on unifying the concepts of scale instrumentation and stochastics in the development of multiphase transport theory a generalized theory of multiphase transport is presented which combines the concepts of scale instrumentation stochastics and time series with the development of transport equations by defining the filtering process as a convolution of a measure p with a field property we are able to exploit the fourier transform to place plausible restrictions on an instrument in frequency space so as to make its measurement relevant to its physical environment an ideal instrument is defined which filters out high frequency noise corresponding to short distances and yet does not alter the structure of low frequencies using ideal instruments we successively filter out lower frequency noise in a multiscale multiphase environment formulas are developed to relate the autocorrelation of a field property on one scale of motion to that on any other scale while taking into account the types of instruments used in the measuring process an equation relating the integral scale on one scale of motion to the integral scale on any other scale of motion is developed power spectra are developed which relate spectra on different scales to the measuring instrument used by successively applying filtering theorems a hierarchy of multiscale transport equations is developed filtered properties in the transport equations are mass averages different properties are allowed to be measured by different instruments and different instruments are allowed on different scales of motion and for different phases the concept of a wide sense stationary ergodic process is introduced to develop mass average autocorrelations and spectra over scales of motion as a function of measuring devices

multi scale modelling of flow in periodic solid structures through spatial averaging this paper presents spatially averaged navier stokes equations for modelling macro scale flow in devices with periodic solid structures such as fin and tube arrays the properties of steady and unsteady periodically developed flow are investigated to assess different strategies for determining the closure terms in the macro scale flow equations it is shown that the spatial averaging technique requires an appropriate weighting function to ensure that the closure terms are spatially constant for periodically developed flow moreover through an appropriate choice of the weighting function the closure terms can be obtained by solving a local closure problem on a unit cell of the periodic structures the theoretical framework of this paper is applied as multi scale modelling technique for flow through a cylindrical tube array this case study illustrates the advantages of the weighted spatial averaging technique over the volume averaging technique

relative index theorems and supersymmetric scattering theory we discuss supersymmetric scattering theory and employ krein s theory of spectral shift functions to investigate supersymmetric scattering systems this is the basis for the derivation of relative index theorems on some classes of open manifolds as an example we discuss the de rham complex for obstacles in n and asymptotically flat manifolds it is shown that the absolute or relative euler characteristic of an obstacle in n may be obtained from scattering data for the laplace operator on forms with absolute or relative boundary conditions respectively in the case of asymptotically flat manifolds we obtain the chern gauss bonnet theorem for thel2 euler characteristic

on measurement scale and scaling the relationships between measurement scale and scaling are discussed and analyzed in detail it is pointed out that only certain fundamental properties can be measured and that these properties do not include constitutive variables the interrelationships between scale and measurement are shown to manifest themselves in two types of heterogeneity which we label as explicit and implicit an important issue that is addressed for the first time is the necessity of scaling instruments when scaling laboratory experiments up to field scale or vice versa explicit constraints that scaled instruments must satisfy are derived for the frequency response of the instrument it is shown that linear interpolators can be viewed as filters instruments but that these interpolators don t preserve the scale constraints of the instrument used in making the field measurements the relationship between signal conditioning and the instrument s window is isolated and shown to be a multiple convolution over the spatial coordinates corresponding to the window followed by a temporal convolution for the signal conditioning

vacuum expectation values of products of chiral currents in 3 1 dimensions an algebraic rule is presented for computing expectation values of products of local nonabelian charge operators for fermions coupled to an external vector potential in 3 1 space time dimensions the vacuum expectation value of a product of four operators is closely related to a cyclic cocycle in noncommutative geometry of alain connes the relevant representation of the current is constructed using kirillov s method of coadjoint orbits

