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

spatial variability of soil moisture and the scale issue a geostatistical approach we study the standard deviation of water saturation sds as function of the mean saturation s by a stochastic model of unsaturated flow which is based on the first order solution of the three dimensional richards equation the model assumes spatially variable soil properties following a given geostatistical description and it explicitly accounts for the different scales involved in the determination of the spatial properties of saturation the extent l i e the domain size the spacing among measurements and the dimension l associated to the sampling measurement it is found that the interplay between those scales and the correlation scale i of the hydraulic properties rules the spatial variability of saturation a scale effect manifests for small to intermediate l i for which sds increase with the extent l this nonergodic effect depends on the structural and hydraulic parameters as well as the scales of the problem and it is consistent with a similar effect found in field experiments in turn the influence of the scale l is to decrease the saturation variability and increase its spatial correlation although the solution focuses on the medium heterogeneity as the main driver for the spatial variability of saturation neglecting other important components it explicitly links the spatial variation of saturation to the hydraulic properties of the soil their spatial variability and the sampling schemes it can provide a useful tool to assess the impact of scales on the saturation variability also in view of the several applications that involve the saturation variability

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

statistical theory of groundwater flow and transport pore to laboratory laboratory to formation and formation to regional scale the statistical approach has been applied increasingly to groundwater flow problems in the last decade as is illustrated in figure 3 by the cumulative number of articles published in this field in water resources reseach this development has been motivated by the recognition of the fact that porous formations are heterogeneous i e with properties which vary in an irregular manner in space flow domains are characterized by the length scale l of their spatial extent and three such scales of a fundamental nature are introduced the laboratory the local and the regional scale heterogeneity is characterized by the spatial correlation scale i of the property of interest the three scales corresponding to the above ones being the pore scale the log hydraulic conductivity and the log transmissivity integral scales the medium properties and related flow variables are regarded as random space functions which satisfy two basic requirements they enjoy some type of stationarity and i l an additional scale d is the measurement or computational scale characterizing the size of the measurement device of a flow variable or the element over which the variable is averaged for computational purposes in both cases the interest resides in the space average of the flow variable over a volume or area of length scale d the primary aim of the theory of flow and transport through porous media is to determine the statistical moments of the space averaged variables given the statistical structure of the spatially variable property the main objective of the study is to show that flow and transport problems at the three fundamental scales can be treated by a unified statistical approach along this line the specific aspects of each scale are examined separately and areas of interest for future research are indicated in the concluding remarks it is submitted that the statistical approach to groundwater flow has become a comprehensive theory beyond the stage of an ad hoc modeling technique

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

a simple and efficient space domain implementation of the turning bands method given a correlation function c x with x in rn the turning bands method tbm for n 2 and 3 constructs realizations of an n dimensional and stationary process y n o c x from appropriately summed line processes therefore an implementation of tbm calls naturally for fast and accurate generations of line realizations these have generally been generated by a spectral approach since the fourier transform of the line covariances is linked in a very simple fashion to the n dimensional fourier transform of c x however we show that in many instances a space domain implementation of tbm with line realizations obtained from a circulant embedding method has advantages over a spectral implementation in that without increase in computational costs 1 the line realizations display exactly the required covariance structure 2 knowledge of the spectral density of the process y is not required and 3 fine tuning of line process parameters is not needed

the extended thiem s solution including the impact of heterogeneity 1 in this study we present a formula for the hydraulic head describing the mean drawdown of a three dimensional steady state pumping test in heterogeneous anisotropic porous media effectively by modeling the hydraulic conductivity k x as spatial random function and using the upscaling method coarse graining we succeed in deriving a closed form solution hefw r which we understand as an extension of thiem s formula to heterogeneous media the solution hefw r does not only depend on the radial distance r but accounts also for the statistics of k x namely geometric mean kg variance 2 horizontal correlation length l and anisotropy ratio e we perform a sensitivity analysis on the parameters of hefw r and implement an inverse estimation strategy using numerical pumping tests we show the applicability of hefw r on the interpretation of drawdown data this will be done for both an ensemble of as well as for single pumping tests making use of the inverse estimation method we find excellent agreement of estimated parameters with initial values in particular for the horizontal correlation length

effective hydraulic conductivity of bounded strongly heterogeneous porous media we develop analytical expressions for the effective hydraulic conductivity ke of a three dimensional heterogeneous porous medium in the presence of randomly prescribed head and flux boundaries the log hydraulic conductivity y forms a gaussian statistically homogeneous and anisotropic random field with an exponential autocovariance by effective hydraulic conductivity of a finite volume in such a field we mean the ensemble mean expected value of all random equivalent conductivities that one could associate with a similar volume under uniform mean flow we start by deriving a first order approximation of an exact expression developed in 1993 by neuman and orr we then generalize this to strongly heterogeneous media by invoking the landau lifshitz conjecture upon evaluating our expressions we find that ke decreases rapidly from the arithmetic mean ka toward an asymptotic value as distance between the prescribed head boundaries increases from zero to about eight integral scales of y the more heterogeneous is the medium the larger is ke relative to its asymptote at any given separation distance our theory compares well with published results of spatially power averaged expressions and with a first order expression developed intuitively by kitanidis in 1990

generalized area operators on hardy spaces abstract we show that if 0 0 and 1 p 0 then the operatorgfmaps the hardy sobolev spacehp intolp d if and only if the functiong d 1 z belongs to the morrey spacelp 1 p in casep 1 this condition is equivalent to the condition that t i c i 1 for all arcsicontained in the circle wheret i is the tent overicontained in the unit disk

