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

the truncated complex moment problem let 2n denote a sequence of complex numbers 00 01 10 0 2n 2n 0 00 0 ij ji and let k denote a closed subset of the complex plane c the truncated complex k moment problem for entails determining whether there exists a positive borel measure on c such that ij zizj d 0 i j 2n and supp k for k kp a semi algebraic set determined by a collection of complex polynomials p pi z z mi 1 we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix m n and the localizing matrices mpi we prove that there exists a rankm n atomic representing measure for 2n supported in kp if and only if m n 0 and there is some rankpreserving extension m n 1 for which mpi n ki 0 where deg pi 2ki or 2ki 1 1 i m

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

stochastic modeling of unsaturated flow in heterogeneous media with water uptake by plant roots tests of the parallel columns model under two dimensional flow conditions in the first paper of this series rubin and or 1993 we developed a stochastic model predicting spatial moments of soil water matric potential and saturation s under conditions of unsaturated steady state flow as function of the spatial moments of log saturated hydraulic conductivity y capillarity index and plant rooting depth in this study we investigate the ability of the proposed parallel columns model to reconstruct two dimensional flow solutions and apply the model to spatial estimation using the geostatistical formalism a finite difference numerical scheme was developed to solve the two dimensional richards equation in heterogeneous flow domain and obtain detailed images of the spatial distribution of and s and their moments two dimensionally based images representing true distributions of and s in a vertical cross section x z plane of laterally heterogeneous soil were used in a synthetic spatial estimation problem for comparison with the one dimensionally based solution and with kriging estimates the results show that 1 the means of and s were practically identical for the one and two dimensional solutions 2 variance obtained from the two dimensional solution was smaller than the one dimensional whereas s variance was larger in the two dimensional case 3 a smoothing of under two dimensional flow regime resulted in spatial covariance function of a bessel type whereas two dimensional spatial covariance of s exhibited a steep decay near its origin denoting spatial discontinuity 4 the least smoothing of was observed near the surface where nearly one dimensional conditions exist 5 during transient flow spatial covariances assume their steady state form quite rapidly 6 spatial covariances predicted by the parallel columns model provided good estimates at unmeasured locations with conservative confidence regions for estimation and less conservative for s estimation and 7 analytically predicted spatial covariances provide a means for network design

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

estimating unsaturated soil hydraulic properties from tension disc infiltrometer data by numerical inversion tension disc infiltrometers are becoming increasingly popular devices for in situ measurement of the unsaturated hydraulic properties of soil tension infiltration data are generally used to evaluate the parametersks and ain gardner s exponential model of the unsaturated hydraulic conductivity either two measurements using different disc diameters or measurements with a single disc but using multiple pressure heads are then used in this paper we describe a parameter estimation procedure which combines the levenberg marquardt nonlinear parameter optimization method involving weighted least squares with a quasi three dimensional numerical model which solves the variably saturatedflow equation by numerical inversion of richards equation the unknown parameters in van genuchten s model of the unsaturated soil hydraulic properties are estimated from observed cumulative infiltration data during transient waterflow additional measurements of the pressure head or water content as well as a penalty function for constraining the unknown parameters to remain in some feasible region bayesian estimation can be optionally included into the parameter estimation procedure the problem of optimal sampling design that is selecting the best points in space and time for making measurements is addressed by studying the sensitivity of the objective function to changes in the optimized hydraulic parameters we calculate objective functions based on available cumulative infiltration pressure head and water content measurements and also on several combinations of these data the behavior of the objective function in three dimensional parameter space is evaluated by means of a series of two dimensional response surfaces the utility of the parameter estimation procedure is demonstrated using numerically generated data the sensitivity of the procedure to different initial estimates of the model parameters is also discussed

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

scaling of constitutive relationships in unsaturated heterogeneous media a numerical investigation numerical models of groundwater flow in the unsaturated zone require relative conductivity and moisture retention curves at a discretization scale much larger than that at which these constitutive relationships are measured this study seeks to identify simple scaling laws linking parameters of block scale constitutive relationships with the parameters of their measurement scale counterparts represented here using gardner russo models the two parameters of these models saturated conductivity and pore size distribution parameter are viewed as cross correlated regionalized variables with isotropic or anisotropic spatial correlation structures and are simulated using the sequential gaussian method upscaled constitutive curves for steady gravity drainage through three dimensional heterogeneous domains are then constructed by repeating numerical flow experiments for a series of applied infiltration rates and plotting the resulting upscaled relative conductivities and average water saturations against average matric potentials the experiments reveal clear log linear relationships between upscaled relative conductivities and average matric potentials the slopes of which are used to define an upscaled pore size distribution parameter the reciprocal of this quantity which represents an upscaled characteristic capillary length is very well approximated by a spatial power average of its measurement scale values furthermore the averaging exponent is found to be the same as that associated with the upscaling of saturated conductivities over the same flow domains when average matric potentials are normalized using this upscaled characteristic capillary length relative conductivity and moisture saturation values for widely different flow field realizations collapse along well defined curves however for wet conditions the form of these curves differs from that of the models assumed at the measurement scale

spline smoothing the equivalent variable kernel method on considere une approche spline pour la regression non parametrique et l estimation de courbe on montre que dans un certain sens le lissage spline correspond approximativement au lissage par une methode du noyau avec une largeur de bande dependant de la densite locale des points du plan de regression