an evaluation of mathematical models of the transport of biologically reacting solutes in saturated soils and aquifers three different conceptual frameworks have been adopted in the past for the development of mathematical models of bacterial growth and biologically reacting solute transport in saturated porous media two schools of thought are based upon assuming that the pore scale geometrical configuration of the attached bacteria consists of biofilms or microcolonies the third school of thought represents the traditional approach where pore scale processes are neglected and the bacteria are assumed to respond to the macroscopic bulk fluid substrate concentration on the basis of a schematic block diagram representation of a saturated porous medium hosting a microbial population it is shown that these frameworks share a common theoretical foundation and that they differ only by the choice of particular constitutive equations for several transfer parameters using one possible option in this respect we derive a mathematical model that involves no unwarranted assumption about the distribution of the microorganisms in the pore space the governing equations of this latter model are shown to be formally identical to those obtained by f j molz et al 1986 using the concept of microcolony and to those that would result from adopting a simple form of biofilm model to describe bacterial growth in the pore space some of the consequences of this formal similarity between macroscopic transport equations obtained in different conceptual frameworks are discussed from an operational standpoint and in terms of model validation

computationally efficient algorithms for parameter estimation and uncertainty propagation in numerical models of groundwater flow finite difference and finite element methods are frequently used to study aquifer flow however additional analysis is required when model parameters and hence predicted heads are uncertain computational algorithms are presented for steady and transient models in which aquifer storage coefficients transmissivities distributed inputs and boundary values may all be simultaneously uncertain innovative aspects of these algorithms include a new form of generalized boundary condition a concise discrete derivation of the adjoint problem for transient models with variable time steps an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation and a new efficient first order second moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values the techniques are presented in matrix form and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations details of matrix structures are provided for a two dimensional linear triangular finite element model

sensitivity analysis for steady state groundwater flow using adjoint operators adjoint sensitivity theory is currently being considered as a potential method for calculating the sensitivity of nuclear waste repository performance measures to the parameters of the system for groundwater flow systems performance measures of interest include piezometric heads in the vicinity of a waste site velocities or travel time in aquifers and mass discharge to biosphere points the parameters include recharge discharge rates prescribed boundary heads or fluxes formation thicknesses and hydraulic conductivities the derivative of a performance measure with respect to the system parameters is usually taken as a measure of sensitivity to calculate sensitivities adjoint sensitivity equations are formulated from the equations describing the primary problem the solution of the primary problem and the adjoint sensitivity problem enables the determination of all of the required derivatives and hence related sensitivity coefficients in this study adjoint sensitivity theory is developed for equations of two dimensional steady state flow in a confined aquifer both the primary flow equation and the adjoint sensitivity equation are solved using the galerkin finite element method the developed computer code is used to investigate the regional flow parameters of the leadville formation of the paradox basin in utah the results illustrate the sensitivity of calculated local heads to the boundary conditions alternatively local velocity related performance measures are more sensitive to hydraulic conductivities

index formulas for generalized wiener hopf operators and boson fermion correspondence in 2n dimensions the kernels of operators associated with special chiral gauge trans formations kinks in the 2 dimensional dirac theory are explicitly deter mined the result is used to obtain index formulas for fredholm operators corresponding to continuous chiral gauge transformations moreover the fock space quadratic forms corresponding to the kinks are proved to converge to the dirac field as the kink size goes to zero it is also shown that for n i 2 mod4 the majorana field can be reached in a similar fashion