modeling the large scale dynamics of saturated groundwater flow using spatial filtering theory 1 theoretical development we develop a model and a framework for understanding the explicitly resolved large scale dynamics of groundwater flow in fully saturated heterogeneous porous media the approach is based upon the large eddy simulation les methodology from geophysical fluid dynamics the idea behind the les approach is to explicitly resolve the larger scales of flow and provide a closure model for the effect of subgrid scales upon resolved scales the technical foundation of the method rests on a spatial filtering operation which allows one to precisely define the scale of the model we spatially filter darcy s law and the continuity equation to produce a model for the resolved scale dynamics we develop les approximations for the resulting flow equations and a closure model for unresolved subgrid scale terms we model the subgrid terms using resolved scale quantities only and thus do not require subgrid scale information the accuracy of these closure models depends upon the scale at which hydraulic conductivity is measured by field tests we characterize the approximations used to model subgrid scale effects asymptotically with a small parameter the dimensionless filter width defined as the ratio of the filter width to the dominant length scale of the explicitly resolved variables

upscaling of permeability of anisotropic heterogeneous formations 1 the general framework in numerical simulations of flow through heterogeneous formations the domain is partitioned into numerical elements the solution requires assigning physical properties to each numerical block the process of transferring information from the scale of actual heterogeneity to that of the numerical elements is known as upscaling we consider steady one phase flow and the only property of interest is the permeability which is regarded as a random space function and the same is true for the upscaled conductivity in this paper we establish the necessary and sufficient conditions to be satisfied by upscaling the necessary conditions are expressed with the aid of the global response of the formation e g the total flux caused by a constant pressure head drop applied on the boundaries the requirement is that the expected value and the variance of the total flux are the same in the actual and upscaled domains these are supplemented by local conditions which stipulate that space averages of flow variables over the numerical blocks in the upscaled domain tend to those in the actual formation when the elements size tends to 0 the result of this analysis is that the upscaled permeability must be such as to ensure the equality in a statistical sense between the space averaged dissipation in the two media by assuming an unbounded domain average uniform flow and stationarity simplified relationships between the moments of the dissipation in the upscaled and actual media are derived it is shown that the effective permeability of the two media is the same the approach is illustrated for stratified formations for which exact composition rules are possible even in this simple case the upscaled permeability is a tensor characterized at second order by two mean components and three covariance functions

three dimensional unsaturated flow modeling using cellular automata 1 the parabolic partial differential equation describing fluid flow in partially saturated porous media richards equation is highly nonlinear due to pressure head dependencies in the specific soil moisture capacity and relative hydraulic conductivity terms in order to solve richards equation several numerical techniques have been developed which starting from the discretization of the partial differential equation produced even more accurate models leading to complex and computationally expensive simulations for large scale systems a three dimensional unsaturated flow modeling developed through a simulation environment based on cellular automata ca is described in this paper the proposed model represents an extension of the original computational paradigm of cellular automata because it uses a macroscopic ca approach where local laws with a clear physical meaning govern interactions among automata this ca structure aimed at simulating a large scale system is based on functionalities capable of increasing its computational capacity both in terms of working environment and in terms of the optimal number of processors available for parallel computing the model has been validated with reference multidimensional solutions taken from benchmarks in literature showing a good agreement even in the cases where nonlinearity is very marked furthermore some analyses have been carried out considering quantization techniques aimed at transforming the ca model into an asynchronous structure the use of these techniques in a three dimensional benchmark allowed a considerable reduction in the number of local interactions among adjacent automata without changing the efficiency of the model especially when simulations are characterized by scarce mass exchanges finally from a computational point of view the higher efficiency values were achieved running the model on a parallel architecture obtaining a high speedup very close to the optimal with the maximum number of processors available

is transmissivity a meaningful property of natural formations conceptual issues and model development 1 at regional scale it is common to model groundwater flow as 2 d in the x y horizontal plane by integrating the full 3 d equations over the vertical furthermore adopting the dupuit assumption results in the local transmissivity t as a formation property equal to the vertically integrated hydraulic conductivity k in practice the related block transmissivity tb defined for a volume of area square of side l in the horizontal plane and height d is the property of interest however most aquifers are of a heterogeneous 3 d structure and y lnk is commonly modeled as a normal and stationary random function which is characterized by the variance y2 the horizontal i and vertical iv integral scales the dupuit assumption is generally not obeyed for formations of 3 d spatially variable y and transmissivity is no more a meaningful property independent of flow conditions useful generalizations of local and block transmissivity are possible for steady mean uniform flow in the horizontal direction and formations of constant thickness in that case t and tb become random stationary variables characterized by their mean variance and integral scales these moments are determined for the first time in an analytical form or by a few quadratures by adopting a first order approximation in y2 and they depend on the ratio d iv e iv i and l i the block conductivity expected values are compared with the numerical solutions of dykaar and kitanidis 1993 and the agreement is very good for y2 1 the main conclusion of the study is that for this simple flow configuration and for common parameter values tb is practically deterministic and equal to keff e d where keff is the effective conductivity in uniform flow in an unbounded formation at regional scale tb may be regarded as a local property which changes slowly in the horizontal plane analysis of numerous field data shows that this variation is also random and characterized by integral scales ireg of the order of kilometers the separation of scales makes possible to regard the local tb as determined along the lines of the present study in a support volume of extent of a few d as a point value at the regional scale practical implications and topics for future investigations are outlined

stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 2 impervious boundary the spatial variability of the head h in a two dimensional flow in the vicinity of an impervious boundary is investigated this study is a continuation of a previous one rubin and dagan 1988 which dealt with the influence of a given head boundary by representing h and the log transmissivity y as space random function and by solving the flow equation first order closed form analytical expressions for the head variogram and the head log transmissivity y covariance are developed these expressions offer the means to evaluate the influence of the boundary on the joint y h random field the assumption of unbounded domain is examined and it is found to be a good approximation at distances larger than three log transmissivity integral scales from the boundary by using conditional probabilities a simple method to estimate the statistical moments close to the boundary is presented it requires the knowledge of the moments of y and h in the unbounded domain only the method is compared with the exact analytical solution showing favorable results

