numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media a numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented the method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield upon appropriate averaging of the fine scale velocity field the coarse scale or equivalent grid block permeability when the region over which this coarse scale permeability is computed constitutes a representative elementary volume rev the resulting equivalent permeability may be interpreted as the effective permeability of the region solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle based finite element numerical procedure which allows for the modeling of geometrically complex features the specification of periodic boundary conditions is shown to yield symmetric positive definite equivalent permeability tensors in all cases the method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields the applicability and limitations of the method for these more general heterogeneity fields are discussed

effective permeability and dispersion in locally heterogeneous aquifers the global effective permeability and hydrodynamic dispersion tensors due to steady flow in locally heterogeneous and statistically anisitropic aquifers are analysed using the self consistent approximation for small perturbations the results for the statistically anisotropic case are shown to be consistent with those derived using stochastic continuum theory however for larger perturbations the stochastic continuum theory has not yet produced rigorous results that can be compared with those in this paper

multifractal anisotropic scaling of the hydraulic conductivity 1 we analyzed the scaling properties of the hydraulic conductivity k at three sites in northern america made borden and cape cod we found that k at all sites exhibits multifractality fractal and multiscaling in both the vertical and horizontal directions though the multiscaling was within a range smaller than that of the maximum distance between measurements in the vertical direction the k data for made borden and cape cod were multiscaling from 0 15 to 1 35 m 0 05 to 0 5 m and 0 15 to 0 9 m respectively they were multiscaling in the horizontal direction from 9 to 45 m 1 to 10 m and 1 to 17 m respectively the multiscaling was also anisotropic evidence of scaling was poorest for the horizontal direction of the made site and it spanned half an order of magnitude such results compel one to treat multifractality in the horizontal direction of made as supported by heuristic arguments rather than by pure statistical evaluation we fitted a multifractal model to the data and estimated its parameters we found the underlying statistics of all data to be non gaussian and the model capable of reproducing the probability distribution of k data especially the negative skewness of log k we also generated two dimensional isotropic multifractal fields illustrating the role of the parameters of the selected multifractal model

steady state groundwater flow simulation with imprecise parameters a methodology based on fuzzy set theory is developed to incorporate imprecise parameters into steady state groundwater flow models in this case fuzzy numbers are used to represent parameter imprecision as such they are also used as a measure for the uncertainty associated with the hydraulic heads due to the imprecision in the input parameters the imprecise input parameters may come from indirect measurements subjective interpretation and expert judgment of available information in the methodology a finite difference method is combined with level set operations to formulate the fuzzy groundwater flow model this fuzzy modeling technique can handle imprecise parameters in a direct way without generating a large number of realizations two numerical solution methods are used to solve the fuzzy groundwater flow model the groundwater model operator method proposed in this methodology and the iterative algorithm based on conventional interval arithmetics the iterative method is simple but may overestimate the uncertainty of hydraulic heads the groundwater model operator method not only provides the hull of the solution set for the hydraulic heads but also considers the dependence of hydraulic head coefficients which are functions of imprecise parameters sensitivity analysis shows that the dependence of hydraulic head coefficients has a critical impact on the model results and neglecting this dependence may result in significant overestimation of the uncertainty of hydraulic heads a numerical model based on the methodology is tested by comparing it with the analytical solution for a homogeneous radial flow problem it is also applied to a simplified two dimensional heterogeneous flow case to demonstrate the methodology

a synthesis of approaches to upscaling of hydraulic conductivities simulation of flow through heterogeneous media often requires discretizing the flow domain into blocks and assigning an equivalent block conductivity value to each one of them the process of defining block conductivities from point values is termed upscaling a number of approaches to upscaling are available most of which consider the uncertainty associated with any natural property so that they cast the problem in a stochastic frame recently indelman and dagan 1993a b provided a general stochastic methodology to upscaling in heterogeneous anisotropic formations by means of the dissipation energy function unfortunately they did not provide any practical method to compute block values from point ones the objective of this work is twofold first we analyze different practical approaches to compute block conductivities and find that all of them provide very similar results in terms of actual computed values second we check that all approaches verify approximately a number of conditions stated by indelman and dagan 1993a specifically we show analytically that for regular blocks the methodologies of both rubin and gomez hernandez 1990 and desbarats 1992 which we call practical methodologies satisfy the condition that the effective conductivity obtained from a field where the elementary conductivities are defined over a certain support we call this the actual formation is identical to that obtained from the same field with conductivities defined at a larger support upscaled formation the analysis is carried out by working with the logarithm of block conductivities and using a small perturbation expansion and thus is strictly valid for small variances on the other hand we show numerically that the two methodologies satisfy approximately an important condition stated in terms of the dissipation energy that block averaged dissipation values computed are indeed very close to the true dissipation values in each block the agreement is even better if we consider statistical moments instead of point values as an important conclusion we should note that all practical methodologies considered in this work perform equally well and more important constitute a simple way to treat an otherwise very complex problem

impact of sampling volume on the probability density function of steady state concentration 1 in recent years statistical theory has been used to compute the ensemble mean and variance of solute concentration in aquifer formations with second order stationary velocity fields the merit of accurately estimating the mean and variance of concentration however remains unclear without knowing the shape of the probability density function pdf in a setup where a conservative solute is continuously injected into a domain the concentration is bounded between zero and the concentration value in the injected solution at small travel distances close to the fringe of the plume an observation point may fall into the plume or outside so that the statistical concentration distribution clusters at the two limiting values obviously this results in non gaussian pdf s of concentration with increasing travel distance the lateral plume boundaries are smoothed resulting in increased probability of intermediate concentrations likewise averaging the concentration in a larger sampling volume as typically done in field measurements leads to higher probabilities of intermediate concentrations we present semianalytical results of concentration pdf s for measurements with point like or larger support volumes based on stochastic theory applied to stationary media to this end we employ a reversed auxiliary transport problem in which we use analytical expressions for first and second central spatial lateral moments with an assumed gaussian pdf for the uncertainty of the first lateral moment and gauss like shapes in individual cross sections the resulting concentration pdf can be reasonably fitted by beta distributions the results are compared to monte carlo simulations of flow and steady state transport in 3 d heterogeneous domains in both methods the shape of the concentration pdf changes with distance to the contaminant source near the source the distribution is multimodal whereas it becomes a unimodal beta distribution far away from the contaminant source the semianalytical and empirical pdf s differ slightly which we contribute to the numerical artifacts in the monte carlo simulations but also to hard assumptions made in the semianalytical approach our results imply that geostatistical techniques for interpolation and other statistical inferences based on gaussian distributions such as kriging and cokriging may be feasible only far away from the contaminant source for calculations near the source the beta like distribution of concentration should be accounted for

