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

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

optimal bounds on the effective bulk modulus of polycrystals the classical problem of bounding the effective bulk modulus of isotropic polycrystalline aggregates built from a single anisotropic phase is studied for three dimensional composites we find that the voigt and reuss bounds are each attained by multi layered laminate geometries and hence are optimal for two dimensional composites the reuss lower bound is proved to be optimal while a new optimal upper bound is derived

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

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

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

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

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

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

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

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

simulation of contaminant transport in three dimensions 1 the alternating direction galerkin technique the alternating direction galerkin technique for the simulation of advective dispersive transport in three dimensions is developed the technique is designed for high efficiency in handling the large and detailed grids that are often required in simulations of natural groundwater systems it is comparable to an alternating direction finite difference scheme except for two advantages first the capability to handle certain curvilinear grids that may conform to either a flow net geometry or a natural stratigraphy and second the option to enhance solution accuracy through a choice between a finite difference or a finite element representation of the time derivative criteria for controlling numerical dispersion and accuracy can be applied easily three options for the time stepping algorithm are developed and analyzed for stability and their accuracy is investigated for simple flow systems the technique is shown to be far more efficient than a conventional three dimensional finite element model in a companion paper an application to a field scale contaminant transport problem is described

pseudopotential functions in construction of flow nets for contaminant transport modeling construction of flow nets in most hydrogeologic field problems is complicated by nonexistence of orthogonality between streamlines and hydraulic head lines in anisotropic porous media the nonexistence of orthogonality can be circumvented by transformation of the anisotropic porous medium into an isotropic one flow nets are valuable in generation of a numerical solution grid for a contaminant transport equation formulated along the direction of flow and the direction orthogonal to flow effect of flow domain transformation on contaminant transport simulation is not known the flow domain transformation can be avoided by constructing flow nets with streamlines and pseudopotential lines the pseudopotential lines are based on the theory of pseudopotential function a discussion of the theory in isotropic porous media exists in literature to my knowledge this is the first time that the theory is being extended to anisotropic porous media an integration procedure is used to evaluate values of pseudopotential function and hydraulic head along a streamline a comparison of flow nets based on pseudopotential lines and hydraulic head lines indicates that pseudopotential lines are more useful than hydraulic head lines in simulation of contaminant transport in saturated groundwater flow systems

asymptotic estimates on uncertainty of the elastic moduli of completely random trigonal polycrystals because of the uncertain irregular microgeometry a random polycrystalline aggregate the shape and crystalline orientations of the constituent anisotropic grains of which are uncorrelated may have elastic moduli scattered over some though small intervals while the conventional macroscopic homogeneity and isotropy hypotheses for it may be considered only as approximate with corresponding uncertainty our formal bounds are proposed to provide the estimates on those uncertainties in asymptotic sense explicit expressions for the aggregates of trigonal crystals classes 3m 32 3m are derived

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

an alternating direction galerkin technique for simulation of contaminant transport in complex groundwater systems a general contaminant transport simulation technique based on a galerkin finite element representation in space and an alternating direction time stepping scheme is developed the formulation which is in terms of natural coordinates following the principal directions of hydraulic conductivity of the medium decouples the spatial components of the equations leading to tridiagonal matrices the numerical accuracy can be easily controlled in each principal direction through the well established peclet and courant numbers results are found to be comparable to those from the classic finite element scheme but the cost of computation is generally much less because the computational effort varies only linearly with the grid size the technique is thus well suited for transport simulations with complex layered anisotropic groundwater systems requiring large and detailed grids

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

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

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

simulation of contaminant plumes with large dispersive contrast evaluation of alternating direction galerkin models the evolution of narrow sharply defined contaminant plumes corresponding in shape to those often observed in the field is examined it is shown that the transverse dispersion mechanism consistent with such shapes can be as low as of the order of molecular diffusion and that dispersivity contrasts of as much as 4 orders of magnitude are possible even at the lowest physically realistic values of transverse dispersion parameters however transverse mass spreading is capable of significantly retarding the advance of the plume front the principal direction and alternating direction galerkin models together are found to be capable of handling all cases of dispersive contrast including those with dispersivity ratio of infinity the models are formulated in curvilinear coordinates with certain restrictions on the element deformation a three way comparison between principal direction alternating direction galerkin and conventional finite element models is performed with respect to accuracy and efficiency the effects of longitudinal and transverse numerical dispersion occuring in the various models are examined and related to the discretization visual indications of the presence of numerical error are discussed

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

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

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

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

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

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

mirror symmetry for orbifold hurwitz numbers we study mirror symmetry for orbifold hurwitz numbers we show that the laplace transform of orbifold hurwitz numbers satisfy a dierential recursion which is then proved to be equivalent to the integral recursion of eynard and orantin with spectral curve given by the r lambert curve we argue that the r lambert curve also arises in the innite framing limit of orbifold gromov witten theory of c 3 z rz finally we prove that the mirror model to orbifold hurwitz numbers admits a quantum curve

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

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

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

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

on the gromov witten donaldson thomas correspondence and ruan s conjecture for calabi yau 3 orbifolds for any toric calabi yau 3 orbifold with transverse a singularities we prove the gromov witten donaldson thomas correspondence and ruan s crepant resolution conjecture in all genera

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

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

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 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

estimation of overall properties of random polycrystals with the use of invariant decompositions of hooke s tensor in the paper the theoretical analysis of bounds and self consistent estimates of overall properties of linear random polycrystals composed of arbitrarily anisotropic grains is presented in the study two invariant decompositions of hooke s tensors are used the applied method enables derivation of novel expressions for estimates of the bulk and shear moduli which depend on invariants of local stiffness tensor with use of these expressions the materials are considered for which at the local level constraints are imposed on deformation or some stresses are unsustained