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

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

the turning bands method for simulation of random fields using line generation by a spectral method the turning bands method tbm for the simulation of multidimensional random fields is presented these fields commonly occur in the monte carlo simulation of hydrologic processes particularly groundwater flow and mass transport the general tbm equations for two and three dimensional fields are derived with particular emphasis on the more complicated two dimensional case for stationary two dimensional fields the unidimensional line process is generated by a simple spectral method a technique which can be generally applied to any two dimensional covariance function and which is easily extended to anisotropic and areal averaged processes theoretically and by example the tbm is shown to be ergodic even for a finite number of lines and it is demonstrated that it rapidly converges to the true statistics of the field guidelines are presented for the selection of model parameters which will be helpful in the design of simulation experiments the tbm is compared to other methods in terms of cost and accuracy demonstrating that the tbm is as accurate as and much less expensive than multidimensional spectral techniques and more accurate than the most expensive approaches which use matrix inversion such as the nearest neighbor approach the unidimensional spectral technique presented here permits for the first time the inexpensive and accurate tbm simulation of any proper two dimensional covariance function and should be of some help in the stochastic analysis of hydrologic processes

a geostatistical approach to the conditional estimation of spatially distributed solute concentration and notes on the use of tracer data in the inverse problem this paper presents a theoretical framework for deriving the moments of the concentration based on the lagrangian approach and using a stochastic framework conditional to measurements of conductivity and hydraulic head the method consists of deriving the spatial correlations between concentration and travel time and hydrologic variables such as the conductivity and the hydraulic head these correlations allow the conditioning of the moments of the concentration on measurements by conditioning the concentration the uncertainty associated with its estimation can be reduced substantially consequently difficulties associated with estimation of the extent of contamination can be alleviated the theoretical framework and derivations may also be used to condition the moments of the conductivity on tracer data such as concentrations travel times and displacements an application of such an approach would require a configuration of sources and samplers we show that measured concentration is inferior to travel time and displacements in terms of efficient conditioning

some new results and generalizations in metric fixed point theory abstract the main purpose of this paper is the study of the generalization of some results given in m berinde v berinde on a general class of multi valued weakly picard mappings j math anal appl 326 2007 772 782 and references therein some generalizations of the mizoguchi takahashi fixed point theorem kannan s fixed point theorems and chatterjea s fixed point theorems are established by using our new fixed point theorems

error in unsaturated stochastic models parameterized with field data 1 we use monte carlo error analysis to illustrate the impact of measurement errors in field estimated hydraulic properties on predictions made with 1d and 3d unconditional stochastic models of unsaturated flow and transport monte carlo simulations are conducted across a series of simplified artificial realities completely described by the gardner russo parametric model the mean values of properties are varied between simulations to elucidate the relationship between true properties and prediction errors hydraulic properties are reestimated by simulating tension infiltrometer measurements in the presence of small simple errors two types of observation error are considered along with one inversion model error resulting from poor contact between the instrument and the medium errors in the spatial statistics of hydraulic properties cause critical stochastic model assumptions to be violated limiting the usable parameter space for model predictions even where critical assumptions are valid stochastic model predictions show significant error and the magnitude and pattern of error changes with the true property means the flow conditions and the type of measurement error mean velocities may show errors up to an order of magnitude the velocity variance is overestimated by up to three orders of magnitude during 3d flow and eight orders of magnitude during 1d flow the 1d velocity integral scale is underestimated by as much as five orders of magnitude the estimates for 1d longitudinal macrodispersivity are surprisingly robust and show relatively small error across most of the parameter space

equivalence of bicontinuum and second order transport in heterogeneous soils and aquifers the mathematical equivalence of a closed second order transport equation for reactive solutes in saturated heterogeneous porous media e g soils and aquifers and a two region mobile mobile formulation is demonstrated in two ways 1 by averaging the bicontinuum equations and 2 by transforming the second order equation to canonical form the derivation of this equivalence is limited to media with heterogeneities in a plane orthogonal to the flow direction which represents a bidirectional generalization of the stratified case results indicate how the parameters of the bicontinuum formulation can be obtained from statistical descriptions of this heterogeneity and establish a stochastic interpretation for the bicontinuum equations one dimensional analytical solutions of the equivalent transport equations are compared to a three dimensional numerical simulation of transport in a spatially autocorrelated lognormal random conductivity field results indicate that the upscaled equations provide good approximations of the dynamics of the mean resident concentration the flux averaged concentration and their variances

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

stochastic analysis of unsaturated steady flows above the water table steady flowtakes place into a three dimensional partially saturated porous medium where due to their spatial variability the saturated conductivity ks and the relative conductivity kr are modeled as random space functions rsf s as a consequence the flow variables fvs i e pressure head and specific flux are also rsfs the focus of the present paper consists into quantifying the uncertainty of the fvs above the water table the simple expressions most of which in closed form of the second order moments pertaining to the fvs allow one to follow the transitional behavior from the zone close to the water table where the fvs are non stationary till to their far field limit where the fvs become stationary rsfs in particular it is shown how the stationary limits and the distance from the water table at which stationarity is attained depend upon the statistical structure of the rsfs ks kr and the infiltrating rate the mean pressure head k l has been also computed and it is expressed as k l 0 1 being a characteristic heterogeneity function which modifies the zero order approximation 0 of the pressure head valid for a vadose zone of uniform soil properties to account for the spatial variability of ks and kr two asymptotic limits i e close near field and away far field from the water table are derived into a very general manner whereas the transitional behavior of between the near far field can be determined after specifying the shape of the various input soil properties besides the theoretical interest results of the present paper are useful for practical purposes as well indeed the model is tested against to real data and in particular it is shown how it is possible for the specific case study to grasp the behavior of the fvs within an environment i e the vadose zone close to the water table which is generally very difficult to access by direct inspection