multilevel upscaling through variational coarsening 1 a new efficient multilevel upscaling procedure for single phase saturated flow in porous media is presented while traditional approaches to this problem have focused on the computation of an upscaled hydraulic conductivity here the coarse scale model is created explicitly from the fine scale model through the application of operator induced variational coarsening this technique which originated with robust multigrid solvers has been shown to accurately capture the influence of fine scale heterogeneous structure over the complete hierarchy of coarse scale models that it generates moreover implicit in this hierarchy is the construction of interpolation operators that provide a natural and complete multiscale basis for the fine scale problem thus this new multilevel upscaling methodology is similar to the multiscale finite element method and indeed it attains similar accuracy in computations of the fine scale hydraulic head and coarse scale normal flux on a variety of problems yet it is an order of magnitude faster on the examples considered here

a note on well boundary condition for flow through heterogeneous formations 1 in modeling flow toward wells it is mathematically convenient to replace the well by a singularity line along its axis in the case of homogeneous aquifers confined flow and fully penetrating wells the strength of the sources is constant and the common condition of constant head on the well is satisfied exactly it also leads to constant flux on the well envelope in the case of aquifers of spatially variable conductivity the constant head boundary condition can be satisfied accurately if the well radius is much smaller than the log conductivity horizontal integral scale by selecting a variable source strength that is proportional to the local conductivity along the well indelman et al 1996 fiori et al 1998 indelman 2003a 2003b in this case the equivalent conductivity tends to the arithmetic mean near the well it changes with distance approaching from above the effective conductivity in uniform horizontal mean flow far from the well selecting inadvertently a constant source strength violates the condition of constant well head the equivalent conductivity tends to the harmonic mean near the well and approaches from below the effective one in uniform mean flow the equivalent conductivity is derived for both types of sources for stratified formations as well as for aquifers of a three dimensional heterogeneous structure the implications on interpretation of head measurements by the two models is illustrated by a few examples

upscaling of permeability of anisotropic heterogeneous formations 3 applications the general results of indelman and dagan this issue a b are applied to some cases of interest in applications the exact solution of upscaling of stratified formations is used in order to assess the range of applicability of the first order analysis the explicit expressions of the two first moments of the upscaled permeability are derived for media of two dimensional isotropic and three dimensional axisymmetric actual log permeability of gaussian covariance and for parallelepipedic rectangular for two dimensional averaging elements the effects of the actual log permeability covariance and of the anisotropy of partition elements on the upscaled permeability statistics are analyzed the asymptotic behavior of the upscaled permeability for large and small blocks is discussed for general heterogeneity the strong dependence of the small element limit on the properties of actual heterogeneity and on the shape of the elements is illustrated by typical examples in contrast the homogenization of the upscaled media by large blocks depends primarily on the volume area for two dimensions of the partition element relative to the one based on the correlation scale of the log permeability

stochastic modeling of groundwater flow by unconditional and conditional probabilities the inverse problem the inverse problem is defined here as follows determine the transmissivity at varius points given the shape and boundary of the aquifer and recharge intensity and given a set of measured log transmissivity y and head h values at a few points the log transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function pdf the solution of the inverse problem is the conditional normal pdf of y conditioned on measured h and y which is expressed in terms of the unconditional joint pdf of y and h the problem is reduced to determining the unconditional head log transmissivity covariance and head variogram for a selected y covariance which depends on a few unknown parameters this is achieved by solving a first order approximation of the flow equations the method is illustrated for an exponential y covariance and the effect of head and transmissivity measurements upon the reduction of uncertainty of y is investigated systematically it is shown that measurement of h has a lesser impact than those of y but a judicious combination may lead to significant reduction of the predicted variance of y possible applications to real aquifers are outlined

stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 1 constant head boundary the spatial variability of log transmissivity y and head h in aquifers assumes a major role in determining the solutions of the direct flow problem as well as of the inverse problem investigation of the joint y h variability has been cast in the past in numerical and analytic frames analytic solutions for the y h statistical moments in two dimensions assumed the flow to occur in an unbounded domain in this study exact analytical expressions for the y h moments are derived which account for the presence of a given head boundary in a half plane this result is achieved by a first order approximation and for an exponential y covariance the result offers the opportunity to examine the effects of boundaries on the semivariogram of h and on the y h cross covariance the assumption of unbounded domain is examined and shown to be accurate at distances larger than two y integral scales from the boundary by using conditional probabilities a general and simple method to estimate the moments close to the boundary is presented it requires the knowledge of the moments of y and h in the unbounded domain only the method is compared with the exact analytic solution showing excellent results

a stochastic approach to the problem of upscaling of conductivity in disordered media theory and unconditional numerical simulations a general problem in groundwater modeling is how to assign conductivity values to blocks of a numerical model in this study a method is proposed to upscale the conductivity measurements observed at a given scale to block conductivity values for arbitrary block size assuming a multilognormal distribution for the point conductivities the block conductivity appears as a random function whose distribution conditional to point measurements can be derived this conditional distribution allows filling the flow simulator blocks with equiprobable realizations of block conductivities locally conditioned to actual data these realizations could also be conditioned to conductivity values measured at any scale if the size of the measurement support is known the analytical results are compared to results obtained through a monte carlo analysis and a good agreement was found even for variances of ln t larger than 1

a stable computation of log derivatives from noisy drawdown data pumping tests interpretation is an art that involves dealing with noise coming from multiple sources and conceptual model uncertainty interpretation is greatly helped by diagnostic plots which include drawdown data and their derivative with respect to log time called log derivative log derivatives are especially useful to complement geological understanding in helping to identify the underlying model of fluid flow because they are sensitive to subtle variations in the response to pumping of aquifers and oil reservoirs the main problem with their use lies in the calculation of the log derivatives themselves which may display fluctuations when data are noisy to overcome this difficulty we propose a variational regularization approach based on the minimization of a functional consisting of two terms one ensuring that the computed log derivatives honor measurements and one that penalizes fluctuations the minimization leads to a diffusion like differential equation in the log derivatives and boundary conditions that are appropriate for well hydraulics i e radial flow wellbore storage fractal behavior etc we have solved this equation by finite differences we tested the methodology on two synthetic examples showing that a robust solution is obtained we also report the resulting log derivative for a real case