nonlocal reactive transport with physical and chemical heterogeneity localization errors the origin of nonlocality in macroscale models for subsurface chemical transport is illustrated it is argued that media that are either nonperiodic e g media with evolving heterogeneity or periodic viewed on a scale wherein a unit cell is discernible must display some nonlocality in the mean a metaphysical argument suggests that owing to the scarcity of information on natural scales of heterogeneity and on scales of observation associated with an instrument window constitutive theories for the mean concentration should at the outset of any modeling effort always be considered nonlocal the intuitive appeal to nonlocality is reinforced with an analytical derivation of the constitutive theory for a conservative tracer without appeal to any mathematical approximations deng et al 1993 present a first order nonlocal eulerian theory for transport of a conservative solute in an infinite nondeforming domain under steady flow conditions hu et al this issue extended these results to account for nonequilibrium linear sorption with random partition coefficient kd but deterministic constant reaction rate kr these theories are localized herein and comparisons are made between the fully nonlocal fnl nonlocal in time nlt and fully localized fl theories for conservative transport there is little difference between the first order fl and fnl models for spatial moments up to and including the third however for conservative transport the first order nlt model differs significantly from the fnl model in the third spatial moments for reactive transport all spatial moments differ between the fnl and fl models the second transverse horizontal and third longitudinal horizontal moments for the nlt model differ from the fnl model these results suggest that localized first order transport models for conservative tracers are reasonable if only lower order moments are desired however when the chemical reacts with its environment the localization approximation can lead to significant error in all moments and a fnl model will in general be required for accurate simulation

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

an evaluation of boundary conditions for one dimensional solute transport 1 mathematical development the laplace transform method is employed to obtain the solution to several boundary value problems in which mixing occurs in reservoirs attached to a porous medium flow is assumed to be uniformly one dimensional throughout the porous medium solutions are obtained for both continuous and discontinuous concentrations at the reservoir medium boundary and for resident and flux averaged concentrations a mass balance conducted on each solution shows that neither continuity condition can be proved generally superior an evaluation of the parametric sensitivity of the solutions is also conducted and shows that the distinction between the continuity conditions and the flux and resident concentrations is only important at small peclet numbers the only exception to this is the distinction between flux averaged versus resident continuous concentrations in a finite solution domain in this case the flux concentrations were found to be insensitive to the volume of the downstream reservoir and consequently the flux and resident concentrations can differ substantially for larger reservoirs in consideration of the general behavior of the solutions three conceptual problems were identified which must be investigated by physical modeling 1 determination of the appropriate solution at small peclet number 2 the possibility that dispersive mass flux occurs from the upstream reservoir under conditions of continuous concentration and 3 the use of the flux transformation for continuous concentration in a finite solution domain the physical modeling is discussed in a companion paper

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

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

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

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

generalized particle statistics in two dimensions examples from the theory of free massive dirac field in the framework of algebraic quantum field theory we analyze the anomalous statistics exhibited by a class of automorphisms of the observable algebra of the two dimensional free massive dirac field constructed by fermionic gauge group methods the violation of haag duality the topological peculiarity of a two dimensional space time and the fact that unitary implementers do not lie in the global field algebra account for strange behaviour of statistics which is no longer an intrinsic property of sectors since automorphisms are not inner we exploit asymptotic abelianness of intertwiners in order to construct a braiding for a suitable c tensor subcategory of end fancyscript a we define two inequivalent classes of path connected bi asymptopias selecting only those sets of nets which yield a true generalized statistics operator

large scale averaging analysis of single phase flow in fractured reservoirs the method of large scale averaging is introduced to derive and analyze the most general form of the dual porosity model of single phase flow in naturally fractured reservoirs the dual porosity model contains the usual equations based on darcy s law and the coupling terms representing the fluid transfer between the matrix and the fractures a transient closure problem is developed to obtain and analyze the fracture and matrix permeability tensors and the fluid transfer terms the spatial averaging theorem is presented from a mathematical point of view and proved rigorously by the distribution theory the problem of wellposedness of the dual porosity model is also considered the techniques developed here are not restricted to either regular geometric fractures or spatially periodic reservoirs