a multiscale reduced basis method for parametrized elliptic partial differential equations with multiple scales we present a technique for solving parametrized elliptic partial differential equations with multiple scales the technique is based on the combination of the reduced basis method c prud homme d rovas k veroy y maday a t patera g turinici reliable real time solution of parametrized partial differential equations reduced basis output bound methods journal of fluids engineering 124 1 2002 70 80 and the multiscale finite element method t y hou x h wu a multiscale finite element method for elliptic problems in composite materials and porous media journal of computational physics 134 1 1997 169 189 to treat problems in which the differential coefficient is characterized by a large number of independent parameters for the multiscale finite element method a large number of cell problems has to be solved at the fine local mesh for each new configuration of the differential coefficient in order to improve the computational efficiency of this method we construct reduced basis spaces that are adapted to the local parameter dependence of the differential operator the approximate solutions of the cell problems are computed accurately and efficiently via performing galekin projection onto the reduced basis spaces and implementing the offline online computational procedure therefore a large number of similar computations at the fine local mesh can be carried out with lower computational cost for each new configuration of the differential coefficient numerical results are provided to demonstrate the accuracy and efficiency of the proposed approach

stochastic modeling of large scale transient unsaturated flow systems a new framework for modeling large scale transient unsaturated flow systems in spatially variable soils is proposed in order to overcome the problem of limited information about the local details of spatial soil variability a stochastic approach which assumes that local soil properties are realizations of three dimensional random fields is followed for derivation of a large scale model representation structure the three dimensionality of the local flow and the nonlinear dependence of the local flow output on the local soil properties are considered the large scale model structure is derived by averaging the local governing flow equation over the ensemble of realizations of the underlying soil property random fields the resulting mean model representation is in the form of a partial differential equation in which averaged or effective model parameters occur these effective model parameters i e effective hydraulic conductivity and effective specific moisture capacity are evaluated using a quasi linearized fluctuation equation and a spectral representation of stationary processes the large scale model representation considers the large scale effects of soil property variability and has relatively few parameters the general conclusions of this study are that soil property variability produces large scale hysteresis and anisotropy of the effective parameters the potential theoretical and practical ramifications of these results in the area of unsaturated flow modeling need to be investigated the general stochastic modeling framework developed here is applicable not only to unsaturated flow but also to other distributed parameter systems e g saturated flow and transport geothermal and oil reservoir modeling

stochastic analysis of one dimensional steady state unsaturated flow a comparison of monte carlo and perturbation methods the stochastic nature of moisture content in a soil profile under steady state unsaturated infiltration is examined where the saturated hydraulic conductivity is taken as a stationary stochastic process two different techniques are employed in determining the stochastic output these are monte carlo simulations and an analytic derivation from a first order perturbation solution in comparing the two techniques the liability of the perturbation expressions is investigated if the monte carlo results are assumed to exactly represent the stochastic nature of the moisture content in four test examples a relatively good comparison is obtained between the results of the two techniques the analytic nature of the perturbation solution then readily provides information concerning the general stochastic properties of the moisture content and in this way additional monte carlo runs are avoided from this analysis it is shown that the moisture content is a stationary stochastic process at distances far from boundary regions the distance from the boundary to regions of stationarity is dependent upon soil properties and the boundary conditions imposed

effects of first order approximations on head and specific discharge covariances in high contrast log conductivity the hydraulic head and the specific dischargefluctuations depend nonlinearly on the hydraulic conductivity however the methods most commonly used in the stochastic analysis of groundwaterflow are based upon the linearization of these relations in this paper we apply a numerical spectral approach to investigate the range of validity of the small perturbation approximation for head and specific discharge moments in two dimensionalfinite domains wefind that the small perturbation approximation tends to underestimate the variance of large scale head and specific dischargefluctuations and error increases with increasing log conductivity variance and increasing domain size the headfluctuations do not appear ergodic even when the small perturbation approximation predicts they will be ergodic the specific dischargefluctuations on the other hand do appear ergodic the small perturbation approximation performs well in estimating specific discharge variance in the longitudinal direction but significantly underestimates transverse specific discharge variance

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 comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow this paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity the ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant seven geostatistical methods fast fourier transform ff fractal simulation fs linearized cokriging lc linearized semianalytical ls maximum likelihood ml pilot point pp and sequential self calibration ss were compared on four synthetic data sets each data set had specific features meeting or not classical assumptions about stationarity amenability to a geostatistical description etc the comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km four of the methods ls ml pp and ss were identified as being approximately equivalent for the specific problems considered the magnitude of the variance of the transmissivity fields which went as high as 10 times the generally accepted range for linearized approaches was not a problem for the linearized methods when applied to stationary fields that is their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods nonstationarity of the true transmissivity field or the presence of anomalies such as high permeability fracture zones was however more of a problem for the linearized methods the importance of the proper selection of the semivariogram of the log10 t field or the ability of the method to optimize this variogram iteratively was found to have a significant impact on the accuracy and precision of the travel time predictions use of additional transient information from pumping tests did not result in major changes in the outcome while the methods differ in their underlying theory and the codes developed to implement the theories were limited to varying degrees the most important factor for achieving a successful solution was the time and experience devoted by the user of the method