nonlocal properties of nonuniform averaged flows in heterogeneous media the properties of nonuniform average potential flows in media of stationary random conductivity are studied the mathematical model of average flow is derived as a system of governing equations to be satisfied by mean velocity and mean head the averaged darcy s law determines the effective conductivity as an integral operator of the convolution type relating the mean velocity to the mean head gradient in a nonlocal way in fourier domain the mean velocity is proportional to the mean head gradient the coefficient of proportionality is referred to as the effective conductivity tensor and is derived by perturbation methods in terms of functionals of the conductivity spatial moments it is shown that the effective conductivity cannot be defined uniquely for potential flows however this nonuniqueness does not affect the spatial distribution of the mean head and velocity analytical expressions of the effective conductivity tensor are derived for two and three dimensional flows and for exponential and gaussian correlations of isotropic conductivity the fundamental solution of the governing equations in effective media mean green function is calculated for the same cases two new asymptotic models of the averaged darcy s law are developed to be applicable to large and small scales of heterogeneity several asymptotic expansions of tue two dimensional and three dimensional mean green functions are derived for exponential and gaussian correlations

stochastic analysis of three dimensional flow in a bounded domain a commonly accepted first order approximation of the equation for steady state flow in a fully saturated spatially random medium has the form of poisson s equation this form allows for the advantageous use of green s functions to solve for the random output hydraulic heads in terms of a convolution over the random input the logarithm of hydraulic conductivity a solution for steady state three dimensional flow in an aquifer bounded above and below is presented consideration of these boundaries is made possible by use of green s functions to solve poisson s equation within the bounded domain the medium hydraulic conductivity is assumed to be a second order stationary random process as represented by a simple three dimensional covariance function upper and lower boundaries are taken to be no flow boundaries the mean flow vector lies entirely in the horizontal dimensions the resulting hydraulic head covariance function exhibits nonstationary effects resulting from the imposition of boundary conditions comparisons are made with existing infinite domain solutions

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

upscaling methods for a class of convection diffusion equations with highly oscillating coefficients this paper investigates the upscaling method to the following parabolic equation tc uc d c f x t which stems from the application of solute transport in porous media because of the highly oscillating permeability of the porous media the darcy velocity u hence the dispersion tensor d has many scales with high contrasts thus how to calculate the macro scale equivalent coefficients of the above equation becomes the target of this paper a new upscaling method is proposed and studied via comparing with another upscaling method which was proposed in z chen w deng h ye discrete contin dyn syst 13 2005 941 960 the two different equivalent coefficients computing formulations are based on the solutions of two different cell local problems which one utilizes the elliptic operator with terms of all orders while the other only uses the second order term error estimates between the equivalent coefficients and the homogenized coefficients are given under the assumption that the oscillating coefficients are periodic which is not required by the method numerical experiments are carried out for the periodic coefficients to demonstrate the accuracy of the proposed method moreover the upscaling method is applied to solve the solute transport in a porous medium with a random log normal relative permeability the results show the efficiency and accuracy of the proposed method

analysis of variance method for the equivalent conductivity of rectangular blocks we use an analysis of variance anova decomposition of the log hydraulic conductivity f x ln k x as the sum of an average value f plus main effects and interactions of various orders to evaluate the equivalent conductivity of rectangular blocks in d dimensional space some of the anova components are dealt with exactly the effect of the other components is approximated through calibration to the effective conductivity keff under ergodic conditions our analysis applies to both isotropic and anisotropic lognormal k fields we evaluate the theoretical findings through analytical and numerical comparisons with exact results and previously proposed approximations we find that the anova formula is nearly unbiased although its performance is generally inferior to numerical renormalization advantages of the anova formula are that it is less computationally demanding and explicitly shows the dependence of the block conductivity kb on the various components of fluctuation of f inside the block such dependence is more complex than previously assumed considering the anova formula to be unbiased we derive the bias of the geometric mean of the cardwell parsons bounds for three dimensional rectangular blocks

an application of the geostatistical approach to the inverse problem in two dimensional groundwater modeling the geostatistical approach to the estimation of transmissivity from head and transmissivity measurements is developed for two dimensional steady flow the field of the logarithm of transmissivity log transmissivity is represented as a zero order intrinsic random field its spatial structure is described in this application through a two term covariance function that is linear in the parameters 1 and 2 linearization of the discretized flow equations allows the construction of the joint covariance matrix of the head and log transmissivity measurements as a linear function of 1 and 2 in this particular application the coefficient matrices are calculated numerically in a noniterative fashion maximum likelihood estimation is employed to estimate 1 and 2 as well as additional parameters from measurements linear estimation theory cokriging then yields point or block averaged estimates of transmissivity the approach is first applied to a test case with favorable results it is shown that the application of the methodology gives good estimates of transmissivities it is also shown that when the transmissivities are used in a numerical model they reproduce the head measurements quite well results from the application of the methodology to the jordan aquifer in iowa are also presented

monte carlo method via a numerical algorithm to solve a parabolic problem this paper is intended to provide a numerical algorithm consisted of the combined use of the finite difference method and monte carlo method to solve a one dimensional parabolic partial differential equation the numerical algorithm is based on the discretize governing equations by finite difference method due to the application of the finite difference method a large sparse system of linear algebraic equations is obtained an approach of monte carlo method is employed to solve the linear system numerical tests are performed in order to show the efficiency and accuracy of the present work