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

use of power averaging for quantifying the influence of structure organization on permeability upscaling in on lattice networks under mean parallel flow with the local permeability distribution variance s 2 is nonnegligible but remains small it is equal to 0 09 for sparse networks and 0 14 for dense networks representing 4 5 and 7 respectively of the full possible range of w values power averaging is not strictly valid but gives an estimate of upscaling at a few percent here w depends slightly on the local permeability distribution shape beyond its variance but mostly on the morphological network structures most of the morphological control on upscaling for on lattice structures is local and topological and can be explained by the dependence on the average number of neighbor by points effective coordination number within the flowing structure backbone

spartan gibbs random field models for geostatistical applications typographical errors in certain mathematical formulas that appear in the article spartan gibbs random field models for geostatistical applications siam j sci comput 24 2003 pp 2125 2162 are corrected

gevrey smoothness of invariant tori for analytic nearly integrable hamiltonian systems under russmann s non degeneracy condition abstract in this paper we prove gevrey smoothness of the persisting invariant tori for small perturbations of an analytic integrable hamiltonian system with russmann s non degeneracy condition by an improved kam iteration method with parameters

eulerian kinematics of flow through spatially periodic models of porous media for spatially periodic models of porous media and incompressible fluids we consider here in detail thephysical interpretation of averaged velocity fields in terms of a purely continuum level eulerian seepage flux relation involving the mass flow across an arbitrarily oriented macroscale surface element we prove that the appropriately averaged microscale velocities do indeed satisfy a macroscale eulerian flux relation albeit to some approximation provided that the surface element is large compared with the pore lengthscale microscale a quantitative discussion of an intermediate scale lying between the pore scale and a characteristic macroscale is achieved through the use of locally periodic velocity fields within the context of seepage velocity kinematics our developments provide a conceptual foundation that renders continuum theories of flow of incompressible fluids through periodic porous media entirely selfcontained on the macroscale free from any details of the discrete microscale structure inaccessible to macroscopic observers from which this continuum theory sprang the error bounds and scaling laws derived are of special interest when the macroscale microscale disparity is not very wide with reference to current rational mechanical theories of immiscible mixtures our analysis provides aconstructive existence proof at least for periodic models of porous media for phase specific velocities heretofore such velocities were usually accepted in a lagrangian sense on an axiomatic basis thus clarification and quantification of thekinematical aspects of the continuum hypothesis for certain classes of multiphase systems can be effecteda priori by our arguments independently of any subsequent macroscaledynamical modelling our analysis whereby we systematically proceed from a discrete picture of the underlying geometry to a purely continuum picture may be likened to that achieved in classical statistical mechanics as in that case we hope that the general results obtained transcend the underlying discrete i e spatially periodic model of the subcontinuum structure

macroscopic balance equations in soils and aquifers the case of space and time dependent instrumental response macroscopic balance equations for mass and linear momentum in soils and aquifers are derived in the relativist concept under the condition that all physical variables are associated with a single instrumental weighting function that depends on both position and time the resulting balance equations differ in concept from expressions obtained recently in the representative elementary volume rev approach by exhibiting additional terms that depend on the time and space derivatives of the instrumental weighting function these terms indicate that mass and linear momentum balance at the macroscopic scale are not preserved unless the weighting function has space time invariance properties over the period and spatial domain of measurement

invariant tori for gevrey smooth hamiltonian systems under russmann s non degeneracy condition abstract we consider gevrey smooth perturbation of a completely integrable gevrey smooth hamiltonian under russmann s non degeneracy condition and obtain a family of invariant tori using an improved kam iteration they are gevrey smooth in the sense of whitney with a gevrey index depending on the gevrey class of the hamiltonian and on the exponent in the diophantine condition

characterization of permeability anisotropy using wavelet analysis 1 wavelet analysis is an image analysis technique that can extract local information at multiple scales because of this capability wavelet analysis can be used to identify dominant scales in statistically heterogeneous and anisotropic random fields we develop and test a wavelet analysis method for identifying dominant scales and orientations in permeability fields and for identifying boundaries between regions with different dominant orientations we evaluate three different wavelets fully anisotropic morlet wavelet mexican hat wavelet and cauchy wavelet and show that the morlet wavelet is the most effective of these three wavelets in identifying dominant orientations we also investigate the use of several different quantitative wavelet measures in identifying dominant scales and orientations in permeability fields the technique is demonstrated using both a synthetic data set with known characteristics and a laboratory collected permeability data set from massillon sandstone