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

spatial analysis of hydraulic conductivity measured using disc infiltrometers spatial variability of surface hydraulic properties and the extrinsic e g traffic cropping etc and intrinsic e g soil type pore size distribution etc factors associated with these properties are important for infiltration and runoff processes in agricultural fields disc infiltrometers measured infiltration at 296 sites arranged on two parallel transects to examine and differentiate the factors contributing to spatial structure under different field conditions these measurements were made in the corn rows no track interrows and wheel track interrows of the field using four different soil water tensions 0 30 60 and 150 mm unsaturated hydraulic conductivity k and saturated hydraulic conductivity k were maximum in the corn rows and minimum in wheel track inter rows with no track interrows intermediate exponents cz parameters of k and k relationships k k exp av for corn rows and no track interrows were not significantly different from each other but were significantly different from q for the wheel track interrows a t p 0 01 level spatial variability o f k and k values showed some pseudoproportional effect in nugget variance for all three field conditions no track interrows clearly showed an inverse trend for semivariogram o f k with changing tension q values whereas differences were found for corn rows and wheel traffic interrows the spatial structure of y for all three field conditions were mostly white noise under corn rows in addition to random variation a small five row periodic variation at th e p 0 20 level matching the five row traffic configuration was discovered the spatial structure of a was influenced by soil type for the no track interrows spatial structure was absent in wheel track interrows indicating the destruction of pore structure due to compaction

stochastic analysis of unsaturated steady state flow through bounded heterogeneous formations the effect of heterogeneity in saturated hydraulic conductivity ks and in a pore scale distribution parameter on unsaturated steady state flow in bounded domains is studied the properties are assumed to be random space functions having stationary means and covariances an analytic model based on small perturbation approximation in ks and for predicting the spatial moments of the pressure head from the moments of the input variables is developed the solution applies to the entire unsaturated domain without invoking the unit mean hydraulic gradient assumption under conditions of small scale variability the solution simplfies considerably resulting in closed form expressions for the moments of the pressure head these expressions are tested in synthetically generated random fields of and ks having prescribed statistical properties

alternative analyses of hydraulic data from disc tension infiltrometers hydraulic conductivity values were compared based on alternative analyses of data from disc tension infiltrometers the first method was based on a single disc and tension and depends on the estimate of sorptivity and steady state flow a second method used steady state flow measurements for two different disc radii 52 and 118 mm a third method used a single disc with multiple tensions from which steady state flow was obtained at three or more tensions which could be at the same location a fourth method used a single disc with two tensions from which steady state flow was obtained at two tensions finally values based on soil cores were compared the results show reasonable agreement between methods for the hydraulic conductivity with the largest differences for data collected for zero tension for the most part there were no significant differences in hydraulic conductivity due to the disc infiltrometer radius a single disc method with multiple tensions more than 3 points and large disc radius gave results which were the most stable accurate and repeatable

stochastic analysis of unsaturated flow in heterogeneous soils 1 statistically isotropic media steady unsaturated flow with vertical mean infiltration through unbounded heterogeneous porous media is analyzed using a perturbation approximation of the stochastic flow equation which is solved by spectral representation techniques the hydraulic conductivity k is related to the capillary pressure head by k ks exp where ks is the saturated conductivity and is a soil parameter a general formulation is presented for the case with ks and represented as statistically homogeneous spatial random fields in part 1 solutions are developed assuming is constant and representing ks variability by one dimensional and three dimensional isotropic random fields results are obtained for head variances and covariance functions effective hydraulic conductivities variances of the unsaturated hydraulic conductivity flux variances and variance of pressure gradient when the parameter is relatively large corresponding to coarse textured soils the head variance decreases and all of the results demonstrate a trend toward gravitationally dominated one dimensional vertical flow the effective conductivity is dependent on the correlation scale of ln ks and the mean hydraulic gradient

implementation of the three dimensional turning bands random field generator numerical techniques to generate replicates of spatially correlated random fields are often used to synthesize sets of highly variable physical quantities in stochastic models of naturally heterogeneous systems within the realm of hydrologic research for example such tools are widely used to develop hypothetical rainfall distributions hydraulic conductivity fields fracture set properties and other surface or subsurface flow parameters the turning bands method is one such algorithm which generates two and three dimensional fields by combining values found from a series of one dimensional simulations along lines radiating outward from a coordinate origin previous work with two dimensional algorithms indicates that radial lines evenly spaced about the unit circle lead to enhanced convergence properties the same can be said for the three dimensional models but it is more difficult to choose an arbitrary number of evenly spaced lines about the unit sphere the current investigation shows that the use of larger numbers of randomly oriented lines 100 can enhance the performance of the three dimensional algorithm this improved performance is needed to effectively simulate problems characterized by full three dimensionality and or anisotropy in either monte carlo or single realization applications use of a large number of lines will also reduce the presence of a distortion effect manifested as linelike patterns in the field increased computational costs can be reduced by employing a fast fourier transform technique to generate the line processes