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

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 1 theory and computational approach we consider the effect of measuring randomly varying hydraulic conductivitiesk x on one s ability to predict numerically without resorting to either monte carlo simulation or upscaling steady state flow in bounded domains driven by random source and boundary terms our aim is to allow optimum unbiased prediction of hydraulic heads h x and fluxes q x by means of their ensemble moments h x c and q x c respectively conditioned on measurements of k x these predictors have been shown by neuman and orr 1993a to satisfy exactly an integrodifferential conditional mean flow equation in which q x c is nonlocal and non darcian here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter y which represents the standard estimation error of ln k x and show how to solve these equations to first order in y2 by finite elements on a rectangular grid in two dimensions in the special case where one treats k x as if it was locally homogeneous and mean flow as if it was locally uniform one obtains a localized darcian approximation q x c kc x h x c in which kc x is a space dependent conditional hydraulic conductivity tensor this leads to the traditional deterministic darcian steady state flow equation which however acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique it further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database localized equations yield no information about predictive uncertainty our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux obtained by traditional inverse methods are generally smaller often considerably so than measures of corresponding predictive uncertainty which can be assessed only by means of stochastic models such as ours we present a detailed comparison between finite element solutions of nonlocal and localized moment equations and monte carlo simulations under superimposed mean uniform and convergent flow regimes in two dimensions paper 1 presents the theory and computational approach and paper 2 guadagnini and neuman this issue describes unconditional and conditional computational results

comparison of fickian and temporally nonlocal transport theories over many scales in an exhaustively sampled sandstone slab 1 it is not a compelling argument solely on the basis of a better fit to solute breakthrough curve btc data that a temporally nonlocal model is necessary to simulate transport in an advection dominated system one may counter that the classical advection dispersion equation ade is a valid model at some small scale and that the detailed hydraulic conductivity k data must be well represented then the nonlocality is only a result of upscaling and loss of information but is the nonlocal model demonstrably necessary at all scales we examine the experiment conducted by klise et al 2008 in which a 30 5 30 5 cm slab of relatively homogeneous cross bedded sandstone was exhaustively sampled for k the slab was sealed saturated with potassium iodide and x rayed 10 times while being flushed with fresh water the 8 649 air permeameter measurements were down and upscaled to make finer and coarser grids on which the velocity field was solved and the ade applied the optimized parameters in the ade were found to scale predictably most notably the longitudinal dispersivity which grew linearly with upscaling but at all levels of up and downscaling including the original k measurement scale of 0 33 cm the ade did not adequately represent the late time tails the temporally nonlocal time fractional ade t fade was applied and the optimized parameters and the immobile capacity did not depend on scale the better fit provided by the t fade in the late btc tails did not bring about a sacrificed fit elsewhere in the btc furthermore the optimized ade and t fade solutions do not converge at the smallest scale directly implying that the temporal nonlocality is a necessary model component we conclude that the logical inference if the ade is valid in heterogeneous material then there is tailing in the btc is not a proof that the reverse is true we provide a clear counterexample a corollary is that a mismatch between data and a discretized solution to the ade does not imply that more data will improve fits or predictive ability

upscaling of permeability of anisotropic heterogeneous formations 2 general structure and small perturbation analysis the general methodology developed in part 1 indelman and dagan this issue of this study is applied to the detailed analysis of upscaled permeability first the structure of the dissipation function for the general conductivity a tensor of stationary random components is examined with the aid of dimensional analysis it is shown that for arbitrary shapes of the numerical elements the upscaled permeability has also this general structure and the numbers of unknown parameters and equations match this result suggests that the upscaling problem has an unique solution in the particular case of scalar permeability of isotropic covariance of the actual formation it is shown that a similar upscaled permeability is possible only for spherical circular numerical elements otherwise the upscaled permeability has to be a tensor of anisotropic covariance if the actual formation has a scalar permeability of axisymmetric covariance upscaling preserves the last property only for axisymmetric partition elements i e for a sphere cylinder and ellipsoid explicit expressions for the first moments of the upscaled permeability mean covariance are derived at first order in the log permeability variance the detailed computations for a scalar permeability of axisymmetric covariance and for axisymmetric numerical elements lead to simple results the upscaled permeability expected values are components of an axisymmetric tensor whereas the fluctuations are determined by a scalar random space function of anisotropic covariance the general case of upscaling by a first order approximation is examined in appendix b these results will be applied to a few particular cases in part 3 indelman this issue

correlation structure of flow variables for steady flow toward a well with application to highly anisotropic heterogeneous formations the study a continuation of that of indelman et al 1996 aims at deriving the second order moments of flow variables such as hydraulic head its gradient and the specific discharge for steady flow toward a fully penetrating well in a confined heterogeneous aquifer the log conductivity y ln k is modeled as a three dimensional stationary function of gaussian correlation of anisotropy ratio e by using first order approximations in 2y and e we derive the variance and the vertical integral scale of the piezometric head h of its radial gradient er and of the radial component of the specific discharge qr owing to the nonuniformity of the average flow these quantities are functions of the distance from the well it is shown that the variances of the head 2h and of its gradient 2er as well as the crossvariance e y between er and y vanish at the well whereas the discharge variance 2qr tends to the product between the log conductivity variance 2y and the squared mean discharge qr 2 this behavior pertains to a stratified formation surrounding the well far from the well 75 horizontal y integral scales i the head variance approaches a constant value for r 10i the moments 2er 2qr and ery tend to the corresponding values for uniform flow but with the local mean head gradient replacing the constant one the head vertical integral scale grows indefinitely with r whereas the vertical integral scale of the flux is larger by one log conductivity vertical scale than the one prevailing in uniform flow this latter property is explained by the presence of the source line which increases the correlations in the vertical direction the present results may be used in identifying the log conductivity statistical parameters from flowmeter velocity measurements in piezometers surrounding pumping or injecting wells

