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

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

uncertainty analysis of transient unsaturated flow in bounded domain 1 this paper analyzes the head variance during transient flows in unsaturated porous media the analysis is based on a first order taylor series expansion of the discretized richards equation and stochastic conceptualization of hydrologic properties in space effects of boundary conditions and flow conditions drying or wetting were examined for different soil models results of the study indicate that a constant flux boundary condition affected the pressure head variance distribution close to the surface originating a decrease of the values at early times during wetting scenarios on the other hand a constant head boundary condition strongly affected the variance profiles during both transient and steady flows also far from the boundary

a matrix version of the wielandt inequality and its applications to statistics abstract suppose that a is an n n positive definite hermitian matrix let x and y be n p and n q matrices respectively such that x y 0 the present article proves the following inequality x ay y ay y ax 1 n 1 n 2 x ax where 1 and n are respectively the largest and smallest eigenvalues of a and m stands for a generalized inverse of m this inequality is an extension of the well known wielandt inequality in which both x and y are vectors the inequality is utilized to obtain some interesting inequalities about covariance matrix and various correlation coefficients including the canonical correlation multiple and simple correlations some applications in parameter estimation are also given

effective permeability and dispersion in locally heterogeneous aquifers the global effective permeability and hydrodynamic dispersion tensors due to steady flow in locally heterogeneous and statistically anisitropic aquifers are analysed using the self consistent approximation for small perturbations the results for the statistically anisotropic case are shown to be consistent with those derived using stochastic continuum theory however for larger perturbations the stochastic continuum theory has not yet produced rigorous results that can be compared with those in this paper

analytical decomposition of the nonlinear unsaturated flow equation an analytical procedure for the solution of the nonlinear unsaturated flow equation is proposed the horizontal and vertical flow equations subject to constant boundary conditions and continuous wetting are approximated using analytical decomposition series the results are verified with the philip 1955 and the parlange 1971 classical solutions and with experimental data a convergence theorem with proof and discussion on the numerical accuracy of decomposition series are included applications to the modeling of infiltration subject to hysteresis in the soil water functional relationships soil heterogeneity and time variable point rainfall are also discussed other potential applications in water management models under more realistic conditions are possible the methodology offers advantages over the common small perturbation solutions the possibility to study nonstationarity in the random quantities statistical separability from the physics of the problem i e independence between the system parameters and the input quantities rather than forced through closure approximations and the construction of an analytical series solution that converges uniformly to the true nonlinear solution finally because of stability and computational economy with respect to linearized numerical solutions an analytical solution appears promising

influence of soil structure and root water uptake strategy on unsaturated flow in heterogeneous media 1 we analyze the combined effects of the spatial variability of soil hydraulic properties and the water uptake by plant roots on unsaturated water flow for this analysis we use a simplified macroscopic root water uptake model which is usually applied only for homogeneous or layered soil and therefore we also determine whether it is applicable for multidimensional heterogeneous media analytical solutions for mean and variance of pressure head first order second moment approximations in layered media and numerical solutions of two dimensional 2 d autocorrelated multi gaussian and non multi gaussian parameter fields are analyzed for steady state and transient flow conditions for non gaussian topological features that have little influence on the mean and the variance of the pressure field if root water uptake is ignored we test whether the influence is significant if root water uptake is accounted for the results reveal that in field structures with large patches of coarse material local regions with pressure head values at the wilting point develop these are surrounded by wet material without a compensation mechanism for local stress the global transpiration demand is not met if local wilting occurs various compensation mechanisms are tested that depend respectively on the saturation the relative conductivity or a strategy where the deficit in the global uptake rate is equally distributed to unstressed locations the strategies lead to a global actual transpiration rate at the potential value and attenuate the formation of locally wilted areas wilted regions can however still occur and may be an artifact of the simplified model concept as root soil interactions are neglected therefore simplified macroscopic models for root water uptake should be used with caution in heterogeneous media