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

minimum relative entropy forward probabilistic modeling the pioneering work of jaynes in bayesian maximum entropy methods has been successfully explored in many disciplines the principle of maximum entropy pme is a powerful and versatile tool of inferring a probability distribution from constraints that do not completely characterize that distribution minimum relative entropy mre is a method which has all the important attributes of the maximum entropy approach with the advantage that prior information may be easily included in this paper we use mre to determine the prior probability density function pdf of a set of model parameters based on limited information the resulting pdf is used in monte carlo simulations to provide expected values in field variables such as concentration and confidence limits we compare the probabilistic results from a traditional advection dispersion ade model based on volumetric averaging concepts with that of a model based on the assumption that the hydraulic conductivity is a stationary stochastic process the results suggest that although naff s 1990 model satisfies the observed data to a better degree than ade model the upper and lower confidence bands about the mean value are larger than the ade results this result we attribute to the fact that naff s 1990 model simply contains more parameters each of which is unknown and has to be estimated there is no statistical difference between the expected values of second spatial moments for the two models the examples presented in this paper illustrate problems associated with assigning gaussian pdfs as priors in a probabilistic model first such an assumption for the input parameters actually injects more information into the problem than may actually exist whether consciously or unconsciously this fact is borne out by comparison with minimum relative entropy theory second the output parameters as suggested from the monte carlo analysis cannot be assumed to be gaussian distributed even when the prior pdf is gaussian in form in a practical setting the significance of this result and the approximation of gaussian form would depend on the toxicity and environmental standards that apply to the problem

finite analytic numerical method for unsteady two dimensional navier stokes equations abstract the main purpose of this paper is to develop a finite analytic fa numerical solution for unsteady two dimensional navier stokes equations the fa method utilizes the analytic solution in a small local element to formulate the algebraic representation of partial differential equations in this study the combination of linear and exponential functions that satisfy the governing equation is adopted as the boundary function thereby improving the accuracy of the finite analytic solution two flows one a starting cavity flow and the other a vortex shedding flow behind a rectangular block are solved by the fa method the starting square cavity flow is solved for reynolds numbers of 400 1000 and 2000 to show the accuracy and stability of the fa solution the fa solution for flow over a rectangular block h h 4 predicts the strouhal number for reynolds numbers of 100 and 500 to be 0 156 and 0 125 details of the flow patterns are given in addition to streamlines and vorticity distribution rest streamlines are given to illustrate the vortex motion downstream of the block

heavy tailed log hydraulic conductivity distributions imply heavy tailed log velocity distributions 1 equations of contaminant transport describing non gaussian dispersion of solute in heterogeneous porous media have been developed by several authors e g berkowitz and scher 1995 1998 benson 1998 2001 berkowitz et al 2000 and baeumer et al 2005 the equations assume that solute particle movement in time and or space can be described by heavy tailed probability distributions e g levy stable distributions this assumption requires a mechanism for solute particles to move through heterogeneous porous media in a manner that can be approximated with heavy tailed probability distributions this work investigates whether a heavy tailed levy stable distribution of increments in log hydraulic conductivity results in a heavy tailed distribution of increments in log velocity a heavy tailed distribution of increments in log velocity would be a mechanism for solute particle movement in time or space that can be described by a heavy tailed probability distribution a computer algorithm was written to generate realistic hydraulic conductivity fields the algorithm modifies the spectral synthesis method for generating fractional brownian motion i e saupe 1988 and brewer and wheatcraft 1994 and produces self similar hydraulic conductivity fields with a heavy tailed levy stable distribution of increments in log hydraulic conductivity a monte carlo approach was adopted in which 100 hydraulic conductivity fields were generated for each of the following values of 0 8 1 1 1 4 and 1 7 the usgs finite difference groundwater code modflow was used to calculate the velocity fields and the increments in longitudinal log hydraulic conductivity and log velocity from the fields were analyzed to determine whether they were consistent with a heavy tailed levy stable probability density function pdf after determining that the distributions of increments in log hydraulic conductivity and log velocity were consistent with a levy stable pdf the tail parameters describing each pair of hydraulic conductivity and velocity fields were estimated with the nolan 1997 maximum likelihood estimator the relationship between the two tail parameters was then determined there are three conclusions of this research first mandelbrot and pp plots confirm that levy stable distributions of increments in log hydraulic conductivity give rise to levy stable distributions of increments in log velocity second the tail parameter for the increments in log velocity is typically smaller indicating a heavier probability tail than the corresponding tail parameter for the increments in log hydraulic conductivity third there is a positive statistical relation between the two tail parameters

identification of large scale spatial trends in hydrologic data it is often useful to distinguish different scales of variability in hydrologic properties such as hydraulic conductivity in the simplest two scale case large scale variations can be viewed as a trend while small scale fluctuations about this trend can be viewed as a random residual this paper describes a method for estimating spatial trends from scattered field measurements the basic concept is to treat both the trend and the residual as stationary random functions these functions are distinguished by their spatial spectral or covariance properties which may be estimated from available data or simply hypothesized we present two versions of a general algorithm for estimating spatial trends 1 a discrete version which is useful in practical applications where data are limited and irregularly spaced and 2 a continuous version which can be used to study the effects of using incorrect spectral parameters applications of the discrete algorithm to both synthetically generated data and field measurements yield satisfactory trend estimates an analysis based on the continuous algorithm shows that the estimation error lower bound for these applications depends on two dimensionless ratios that is the scale disparity ratio of the trend and residual correlation scales and the signal to noise ratio ratio of the trend and residual variances these ratios may be used to evaluate the feasibility of trend estimation before field samples are actually collected

more on stochastic models in a recent review article g sposito et al 1986 examined the various stochastic theories which are concerned with transport of solutes in porous media in this short note we expand on their discussion to include several topics which had been omitted we begin by looking at two definitions of probability theory and their relation to the concept of an ensemble an rev ensemble of soils is defined and examined the concept of ergodicity is reviewed and it is pointed out that most stochastic models are theoretically unverifiable the relationship between scale of measurement and stochasticity is briefly reviewed and an equation that combines the two concepts is presented

estimating the variance and integral scale of the transmissivity field using head residual increments a modification of previously published solutions regarding the spatial variation of hydraulic heads is discussed whereby the semivariogram of increments of head residuals termed head residual increments hris are related to the variance and integral scale of the transmissivity field a first order solution is developed for the case of a transmissivity field which is isotropic and whose second order behavior can be characterized by an exponential covariance structure the estimates of the variance y2 and the integral scale of the log transmissivity field are then obtained via fitting a theoretical semivariogram for the hri to its sample semivariogram this approach is applied to head data sampled from a series of two dimensional simulated aquifers with isotropic exponential covariance structures and varying degrees of heterogeneity y2 0 25 0 5 1 0 2 0 and 5 0 the results show that this method provided reliable estimates for both and y2 in aquifers with the value of y2 up to 2 0 but the errors in those estimates were higher for y2 equal to 5 0 it is also demonstrated through numerical experiments and theoretical arguments that the head residual increments will provide a sample semivariogram with a lower variance than will the use of the head residuals without calculation of increments

mechanics of heterogeneous porous media with several spatial scales by extending the theory of homogenization we consider a heterogeneous porous medium whose material structure is characterized by multiple periodicity over several disparate length scales because for geological and engineering applications the driving force e g the pressure gradient and the global motion of primary interest is usually at a scale much larger than the largest of structural periodicity we derive the phenomenological equations of motion by using the perturbation technique of multiple scales all the effective coefficients are defined by boundary value problems on unit cells in smaller scales and no constitutive assumptions are added aside from the basic equations governing the mechanics of the pore fluid and the solid matrix in part i the porous matrix is assumed to be rigid the case of three scales is treated first symmetry and positiveness of the effective darcy permeability tensor are proven extensions to four and more scales are then discussed in part ii we allow the solid matrix to be deformable and deduce the equations of consolidation of a two phase medium the coupling of fluid flow and the quasi static elastic deformation of the matrix is considered when there are three disparate length scales effective coefficients of various kinds are deduced in terms of cell problems on the scale of the pores several general symmetry relations as well as specific properties of a medium composed of layers of porous matrix are discussed