transmissivity upscaling in numerical aquifer models of steady well flow unconditional statistics 1 numerical solution of regional scale aquifer flow requires discretizing the transmissivity t typically the numerical element scale r and the transmissivity integral scale i are both of the order of hundreds of meters then the upscaled block transmissivity is random and its statistical moments depend on those of t on flow conditions and on r i modeling y ln t as a two dimensional normally distributed stationary random field we derive unconditional statistics of the upscaled ln accurate to the first order in y2 for strongly nonuniform flow in a circular block of radius r centered on a well of radius rw in an unbounded domain after a proper definition it was found that the ensemble mean is approximately but not exactly equal to ln tg the geometric mean and that the variance ratio 2 y2 which depends on rw i and the shape of the correlation drops slowly from unity for r i 0 to approximately 0 6 for r i 10 hence the variance of the upscaled transmissivity in radial flow is larger than that determined previously for uniform flow additionally defining the equivalent teq as a deterministic value for which the solution of the flow problem renders directly the mean upscaled head drop and specific discharge we find that in radial flow teq th the harmonic mean and grows slowly with increasing r i whereas for mean uniform flow teq tg application of the procedure is illustrated for an example of aquifer with selected values of parameters

bergman projections and operators on hardy spaces we show that toeplitz like operators of the formtsuf ps uf wherepsis a weighted bergman projection are bounded on the hardy spaceshp for 1 p for certain symbols udefined on the unit disk in particular tsuis bounded ifuis of the formu h g wherehis a bounded harmonic function andg is the green potential of a borel measure satisfying the condition that 1 z d z is a carleson measure ifuis of the formu h g where is a positive measure then these two conditions are also necessary thattsube bounded onhp we also consider the cases 0 p 1

representation of grid block permeability in coarse scale models of randomly heterogeneous porous media the two basic approaches for the assignment of coarse scale grid block permeabilities from highly detailed permeability descriptions are sampling where the permeability of the coarse scale grid block is simply specified as the fine scale permeability evaluated at the grid block center no averaging is performed and scaleup or homogenization where the coarse scale permeability is determined through some procedure which appropriately averages the fine scale data the intent of this paper is to quantitatively compare these two approaches through extensive numerical calculations for the modeling of single phase flow through two dimensional realizations of randomly heterogeneous porous media with heterogeneity quantified through the correlation length and the variance of the log of permeability 2 several different flow scenarios and two scaleup techniques local and global are considered over a range of l and it is seen that for flows driven by smoothly varying boundary conditions a sampling procedure is more accurate than global scaleup when the length scale of a typical grid block is less than l independent of for flows driven by source terms by contrast more resolution is required for a sampling approach to be preferable to global scaleup particularly at high variances local scaleup is shown to be preferable to either sampling or global scaleup in essentially all cases though its use for more general permeability fields requires additional numerical calculations

analysis of the spatial structure of properties of selected aquifers data from 31 regional aquifers are analyzed to identify the horizontal spatial correlation structure of transmissivity hydraulic conductivity and storage coefficient three parameters are estimated the variance of small scale variability or nugget and the variance and integral scale range of an exponential covariance function the assumption of normality of the aforementioned geohydrologic parameters and their logarithmic transformations is also examined results are obtained which indicate that the logarithms of these properties generally pass normality tests

steady flow toward wells in heterogeneous formations mean head and equivalent conductivity we consider steady flow of water in a confined aquifer toward a fully penetrating well of radius rw figure 1 the hydraulic conductivity k is modeled as a three dimensional stationary random space function the two point covariance of y in k kg is of axisymmetric anisotropy with i and i the horizontal and vertical integral scales respectively and kg the geometric mean of k unlike previous studies which assumed constant flux the well boundary condition is of given constant head figure 1 the aim of the study is to derive the mean head h and the mean specific discharge q as functions of the radial coordinate r and of the parameters y2 e i i and rw i an approximate solution is obtained at first order in y2 by replacing the well by a line source of strength proportional to k and by assuming ergodicity i e equivalence between space averages over the vertical and h q ensemble means an equivalent conductivity keq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge q as the actual one for the given head hw in the well and a given head in a piezometer at distance r from the well this definition corresponds to the transmissivity determined in a pumping test by an observer that measures hw andq the main result of the study is the relationship 19 keq ka 1 kefu where ka is the conductivity arithmetic mean and kefu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer the weight coefficient 10 has the simple approximate expression ln r i in r rw near the well 0 and keq ka which is easily understood since for rw i 1 the formation behaves locally like a stratified one in contrast far from the well 1 and keq kefu the flow being slowly varying there since ka kefu our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well however the difference between ka and kefu is small for highly anisotropic formations for which e 1 a nonlocal effective conductivity which depends only on the heterogeneous structure is derived in appendix a along the lines of indelman and abramovich 1994

embedding bergman spaces into tent spaces let a p omega denote the bergman space in the unit disc mathbb d of the complex plane induced by a radial weight omega with the doubling property int r 1 omega s ds le c int frac 1 r 2 1 omega s ds the tent space t q s nu omega consists of functions such that begin aligned begin aligned vert f vert t q s nu omega q int mathbb d left int vargamma zeta f z s d nu z right frac q s omega zeta da zeta infty quad 0 q s infty end aligned end aligned here vargamma zeta is a non tangential approach region with vertex zeta in the punctured unit disc mathbb d setminus 0 we characterize the positive borel measures nu such that a p omega is embedded into the tent space t q s nu omega where 1 frac s p frac s q 0 by considering a generalized area operator the results are provided in terms of carleson measures for a p omega