application of the finite difference heterogeneous multiscale method to the richards equation 1 this paper extends the finite difference heterogeneous multiscale method fdhmm to simulate transient unsaturated water flow problems in random porous media the numerical method is based on the use of two different schemes for the original equation at different grid levels which allows numerical results at a lower cost than solving the original equations the main feature of fdhmm is that the necessary data for the macroscopic model are supplied by solving the microscopic model on a sparse spatial domain the generated code is verified by applying the linearization model of the richards equation considering two different constitutive relationships this method is applied to several test examples with different soil textures and boundary conditions both the dirichlet and the periodic boundary conditions are considered for solving the local microscopic model when the water flow in heterogeneous unsaturated soils is simulated by fdhmm the numerical experiments demonstrate that fdhmm can effectively simulate the transient unsaturated water flow in the specific soils the numerical experiments also demonstrate that fdhmm can achieve accurate global mass balance and is a globally convergent algorithm and the spatial correlation length of random coefficients under the specific standard deviation has relatively little influence on the accuracy of the method

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

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

stochastic modeling of large scale flow in heterogeneous unsaturated soils a simulation experiment is used to test the stochastic theory of unsaturated flow proposed by mantoglou and gelhar 1987a b c tension means and variances derived from this theory are compared to tension distributions obtained from a detailed three dimensional model developed by ababou 1988 the synthetically generated unsaturated hydraulic conductivity functions used in the detailed model vary significantly over relatively small distances results from a simulated infiltration event indicate that the stochastic theory s predictions reproduce the overall trend of the heterogeneous tension distribution deviations from the predicted mean generally lie within confidence intervals derived from the predicted variance the stochastic theory predicts less vertical moisture movement and somewhat more horizontal spreading than a comparable deterministic analysis this reflects the fact that the effective hydraulic conductivity function used in the stochastic approach is anisotropic the magnitude of this anisotropy increases with increasing tension

normal forms symmetry and linearization of dynamical systems we discuss how the presence of a suitable symmetry can guarantee the perturbative linearizability of a dynamical system or a parameter dependent family via the poincare normal form approach we discuss this formally at first and later pay attention to the convergence of the linearizing procedure we also discuss some generalizations of our main result

uncertainty propagation with numerical models for flow and transport in the unsaturated zone uncertainty propagation with numerical models for flow and transport in the unsaturated zone is studied using vector state space methods the output uncertainty is the result of spatially randomly varying parameters describing the soil hydraulic properties an efficient state covariance propagation method is proposed based on first order linearization of the system dynamics the approach has been previously used with groundwater flow models and this paper contributes an original generalization for unsaturated flow including solute transport comparisons are made with monte carlo simulations of infiltration in two soils with distinctly different soil properties for coefficients of variation of the parameters less than 0 3 the two methods agree well the linearization method is found to be faster due to its analytical nature the results are a promising first step for future research on parameter estimation and optimization using the state space linearization approach

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

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

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

dual homogenization of immiscible steady two phase flows in random porous media 1 we develop a dual conductivity resistivity homogenization for immiscible steady two phase flow in heterogeneous spatially correlated random porous media the objective is to obtain macroscale equations for an equivalent homogeneous medium we assume that the flow is governed locally by the generalized darcy equations two formulations are developed one in terms of capillary pressure pc and the other in terms of saturation s in each case the perturbation approach is used to express average and fluctuation equations and the closure problems are solved using the wiener kinchine spectral theory the macroscale permeability kea and resistivity rea tensors are determined for each fluid a which also yield the relative permeabilities these dual results provide a physically consistent correction of the perturbation based tensors the corrected macroscale tensors are then always positive symmetric and reciprocal to each other we also develop similarly the homogenization of the capillary pressure curve pc x and of its reciprocal x pc where is the wetting fluid content finally these general analytical results are applied to the special case where pc 0 and or 0 which can be interpreted as a mean capillary equilibrium

