prediction of steady state flow of real gases in randomly heterogeneous porous media abstract we consider steady state flow of real gases through bounded randomly heterogeneous porous media such flow is described by a nonlinear partial differential equation with the random coefficient medium s permeability and source terms subject to randomly prescribed boundary conditions prior to applying stochastic analysis the problem is linearized by means of the kirchhoff transformation which allows us to obtain the exact expressions for an effective upscaled gas permeability in particular for one two and three dimensional mean uniform flows in infinite statistically homogeneous and isotropic domains the resulting effective permeability is given by harmonic geometric and arithmetic averages respectively the influence of statistical anisotropy of the random permeability field and domain s boundaries on the effective gas permeability is also investigated

the foundations of confounding in epidemiology abstract a statistically coherent view of confounding motivated by the over controversy over the proper control of confounding in the presence of prior knowledge is presented confounding by a covariate c in the presence of data on c is distinguished from confounding in the absence of data on c a covariate c is defined to be a nonconfounder in the absence of data on c if the population parameter of interest can be unbiasedly estimated asymptotically absent data on c under this definition c may be a confounder for some parameters of interest and a nonconfounder for others if c is a confounder for a parameter of interest that has a causal interpretation we call c a causal confounder when data on c are available c is defined to be a nonconfounder for a particular parameter of interest if and only if inference on the parameter of interest does not depend on the data through c bayesians frequentists and pure likelihoodists will in general agree on the prior knowledge necessary to render c a non confounder in particular c will in general be a nonconfounder precisely when the crude data ignoring c are s sufficient for the parameter of interest the intuitive view held by many practicing epidemiologists that confounding by c represents a bias of the unadjusted crude estimator is in a sense correct provided inference is performed conditional on approximate ancillary statistics that measure the degree to which associations in the data differ due to sampling variability from those population associations known a priori

estimation of a structural parameter in the presence of a large number of nuisance parameters summary when the number of nuisance parameters increases in proportion to the sample size the cramer rao bound does not necessarily give an attainable lower bound for the asymptotic variance of an estimator of the structural parameter the present paper presents a new lower bound under a criterion called information uniformity the bound is expressed as the inverse of the sum of the partial information and a certain nonnegative term which is derived by differential geometrical considerations the optimal estimating function meeting this lower bound when it exists is also obtained in a decomposed form the first term is the modified score function and the second term is roughly speaking given by the normal component of the mixture covariant derivative of some random variable furthermore special versions of these results are given in concise form and these are then applied to elucidate the efficiency of some examples

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

nonparametric estimation of the mean function for recurrent event data with missing event category recurrent event data frequently arise in longitudinal studies when study subjects possibly experience more than one event during the observation period often such recurrent events can be categorized however part of the categorization may be missing due to technical difficulties if the event types are missing completely at random then a complete case analysis may provide consistent estimates of regression parameters in certain regression models but estimates of the baseline event rates are generally biased previous work on nonparametric estimation of these rates has utilized parametric missingness models in this paper we develop fully nonparametric methods in which the missingness mechanism is completely unspecified consistency and asymptotic normality of the nonparametric estimators of the mean event functions accommodate nonparametric estimators of the event category probabilities which converge more slowly than the parametric rate plug in variance estimators are provided and perform well in simulation studies where complete case estimators may exhibit large biases and parametric estimators generally have a larger mean squared error when the model is misspecified the proposed methods are applied to data from a cystic fibrosis registry

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

boundary element solution for stochastic groundwater flow random boundary condition and recharge this paper presents an integral equation technique for solving stochastic boundary value problems in groundwater flow the aquifer considered has deterministic hydraulic conductivity but is subject to random boundary condition and domain recharge using the distribution of fictitious sources and dipoles stochastic integral equations for the mean and covariance of head and flux are derived an iterative boundary element technique is applied for numerical solution two one dimensional examples are examined and compared with exact solution a two dimensional problem is then presented

efficient estimation from right censored data when failure indicators are missing at random the kaplan meier estimator of a survival function is well known to be asymptotically efficient when cause of failure is always observed it has been an open problem however to find an efficient estimator when failure indicators are missing at random lo showed that nonparametric maximum likelihood estimators are inconsistent and this has led to several proposals of ad hoc estimators none of which are efficient we now introduce a sieved nonparametric maximum likelihood estimator and show that it is efficient our approach is related to the estimation of a bivariate survival function from bivariate right censored data 1 introduction suppose that we wish to estimate a survival distribution based on right censored data when cause of failure is always observed the method of nonparametric maximum likelihood leads to the well studied kaplan meier 1958 estimator which has many desirable properties including asymptotic efficiency wellner 1982 in this paper we address the problem of finding an asymptotically efficient estimator when cause of failure information is missing for some individuals