conditional stochastic averaging of steady state unsaturated flow by means of kirchhoff transformation we consider the effect of measuring randomly varying soil hydraulic properties on one s ability to predict steady state unsaturated flow subject to random sources and or initial and boundary conditions our aim is to allow optimum unbiased prediction of system states pressure head water content and fluxes deterministically without upscaling and without linearizing the constitutive characteristics of the soil it has been shown by neuman et al 1999 that such prediction is possible by means of first ensemble moments of system states and fluxes conditioned on measured values of soil properties when the latter scale in a linearly separable fashion as proposed by vogel et al 1991 the uncertainty associated with such predictions can be quantified by means of the corresponding conditional second moments the derivation of moment equations for soils whose properties do not scale in the above manner requires linearizing the corresponding constitutive relations which may lead to major inaccuracies when these relations are highly nonlinear as is often the case in nature when the scaling parameter of pressure head is a random variable independent of location the steady state unsaturated flow equation can be linearized by means of the kirchhoff transformation for gravity free flow linearization is also possible in the presence of gravity when hydraulic conductivity varies exponentially with pressure head for the latter case we develop exact conditional first and second moment equations which are nonlocal and therefore non darcian we solve these equations analytically by perturbation for unconditional vertical infiltration and compare our solution with the results of numerical monte carlo simulations our analytical solution demonstrates in a rigorous manner that the concept of effective hydraulic conductivity does not apply to ensemble averaged unsaturated flow except when gravity is the sole driving force

porous media upscaling in terms of mathematical epistemic cognition in this work we revisit the meaning of the term solution with respect to a mathematical model representing a physical media upscaling system a central aim is to combine the science of mind with human mathematical ideas to solve real world upscaling problems we propose that in certain cases a nonconventional epistemic cognition solution which assumes that the model describes incomplete knowledge about nature and focuses on conceptual mechanisms and thinking processes can lead to a more realistic upscaling analysis than a conventional ontologic approach which assumes that the model describes nature as is and focuses on form manipulations in the present work we apply the epistemic cognition approach to the solution of the two dimensional bounded porous media upscaling problem a formal framework is presented and implementation issues related to the epistemic cognition methodology are considered numerical experiments are presented involving effective conductivities in bounded two dimensional sp

analysis of flow through heterogeneous random aquifers by the method of embedding matrix 1 steady flow the statistical structures of the hydraulic conductivity head gradient specific discharge and head fields are analyzed with the aid of a simplified model consisting of a collection of spherical in three dimensions and cylindrical in two dimensions blocks of different conductivity and diameter set at random and independently in space results of a previous work for the effective conductivity and unbounded formations dagan 1979 are recovered and given a rational derivation the influence of the aquifer boundary and nonuniform average flows are examined for the first time the growth of the head variance with distance from boundary in two dimensional flows is compared with that of the monte carlo simulations of smith and freeze 1979

upscaling transmissivity under radially convergent flow in heterogeneous media most field methods used to estimate transmissivity values rely on the analysis of drawdown under convergent flow conditions for a single well in a homogeneous and isotropic aquifer and under steady state flow conditions drawdown s is directly related to the pumping rate q through transmissivity t in real nonhomogeneous aquifers s and q are still directly related now through a value called equivalent transmissivity teq in this context teq is defined as the value that best fits thiem s equation and would for example be the transmissivity assigned to the well location in the classical interpretation of a steady state pumping test this equivalent or upscaled transmissivity is clearly not a local value but is some representative value of a certain area surrounding the well in this paper we present an analytical solution for upscaling transmissivities under radially convergent steady state flow conditions produced by constant pumping from a well of radius rw in a heterogeneous aquifer based upon an extension of thiem s equation using a perturbation expansion we derive a second order expression for teq given as a weighted average of the fluctuations in log t throughout the domain this expression is compared to other averaging formulae from the literature and differences are pointed out teq depends upon an infinite series which may be expressed in terms of coefficients of the finite fourier transform of the log transmissivity function sufficient conditions for convergence of this series are examined finally we show that our solution agrees with existing analytical ones to second order and test the solution with a numerical example

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 reassessment of the groundwater inverse problem this paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a possibly infinite basis function expansion with random coefficients the unknown parameter function is related to the measurements used for estimation by a forward operator which describes the measurement process in the particular case considered here the parameter of interest is the large scale log hydraulic conductivity the measurements are point values of log conductivity and piezometric head and the forward operator is derived from an upscaled groundwaterflow equation the inverse algorithm seeks the most probable or maximum a posteriori estimate of the unknown parameter function when the measurement errors and parameter function are gaussian and independent the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness of fit and prior terms when the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate this constraint tends to make the problem well posed by limiting the range of admissible solutions the gaussian maximum a posteriori problem may be solved with variational methods using functional generalizations of gauss newton or gradient based search techniques several popular groundwater inverse algorithms are either special cases of or variants on the functional maximum a posteriori algorithm these algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use linear versus nonlinear the accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a bayesian extension of the cramer rao lower bound on the estimation error covariance this bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements

three dimensional numerical analysis of steady state pumping tests in heterogeneous confined aquifers 1 we study three dimensional steady state flow field generated by a fully penetrating well extracting a constant water discharge from a confined aquifer of constant thickness the hydraulic conductivity k is modeled as an axisymmetric stationary random space function and the flow equation is solved by using the seven node finite volume numerical scheme in view of application in aquifer characterization we analyze how the equivalent conductivity keq varies as a function of the distance from the extraction well the numerical simulations showed that the boundary condition at the well exerts a large impact on keq and that the first order solution in y2 obtained by indelman et al 1996 is valid up to y2 0 5 for higher values of y2 the first order solution overestimates keq with differences that for a given y2 reduce progressively as the formation becomes more anisotropic a simple inverse procedure based on keq obtained through cross well interference is finally proposed and tested with the objective of estimating the parameters of the geostatistical model of variability of the hydraulic conductivity a few parametric examples showed that a suitable interpretation of pumping tests can be used to safely estimate parameters such as the geometric mean of the hydraulic conductivity its variance and although with larger uncertainty the horizontal integral scale while the inference of the anisotropy ratio is highly uncertain and error prone