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

an efficient method for simulating steady unsaturated flow in random porous media using an analytical perturbation solution as initial guess to a numerical model numerical simulation of flow through multidimensional heterogeneous soils under unsaturated conditions is a computationally intensive task the governing unsaturated flow equation is nonlinear the degree of nonlinearity depends on the unsaturated hydraulic properties of the soil and the degree of heterogeneity in this paper we present a highly efficient method to simulate unsaturated steady state flow through random porous media our method utilizes the analytical approximate solution derived by a perturbation spectral method as an initial guess solution for a numerical model to simulate two dimensional vertical infiltration problems it is found that this approach which we call asigning reduces the required cpu time by one to two orders of magnitude asigning is demonstrated to operate successfully under a wide variety of boundary conditions which may substantially deviate from those imposed on the initial guess solution a large range of mean and variances in the independent variables ln ks and or alternatively in has been tested and it is shown that the method works well for variances of the unsaturated hydraulic conductivity ln2k 5 and average 0 1 cm 1

characteristics and controlling factors of dispersion in bounded heterogeneous porous media 1 in this study we present first order in terms of the log conductivity variance analytical solutions to displacement covariances xij and macrodispersion coefficients dij for transport of conservative solutes in two dimensional bounded heterogeneous porous media these solutions are presented as infinity series and are explicit functions of the statistics of the log hydraulic conductivity using numerical examples the convergence of these solutions in terms of the number of terms in truncated finite summations is first investigated and the accuracy of these solutions is examined by comparing them with results from monte carlo simulations and the semianalytical solutions of osnes 1998 through several examples the general features of time dependent xij and dij are discussed unlike in unbounded domains the longitudinal macrodispersivity d11 for bounded domains does not approach an asymptotic value at large dimensionless time but instead increases quickly near the downstream constant head boundary in addition d11 for bounded domains is always larger than that in unbounded domains and accordingly the transverse macrodispersivity d22 in bounded domains is smaller than that in unbounded domains these differences stem from the lateral no flow boundaries in our bounded domain models we also investigated the effect of domain sizes on xij and dij our study shows that both xij and dij depend not only on the dimensionless size of the domain but also on its aspect ratio the dependence of xij and dij on the particles initial location has also been investigated at detail our results indicate that while this dependence is very small for x11 and d11 the release location has a significant impact on both x22 and d22 our solutions to both displacement covariances and macrodispersivity are compared against those derived from tracer test data at the borden site the comparison shows that our solutions are quite close to observed data indicating that they may be applicable to predict solute transport at the field scale

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

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

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

an inverse model for three dimensional flow in variably saturated porous media geostatistical theory has shown promise in dealing with issues of stability uniqueness and identity of estimates inherent in inverse problems of subsurface flow here the geostatistical method is extended to three dimensional unsteady flow in variably saturated porous geological media the vadose zone that are modeled using the richards equation and the van genuchten mualem constitutive equations the saturated hydraulic conductivity and n parameters of this relationship are treated as spatially correlated statistically independent stochastic processes for representing heterogeneity of porous media for given covariance functions of the parameters the adjoint state sensitivity method is used to calculate first order approximations for covariances of capillary pressure and moisture content and cross covariances between capillary pressure moisture content and the hydraulic properties these covariances and cross covariances are then used in a successive linear estimator sle to estimate the conditional means of the heterogeneous hydraulic property fields based on measurements of pressure and moisture content data a sequential conditioning approach for our sle was also applied to data sets collected at different sampling times during a transient infiltration event this approach has the benefit of reducing the size of the matrices and so helps avoid numerical stability problems on the basis of our study pressure and moisture content data sets collected at later times of an infiltration event or during steady state flow were found to provide better estimates smaller mean square error compared to the true field of the hydrological parameters of the vadose zone than data from very early times