invariant tori for nearly integrable hamiltonian systems with degeneracy does not fall into a hyperplanethrough the origin the condition 1 3 is the sharpest one we have not seenits proof yet in this paper we will give a nondegeneracy condition by meansof the derivatives of h p which is equivalent to the condition 1 3 in analyticcase and under this nondegeneracy condition we obtain the russmann s results for analytic case furthermore the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the nite orderderivatives of h p main resultstheorem a suppose that h h p f p q is analytic in t

evaluation of sampling in the subsurface the evaluation of sampling is presented in the form of a sampling error defined as a difference between the true average and the sampled average of the measured process over the sampling domain the sampling error is quantified as a product between the process point variance defined at the scale of a sampling device s support volume and the sampling variance function which quantifies the reduction of uncertainty due to the sampling activity the sampling variance function is expressed analytically as a function of the scale of fluctuation of the measured process for a single sampling point the sampling variance function defines the reduction in the process point variance as a result of sampling over a certain support volume the presented evaluation of the sampling performance can be used for designing the monitoring activity and for measurement conditioning of stochastic theories the presented examples demonstrate the use of sampling evaluation for the purpose of conditioning on the concentration measurements

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

fermion current algebras and schwinger terms in 3 1 dimensions we discuss the restricted linear group in infinite dimensions modeled by the schatten class of rank 2p 4 which contains the 3 1 dimensional analogs of the loop groups and is closely related to yang mills theory with fermions in 3 1 dimensions we give an alternative to the construction of the highest weight representation of this group found by mickelsson and rajeev our approach is close to quantum field theory with the elements of this group regarded as bogoliubov transformations for fermions in an external yang mills field though these cannot be unitarily implemented in the physically relevant representation of the fermion field algebra we argue that they can be implemented by sesquilinear forms and that there is a regularized product of forms providing an appropriate group structure on the lie algebra level this gives an explicit non perturbative construction of fermion current algebras in 3 1 space time dimensions which explicity shows that the wave function renormalization required for a consistent definition of the currents and their lie bracket naturally leads to the schwinger term identical with the mickelsson rajeev cocycle though the explicit form of the schwinger term is given only for the casep 2 our arguments apply also to the restricted linear groups modeled by schatten classes of rank 2p 6 8 corresponding to current algebras in d 1 dimensions d 5 7

a comparison of methods to determine measurement support volumes we examine methods that can be used to identify the spatial filter function associated with the measurement of quantities on two scales we divide the methods into two general classes deconvolution methods and sensitivity based methods we briefly describe the two classes of methods provide examples of each from the literature and introduce a new sensitivity based method inspired by backus gilbert model resolution ideas we compare the approaches by examining the filter function that relates smaller scale grid block parameters to parameters determined by a theis model interpretation of drawdown data in general we find that sensitivity approaches are easier to implement numerically and are less subject to the problems of identifiability stability and uniqueness associated with deconvolution methods

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

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

what does an instrument measure empirical spatial weighting functions calculated from permeability data sets measured on multiple sample supports with the aid of linear filter theory we analyze 13 824 permeability measurements to empirically address the question what does an instrument measure by measure we mean the sample support or sample volume associated with an instrument as well as how the instrument spatially weights the heterogeneities comprising that sample support although the theoretical aspects of linear filter analysis are well documented physical data for testing the filtering behavior of an instrument particularly in the context of porous media flow are rare to nonexistent our exploration makes use of permeability data measured with a minipermeameter on a block of berea sandstone data were collected according to a uniform grid that was resampled with tip seals of increasing size i e increasing sample support spatial weighting filter functions characterizing the minipermeameter measurements were then calculated directly from the permeability data sets in this paper we limit our presentation to one of the six rock faces consisting of 2304 measurements as the general results for each rock face are similar we found that the empirical weighting functions are consistent with the basic physics of the minipermeameter measurement they decay as a nonlinear function of radial distance from the center of the tip seal consistent with the divergent flow geometry imposed by the minipermeameter the magnitude of the weighting function decreases while its breadth increases with increasing tip seal size reflecting the increasing sample support we further demonstrate both empirically and theoretically that nonadditive properties like permeability are amenable to linear filter analysis under certain limiting conditions i e small variances specifically the weighting function is independent of the power average employed in its calculation e g arithmetic versus harmonic average finally we examine the implications of these results for other instruments commonly employed in hydraulic testing e g slug and pump tests