characterizing polynomials with roots in a semialgebraic set let p spl isin r x be a real valued polynomial and s spl sube c a set defined by polynomial inequalities g sub k z z spl ges 0 for some polynomials g sub k in c z z we provide a necessary and sufficient condition on the coefficients of p for all the zeros of p to be in s

homogenization of richards equation in permeability fields with different connectivities 1 large scale modeling of transient flow in the unsaturated zone is important for the estimation of the water budget and solute transport in the vadose zone upscaled flow models need to capture the impact of small scale heterogeneities which are not resolved by the model on large scale flow we perform upscaling of the richards equation in heterogeneous porous media with continuous distributions of the soil hydraulic parameters using homogenization theory and stochastic averaging techniques we restrict the analysis to flow regimes in which the capillary equilibrium assumption holds on the small scale in order to account for effects of capillary entry pressure we apply the brooks corey model for the soil retention and relative permeability curves and consider leverett scaling for the coupling of intrinsic permeability and entry pressure for this model we derive and analyze the ensemble averaged parameter functions for the macroscopic flow equations the effects of a definite entry pressure vanish with increasing variance of the log intrinsic permeability we compare the statistically averaged parameter functions to numerically calculated effective functions for parameter fields with different connectivity properties these results illustrate that soils with well connected coarse materials differ in the relative permeability from those with well connected fine materials or those without particular connectedness

a moment approach to analyze zeros of triangular polynomial sets let i be a zero dimensional ideal of r x 1 x n such that its associated set g of polynomial equations g i x 0 for all i 1 n is in triangular form by introducing multivariate newton sums we provide a numerical characterization of polynomials in i we also provide a necessary and sufficient numerical condition for all the zeros of g to be in a given set k c c n without explicitly computing the zeros in addition we also provide a necessary and sufficient condition on the coefficients of the g i s for g to have a only real zeros b to have only real zeros all contained in a given semi algebraic set k c r n in the proof technique we use a deep result of curto and fialkow 2000 on the k moment problem and the conditions we provide are given in terms of positive definiteness of some related moment and localizing matrices depending on the g i s via the newton sums of g in addition the number of distinct real zeros is shown to be the maximal rank of a related moment matrix

parallel computation of flow in heterogeneous media modelled by mixed finite elements abstract in this paper we describe a fast parallel method for solving highly ill conditioned saddle point systems arising from mixed finite element simulations of stochastic partial differential equations pdes modelling flow in heterogeneous media each realisation of these stochastic pdes requires the solution of the linear first order velocity pressure system comprising darcy s law coupled with an incompressibility constraint the chief difficulty is that the permeability may be highly variable especially when the statistical model has a large variance and a small correlation length for reasonable accuracy the discretisation has to be extremely fine we solve these problems by first reducing the saddle point formulation to a symmetric positive definite spd problem using a suitable basis for the space of divergence free velocities the reduced problem is solved using parallel conjugate gradients preconditioned with an algebraically determined additive schwarz domain decomposition preconditioner the result is a solver which exhibits a good degree of robustness with respect to the mesh size as well as to the variance and to physically relevant values of the correlation length of the underlying permeability field numerical experiments exhibit almost optimal levels of parallel efficiency the domain decomposition solver doug http www maths bath ac uk parsoft used here not only is applicable to this problem but can be used to solve general unstructured finite element systems on a wide range of parallel architectures

fixed points for φ ψ p weakly contractive mappings in metric spaces with w distance the fixed point theorems presented in this paper generalize recent results of dutta and choudhury 3 branciari 2 banach 1 rhoades 9 khan et al 6 kada 5 and rhoades and abbas 10

aquifer operator scaling and the effect on solute mixing and dispersion 1 since aquifer parameters may have statistical dependence structures that are present across a huge range of scales the concepts of fractional brownian motion fbm have been used in both analytic and numerical stochastic settings most previous models have used isotropic scaling characterized by a single scalar hurst coefficient any real world anisotropy has been handled by an elliptical stretching random k field we define a d dimensional extension of fbm in which the fractional order integration may take on different orders in the d primary possibly nonorthogonal scaling directions and the degree of connectivity and long range dependence is freely assigned via a probability measure on the unit sphere this approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of k in certain directions it also allows for the representation of fracture networks in a continuum setting the eigenvectors of the scaling matrix describe the primary fracture scaling directions and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions in a numerical experiment the motion of solutes through 2 d operator fractional gaussian fields depends very much on transverse hurst coefficients transverse scaling in the range of fractional gaussian noise engenders greater plume mixing and a transition to a fickian regime higher orders of integration in the range of fractional brownian motion are associated with thicker layering of aquifer sediments and more preferential unmixed transport therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport

a composite l1 parameter estimator for model fitting in groundwater flow and solute transport simulation this paper proposes a composite l1 parameter estimator to solve the inverse problems in groundwater flow and solute transport modeling the estimator is formulated using a weighted l1 norm as the error measure between the vector of the observed hydraulic heads and solute concentrations and the corresponding vector of the heads and concentrations computed from a finite element model for steady state groundwater flow and solute transport in a two dimensional areal confined aquifer system the parameters currently considered are the hydraulic conductivities the dispersivities the porosities of the aquifer system and the solute source concentration s the gradients of the state variables with respect to the parameters are computed analytically using a formulation stemming from the finite element model for the state variables the solutions to two hypothetical groundwater parameter estimation problems are presented to illustrate that the proposed estimator does have the property of being fairly accurate and more significantly being robust in handling observation data containing certain outliers