some further matrix extensions of the cauchy schwarz and kantorovich inequalities with some statistical applications abstract the well known cauchy schwarz and kantorovich inequalities may be expressed in terms of vectors and a positive definite matrix we consider what happens to these inequalities when the vectors are replaced by matrices the positive definite matrix is allowed to be positive semidefinite singular and the usual inequalities are replaced by lowner partial orderings some examples in the context of linear statistical models are presented

a combined parameter scaling and inverse technique to upscale the unsaturated hydraulic parameters for heterogeneous soils 1 determining a large number of soil hydraulic parameters for heterogeneous soils remains a challenge because inverting for too many parameters can lead to parameter values that are nonunique furthermore such inversions may need very long simulation times for example months or more when inverting field scale problems in this research a combined parameter scaling and inverse technique cpsit is proposed to upscale hydraulic parameters from the local scale to the field scale the cpsit approach includes two steps 1 parameter scaling and 2 inverse modeling in step 1 the number of parameters to be estimated at field scale fs is reduced by applying parameter scaling whereby a heterogeneous soil is treated as a composition of multiple equivalent homogeneous media ehms in step 2 the fs parameters for the reference ehm are determined using the inverse technique and observations from well designed field experiments the advantages of the cpsit approach are that the number of parameters to be inverted is reduced by a factor of the number m of ehms and the simulation time is reduced by a factor of about m2 the cpsit approach was tested by upscaling the hydraulic parameters using a field injection experiment at the hanford site results show that when the cpsit upscaled parameters were used to simulated flow the mean squared residual was reduced by 83 2 relative to that when the local scale parameters were used

stochastic analysis of field measured unsaturated hydraulic conductivity unsaturated hydraulic conductivity k values as a function of soil water pressure head h were measured in the soil at 75 cm depth at 70 different sites separated from one another by a distance of l m along a horizontal transect k field was viewed as a random function of spatial location x field data were analyzed 1 to examine the isotropy and stationarity of k 2 to check the ergodicity of k in the mean and covariance functions and 3 to characterize the distribution properties of k by estimating the higher order correlations that is third and fourth cumulants the mean functions were estimated by averaging over h and x the covariance function was studied to investigate its spatial origin dependency logs and square roots of k were used for estimating the third and fourth cumulants results showed that spatial covariance functions are anisotropic and both lag and origin dependent that is spatially nonhomogeneous because the stationarity statistical homogeneity of k is scale dependent which was indicated by the identification of locally stationary covariance regions the ergodic properties of k are also scale dependent at smaller spatial scales results related to the distribution characteristics of k indicated that although ln k is marginally gaussian distributed in the context of spatial stochastic processes the random field of ln k is not gaussian because the third and fourth cumulants of the field are still significantly different from zero and have the same order of magnitude as the first and second cumulants the square root transformation however resulted in a random field that is approximately gaussian although marginal distributions of k remained skewed analyses of ln k and k indicated that better transformations which would result in both marginal and joint gaussian behavior for the random field of k are needed

an iterative geostatistical inverse method for steady flow in the vadose zone an iterative geostatistical inverse approach is developed to estimate conditional effective unsaturated hydraulic conductivity parameters soil water pressure head and degree of saturation in heterogeneous vadose zones this approach is similar to the classical cokriging technique and it uses a linear estimator that depends on covariances and cross covariances of unsaturated hydraulic parameters soil water pressure head and degree of saturation the linear estimator is however improved successively by solving the governing flow equation and by updating the residual covariance and cross covariance functions in an iterative manner as a result the nonlinear relationship between unsaturated hydraulic conductivity parameters and head is incorporated in the estimation and the estimated fields are approximate conditional means the ability of the iterative approach is demonstrated through some numerical examples