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

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

conditioning likelihood and coherence a review of some foundational concepts 2000 conditioning likelihood and coherence a review of some foundational concepts journal of the american statistical association vol 95 no 452 pp 1340 1346

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

a geostatistical approach to the inverse problem in groundwater modeling steady state and one dimensional simulations the problem of estimating hydrogeologic parameters in particular permeability from input output measurements is reexamined in a geostatistical framework the field of the unknown parameters is represented as a random field and the estimation procedure consists of two main steps first the structure of the parameter field is identified i e mathematical representations of the variogram and the trend are selected and their parameters are established by using all available information including measurements of hydraulic head and permeability second linear estimation theory is applied to provide minimum variance and unbiased point estimates of hydrogeologic parameters kriging structure identification is achieved iteratively in three substeps structure selection maximum likelihood estimation and model validation and diagnostic checking the methodology was extensively tested through simulations on a simple one dimensional case the results are remarkably stable and well behaved the estimated field is smooth while small scale variability is statistically described as the quality of measurements improves the procedure reproduces more features of the original field the results are also shown to be rather insensitive to deviations from assumptions about the geostatistical structure of the field

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

semiparametric estimators for the regression coefficients in the linear transformation competing risks model with missing cause of failure summary we consider the problem of estimating the regression coefficients in a competing risks model where the relationship between the cause specific hazard for the cause of interest and covariates is described using linear transformation models and when cause of failure is missing at random for a subset of individuals using the theory of robins et al 1994 for missing data problems and the approach of chen et al 2002 for estimating regression coefficients for linear transformation models we derive augmented inverse probability weighted complete case estimators for the regression coefficients that are doubly robust simulations demonstrate the relevance of the theory in finite samples

flow and transport in highly heterogeneous formations 1 conceptual framework and validity of first order approximations 1 flow of uniform mean velocity u takes place in a formation of spatially variable random conductivity k x advective transport of a plume of an inert solute is investigated by the lagrangean approach the aim of the study is to determine the spatial moments of the plume i e of fluid particle trajectories for highly heterogeneous aquifers for which y 1 where y ln k a multi indicator model of the permeability structure which is different from the common multi gaussian one is proposed the formation is modeled as a collection of n blocks of different k j the structure is defined by the distribution of k j the blocks shape and the coordinates of their centroids the following simplifications are adopted the blocks are inclusions of a regular shape circles spheres for isotropic media investigated here defined by the radius a and the inclusions are not overlapping and their centroids are distributed uniformly and independently in space at the continuous limit the model is characterized by the joint pdf f y a the model is shown to be quite general and to comprise binary bimodal indicator variograms and unimodal distributions of y as particular cases the study is focused on the latter case with y normal n y y2 and stationary covariance of given integral scale iy these are the parameters commonly estimated for sedimentary formations this leaves still freedom in selecting the pdf f a the simple model selected for semianalytical and numerical analysis is that of inclusions of radius r and volume fraction n submerged in a matrix of effective conductivity kef the latter represents the effect of inclusions of much smaller radius which appear as a nugget in the log conductivity two point covariance an approximate analytical solution of the flow is obtained by using a self consistent approximation while a fully numerical one is derived in part 3 jankovic et al 2003a transport is solved by particle tracking and the time dependent spatial moments trajectories variance skewness kurtosis are presented in part 2 fiori et al 2003 in the self consistent approximation the asymptotic longitudinal macrodispersivity l which is a function of y shows strong nonlinear effects inclusions of large positive y lead to a finite l whereas l grows unbounded for those of negative y this effect is not captured by the common first order approximation in y which is symmetrical and overestimates l for y 0 and underestimates it for y 0 as a result the second spatial moment is predicted accurately by the first order approximation by cancellation of errors provided that f y is symmetrical however the transient regime and higher order moments are not captured by the first order approximation

geometrical theory of asymptotic ancillarity and conditional inference summary differential geometry is applied to the problems of defining higher order asymptotic ancillarity and of obtaining the asymptotic conditional distribution of an efficient estimator in multiparameter curved exponential families it is shown that a fundamental role is played in the asymptotic theory of estimation by a one parameter family of affine connexions and curvatures of subspaces asymptotic ancillary statistics of higher order are explicitly constructed with the help of the geometry the conditional distribution of an estimator is given in terms of the exponential curvature of the model and the mixture curvature of the ancillary subspaces associated with the estimator

the use of field theoretic methods for the study of flow in a heterogeneous porous medium the effects of heterogeneities on the steady state flow of a single fluid in a porous medium are examined it is argued that incomplete knowledge of the permeability requires the use of a stochastic model of the system it is shown that the problem may be written as a field theory which allows a perturbation series to be expressed by diagrammatic means this allows the calculation of effective permeability the mean pressure and the pressure variance the method as well as recovering familiar results gives a formal means of improving the approximation and approaching more complex systems