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

what does a slug test measure an investigation of instrument response and the effects of heterogeneity 1 we consider parameters determined by the inversion of slug test head recovery data with the homogeneous parameter model of cooper et al 1967 to be weighted spatial averages of transmissivity and storage defined at a smaller scale we quantify the spatial averaging using a power average spatial filter expression we determine the form of the filter function and the power exponent using numerically simulated slug test data the filter function that describes how smaller scale transmissivity is weighted by slug tests displays an approximate 1 r2 behavior with r the radial distance from the well the radius of the cylinderical volume that is averaged by the slug test is inversely proportional to the square root of the storage coefficient larger averaging volume with smaller storage the power exponent grows from 0 19 to 0 345 as the ratio of the characteristic scale of the heterogeneity to the characteristic scale of the averaging volume grows although a power exponent of zero corresponding to geometric averaging provides good results for most simulations our results show that while slug tests are useful to estimate transmissivity they have dubious value for estimating storage coefficients we find that the transmissivity estimate is unbiased and does not appear to be strongly influenced by storage properties the storage coefficient estimate is however strongly influenced by the transmissivity and is biased we investigate the interaction between storage coefficient and transmissivity by examining an analytical slug test model that contains two annular zones each with distinct transmissivity and storage coefficient

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

spatial averaging of transmissivity in heterogeneous fields with flow toward a well steady state nonuniform flow in a heterogeneous medium is investigated using a combined numerical empirical approach the objective of the study is to determine the effective transmissivity of a square field with a constant head external boundary and a constant rate well at its center transmissivity at the point scale is modeled as an isotropic and multivariate lognormal spatial random function block transmissivities are defined using an empirical scaling up of point support values within the field the scaling process is a weighted spatial geometric averaging where log transmissivities are weighted by the inverse square of their distance from the well block transmissivities thus obtained and true effective values calculated using a numerical flow model are compared for realizations a finely discretized transmissivity field the principal finding of this study is that effective transmissivities obtained from the numerical flow model and block values obtained by spatial averaging are in excellent agreement for low to moderate variances of log transmissivity and moderately eccentric field geometries the geostatistical model for point scale transmissivity and the deterministic spatial averaging law are combined in order to create a geostatistical model for transmissivity at the block scale exact expressions are derived for the ensemble moments of block averaged transmissivities and well drawdowns in both the nonconditional case and the conditional case when transmissivity at the well bore is known the ensemble mean of block averaged transmissivity is found to decrease from the ensemble arithmetic mean toward the ensemble geometric mean as the field size becomes large compared to the integral range of spatial correlation this result contradicts earlier published work numerical results presented in this study also disagree with the analytical results of a recent stochastic analysis of specific discharge in radial systems

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

birkhoff kolmogorov arnold moser tori in convex hamiltonian systems for the hamiltonian systems of kam type it is proved that some lower dimensional invariant tori always exist in the resonance gaps although those maximum tori can not survive small perturbations in the generic case