stochastic analysis of three phase flow in heterogeneous porous media 1 spectral perturbation approach stochastic analysis of steady state three phase water oil and air flow in heterogeneous porous media was performed using the perturbation theory and spectral representation technique the governing equations describing flow of water oil and air are coupled and nonlinear the key stochastic input variables are intrinsic permeability k and the soil grain size distribution index three different stochastic combinations of these two input parameters were considered in the analysis the spectral perturbation approach was used to develop closed form expressions to describe stochastic variability of key output processes such as capillary and individual phase pressures the second paper of this two part series abdin and kaluarachchi this issue will discuss the applicability and the validity of the spectral perturbation technique to study the stochastic behavior of three phase flow using numerical simulations

bayesian inversion of mualem van genuchten parameters in a multilayer soil profile a data driven assumption free likelihood function this paper introduces a hierarchical simulation and modeling framework that allows for inference and validation of the likelihood function in bayesian inversion of vadose zone hydraulic properties the likelihood function or its analogs objective functions and likelihood measures are commonly assumed to be multivariate gaussian in form however this assumption is not possible to verify without a hierarchical simulation and modeling framework in this paper we present the necessary statistical mechanisms for utilizing the hierarchical framework we apply the hierarchical framework to the inversion of the vadose zone hydraulic properties within a multilayer soil profile conditioned on moisture content observations collected in the uppermost four layers the key result of our work is that the goodness of fit validated likelihood function form provides empirical justification for the assumption of multivariate gaussian likelihood functions in past and future inversions at similar sites as an alternative the likelihood function need not be assumed to follow a parametric statistical distribution and can be computed directly using nonparametric methods the nonparametric methods are considerably more computationally demanding and to demonstrate this approach we present a smaller dimension synthetic case study of evaporation from a soil column the main drawback of our work is the increased computational expense of the inversion

sensitivity analysis of flow in unsaturated heterogeneous porous media theory numerical model and its verification sensitivity analysis is one of the tools available for analyzing the effects of parameter uncertainty and soil heterogeneity on the transport of moisture in the unsaturated similar porous media direct differentiation of the discretized richards equation with respect to parameters defining spatial variability leads to linear systems of equations for elementary sensitivities that are readily solved in conjunction with the original equation these elementary sensitivities can be easily transformed into approximations of functional sensitivities and into sensitivities of boundary fluxes a numerical implementation of this technique in one space dimension yields results that are consistent with exact analytical solutions and with numerical perturbation calculations the effects of a given heterogeneity can be modeled adequately provided that the maximum relative change of the scale factor from one grid point to the next not exceed a number on the order of unity

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

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

the quasilinear approximation for unsaturated porous media flow this paper gives a review of the quasilinear approximation an exponential relationship between hydraulic conductivity and moisture potential and the uses this has been put to since the time of its creation one of the major attractions of this approximation is that under steady regimes the highly nonlinear richards equation governing unsaturated flow can be linearized this simplification has enabled analytic and semianalytic solutions to be found for many unsaturated flow problems these are reviewed and the current research areas in which the quasilinear approximation is being used are discussed

state dependent anisotropy comparisons of quasi analytical solutions with stochastic results for steady gravity drainage anisotropy in large scale unsaturated hydraulic conductivity of layered soils changes with the moisture state here state dependent anisotropy is computed under conditions of large scale gravity drainage soils represented by gardner s exponential function are perfectly stratified periodic and inclined analytical integration of darcy s law across each layer results in a system of nonlinear equations that is solved iteratively for capillary suction at layer interfaces and for the darcy flux normal to layering computed fluxes and suction profiles are used to determine both upscaled hydraulic conductivity in the principal directions and the corresponding state dependent anisotropy ratio as functions of the mean suction three groups of layered soils are analyzed and compared with independent predictions from the stochastic results of yeh et al 1985b the small perturbation approach predicts appropriate behaviors for anisotropy under nonarid conditions however the stochastic results are limited to moderate values of mean suction this limitation is linked to a taylor series approximation in terms of a group of statistical and geometric parameters two alternative forms of the taylor series provide upper and lower bounds for the state dependent anisotropy of relatively dry soils