geometrical theory of higher order asymptotics of test interval estimator and conditional inference higher order asymptotic powers of one sided and two sided tests both conditional and unconditional are evaluated for a one parameter curved exponential family of distributions by using differential geometrical notions higher order asymptotic powers and the expected size of one sided and two sided interval estimators both conditional and unconditional are also obtained the tests and interval estimators which are third order most powerful at any arbitrary specified one point are explicitly designed with the help of the approximate ancillary statistic the characteristics of the conditional inference are elucidated by proving the equivalence up to the third order of the conditional inference and likelihood ratio inference geometrical notions such as curvatures and angles play a fundamental role in the present theory

nonparametric analysis of competing risks data with event category missing at random summary in competing risks setup the data for each subject consist of the event time censoring indicator and event category however sometimes the information about the event category can be missing as for example in a case when the date of death is known but the cause of death is not available in such situations treating subjects with missing event category as censored leads to the underestimation of the hazard functions we suggest nonparametric estimators for the cumulative cause specific hazards and the cumulative incidence functions which use the nadaraya watson estimator to obtain the contribution of an event with missing category to each of the cause specific hazards we derive the propertied of the proposed estimators optimal bandwidth is determined which minimizes the mean integrated squared errors of the proposed estimators over time the methodology is illustrated using data on lung infections in patients from the united states cystic fibrosis foundation patient registry

an efficient deterministic probabilistic approach to modeling regional groundwater flow 1 theory an efficient probabilistic model is developed and cascaded with a deterministic model for predicting water table elevations in regional aquifers the objective is to quantify model uncertainty where precise estimates of water table elevations may be required the probabilistic model is based on the two point probability method which only requires prior knowledge of uncertain variables mean and coefficient of variation the two point estimate method is theoretically developed and compared with the monte carlo simulation method the results of comparisons using hypothetical determinisitic problems indicate that the two point estimate method is only generally valid for linear problems where the coefficients of variation of uncertain parameters for example storage coefficient and hydraulic conductivity is small the two point estimate method may be applied to slightly nonlinear problems with good results provided coefficients of variation are small in such cases the two point estimate method is much more efficient than the monte carlo method provided the number of uncertain variables is less than eight

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

on second order optimality of the observed fisher information the realized error of an estimate is determined not only by the efficiency of the estimator but also by chance for example suppose that we have observed a bivariate normal vector whose expectation is known to be on a circle then intuitively the longer that vector happens to be the more accurately its angle is likely to be estimated yet this chance though its information is contained in the data cannot be accounted for by the variance of the estimate one way to capture it is by the direct estimation of the realized error in this paper we will demonstrate that the squared error of the maximum likelihood estimate to the extent to which it can be estimated can be most accurately estimated by the inverse of the observed fisher information in relation to this optimality we will also study the properties of several other estimators including the inverse of the expected fisher information the sandwich estimators the jackknife and the bootstrap estimators unlike the observed fisher information these estimators are not optimal

successive improvement of the order of ancillarity summary it is shown by an argument similar to one of welch 1947 that it is under usual regularity conditions in general possible to improve the order of ancillarity successively this means that when an asymptotic ancillary has been constructed such that its distribution is independent of the parameter apart from terms of order o n k then a new ancillary may be constructed with a distribution which depends on the parameter only through terms of order o nk22 it is shown how the new ancillary may be constructed explicitly when the expansion is of the edgeworth type more specifically when it is a normal density multiplied by linear combinations of hermite polynomials

differential geometry of curved exponential families curvatures and information loss the differential geometrical framework is given for analyzing statistical problems related to multi parameter families of distributions the dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint the duality connected by the legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures the second order information loss is calculated for fisher efficient estimators and is decomposed into the sum of two non negative terms one is related to the exponential curvature of the statistical model and the other is related to the mixture curvature of the estimator only the latter term depends on the estimator and vanishes for the maximum likelihood estimator a set of statistics which recover the second order information loss are given the second order efficiency also is obtained the differential geometry of the function space of distributions is discussed

weak and strong uniform consistency of kernel regression estimates we study the estimation of a regression function by the kernel method under mild conditions on the window the bandwidth and the underlying distribution of the bivariate observations x i y i we obtain the weak and strong uniform convergence rates on a bounded interval these results parallel those of silverman 1978 on density estimation and extend those of schuster and yakowitz 1979 and collomb 1979 on regression estimation