a theoretical study of the estimation of the correlation scale in spatially variable fields 2 nonstationary fields a nonstationary field may be viewed as a process composed of two components a stochastic component characterized by a relatively small scale high frequency variation hfv and a deterministic drift component characterized by a relatively large scale low frequency variation lfv in this case the estimation of the variogram or the covariance function and its associated correlation scale requires the prior estimation of the drift in this study the effects of the type of drift function the size of the integral scale and the variances associated with the two components on the estimation of the correlation scale were analyzed using a theoretically generated master population of points representing the nonstationary field a set of nonstationary fields was generated by superimposing different drift functions with a given variance l2 on stationary fields with a given integral scale and variance h2 results showed that the shape of the estimated variograms and their associated estimated correlation scales the fractal dimensions and the estimates of the covariance function provided by the parametric estimation procedures were affected by the type of the imposed drift function by the integral scale of the stochastic component and by the ratio between the variances of the two components both the maximum likelihood ml and the restricted maximum likelihood rml estimation procedures may fail to correctly identify and separate the two components when their respective variances are of the same order of magnitude and when the correlation scale or spatial extent of the stochastic component is small compared with the typical size of the field domain in these situations the fractal dimension analysis also may not indicate the dominance of the lfv over the hfv even when l2 2 h2 the model validation tests also may fail to indicate inconsistency between the estimates of the parameters of the structural model and the data when l2 h2 in general the ml and the rml estimation procedures overestimated the integral scale j and the variance c o of the underlying process whereas the iterative generalized least squares and the ordinary least squares procedures coupled with the minimum mean square deviation msd criterion for variogram model fitting underestimated the value of the integral scale however results of model validation tests suggested that these estimates were accurate and consistent with the data and almost indistinguishable in terms of the values of the akaike information criterion consequently different sets of and 0 values may adequately describe a given realization of the nonstationary field to the same level of precision careful examination of the estimated variograms over the entire set of lag classes application of fractal dimension analysis and a validation test of the fitted variogram models are necessary steps for drift detection quantification of the drift however may not be achieved in certain situations

effects of kriging and inverse modeling on conditional simulation of the avra valley aquifer in southern arizona the avra valley aquifer in southern arizona is modeled stochastically at three levels of uncertainty the highest level of uncertainty occurs when log transmissivity estimates are based on measured values of this parameter but without regard to the geographic location of each measurement point the resulting steady state hydraulic heads in the aquifer computed by unconditional simulation with the aid of a multivariate normal random number generator coupled with a finite element model have a relatively large variance this variance can be reduced by conditioning the log transmissivity estimates on the spatial arrangement of the data by means of kriging when conditional simulation of the aquifer is performed at this intermediate level of uncertainty with the aid of the same technique as in the unconditional case the variance of the predicted head values is reduced by a factor of 3 2 the lowest level of uncertainty is achieved when the log transmissivity estimates are further conditioned on data relating to the flow regime such as flow rates and water levels in observation wells by means of a statistical inverse procedure conditional simulation at this level of uncertainty is a novel concept it results in a hydraulic head prediction variance that is 14 3 times lower than the corresponding variance based on kriged log transmissivities the net effect of conditioning by means of kriging and inverse modeling is to reduce the prediction variance by a factor of 46 0 a similar study performed by binsarti 1980 on the cortaro aquifer in southern arizona showed insignificant variance reduction due to kriging but a factor of four reduction after inverse modeling these results indicate that the conditioning effect of the inverse method may be much greater than that of kriging the fact that conditioning can reduce the hydraulic head prediction variance by as much as a factor of 46 implies that one should be cautious when dealing with the results of stochastic aquifer models in which conditioning effects are disregarded

exploiting symmetries in sdp relaxations for polynomial optimization in this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations our special focus is on constrained problems especially when the symmetric group is acting on the variables in particular we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute the geometric quotient

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

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

predictive uncertainty analysis of a saltwater intrusion model using null space monte carlo 1 because of the extensive computational burden and perhaps a lack of awareness of existing methods rigorous uncertainty analyses are rarely conducted for variable density flow and transport models for this reason a recently developed null space monte carlo nsmc method for quantifying prediction uncertainty was tested for a synthetic saltwater intrusion model patterned after the henry problem saltwater intrusion caused by a reduction in fresh groundwater discharge was simulated for 1000 randomly generated hydraulic conductivity distributions representing a mildly heterogeneous aquifer from these 1000 simulations the hydraulic conductivity distribution giving rise to the most extreme case of saltwater intrusion was selected and was assumed to represent the true system head and salinity values from this true model were then extracted and used as observations for subsequent model calibration random noise was added to the observations to approximate realistic field conditions the nsmc method was used to calculate 1000 calibration constrained parameter fields if the dimensionality of the solution space was set appropriately the estimated uncertainty range from the nsmc analysis encompassed the truth several variants of the method were implemented to investigate their effect on the efficiency of the nsmc method reducing the dimensionality of the null space for the processing of the random parameter sets did not result in any significant gains in efficiency and compromised the ability of the nsmc method to encompass the true prediction value the addition of intrapilot point heterogeneity to the nsmc process was also tested according to a variogram comparison this provided the same scale of heterogeneity that was used to generate the truth however incorporation of intrapilot point variability did not make a noticeable difference to the uncertainty of the prediction with this higher level of heterogeneity however the computational burden of generating calibration constrained parameter fields approximately doubled predictive uncertainty variance computed through the nsmc method was compared with that computed through linear analysis the results were in good agreement with the nsmc method estimate showing a slightly smaller range of prediction uncertainty than was calculated by the linear method

some fixed point theorems using weaker meir keeler function in metric spaces with w distance abstract in the present paper we prove some new fixed point theorems for self mappings defined on a complete metric space with a w distance these results extend some previous fixed point theorems in this field to more general contractive conditions in the setting of w distances for selfmappings which satisfy certain weaker meir keeler conditions