solving the steady state groundwater flow equation for finite linear aquifers using a generalized fourier series approach in two dimensional domains a method to solve steady linear groundwater flow problems using generalized fourier series is developed and particularized for multiple fourier series in two dimensional domains it leads to a linear vector equation whose solution provides a finite number of generalized fourier coefficients approximating the hydraulic head field its implementation is shown and two relevant properties are found for the system matrix it is always symmetric and once computed if additional fourier terms are needed for a better approximation of the hydraulic head field previously computed matrix elements remain invariant i e only new rows and columns are added to the system matrix the method is demonstrated in three simple cases with different geometries and transmissivity fields where solutions are compared with analytical and finite element method results thus the method is verified as an alternative to other flow solvers additionally it provides a direct way to obtain the spectral form of the flow equation solution given a spectral representation of transmissivity and can be easily extended to obtain continuous velocity fields and their approximated spectral expressions copyright 2009 john wiley sons ltd

look up table based large finite field multiplication in memory constrained cryptosystems many cryptographic systems use multiplication in the finite field gf 2 sup n for their underlying computations in the recent past a number of look up table based algorithms have been proposed for the software implementation of gf 2 sup n multiplication look up table based algorithms can provide speed advantages but they either require a large memory space or do not fully utilize the resources of the processor on which the software is executed in this work an algorithm for gf 2 sup n multiplication is proposed which can alleviate this problem in each iteration of the proposed algorithm a group of bits of one of the input operands are examined and two look up tables are accessed the groupsize determines the table sizes but does not affect the utilization of the processor resources it can be used for both software and hardware realizations and is particularly suitable for implementations in memory constrained environment such as smart cards and embedded cryptosystems

three dimensional simulation of unstable gravity driven infiltration of water into a porous medium infiltration of water in dry porous media is subject to a powerful gravity driven instability although the phenomenon of unstable infiltration is well known its description using continuum mathematical models has posed a significant challenge for several decades the classical model of water flow in the unsaturated flow the richards equation is unable to reproduce the instability here we present a computational study of a model of unsaturated flow in porous media that extends the richards equation and is capable of predicting the instability and captures the key features of gravity fingering quantitatively the extended model is based on a phase field formulation and is fourth order in space the new model poses a set of challenges for numerical discretizations such as resolution of evolving interfaces stiffness in space and time treatment of singularly perturbed equations and discretization of higher order spatial partial differential operators we develop a numerical algorithm based on isogeometric analysis a generalization of the finite element method that permits the use of globally smooth basis functions leading to a simple and efficient discretization of higher order spatial operators in variational form we illustrate the accuracy efficiency and robustness of our method with several examples in two and three dimensions in both homogeneous and strongly heterogeneous media we simulate for the first time unstable gravity driven infiltration in three dimensions and confirm that the new theory reproduces the fundamental features of water infiltration into a porous medium our results are consistent with classical experimental observations that demonstrate a transition from stable to unstable fronts depending on the infiltration flux

fast normal basis multiplication using general purpose processors for cryptographic applications normal bases have received considerable attention especially for hardware implementation we consider fast software algorithms for normal basis multiplication over the extended binary field gf 2 sup m we present a vector level algorithm which essentially eliminates the bit wise inner products needed in the conventional approach to the normal basis multiplication we then present another algorithm which significantly reduces the dynamic instruction counts both algorithms utilize the full width of the data path of the general purpose processor on which the software is to be executed we also consider composite fields and present an algorithm which can provide further speed ups and an added flexibility toward hardware software codesign of processors for very large finite fields

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

effective groundwater model parameter values influence of spatial variability of hydraulic conductivity leakance and recharge a stochastic simulation approach was used to inspect the influence of spatial variability in aquifer and recharge properties on effective averaged groundwater model parameters a two dimensional unconfined aquifer model was set up for an area similar to that near livermore california monte carlo simulations were generated for different sets of spatial correlation structures assuming stationarity and an exponential semivariogram for each monte carlo realization groundwater flow was simulated and estimates of the means and standard deviations of the hydraulic head were obtained nine different cases were studied involving different correlation lengths of hydraulic conductivity conditional simulations parameter zonation and spatial recharge distributions results indicate that there is no single uniform value of the hydraulic conductivity that reproduces the expected head values spatial averaging to obtain effective values was based on the p norm power norm which ranges from 1 for the arithmetic mean to 1 for the harmonic mean with 0 representing the geometric mean for the unconditional case the 0 4 p norm seems to best reproduce the expected values the 0 2 p norm gave the best result for the conditional simulation case if the geometric mean were used instead the heads would deviate from the expected heads by about 2 m near the wells some value between the arithmetic and the geometric means of riverbed leakance will give results close to the expected head values the arithmetic mean of aerially distributed recharge produced results with small deviations from the expected head values these particular results may only apply to this modeled system as the pumping pattern and magnitudes exhibited strong influences on zones of maximum head deviation the implication of this work is that if an effective hydraulic conductivity is used in a simulation model it must be selected for a particular set of wells and pumping rates in an aquifer with significant pumping centers the best effective value for a two dimensional unconfined flow model is most likely between the geometric and harmonic mean if the wells are turned off the effective hydraulic conductivity for that flow model reverts to the geometric mean

effective dispersion in heterogeneous media under random transient flow conditions 1 under steady state flow conditions solute dispersion in heterogeneous porous media is much smaller in the transverse than in the longitudinal direction this holds particularly for effective dispersion of a plume originating from a point like injection the effective dispersion coefficient describes the actual dispersive mixing of solutes in the aquifer the lack of dispersive transverse mixing may limit considerably natural attenuation of certain contaminants temporal fluctuations of the flow direction enhance horizontal transverse dispersion this has been shown previously for uniform flow and for macrodispersion in stationary media we present a linear stochastic theory for effective dispersion under quasi steady state flow conditions with random temporal fluctuations of the mean flow direction as for macrodispersion effective transverse dispersion proves to be dominated by transient flow effects we compare semianalytical results derived from linear theory to those from particle tracking random walk simulations for a three dimensional test case the parameters of the test case are similar to those obtained at the borden site where the mean transverse flow component fluctuated approximately by 10 linear theory and particle simulations agree well

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

stochastic modeling of vertically averaged concentration uncertainty in a perfectly stratified aquifer solute concentration which results from transport in an aquifer with unknown variability in hydraulic conductivity can be subject to large predictive uncertainty this uncertainty can be modeled using a stochastic approach in which conductivity is represented by a random field with known probabilistic structure in this work we use monte carlo methods of stochastic modeling to study the uncertainty in vertically flux averaged concentration for transport in the simplified case of a perfectly stratified aquifer we assume steady flow parallel to bedding and negligible transverse dispersion we measure uncertainty with the ensemble coefficient of variation of concentration and find it to be large in most conditions examined we consider the validity of an ergodic assumption practical equivalence of ensemble mean and single realization concentrations by investigating the dependence of uncertainty upon the vertical averaging length even for averaging over a depth of 100 times the vertical correlation scale of conductivity uncertainty is large enough around 20 to be considered significant in many predictive situations we conclude that prediction based solely on the ensemble mean concentration would be inappropriate for this simple system and emphasize the importance of the ensemble variance for quantification of predictive uncertainty in stochastic modeling of solute transport

pumping tests in heterogeneous aquifers an analytical study of what can be obtained from their interpretation using jacob s method interpretation of pumping tests to estimate hydraulic parameter values is typically based on the assumption of aquifer homogeneity the applicability of the traditional methods of interpretation in real aquifers can be questioned since the evaluation of the drawdown curves observed at different locations in a single test may not result in one consistent set of hydraulic parameters thus most hydrogeologists tend to look at estimated transmissivities t as some average property of the medium while estimated storativities s are disregarded in some cases particularly when they are obtained from data measured at the pumping well an analytical study of drawdown under radially convergent flow toward a single point in heterogeneous aquifers shows that large time drawdown values form a straight line on a drawdown versus log time plot jacob s method consists of obtaining estimates for t and s from the slope and intercept of this line we find that even in a heterogeneous field these estimates provide valuable information about the aquifer estimated t values for different observation points tend to converge to a single value which corresponds to the effective t derived under parallel flow conditions estimated storativities however display higher variability but the geometric mean of the sest values can be used as an unbiased estimator of the actual s thus it appears that although jacob s method was originally derived for homogeneous media it can provide valuable information in real aquifers