stochastic analysis of the velocity covariance and the displacement covariance tensors in partially saturated heterogeneous anisotropic porous formations considering steady state unsaturated flow in an unbounded domain of heterogeneous formation with three dimensional anisotropic structure first order approximations of the principal and off diagonal components of the velocity covariance tensor were derived for the general case in which the separation vector is inclined to the mean gradient vector j at arbitrary angles taking into account the texture of the soil materials constituting the heterogeneous formation the derived velocity covariance tensors were employed for modeling transport in the vadose zone for the general case in which j is inclined to the principal axes of the formation heterogeneity at arbitrary angles by the use of a lagrangian formulation results of this study suggest that under unsaturated flow for given statistics of the formation properties and given water saturation macrodispersion will diminish in formations of coarser texture when j does not coincide with the principal axes of the heterogeneous formation in this case and in formations of coarser texture the principal axes associated with the principal components of the dimensionless macrodispersion tensor d ij may be deflected in a direction opposite to that of j relative to the principal axes of the formation heterogeneity coarser texture may compensate for the enhanced lateral dispersion and the enhanced deflection of the principal axes of the principal components of d ij due to an increase in the statistical anisotropy of the heterogeneous formation for a formation of given statistics and texture and for a given the longitudinal component of d ij decreases while its transverse components increase when j is more inclined to the longitudinal axis of the formation heterogeneity the latter situation is pertinent to periods of redistribution between successive water applications in which increasing capillary pressure gradients combined with spatial heterogeneity in the formation properties may increase the transverse components of j

determining soil hydraulic properties by parameter estimation on the selection of a model for the hydraulic properties parameter estimation procedures involving solution of the inverse flow problem pertinent to a transient experiment constitute a powerful method of determining the soil hydraulic properties one of the problems associated with these procedures is the selection of an appropriate model to describe the soil hydraulic properties three different models were considered those of van genuchten vg and brooks and corey bc model and the exponential model of gardner for the hydraulic conductivity soil water pressure relationships coupled with a new expression for the soil water content soil water pressure relationship gr model for a given model the soil hydraulic properties were determined from simulated and measured outflow experiments supplemented with water content at a soil water pressure head of 15 000 cm h2o using the parameter estimation procedure of kool et al 1985a and data from two soils a hypothetical sandy loam soil assumed to be described by the vg model and a silt loam soil model validation tests were performed and the most appropriate model was selected from the candidate models by discrimination tests using the akaike information criterion aic in the case of the hypothetical sandy loam the vg model was found to be the most accurate and most consistent with the data the performance of both the bc and the gr models in terms of the aic was less good but very similar to each other in the case of the silt loam soil the vg model with additional parameter m which accounts for the correlation between pores and for the flow path tortuosity m 2 02 rather than a constant m 0 5 was found to be most accurate and most consistent with the data the performance of the bc and the gr models was similar when m was considered as an unknown parameter instead of a constant

analyses of infiltration events in relation to determining soil hydraulic properties by inverse problem methodology at the field scale because of the relatively large number of observation points required to estimate the spatial distributions of hydraulic properties application of inversion procedures must be based on a characteristic of the flow that for a given point in the field can be measured with relative ease and can be used as input for the inversion procedure suggesting the use of infiltration data for this purpose and recognizing the fact that a basic requirement for the design of a transient flow experiment is that the resulting inverse problem be sufficiently well posed to allow solution this paper addresses the question under which circumstances the resulting inverse problem is well posed utilizing the concepts of identifiability uniqueness and stability for this purpose three different models of the soil hydraulic properties were analyzed the main conclusion of this study is that when infiltration data measured at the darcy scale are used as input for the inversion procedure the inclusion of prior information on a single measurable parameter the saturated conductivity ks in the estimation criterion will enhance the likelihood of uniqueness and stability of the inverse solution provided that the structure of the hydraulic model is sufficiently simple however since for a particular situation it is impossible to determine a priori whether the resultant inverse problem is well posed or not this must be carried out only a posteriori by solving the problem several times with different initial parameter estimates accompanied by an analysis of the associated estimation errors