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

the one dimensional approximation for infiltration in heterogeneous soils appendix is available with entire article on microfiche order from american geophysical union 2000 florida avenue n w washington dc 20009 document w91 004 2 50 payment must accompany order by using analytical solutions to the problem of one and two dimensional linearized unsteady infiltration from a strip source in a soil with saturated hydraulic conductivity varying exponentially in space explicit criteria for the validity of the one dimensional approximation are derived in terms of parameters describing the soil type the uniformity of the infiltrating source at the surface and the soil heterogeneity numerical solutions for the linearized infiltration problem are developed for any variation of hydraulic conductivity in space extensive simulations suggest that for a sizable strip source and moderate variation of hydraulic conductivity a set of independent neighboring soil columns represents well the multidimensional infiltration problem the possibility of defining sufficient parameters controlling the accuracy of the one dimensional approximation is discussed

statistical analysis of spatial variability in unsaturated flow parameters core scale estimates of soil parameters of the gardner russo and van gemachten models of the hydraulic conductivity and water retention functions were obtained for 417 undisturbed soil cores taken from a wall of a trench 20 m long 2 5 m deep using a procedure based on inverse problem methodology these estimates were used to evaluate the first two statistical moments of the underlying random space functions rsfs using the restricted maximum likelihood estimation procedure coupled with the weighted least squares procedure to estimate parameters of models of the covariance and the drift functions of the pertinent rsfs the fitted models were used to evaluate the mean and covariance functions of the hydraulic conductivity and water rentention functions for given water saturations covariance functions of log saturated conductivity log ks and the shape parameters of the gardner russo and van genuchten models exhibited statistical anisotropy characterized by aspect ratios that vary between 3 to 4 correlation scales of log ks were larger than those of the shape parameters consequently the product of the variance of log unsaturated conductivity and its correlation scale remained essentially invariant for a considerable range of water saturation the implications of these results regarding stochastic modeling of transport in heterogeneous porous formations and possible applications of the results of this study are discussed briefly

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

explicit analytical solutions for one dimensional steady state flow in layered heterogeneous unsaturated soils under random boundary conditions 1 in this study we directly derive first order analytical solutions to the pressure head moments mean and variance for one dimensional steady state unsaturated flow in randomly heterogeneous layered soil columns under various random boundary conditions we assume that the constitutive relation between the unsaturated hydraulic conductivity and the pressure head follows an exponential model and treat the saturated hydraulic conductivity ks as a random function and the pore size distribution parameter as a random constant unlike the solution given in lu and zhang 2004 in which kirchhoff transformation was used and the solution to pressure head variance was presented as a function of cross covariances related to the intermediate kirchhoff transformed variable the solution to the pressure head variance presented in this paper is an explicit function of the input variabilities in addition we also give analytical solutions to the statistics of the unsaturated hydraulic conductivity and the effective water content these first order analytical solutions are compared with those from monte carlo simulations we also investigated the effect of uncertain boundary conditions the relative contribution of input variabilities to the head variance and the possible errors introduced by treating the correlated field as a random constant in the analytical solutions the results indicate that the uncertain constant head at the bottom of a deep soil column may not have a significant effect on predicting flow statistics in the upper portion of the column furthermore it is found that treating as a random constant is justified when the correlation length of is relatively large as compared to the layer thickness

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

a general theory for matrix root clustering in subregions of the complex plane we consider the general problem of root clustering of a matrix in the complex plane let a in c n times n and s subset c find the largest class of s and an algebraic criterion which is necessary and sufficient for lambda i a in s i 1 2 n we introduce two types of regions which constitute the largest class of s known to date the criterion is presented both for open regions and closed ones the results are used to define a design methodology for control systems moreover all classical results are shown to be special cases of the present theory

multilevel ilu decomposition in this paper the multilevel ilu mlilu decomposition is introduced during an incomplete gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels on these levels some smoothing steps are computed the mlilu decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices convergence rates independent of the number of unknowns are shown numerically for several examples many numerical experiments including unsymmetric and anisotropic problems problems with jumping coefficients as well as realistic problems are presented they indicate a very robust convergence behavior of the mlilu method

an extension of lakzian rhoades results in the structure of ordered b metric spaces via wt distance with an application abstract in this present article we establish two new kinds of nonlinear contraction mappings to obtain fixed point results in the structure of ordered b metric space via wt distance in fact our presented results are extensions of recent theorems due to lakzian rhoades 2019 appl math comput and other existing classical results of fixed point theory furthermore we provide examples to show the validity of our new investigations as an application we apply our new findings to obtain solution of a matrix equation finally we verify the accuracy of our new results numerically