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

fefferman stein inequalities for the mathbb z _2 d dunkl maximal operator in this article we establish the fefferman stein inequalities for the dunkl maximal operator associated with a finite reflection group generated by the sign changes similar results are also given for a large class of operators related to dunkl s analysis

finite analytic numerical solution for two dimensional groundwater solute transport a new numerical method called the finite analytic fa method is used to solve a groundwater solute transport problem the basic idea of the finite analytic method is the incorporation of local analytic solution in the numerical solution of the partial differential equation when the local analytic solution is evaluated at a given nodal point it gives an algebraic relationship between a nodal value in an element and its neighboring nodal points the assemble of the linear system equations results in a tridiagonal matrix like most finite difference method the advantages of using efficient iterative techniques for solving tridiagonal matrices are equally applicable to fa method the automatic localized upstream shift and the analytic property of the fa method eliminates the difficulty of numerical dispersion locally and suppresses the overall numerical dispersion for large peclet number for small peclet number fa method yields excellent results in comparison with the analytic solution for large peclet number fa solutions are oscillation free with some degree of numerical dispersion the results are comparable with those obtained using upstream weighted finite element method

a higher order approximation to effective conductivity in media of anisotropic random structure properties of the effective conductivity tensor keff are studied by deriving the second order terms in its expansion in the variance 2 of normally distributed log conductivity it is shown that for media of anisotropic structure the components of the effective conductivity tensor are expressed by a functional of the log conductivity covariance that is it depends on the shape of the correlation function and not only on anisotropy ratios variance 2 and space dimensions however the trace of keff is independent of the log conductivity autocovariance and for a given mean conductivity depends only on 2 the dependence of the effective conductivity on the correlation structure is illustrated for gaussian and exponential autocovariances of log conductivity and for two and three dimensional flows

a characteristic based finite analytic method for solving the two dimensional steady state advection diffusion equation 1 this paper develops an improved finite analytic fa solution method to the advection diffusion equation ade for solving advection dominated steady state transport problems the fa method solves the ade analytically in localized discrete elements with each element linked through local boundary conditions previous fa methods have suffered from complex solution formulations as well as from numerical dispersion stemming from inaccuracies in the local boundary estimations here we use finite difference approximations of the dispersion terms to reduce solution complexity implement an improved particle tracking scheme to account for velocity variations within each element and use a hermite interpolation scheme to estimate the local boundary conditions the new fa method previous fa methods and a finite difference upwinding method are compared in both homogeneous and heterogeneous velocity fields at different peclet numbers results show the new fa method exhibits little numerical dispersion without undue complexity or computational effort across all tested flow conditions

ensemble solute transport in two dimensional operator scaling random fields 1 motivated by field measurements of aquifer hydraulic conductivity k recent techniques were developed to construct anisotropic fractal random fields in which the scaling or self similarity parameter varies with direction and is defined by a matrix ensemble numerical results are analyzed for solute transport through these two dimensional operator scaling fractional brownian motion ln k fields both the longitudinal and transverse hurst coefficients as well as the radius of isotropy are important to both plume growth rates and the timing and duration of breakthrough it is possible to create operator scaling fractional brownian motion fields that have more continuity or stratification in the direction of transport the effects on a conservative solute plume are continually faster than fickian growth rates highly non gaussian shapes and a heavier tail early in the breakthrough curve contrary to some analytic stochastic theories for monofractal k fields the plume growth rates never exceed a mercado s 1967 purely stratified aquifer growth rate of plume apparent dispersivity proportional to mean distance apparent superstratified growth must be the result of other demonstrable factors such as initial plume size

continuous time stochastic analysis of groundwater flow in heterogeneous aquifers the problem of depth averaged groundwater flow in heterogeneous aquifers is looked at from a stochastic point of view the galerkin finite element version of the linearized stochastic equation governing the flow is solved analytically in time using an eigenvalue eigenvector technique the stochastic solution which is valid for small variance of aquifer log transmissivity relates spatial and temporal variabilities of aquifer head to aquifer heterogeneity stochastic recharge and random initial head the computational effort requires the evaluation of integrals of matrices whose elements are linear functions of the nodal mean heads the solutions are obtained for the exact i e continuous and quasi steady approximation of the mean head aquifer head temporal covariances are evaluated using standard matrix operations once the storage matrix is inverted and the associated generalized eigenvalue problem is solved implication of the level of spatial discretization on the performance of the solution is examined and the influence of aquifer heterogeneity on spatial and temporal variances of hydraulic heads is investigated in the presence of a semipervious boundary

stochastic analysis of spatial variability in two dimensional steady groundwater flow assuming stationary and nonstationary heads two dimensional steady groundwater flow in a confined aquifer with spatially variable transmissivity t is analyzed stochastically using spectral analysis and the theory of intrinsic random functions conditions that ensure a stationary statistically homogeneous head process are derived and using two convenient forms for the covariance function of the ln t process the head covariance function is studied in addition the head variogram is obtained for a particular nonstationary case and the asymptotic head variogram is derived under very general conditions results are compared to those obtained by gelhar 1976 for one and two dimensional phreatic flow and bakr et al 1978 for one and three dimensional confined flow multidimensional flow analysis results in a significantly reduced head variance the head correlation remains high over much greater distances than the ln t correlation the variogram obtained when stationary heads are assumed is identical to that obtained for nonstationary heads for dimensionless lag distances up to 2 times the correlation scale of the log transmissivity the variogram for nonstationary heads continues to grow logarithmically as lag distance increases independent of the form of the input covariance in the nonstationary case the conditions for stationarity are contrasted with the corresponding results obtained for the one and three dimensional cases of gutjahr and gelhar 1981 the head variance calculated from the stationary theory is found to agree with that of previous monte carlo simulations

a new architecture for a parallel finite field multiplier with low complexity based on composite fields a bit parallel structure for a multiplier with low complexity in galois fields is introduced the multiplier operates over composite fields gf 2 sup n sup m with k nm the karatsuba ofman algorithm a karatsuba and y ofmanis 1963 is investigated and applied to the multiplication of polynomials over gf 2 sup n it is shown that this operation has a complexity of order o k sup log23 under certain constraints regarding k a complete set of primitive field polynomials for composite fields is provided which perform module reduction with low complexity as a result multipliers for fields gf 2 sup k up to k 32 with low gate counts and low delays are listed the architectures are highly modular and thus well suited for vlsi implementation