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

approximation of asymptotic dispersivity of conservative solute in unsaturated heterogeneous media with steady state flow the relation between the heterogeneity of hydraulic properties and effective asymptotic transport is studied for microscopically heterogeneous but macroscopically homogeneous unsaturated media with steady flow heterogeneity is described by the scaling of the hydraulic functions m and k the study is based on an analytical approximation of asymptotic dispersivity under the assumption that matric potential is spatially constant for the special case of a water saturated medium the result of the stochastic continuum theory is recovered when applied to several published numerical simulations the approximated asymptotic dispersivities are found to agree well with the numerical values by exploring the heterogeneous media for various hydraulic states the approximation resolves some apparent inconsistencies found in the simulations the cases considered span the entire range from perfect to zero correlation between scaling factors of matric potential and hydraulic conductivity it is demonstrated that this correlation is crucial for the behavior of asymptotic dispersivity with changing flow rate since weak to moderate correlations are often found in soils this result has significant implications for solute transport through heterogeneous soils

stochastic analysis of unsaturated flow one dimensional monte carlo simulations and comparisons with spectral perturbation analysis and field observations a numerical experiment was designed to study the stochastic behavior of one dimensional transient unsaturated flow in a monte carlo setting soil hydraulic properties log saturated hydraulic conductivity ln ks pore size distribution parameter and the specific water capacity c are assumed to be statistically homogeneous random fields described by exponential correlation functions with identical correlation lengths fifty realizations of each soil hydraulic property with statistical properties obtained from a field experiment are generated by using a nearest neighbor model numerical solutions of the one dimensional flow equation are used repeatedly to obtain realizations of soil water pressure head and flux q corresponding to the realizations of in ks and c the dependence of and q on the statistical properties of ln ks and c and the magnitude of the hydraulic head gradient g prevailing at the lower boundary are investigated in addition results of monte carlo analysis and spectral perturbation analyses are compared with field observations the greatest variability of and q are observed when soil hydraulic properties are uncorrelated and have large variances and integral scales and when g at the lower boundary is unity the monte carlo and spectral perturbation analyses tend to agree reasonably well for the flow domains in which ergodicity of local soil hydraulic properties is assured results of the monte carlo and spectral perturbation analyses are also supported by field observations

adaptive neural output feedback control of output constrained nonlinear systems with unknown output nonlinearity this paper addresses the problem of adaptive neural output feedback control for a class of special nonlinear systems with the hysteretic output mechanism and the unmeasured states a modified bouc wen model is first employed to capture the output hysteresis phenomenon in the design procedure for its fusion with the neural networks and the nussbaum type function two key lemmas are established using some extended properties of this model to avoid the bad system performance caused by the output nonlinearity a barrier lyapunov function technique is introduced to guarantee the prescribed constraint of the tracking error in addition a robust filtering method is designed to cancel the restriction that all the system states require to be measured based on the lyapunov synthesis a new neural adaptive controller is constructed to guarantee the prescribed convergence of the tracking error and the semiglobal uniform ultimate boundedness of all the signals in the closed loop system simulations are implemented to evaluate the performance of the proposed neural control algorithm in this paper

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

unsaturated flow in spatially variable fields 2 application of water flow models to various fields the method of modeling water flow during infiltration and redistribution formulated in part 1 of this study has been applied to compute expectations and variances of a few water flow variables and of effective hydraulic properties two spatially variable soils with different degrees of variation have been investigated the expectations and variances are obtained by using the statistical averaging procedure and probability density function pdf of saturated hydraulic conductivity ks the stationarity hypotheses and the requirement that the integral scale of ks is much smaller than the length scale characterizing the field in the x y plane have been adopted an approximate solution which assumes the concept of a wetting front and uniform water content is used for the statistical averaging procedure a comparison of these results with data computed by a more accurate numerical solution to richard s equation shows that the approximate simplified models lead to a quite accurate value of the expectations and variances of the flow variables when the field is sufficiently heterogeneous it is suggested that in spatially variable fields stochastic modeling represents the actual flow phenomena more realistically and provides the main statistical moments mean variances by using simplified flow models which can be used with confidence in applications the field effective hydraulic properties have been defined and derived by using approximate models it is shown that effective properties may be meaningful only under very restricted and special conditions such as steady gravitational flow they do not exist in the general case of infiltration redistribution it is concluded that the traditional deterministic approach for solving the flow equations cannot be justified for solving flow problems in spatially variable fields