spatial bias in field estimated unsaturated hydraulic properties hydraulic property measurements often rely on non linear inversion models whose errors vary between samples in non linear physical measurement systems bias can be directly quantified and removed using calibration standards in hydrologic systems field calibration is often infeasible and bias must be quantified indirectly we use a monte carlo error analysis to indirectly quantify spatial bias in the saturated hydraulic conductivity k sub s and the exponential relative permeability parameter alpha estimated using a tension infiltrometer two types of observation error are considered along with one inversion model error resulting from poor contact between the instrument and the medium estimates of spatial statistics including the mean variance and variogram model parameters show significant bias across a parameter space representative of poorly to well sorted silty sand to very coarse sand when only observation errors are present spatial statistics for both parameters are best estimated in materials with high hydraulic conductivity like very coarse sand when simple contact errors are included the nature of the bias changes dramatically spatial statistics are poorly estimated even in highly conductive materials conditions that permit accurate estimation of the statistics for one of the parameters prevent accurate estimation for the other accurate regions for the two parameters do not overlap in parameter space false cross correlation between estimated parameters is created because estimates of k sub s also depend on estimates of alpha and both parameters are estimated from the same data

nonstationary stochastic analysis of transient unsaturated flow in randomly heterogeneous media in this study a first order nonstationary stochastic model of transient flow is developed which is applicable to the entire domain of a bounded vadose zone in the presence of sink source we derive general equations governing the statistical moments of the flow quantities by perturbation expansions owing to the mathematical complexity of the equations in general we need to solve them numerically the numerical moment equation approach however has the flexibility in handling different boundary conditions flow configurations input covariance structures and soil constitutive relationships the moment equations presented in this study are simpler and easier to solve than those in the literature for transient unsaturated flow we solve these moment equations by the method of finite differences and demonstrate the developed model through some one and two dimensional examples under various transient conditions

beyond thiem a new method for interpreting large scale pumping tests in heterogeneous aquifers 1 the heterogeneous character of aquifers prevents a consistent interpretation of pumping tests based on thiem s formula which has been developed for homogeneous aquifers we present a new mathematical method for interpreting an ensemble and single large scale pumping tests in heterogeneous confined isotropic aquifers by combining a multiscale technique called coarse graining with renormalization group analysis we derive an explicit formula for the coarse grained aquifer transmissivity the coarse grained transmissivity is dependent on the radial distance from the well close to the well it equals the local transmissivity in single pumping tests and the harmonic mean of the heterogeneous transmissivity field in ensemble pumping tests in the far field of the well the coarse grained transmissivity approaches the geometric mean the transition between both regimes is mainly determined by the transmissivities correlation length this relation is used for an inversely estimate of the statistical properties of the transmissivity such as its geometric mean in addition this method allows for determining aquifer structures characterized by a correlation length from a single pumping test while the variance can be estimated from pumping tests with ensemble averaged hydraulic heads

effective hydraulic parameters for steady state vertical flow in heterogeneous soils 1 in hydroclimate and land atmospheric interaction models effective hydraulic properties are needed at large grid scales in this study the effective soil hydraulic parameters of the areally heterogeneous soil formation are derived by conceptualizing the heterogeneous soil formation as an equivalent homogeneous medium and assuming that the equivalent homogeneous soil will approximately discharge the same total amount of flux and produce same average pressure head profile in the formation as compared to previous effective hydraulic property studies a specific feature of this study is that the derived effective hydraulic parameters are mean gradient dependent i e vary across depth although areal soil heterogeneity was formulated as parallel homogeneous stream tubes in this study our results appear to be consistent with the previous findings of mean gradient unsaturated hydraulic conductivity yeh et al 1985a 1985b three widely used hydraulic conductivity models were employed in this study i e the gardner model the brooks and corey model and the van genuchten model we examined the impact of parameter correlation boundary condition surface pressure head and elevation above the water table on the effective saturated hydraulic conductivity and shape parameter the correlation between the saturated hydraulic conductivity ks and the shape parameter increases the effective saturated hydraulic conductivity while it does not affect the effective the effective is usually smaller than the mean value of while the effective ks can be smaller or larger than the mean value depending on no correlation or full correlation between ks and fields respectively an important observation of this study is that gardner and van genuchten functions resulted in effective parameters whereas it is difficult to define effective parameters for the brooks corey model since this model uses a piecewise continuous profile for hydraulic conductivity

stochastic analysis of flow in a heterogeneous unsaturated saturated system 1 in this study we develop a stochastic model for transient unsaturated saturated flow in randomly heterogeneous media with the method of moment equations we first derive partial differential equations governing the statistical moments of the flow quantities by perturbation expansions and then implement these equations under general conditions with the method of finite differences the stochastic model developed is applicable to the entire domain of a bounded multidimensional unsaturated saturated system in the presence of random or deterministic recharge and sink source and in the presence of multiscale nonstationary medium features the unsaturated and saturated zones are coupled through the water table whose position is random in randomly heterogeneous porous media the presence of the water table renders the flow moments strongly nonstationary even in the absence of medium nonstationary features this finding is confirmed with monte carlo simulations the resulting first two moments of flow may be used to approximate confidence intervals for the flow quantities in addition the integrated stochastic flow model provides the prerequisite quantities for realistically studying contaminant transport in unsaturated saturated systems with a stochastic approach

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

the 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

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

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

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

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

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