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

riesz transform and riesz potentials for dunkl transform analogues of riesz potentials and riesz transforms are defined and studied for the dunkl transform associated with a family of weight functions that are invariant under a reflection group the lp boundedness of these operators is established in certain cases

structure of parallel multipliers for a class of fields gf 2 m abstract this paper presents a configuration of parallel multipliers for gf 2 m based on canonical bases the possible parallel multipliers by the proposed configuration are limited to a class of fields gf 2 m however they can be constructed by o m 2 and gates and o m 2 eor gates with the structural modularity this is a desirable feature for the hardware implementation and their operation time is about log m t where m is the dimension of gf 2 m and t is the delay time of an eor gate in order to construct such parallel multipliers we define two types of polynomials of special form over gf 2 one is called all one polynomial denoted by aop and the other is called equally spaced polynomial denoted by esp furthermore we show a necessary and sufficient condition for esps to be irreducible over gf 2 and the uniqueness of the irreducible esps over gf 2 finally we propose the configuration of parallel multipliers for a class of fields gf 2 m based on irreducible aops and esps over gf 2

asymptotic dispersion for two dimensional highly heterogeneous permeability fields under temporally fluctuating flow 1 temporal fluctuations of water flux have been investigated as a mechanism that strongly enhances transverse dispersion in heterogeneous media unfortunately most results have been obtained by linear stochastic theories on permeability fields of limited variability worse results are inconsistent regarding the impact of fluctuations on longitudinal dispersion which motivates our work to find the effect of temporal velocity fluctuations on macrodispersion we perform numerical monte carlo simulations for highly variable permeability fields of up to 800 correlation lengths we find that fluctuations longitudinal to the main flow direction hardly modify macrodispersion because they do not alter the flow lines fluctuations transverse to the main flow direction not only increase transverse dispersion which is well known but also reduce the longitudinal macrodispersion in a significant and consistent way which contradicts previous findings the reduction of the longitudinal dispersion is comparable to the increase of transverse dispersion most surprisingly for high heterogeneity temporal fluctuations cause total longitudinal plus transverse macrodispersion to drop with respect to the steady state one enhancement of the transverse macrodispersion comes from both the increase of the transverse velocity variability and lagrangian correlation reduction of the longitudinal macrodispersion results from the reduction of the lagrangian correlation of the longitudinal velocity that is temporal fluctuations reduce longitudinal spreading both by breaking the fastest velocity paths on the plume front and by letting solute bypass the low permeability zones that tend to block or trap the solute in steady state flow conditions

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

flow in heterogeneous media displaying a linear trend in the log conductivity the purpose of this study is to derive a general solution for the problem of flow in nonstationary geological formations where the nonstationary manifests itself in the form of a spatial trend in the mean log conductivity a stochastic frame of reference is adopted to account for the spatial variability of the hydraulic conductivity for a complete stochastic description we derive the expected values and spatial covariances of the hydraulic head and the fluid flux vector as well as a relation between the expected values of the head and the fluxes these expressions are obtained using a perturbation expansion of the log conductivity about its nonstationary mean and they are correct to the first order in the variance of the log conductivity the expressions we derive are applicable for any space dimensionality and for arbitrary orientation of the trend in space a general methodology is outlined for derivation of these expressions for any type of spatial covariance and for demonstration explicit results are obtained for a gaussian isotropic covariance 16 refs 6 figs

stochastic analysis of the concentration variability in a three dimensional heterogeneous aquifer a spectrally based perturbation approach is used to evaluate the concentration variability for a steady concentration field in a three dimensional statistically homogeneous and anisotropic aquifer the analysis assumes small locally stationary concentration perturbations and consequently is valid only after mean displacements that are large compared to the scale of aquifer heterogeneity it is shown that the concentration variance is proportional to the square of the local mean concentration gradient and the variance and the correlation scales of log hydraulic conductivity ln k and is inversely proportional to the local dispersivity the concentration covariance function is highly anisotropic with the largest correlation length aligned to the mean flow direction another important finding is the sensitivity of the longitudinal persistence of the concentration field to the high wave number behavior of the input ln k spectrum the commonly employed exponential type covariance function which corresponds to a nondifferentiable random field results in extremely large concentration correlation lengths along the flow direction input spectra corresponding to differentiable fields with more rapid high wave number cutoffs produce a significant reduction of the longitudinal correlation length

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

macrodispersion in a radially diverging flow field with finite peclet numbers 1 perturbation theory approach we study the transport behavior of a tracer in a radially diverging heterogeneous velocity field making use of homogenization theory we derive effective transport equations these effective transport equations are very similar to those defined on the local scale however the local transport parameters such as local dispersion coefficients are replaced by effective dispersion coefficients for smoothly varying heterogeneous media explicit results for effective radial dispersion coefficients are derived starting with the purely advective transport behavior infinite peclet numbers we extend our calculations to transport with finite peclet numbers we find that the impact of molecular diffusion on the effective dispersivity differs from the impact of local dispersion including local dispersion leads to effective dispersivities which are constant and equivalent to the effective dispersivities found in uniform flow configurations in contrast effective dispersivities including diffusion are not constant but depend on the radial distance we compare the results found by homogenization theory with those derived by neuweiler et al this issue by standard method of moments

state dependent anisotropy comparisons of quasi analytical solutions with stochastic results for steady gravity drainage anisotropy in large scale unsaturated hydraulic conductivity of layered soils changes with the moisture state here state dependent anisotropy is computed under conditions of large scale gravity drainage soils represented by gardner s exponential function are perfectly stratified periodic and inclined analytical integration of darcy s law across each layer results in a system of nonlinear equations that is solved iteratively for capillary suction at layer interfaces and for the darcy flux normal to layering computed fluxes and suction profiles are used to determine both upscaled hydraulic conductivity in the principal directions and the corresponding state dependent anisotropy ratio as functions of the mean suction three groups of layered soils are analyzed and compared with independent predictions from the stochastic results of yeh et al 1985b the small perturbation approach predicts appropriate behaviors for anisotropy under nonarid conditions however the stochastic results are limited to moderate values of mean suction this limitation is linked to a taylor series approximation in terms of a group of statistical and geometric parameters two alternative forms of the taylor series provide upper and lower bounds for the state dependent anisotropy of relatively dry soils

stochastic analysis of the velocity covariance and the displacement covariance tensors in partially saturated heterogeneous anisotropic porous formations considering steady state unsaturated flow in an unbounded domain of heterogeneous formation with three dimensional anisotropic structure first order approximations of the principal and off diagonal components of the velocity covariance tensor were derived for the general case in which the separation vector is inclined to the mean gradient vector j at arbitrary angles taking into account the texture of the soil materials constituting the heterogeneous formation the derived velocity covariance tensors were employed for modeling transport in the vadose zone for the general case in which j is inclined to the principal axes of the formation heterogeneity at arbitrary angles by the use of a lagrangian formulation results of this study suggest that under unsaturated flow for given statistics of the formation properties and given water saturation macrodispersion will diminish in formations of coarser texture when j does not coincide with the principal axes of the heterogeneous formation in this case and in formations of coarser texture the principal axes associated with the principal components of the dimensionless macrodispersion tensor d ij may be deflected in a direction opposite to that of j relative to the principal axes of the formation heterogeneity coarser texture may compensate for the enhanced lateral dispersion and the enhanced deflection of the principal axes of the principal components of d ij due to an increase in the statistical anisotropy of the heterogeneous formation for a formation of given statistics and texture and for a given the longitudinal component of d ij decreases while its transverse components increase when j is more inclined to the longitudinal axis of the formation heterogeneity the latter situation is pertinent to periods of redistribution between successive water applications in which increasing capillary pressure gradients combined with spatial heterogeneity in the formation properties may increase the transverse components of j

a fast algorithm for the estimation of the equivalent hydraulic conductivity of heterogeneous media fast upscaling of hydraulic conductivity is a recurrent problem in modeling flow through heterogeneous porous media we propose a new renormalization technique it is based on the iterative application of the cardwell and parsons 1945 bounds on elementary groups of cells the combination of the bounds with a heuristic formula allows anisotropy to be taken into account the new technique is tested and compared with other fast techniques among the tested techniques the two most reliable ones are the tensorial renormalization and the new simplified renormalization the numerical efficiency of the simplified renormalization leads us to recommend it when a diagonal tensor of equivalent conductivity is sufficient