first order analysis of uncertainty in numerical models of groundwater flow part 1 mathematical development in part 1 of this paper first and second order analysis of uncertainty is applied to numerical models of groundwater flow the models are cast in state space form with boundary conditions and inputs that are functionally dependent but statistically independent of time using a compact matrix calculus notation first and second order taylor series expansions of the model equations are derived and used to estimate the mean and variance covariance properties of piezometric head predictions given corresponding statistics for aquifer parameters material properties initial conditions boundary conditions and inputs the mathematical results demonstrate that the prediction uncertainty is a function of the magnitude of the parameter uncertainty and sensitivity of the predictions to the parameters furthermore the first order estimate of the piezometric head is identical to the dete ministic result part 2 of this paper to be presented later will illustrate these and other results through numerous applications of the methodology

multiple imputation methods for estimating regression coefficients in the competing risks model with missing cause of failure we propose a method to estimate the regression coefficients in a competing risks model where the cause specific hazard for the cause of interest is related to covariates through a proportional hazards relationship and when cause of failure is missing for some individuals we use multiple imputation procedures to impute missing cause of failure where the probability that a missing cause is the cause of interest may depend on auxiliary covariates and combine the maximum partial likelihood estimators computed from several imputed data sets into an estimator that is consistent and asymptotically normal a consistent estimator for the asymptotic variance is also derived simulation results suggest the relevance of the theory in finite samples results are also illustrated with data from a breast cancer study

an investigation of the validity of first order stochastic dispersion theories in isotropie porous media monte carlo simulations are used to 1 investigate the accuracy of approximations that are implicit in first order stochastic dispersion theories and 2 identify the accuracy limits of first order dispersion theories in isotropic porous media the fickian theory of gelhar and axness 1983 as well as the fickian and non fickian theories of dagan 1984 and neuman and zhang 1990 are investigated all monte carlo simulations are in three dimensions confidence limits of ensemble averaged monte carlo results in isotropic porous media are established for 0 1 y 1 5 these results showed that first order theoretical estimates of the eulerian velocity covariance function are quite accurate for y 1 theoretical estimates of the non fickian longitudinal dispersivity do not deviate significantly from theory for at least y 1 5 theoretical estimation of the transverse dispersivity is limited to y 1 and the fickian longitudinal dispersivity is overestimated by the theory of gelhar and axness 1983 of all first order dispersion theories the theory of dagan 1984 is most robust in estimating the dispersivity tensor

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

likelihood and observed geometries a partir du point de vue de la conditionnalite on etablit un parallele observe a la theorie de chentsov efron amari etc en utilisant l information observee et un tenseur d asymetrie observee

generation of random fields characterized by discrete regions of constant value a new simulator is introduced which provides realizations of parameter fields characterized by randomly sized regions within which the parameter value is constant this generator produces second order stationary random fields with specified mean and correlation structure the structure of the simulated field is significantly different than that which can be obtained by previously existing techniques this new structure provides a valuable simulation tool for discretely varying fields such as required when modeling complex lithologies one application of this method utilizing ovals in two dimensions and ellipsoids in three dimensions is demonstrated for grids in which a finite number of simulation points are to be generated this technique is compared with a recently published turning bands routine and is faster in terms of cpu time for large values of i l where i is the integral scale of the correlation structure and l is the length of the longest side of the simulation grid for small values of i l the turning bands routine is faster conditional simulation based on kriging has been studied for this method the resulting random field varies continuously rather than discretely a second method of conditioning which produces discretely varying random fields is suggested but not examined computationally

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

analysis of flow through heterogeneous random aquifers 2 unsteady flow in confined formations the direct problem of determining the head field in a formation of given stationary isotropic random structure is solved approximately for unsteady flow conditions under restrictive assumptions of sufficiently small conductivity variance and average head gradient slowly varying in time and space closed form expressions are derived for the effective conductivity or storativity and for the head variogram two types of unsteady flow are considered transients and periodic flows in the case of transients it is assumed that initially the head is constant no flow and that ultimately the flow becomes uniform it is found that the effective conductivity and transmissivity are time dependent and drop from the arithmetic mean initially to the steady state value during a relaxation time the latter is much larger for one dimensional flows than for two or three dimensional ones the head variogram also relaxes from the one corresponding to the initial lack of correlation to the steady state variogram during a relaxation time which grows with the lag while in three dimensional structure the relaxation time scales are relatively short and steady state analysis can be used accurately even under transient conditions the tendency to steady state may be very slow in the case of transmissivity because of its large integral scale hence steady state analysis for transients might not be warranted for regional transient flows in the case of periodic flows of seasonal nature the head gradient is assumed to be made up from a steady component and a periodic one if the amplitude of the latter is sufficiently small compared to the steady one steady state value of effective transmissivity and storativity can be adopted the head variogram has also its steady state form for large lags and quasisteady for small lags hence the variogram lies between the steady one steady component of head gradient and the quasisteady one instantaneous head gradient