaminants in groundwater systems mt3dms can be directly applied as the solver through a series of two dimensional examples the validity of the klme approach is evaluated against the classical monte carlo simulations results indicate that under the flow and transport conditions examined in this work the klme approach provides an accurate representation of the mean concentration for the concentration variance the accuracy of the klme approach is good when the conductivity variance is 0 5 as the conductivity variance increases up to 1 0 the mismatch on the concentration variance becomes large although the mean concentration can still be accurately reproduced by the klme approach our results also indicate that when the conductivity variance is relatively large neglecting the effects of the cross terms between velocity fluctuations and local dispersivities as done in some previous studies can produce noticeable errors and a rigorous treatment of the dispersion terms becomes more appropriate