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

upscaling and dispersion for transport in heterogeneous media this paper focuses on upscaling of the transport equation for heterogeneous porous media with random flow we consider the local flow field being a stationary random field and develop an upscaling by the recently developed coarse graining method which is based on filtering procedures in fourier space the coarse graining method is used to obtain an upscaled dispersion tensor which depends on the given length scale of the upscaling we give explicit results for the scale dependent dispersion coefficient in lowest order perturbation theory for finite length scales the upscaled dispersion models the effect of the unresolved subscale flow fluctuations and for a global upscaling the upscaled value agrees with the well known macrodispersion coefficient which is however nearly approached for length scales larger than tenfold of the correlation length

generalized hermite polynomials and the heat equation for dunkl operators based on the theory of dunkl operators this paper presents a general concept of multivariable hermite polynomials and hermite functions which are associated with finite reflection groups on n the definition and properties of these generalized hermite systems extend naturally those of their classical counterparts partial derivatives and the usual exponential kernel are here replaced by dunkl operators and the generalized exponential kernel k of the dunkl transform in the case of the symmetric group s n our setting includes the polynomial eigenfunctions of certain calogero sutherland type operators the second part of this paper is devoted to the heat equation associated with dunkl s laplacian as in the classical case the corresponding cauchy problem is governed by a positive one parameter semigroup this is assured by a maximum principle for the generalized laplacian the explicit solution to the cauchy problem involves again the kernel k which is on the way proven to be nonnegative for real arguments

mastrovito multiplier for all trinomials an efficient algorithm for the multiplication in gf 2 sup m was introduced by mastrovito the space complexity of the mastrovito multiplier for the irreducible trinomial x sup m x 1 was given as m sup 2 1 xor and m sup 2 and gales in this paper we describe an architecture based on a new formulation of the multiplication matrix and show that the mastrovito multiplier for the generating trinomial x sup m x sup n 1 where m spl ne 2n also requires m sup 2 1 xor and m sup 2 and gates however m sup 2 x sup m 2 xor gates are sufficient when the generating trinomial is of the form x sup m x sup m 2 1 for an even m we also calculate the time complexity of the proposed mastrovito multiplier and give design examples for the irreducible trinomials x sup 7 x sup 4 1 and x sup 6 x sup 3 1

using the nonstationary spectral method to analyze flow through heterogeneous trending media this paper describes a nonstationary spectral theory for analyzing flow in a heterogeneous porous medium with a systematic trend in log hydraulic conductivity this theory relies on a linearization of the groundwater flow equation but does not require the stationarity assumptions used in classical spectral theories the nonstationary theory is illustrated with a two dimensional analysis of a linear trend aligned with the mean flow direction in this case closed form solutions can be obtained for the effective hydraulic conductivity head covariance and log conductivity head cross covariance the effective hydraulic conductivity decreases from the geometrical mean as the mean slope of the log conductivity increases trending leads to a reduction of head variance and a structural change in the head covariance and the log conductivity head cross covariance such changes have important implications for measurement conditioning or cokriging methods which rely on the head covariance and log conductivity head covariance the nonstationary spectral analysis is also compared with classical spectral analysis this comparison indicates that the classical spectral method correctly predicts the normalized head covariance in a linear trending media the stationary spectral method fails to capture the qualitative influence of trends on the effective hydraulic conductivity and the log conductivity head cross covariance although the magnitude of the error is relatively small for realistic values of the mean log conductivity slope the stationary and nonstationary results are the same when there is no trend in log conductivity the trending conductivity example illustrates that the nonstationary spectral method has all the capabilities of the classical spectral approach while not requiring as many restrictive assumptions

a space time accurate method for solving solute transport problems solute transport studies frequently rely on numerical solutions of the classical advection diffusion equation unfortunately solutions obtained with traditional finite difference and finite element techniques typically exhibit spurious damping or oscillation when advection dominates recently developed variants of these techniques such as the finite analytic method chen and li 1979 chen and chen 1984 and the optimal test function method celia et al 1989a b c perform well for steady state problems extensions of these methods to the transient case have however not been successful primarily because of inadequate approximations of the temporal derivative the new numerical method proposed in this paper avoids this difficulty by taking the laplace transform of the transient equation the transformed expression behaves like a steady state advection diffusion equation with a first order decay term this expression can be solved with either the finite analytic or optimal test function method and the time dependence recovered with an efficient inverse laplace transform algorithm the result is an accurate and robust transient solution which performs well over a very wide range of peclet numbers we demonstrate this approach by applying the finite analytic method to a laplace transformed one dimensional model problem a comparison with other competing techniques shows that good approximations are required in both space and time in order to obtain accurate solutions to advection dominated problems a good space approximation combined with a poor temporal approximation or vice versa does not give satisfactory results the method we propose provides a balanced space time approximation which works very well for one dimensional problems extensions to multiple dimensions are conceptually straightforward and briefly discussed

approximation of asymptotic dispersivity of conservative solute in unsaturated heterogeneous media with steady state flow the relation between the heterogeneity of hydraulic properties and effective asymptotic transport is studied for microscopically heterogeneous but macroscopically homogeneous unsaturated media with steady flow heterogeneity is described by the scaling of the hydraulic functions m and k the study is based on an analytical approximation of asymptotic dispersivity under the assumption that matric potential is spatially constant for the special case of a water saturated medium the result of the stochastic continuum theory is recovered when applied to several published numerical simulations the approximated asymptotic dispersivities are found to agree well with the numerical values by exploring the heterogeneous media for various hydraulic states the approximation resolves some apparent inconsistencies found in the simulations the cases considered span the entire range from perfect to zero correlation between scaling factors of matric potential and hydraulic conductivity it is demonstrated that this correlation is crucial for the behavior of asymptotic dispersivity with changing flow rate since weak to moderate correlations are often found in soils this result has significant implications for solute transport through heterogeneous soils

effective hydraulic conductivity of three dimensional heterogeneous formations of lognormal permeability distribution the impact of connectivity uniform mean flow takes place in a 3 d heterogeneous formation of normal hydraulic logconductivity y ln k the aim of the study is to derive the dependence of the horizontal kefh and vertical kefv effective conductivities on the structural parameters of hydraulic conductivity and investigate the impact of departure from multi gaussianity on kef by numerical simulations of flow in formations that share the same pdf and covariance of y but differ in the connectivity of classes of y the main result is that for the extreme models of connected and disconnected high y zones the ratio between the effective conductivities in isotropic media is much smaller than in 2 d the dependence of kefh and kefv upon the logconductivity variance and the anisotropy ratio is compared with existing approximations first order landau matheron conjecture self consistent approximation besides the theoretical interest the results offer a basis for empirical relationships to be used in applications