the universal structure of the groundwater flow equations most groundwater flow models are not sufficiently detailed to allow an explicit representation of all dynamical scales instead most models are constructed at such a coarse scale of resolution that unresolved subgrid variability often exists it is therefore important to understand the interaction between unresolved dynamics and explicitly resolved dynamics the concept of universality is central to this interaction and is here related to the problem of rescaling models of groundwater flow we examine when it is possible to construct an accurate model for the explicitly resolved large scales without an explicit description of the subgrid scale dynamics we demonstrate how unresolved subgrid scale dynamics interact with resolved scale dynamics we show that a universally valid resolved scale model can be constructed if the resolved dynamics are sufficiently independent of the details of the subgrid scale dynamics in that event a resolved scale model is composed of a universal structure and accompanying model parameters the model parameters represent the effect of unresolved dynamics upon resolved dynamics we examine two possible model structures for groundwater flow we show theoretically and numerically that a local darcy law is a universally valid resolved scale model if the resolved and subgrid scales of the hydraulic conductivity are separated in scale by a so called spectral gap if however the hydraulic conductivity possesses many scales of variability then a more general nonlocal darcy law is a more appropriate model structure when the nonlocal darcy s law is more appropriate numerical experiments suggest that errors using a local darcy s law with effective parameters are most significant at the smallest resolving scale of the model and are minimal at scales between 8 and 16 times the resolving scale

gevrey smoothness of elliptic lower dimensional invariant tori in hamiltonian systems under russmann s non degeneracy condition in this paper we prove gevrey smoothness of elliptic lower dimensional invariant tori for nearly integrable analytic hamiltonian systems under russmann s non degeneracy condition by an improved kam iteration

discrete time and length averaged solutions of the advection dispersion equation solute concentrations obtained from displacement experiments in porous media frequently represent discrete values as a result of averaging over a finite sampling interval for example effluent curves are made up of time averaged concentrations while volume averaged concentrations are obtained from core samples the discrete concentrations are often described by continuous solutions of macroscopic solute transport equations such as the advection dispersion equation ade the continuous solution is often shifted to describe the average concentration this paper compares continuous and time or length averaged solutions of the one dimensional ade cast in terms of flux averaged and resident concentrations expressions for the time and length averaged concentrations are presented for solute applications described by dirac delta or heaviside functions instantaneous and continuous releases of the solute using four different combinations of solute application and detection modes a temporal and spatial moment analysis was conducted to compare the traditional continuous description with the discrete time or length averaged approach graphical and tabular data are presented to evaluate the accuracy of continuous solutions of the ade for determining transport parameters although significant errors may occur for extreme cases with low dispersion coefficients and large sampling intervals shifting the continuous solution by half the sampling interval generally yields results similar to those obtained with the time or length averaged analysis an advantage of averaged concentrations is that they permit greater flexibility to conduct experiments since averaged concentrations provide an exact description of the data regardless of the sampling interval

the operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers the operational meaning of the representative elementary volume rev concept on which current foundational theories of water movement through porous media are based is analyzed critically it is concluded that the rev concept as applied to real porous media is both unnecessarily restrictive and experimentally unverifiable in its place a relativist concept is proposed in which macroscopic physical variables are defined as convolution products of microscopic properties of a porous medium with weighting functions that represent the appropriate measuring instruments the effects of resolution and precision differences among experimental measuring devices now recognized as critical in the assessment of spatial variability in the properties of soils and aquifers can be incorporated naturally within the relativist concept but are excluded a priori in the rev concept the relativist point of view unlike the rev concept has a clear operational meaning and in principle extends theoretical validity to all local measurements made on porous media it is sufficient mathematically to derive conventional macroscopic balance equations for mass and linear momentum as well as a differential equation for the isothermal transport of water through an unsaturated anisotropic deformable soil or aquifer

linearization of gevrey flows on td with a brjuno type arithmetical condition abstract we show that in the gevrey topology a d torus flow close enough to linear with a unique rotation vector is linearizable as long as satisfies a novel brjuno type diophantine condition the proof is based on the fast convergence under renormalization of the associated gevrey vector field it requires a multidimensional continued fractions expansion of and the corresponding characterization of the brjuno type vectors this demonstrates that renormalization methods deal very naturally with gevrey regularity expressed in the decay of fourier coefficients in particular they provide new linearization results including frequencies beyond diophantine in non analytic topologies