measurement scale network sampling scale and groundwater model parameters the scales at which model parameters are measured with local field tests distributed on a sampling network are examined two scales are defined to characterize the problem 1 the measurement scale associated with the resolution of a single field test and 2 the network scale associated with the separation between samples on a network using a spatial filtering approach it is shown that a network of measurements can only resolve a larger scale component of a parameter field the smaller scale component of the parameter field not seen by measurements here called the subgrid component can only vary on scales larger than the measurement scale and smaller than the network scale these unobserved subgrid scales give rise to the so called closure problem and consequent modeling errors when a significant proportion of the parameter variability is contained in the subgrid scales not only will the closure problem be significant but aliasing errors will also pollute the estimate of the large scale component of the parameter field these concepts are illustrated by performing simple analytical and numerical calculations

statistics of block conductivity through a simple bounded stochastic medium a monte carlo approach is employed to estimate the distribution of an equivalent conductivity measure the block conductivity which characterizes the total flow rate through a two dimensional bounded domain and which is itself a random variable chi square goodness of fit tests indicate that the lognormal distribution is an appropriate choice for the block conductivity distribution for square domains the mean and variance of the block conductivity are seen to be closely approximated using the statistics of local averages of log conductivity

a continuous time inverse operator for groundwater and contaminant transport modeling model identifiability general identifiability conditions are derived for the model coefficients of the spatially discretized continuous time diffusion equation these conditions in turn determine some sufficient and necessary conditions for the existence and uniqueness of the continuous time inverse operator described in a previous article the identifiability conditions are in terms of the initial conditions and the flux vector containing boundary condition information and the eigenspace of the spatially discretized system the derivation of these conditions gives some insight into the location of the solution of the inverse problem in terms of the direct operator under investigation in the space of the laplace transform and limited insight into the stability of the solution near that location

extension of van der corput algorithm to ls sequences the ls sequences of points recently introduced by the author are a generalization of van der corput sequences they are constructed by reordering the points of the corresponding ls sequences of partitions here we present another algorithm which is simpler to compute than the original construction and coincides with the classical one for van der corput sequences this algorithm is based on the representation of natural numbers in base l s moreover when s l these sequences have low discrepancy and can be useful in quasi monte carlo methods

imbedding potentials in tent spaces abstract we obtain characterizations of positive borel measures on b n so that the nonisotropic potential space k lp d is imbedded in the tent space t2q d where 1 q we deduce characterizations for pointwise multipliers of the space h p

transmissivity upscaling in numerical models of steady aquifer flow conditional statistics 1 numerical solution of regional scale aquifer flow requires discretizing the transmissivity t for spatially varying t and by the stochastic approach t is modeled as a random field typically both the numerical element scale r and the log transmissivity integral scale i are of order of hundreds to thousands meters consequently the upscaled block transmissivity is random and its statistical moments depend on those of the fine scale t on flow conditions and on r i modeling y ln t as a two dimensional normally distributed stationary random field we have derived dagan and lessoff water resources research 43 w05431 doi 10 1029 2006wr005235 2007 the unconditional statistics of the upscaled ln accurate to the first order in y2 the log transmissivity variance both cases of mean uniform flow solved previously by indelman and dagan transport in porous media 12 161 183 1993 and of strongly nonuniform flow in a circular block of radius r centered at a well of radius rw were considered such upscaling applies either to elements in regions of natural gradient flow or those surrounding a projected well respectively which are sufficiently far from points of measurement of t the present article extends the analysis to account for the presence of a measurement removed in the center of the element e g result of a pumping test the fine scale conditional log transmissivity yc is modeled as multi gaussian of nonstationary mean and of two point covariance which depend on the measured value and distance from the measurement location upscaling in mean uniform flow is related for instance to the case of a pumping well which is not operative but served to determine t while natural gradient flow conditions prevail in the area it was found that upscaling has a significant effect upon the mean upscaled log transmissivity that changes from the measured value for r i 1 to the effective one y ln tg for r i 6 similarly the conditional log transmissivity variance is greatly reduced compared to the unconditional case it tends to zero for r i 0 or r i and it reaches a maximal value of around 0 15 y2 for r i 1 the second case of upscaling in well flow applies to an element with an operative well at its center which was used to determine the transmissivity tw unlike mean uniform flow it is found that upscaling has a very small effect for the realistic values of rw i 0 10 3 and r i 5 in this case the upscaled transmissivity is practically deterministic and equal to the given tw a summary of the two articles dagan and lessoff 2007 and the present one concludes the paper

acceleration of quasi monte carlo approximations with applications in mathematical finance a procedure to accelerate the convergence of quasi monte carlo approximations is described and investigated the procedure applies a weighted least squares smoothing method to improve the approximations that were obtained using low discrepancy sequences the acceleration procedure is tested computationally on several numerical integration problems including two problems of interest in mathematical finance the results indicate that when halton low discrepancy points are used in the quasi monte carlo method the accelerated sequence often converges a factor of five or more faster than the original quasi monte carlo approximations

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

stochastic identification of transmissivity and effective recharge in steady groundwater flow 2 case study the study is a continuation and extension of a previous work dagan 1985a whose aim was to identify the values of the log transmissivity y for steady flow the common basic assumptions are that y is a normal and stationary random space function the aquifer is unbounded and a first order approximation of the flow equation is adopted the expected value of the water head h as well as the y unconditional autocovariance are supposed to have analytical expressions which depend on a parameters vector the proposed solution of the inverse problem consists of identifying with the aid of the model and of the measurements of y and h and subsequently computing the statistical moments of y conditioned on the same data the additional features of the present study are 1 incorporation of a constant but random effective recharge and its identification and 2 accounting for the fact that estimation is associated with some uncertainty whereas before was assumed to be identified with certainty analytical expressions are derived for the y and h covariances for an exponential autocovariance of y paper 2 rubin and dagan this issue of the study illustrates the applications of the method to a real life case