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

on spectral measure and monte carlo approaches to flow in heterogeneous media a pedagogical approach to the spectral integral method as applied to flow problems is presented the deterministic fourier analog to the random spectral integral is highlighted thus illustrating the close ties between the two the general solution to the deterministic flow problem is used to illustrate that a weakly stationary conductivity field does not in general lead to a weakly stationary head field several alternative monte carlo methods are presented and compared to illustrate the problems associated with dropping terms of the form h from the flow equation where and h are respectively perturbations in the conductivity and head fields

a second order approach for the modeling of dispersive transport in porous media 1 theoretical development a new approach for the treatment of dispersive transport in a simplified porous medium is presented use of a volume averaging approach allows the development of a separate point macroscopic balance equation for the dispersion quantity that appears in the averaged transport equation the new equation is a more general representation which avoids the complications and questions that arise concerning the coefficients associated with the fickian approximation for dispersion constitutive relationships are developed for a number of the source terms arising in the averaged dispersion equation the coupled transport and dispersion equations along with darcy s law and the macroscopic continuity equation constitute a closed second order model for dispersion in a porous medium

stochastic modeling of groundwater flow by unconditional and conditional probabilities 2 the solute transport the problem of solute transport in a heterogeneous formation whose transmissivity or hydraulic conductivity are subject to uncertainty is studied for two and three dimensional flows approximate closed form solutions are derived for the case of a solute pulse in an average uniform flow through a formation of unconditional stationary random transmissivity the solute concentration regarded as a random variable is determined in terms of its expectation and variance and is found to be subject to a high degree of uncertainty the uncertainty is greatly reduced if the solute input zone is large compared to the transmissivity integral scale in any case the concentration expectation does not obey a diffusion type equation in the case of two dimensional flows unless the solute body has traveled a distance larger than a few tens transmissivity integral scales this distance may be exceedingly large in many conceivable applications

modular construction of low complexity parallel multipliers for a class of finite fields gf 2 sup m structures for parallel multipliers of a class of fields gf 2 sup m based on irreducible all one polynomials aop and equally spaced polynomials esp are presented the structures are simple and modular which is important for hardware realization relationships between an irreducible aop and the corresponding irreducible esps have been exploited to construct esp based multipliers of large fields by a regular expansion of the basic modules of the aop based multiplier of a small field some features of the structures also enable a fast implementation of squaring and multiplication algorithms and therefore make fast exponentiation and inversion possible it is shown that if for a certain degree an irreducible aop as well as an irreducible esp exist then from the complexity point of view it is advantageous to use the esp based parallel multiplier

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

the significance of heterogeneity of evolving scales to transport in porous formations flow takes place in a heterogeneous formation of spatially variable conductivity which is modeled as a stationary space random function to model the variability at the regional scale the formation is viewed as one of a two dimensional horizontal structure a constant head gradient is applied on the formation boundary such that the flow is uniform in the mean a plume of inert solute is injected at t 0 in a volume v0 under ergodic conditions the plume centroid moves with the constant mean flow velocity u and a longitudinal macrodispersion coefficient dl may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid for a log conductivity covariance cy of finite integral scale i at first order in the variance y2 and for a travel distance l ut i dl y2ui and transport is coined as fickian ergodicity of the moments is ensured if l i where l is the initial plume scale some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with l without reaching a constant limit anomalous diffusion to model such a behavior previous studies have assumed that cy is stationary but of unbounded integral scale with cy ar 1 0 for large lag r under ergodic conditions it was found that asymptotically dl aul1 i e non fickian behavior and anomalous dispersion the present study claims that an ergodic behavior is not possible for a given finite plume of initial size l since the basic requirement that l i cannot be satisfied for cy of unbounded scale for instance the centroid does not move any more with u but is random figure 1 owing to the large scale heterogeneity in such a situation the actual effective dispersion coefficient dl is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization this is the accessible entity in a given experiment we show that in contrast with dl the behavior of dl is controlled by l and it has the fickian limit dl aul1 figure 3 we also discuss the case in which y is of stationary increments and is characterized by its variogram y then u and dl can be defined only if y is truncated equivalently an infrared cutoff is carried out in the spectrum of y however for a bounded u it is shown that dl depends only on y furthermore for y ar dl aul2l 1 i e dispersion is fickian for 0 1 whereas for 1 2 transport is non fickian since 2 dl cannot grow faster than l ut this is in contrast with a recently proposed model neuman 1990 in which the dispersion coefficient is independent of the plume size and it grows approximately like l1 5

prediction of velocity statistics in three dimensional multi gaussian hydraulic conductivity fields 1 to study statistics of velocity fields in three dimensional heterogeneous multi gaussian saturated hydraulic conductivity fields and the accuracy of their prediction we performed high resolution monte carlo mc analyses the mc analyses included variances of the log hydraulic conductivity in the range of 0 5 y2 3 0 and anisotropy ratios in the range of 0 017 e 1 the statistics of the velocity fields from the mc analyses are compared with analytical solutions of the first and second order approximations of the stochastic flow equation this paper shows that the second order approximations fit significantly better to the univariate statistics of the darcy velocity from the mc analyses for isotropic cases the second order approximations correspond fairly well to the univariate statistics of the velocity for anisotropic cases the accordance is given only for the mean velocity and the variance of the transverse vertical component of the velocity the mc analyses show that the spatial correlation of the velocity decreases more rapidly with increasing y2 this was more pronounced for the anisotropic than for the isotropic case the negative correlations in absolute terms of the transverse velocity components simultaneously decrease with increasing y2 this is in contrast to the first order approximation of the spatial correlations of the velocity it is assumed that the discrepancies between approximate solutions of the stochastic flow equation and the results of the mc analyses are strongly dependent on the nonnormality of the probability density distributions of the velocity

a positive radial product formula for the dunkl kernel it is an open conjecture that generalized bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated dunkl operators in this paper a partial result towards this conjecture is proven namely a positive radial product formula for the non symmetric counterpart of the generalized bessel function the dunkl kernel radial here means that one of the factors in the product formula is replaced by its mean over a sphere the key to this product formula is a positivity result for the dunkl type spherical mean operator it can also be interpreted in the sense that the dunkl type generalized translation of radial functions is positivity preserving as an application we construct dunkl type homogeneous markov processes associated with radial probability distributions

monte carlo studies of flow and transport in fractal conductivity fields comparison with stochastic perturbation theory a monte carlo simulation of flow and transport is employed to study tracer migration in porous media with evolving scales of heterogeneity fractal media transport is studied with both conservative and reactive chemicals in media that possess physical as well as chemical heterogeneity linear kinetic equations are assumed to relate the sorbed phase and the aqueous phase concentrations the fluctuating log conductivity possesses the power law spectrum of a fractional brownian motion fbm chemical heterogeneity is represented as spatially varying reaction rates that also are assumed to obey fbm statistics and may be correlated to the conductivity field the model is based on a finite difference approximation to the flow problem and a random walk particle tracking approach for solving the solute transport equation the model is used to make comparisons with the nonlocal transport equations recently developed by deng et al 1993 and hu et al 1995 1997 the results presented herein support these nonlocal models for a wide range of heterogeneous systems however the infinite integral scale associated with the fractal conductivity has a significant effect on the prediction of the nonlocal theories this suggests that integral scale should play a role in stochastic eulerian perturbation theories the importance of the local scale dispersion depends to a great extent on the magnitude of the local dispersivities the effect of neglecting local dispersion decreases with the decrease in local dispersivity