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

a continuous time inverse operator for groundwater and contaminant transport modeling deterministic case the value of time dependent data in the calibration of numerical models of groundwater flow has been recently highlighted in the literature in this paper the background for a new development in the inverse problem is placed and a new operator for inversion of the spatially discretized diffusion equation is introduced the motivation for the development of an inverse problem solution procedure which uses information in transient data is illustrated with a focused review of recent literature such an approach which implicitly incorporates the continuous time dependency of the flow model is then presented this approach is based upon the new concept of inverting the model after the for example finite element spatial discretization but before the typically finite difference temporal discretization of the model not afterward as has been done in the past the laplace transform is used to replace discretization in time of the system of ordinary differential equations to discretization in the laplace variable of the algebraic system in laplace space the existence of the deterministic solution in the laplace transform space does not require knowledge of the eigenvalues of the diffusion matrix but only the specification of the laplace variables near zero a simple numerical example from the literature is used to generate a set of three simplified test cases upon which the proposed method is tested the implications of the theoretical investigation and the results of the numerical experiments are discussed with a view toward prospects of practical implementation

volume averaging for determining the effective dispersion tensor closure using periodic unit cells and comparison with ensemble averaging in this work we use the method of volume averaging to determine the effective dispersion tensor for a heterogeneous porous medium closure for the averaged equation is obtained by solution of a concentration deviation equation over a periodic unit cell our purpose is to show how the method of volume averaging with closure can be rectified with the results obtained by other upscaling methods under particular conditions although this rectification is something that is generally believed to be true there has been very little research that explores this issue explicitly we show that under certain limiting but mild assumptions the closure problem provides a fourier series solution for the effective dispersion tensor when second order spatial stationarity is imposed on the velocity field the method yields a simple fourier series that converges to an integral form in the limit as the period of the unit cell approaches infinity this limiting result is identical to the quasi fickian forms that have been developed previously via ensemble averaging by deng et al 1993 and recently by fiori and dagan 2000 except in the definition of the averaging operation as a second objective we have conducted a numerical study to evaluate the influence of the size of the averaging volume on the effective dispersion tensor and its volume averaged statistics this second objective is complimentary in many ways to recent research reported by rubin et al 1999 via ensemble averaging and by wang and kitanidis 1999 via volume averaging on the block averaged effective dispersion tensor the variability of the effective dispersion tensor from realization to realization is assessed by computing the volume averaged effective dispersion tensor for an ensemble of finite fields with the same ensemble statistics ensembles were generated using three different sizes of unit cells all three unit cell sizes yield similar results for the value of the mean effective dispersion tensor however the coefficient of variation depends strongly upon the size of the unit cell and our results are consistent with those developed by fiori 1998 from the ensemble averaging perspective this implies that in applications the actual value of the effective dispersion tensor may be significantly different than expected on the basis of unconditioned hydraulic conductivity statistics and this variation should be considered when applying macrodispersion to real world systems

a physical basis for calculating instrument spatial weighting functions in homogeneous systems 1 starting with a streamline based argument the theoretical basis for a spatial weighting function in a steady homogenous and isotropic flow system is developed for both compressible and incompressible flow a physical interpretation of spatial weighting functions in terms of the ratio of steady state energy dissipation rate per unit volume of porous medium to total energy dissipation rate over the entire flow domain is formulated finally applications of the spatial weighting functions are presented based on calculated gas minipermeameter mass flux fields for the conventional surface probe and a new small drill hole probe configuration spatial weighting function distributions indicate that with diverging flow field instruments such as the gas minipermeameter porous medium volumes in the inlet vicinity are heavily weighted with volumes near the seal boundaries shown to be extremely important the technique described allows one to quantify the size and shape of the subvolume of a domain contributing to a hydraulic conductivity or intrinsic permeability measurement for heterogeneous systems it is suggested that the spatial weighting function for a given instrument will be heterogeneity dependent and therefore nonunique however extending the weighting function concept to heterogeneous and or transient systems is an important research priority