random evolution equations in hydrology a number of partial differential equations governing the dynamic behavior of different components of the hydrologic cycle in a region can be treated as abstract evolution equations in appropriate hilbert spaces when the source terms the boundary conditions the initial conditions and or the equation parameters are prescribed as stochastic processes the resulting random evolution equations may be solved in terms of the input stochastic processes if the semigroup associated with the partial differential operator is known this places in the hands of the modeler a very flexible tool to solve a large variety of problems several explicit examples of the methodology are illustrated finally applications of the ito s lemma in hilbert spaces to the numerical solution of stochastic partial differential equations in hydrology are presented

high resolution monte carlo simulation of flow and conservative transport in heterogeneous porous media 1 methodology and flow results in this the first of two papers concerned with the use of numerical simulation to examine flow and transport parameters in heterogeneous porous media via monte carlo methods various aspects of the modelling effort are examined in particular the need to save on core memory causes one to use only specific realizations that have certain initial characteristics in effect these transport simulations are conditioned by these characteristics also the need to independently estimate length scales for the generated fields is discussed the statistical uniformity of the flow field is investigated by plotting the variance of the seepage velocity for vector components in the x y and z directions finally specific features of the velocity field itself are illuminated in this first paper in particular these data give one the opportunity to investigate the effective hydraulic conductivity in a flow field which is approximately statistically uniform comparisons are made with first and second order perturbation analyses the mean cloud velocity is examined to ascertain whether it is identical to the mean seepage velocity of the model finally the variance in the cloud centroid velocity is examined for the effect of source size and differing strengths of local transverse dispersion

permeability estimates of self affine fracture faults based on generalization of the bottleneck concept we propose a method for calculating the effective permeability of two dimensional self affine permeability fields based on generalizing the one dimensional concept of a bottleneck we test the method on fracture faults where the local permeability field is given by the cube of the aperture field the method remains accurate even when there is substantial mechanical overlap between the two fracture surfaces the computational efficiency of the method is comparable to calculating a simple average and is more than two orders of magnitude faster than solving the reynolds equations using a finite difference scheme pacs numbers in many low permeability geological formations flow occurs primarily through fracture networks 19 in order to model such systems and to predict their behavior there is a need for reliable modeling of the hydromechanical behavior of fracture we consider in this note the situation where the shear displacement between the fracture walls strongly affects its permeability because of its relevance this situation has been considered in many recent hydromechanical studies 2 3 10 18 20 27 28 laboratory tests report that the shearing process results in a significant channelization of the flow and an enhancement of the permeability in the direction normal to the shear this behavior is found to be related to the longrange spatial organization of the void space and efforts have been undertaken to modelize such system in order to provide upscaled value for the fracture permeability recently mallikanas and rajaram 14 determined analytically using perturbation analysis of the reynolds equation to the lowest non trivial order the fracture permeability this model however does not take into account the role of contact areas and will fail if they appear the effect of contacts may however be taken into account by introducing an empirical parameter 29 that is strongly influenced by the number and the spatial distribution of the contacts 13 we present in this note a computational method for calculating the permeability of such fracture faults even in the presence of contacts this new method scales linearly with the number of grid points and is more than two orders of magnitude faster than solving the finitedifferenced reynolds equations through lu decomposition

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

stochastic models of subsurface flow infinite versus finite domains and stationarity stochastic solutions of the differential equation describing one dimensional flow through a porous medium with spatially variable hydraulic conductivity which is represented by a stationary statistically homogeneous random process are developed using several techniques the analytical approximations using first order analysis propagation of error covariance differential equations and variogram analysis all yield consistent results which demonstrate the important effects of boundary conditions and conditioning using the theory of intrinsic random functions stochastic solutions are developed for the case when the logarithm of the hydraulic conductivity is a three dimensional stationary random field in contrast to the one dimensional case it is found that the resulting three dimensional head perturbation will be locally stationary under very general conditions results from the one dimensional analytical solutions are found to be in agreement with previous monte carlo simulations for a flow system of finite length the solution based on linearization in the logarithm of hydraulic conductivity proved to be very robust showing reasonable agreement with monte carlo results even for the largest input standard deviation of lnk 4 6

cross correlated random field generation with the direct fourier transform method this paper presents a computer algorithm that is capable of cogenerating pairs of three dimensional cross correlated random fields the algorithm produces random fields of real variables by the inverse fourier transform of a randomized discrete three dimensional spectral representations of the variables the randomization is done in the spectral domain in a way that preserves the direct power and cross spectral density structure two types of cross spectra were examined one type specifies a linear relationship between the two fields which produces the same correlation scales for both variables but different variances the second cross spectrum is obtained from a specified transfer function and the two power spectra and it produces fields with different correlation scales for both models the degree of correlation is specified by the coherency a delay vector can also be specified to produce an out of phase correlation between the two fields the algorithm is very efficient computationally is relatively easy to use and does not produce the lineation problems that can be encountered with the turning bands method perhaps most important this random field generator is capable of co generating cross correlated random fields

stochastic analysis of steady state unsaturated flow in heterogeneous media comparison of the brooks corey and gardner russo models existing stochastic models of unsaturated flow and transport are usually developed using the simple gardner russo constitutive relationship though it is generally accepted that the more complex van genuchten and brooks corey relationships may perform better in describing experimental data in this paper we develop first order stochastic models for gravity dominated flow in second order stationary media with both the brooks corey and the gardner russo constitutive relationships these models also account for the spatial variability in effective water content while the spatial variability is generally neglected in most existing stochastic models analytical solutions are obtained for the case of one dimensional gravity dominated flow on the basis of the solutions we illustrate the differences between results from these two constitutive models through some one dimensional examples it is found that the impacts of the constitutive models on the statistical moments of suction head effective water content unsaturated hydraulic conductivity and velocity depend on the saturation ranges for example the mean head and the mean effective water content for the brooks corey model differ in a great manner with their counterparts for the gardner russo model near the dry and wet limits while the differences are small at the intermediate range of saturation this finding is confirmed with some two dimensional examples it is also found that the brook corey model has certain advantages over the gardner russo model in analyzing unsaturated flow in heterogeneous media for example the stochastic model developed based on the brooks corey function requires the coefficient of variation of head and soil parameter bc to be small 1 whereas that based on the gardner russo function assumes the one point cross covariance of head and olgr to be small 1 illustrative examples reveal that the latter condition may be violated because the one point covariance is found to increase rapidly to beyond unity as the soil becomes dry whereas the former may be readily satisfied

an investigation of eulerian lagrangian methods for solving heterogeneous advection dominated transport problems numerical simulation of solute transport in heterogeneous porous media is greatly complicated by the large velocity and concentration gradients induced by spatial variations in hydraulic conductivity eulerian lagrangian methods for solving the transport equation can give accurate solutions to heterogeneous problems if their interpolation algorithms are properly selected this paper compares the performance of four eulerian lagrangian solvers that rely on linear quadratic cubic spline and taut spline interpolators in each case a tensor product decomposition is used to reduce the general n dimensional interpolation problem to a sequence of n one dimensional problems comparisons of a set of test problems indicate that the linear and taut spline interpolators are dispersive while the quadratic and cubic spline interpolators are oscillatory the cubic and taut spline interpolators give consistently better accuracy than the more conventional linear and quadratic alternatives simulation experiments in two and three dimensional heterogeneous media indicate that the taut spline interpolator which is applied here for the first time to a solute transport problem is able to yield accurate essentially nonoscillatory solutions for high grid peclet numbers the cubic spline interpolator requires significantly less computational effort to achieve performance comparable to the other methods