a unified version of cauchy schwarz and wielandt inequalities abstract let a be an n n positive definite symmetric real matrix with n eigenvalues 1 2 n and let x and y be two n 1 vectors with the angle this paper proves the following inequality x t ay 2 max i j i cos 2 2 j sin 2 2 i cos 2 2 j sin 2 2 2 x t ax y t ay it is a unified version of the cauchy schwarz inequality and the wielandt one

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

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

infiltration in stratified heterogeneous soils relative importance of parameters and model variations 1 this study presents a framework that accounts for the uncertainty relative importance and relative contribution of uncertain and spatially variable parameters and the influence of statistical model assumptions in problems of infiltration in unsaturated heterogeneous formations the relative importance of the saturated hydraulic conductivity k s the van genuchten and 3 parameters was quantified toward the mean pressure head profile and the fluctuations about it the broad implications of our study for site characterizations efforts are the following statistical moments were seen not to be sufficient for producing reliable modeling estimates significant information on the detailed shape of the probability distribution is required in order to produce meaningful predictions when limited data are available optimum allocation of resources is accomplished by concentrating on k s when more extensive sampling can be done both k s and appear to be critical total system performance analyses may not provide the most appropriate modeling estimates the results of the relative importance of the parameters for a specific layer did not coincide with the conclusions drawn from an analysis of the system as a whole

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

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

error analyses of simplified unsaturated flow models under large uncertainty in hydraulic properties the functional relationships between soil hydraulic properties k and are needed for analyzing unsaturated water flow in soils and are generally of a nonlinear form these relationships estimated from field measured data usually show large variability because of measurement errors and natural spatial heterogeneity exhibited by many field soils a probabilistic approach is often employed to account for this uncertainty in parameters whereby the coefficients in the resulting partial differential equation are treated as random quantities usually characterized by their moments the objective of this analysis is to evaluate the errors resulting when simplified solutions based on simple forms of the hydraulic properties are used rather than nonlinear and numerically complicated models as an example of a parametric functional relationship for k we adopt the brooks corey formulation in which a single parameter is considered as random to emulate the spread encountered in practice for data that show a large enough spread and when one is interested in the mean behavior of k or simple flow models such as the green ampt model and the linear model are proposed as alternatives to the more accurate but numerically demanding richards equation to compare models scaled error definitions are introduced and are expressed as a percentage of the saturated hydraulic conductivity of the soil these time dependent errors are then computed under various boundary conditions in general the green ampt solution is found to yield smaller errors than the linear model when compared with solutions from richards equation it is concluded that the use of simplified models is justified when field data exhibit sufficient spread and a tight bound cannot be placed on the parameter estimates

solute spreading in nonstationary flows in bounded heterogeneous unsaturated saturated media 1 it is commonly assumed in stochastic solute advective transport models that either the velocity field is stationary statistically homogeneous or the mean flow is unidirectional in this study using a lagrangian approach we develop a general stochastic model for transport in variably saturated flow in randomly heterogeneous porous media the mean flow in the model is multidirectional and the velocity field can be nonstationary with location dependent statistics the nonstationarity of the velocity field may be caused by statistical nonhomogeneity of medium properties or complex boundary configurations the particle s mean position is determined using the mean lagrangian velocity particle spreading the displacement covariances is expressed in terms of the state transition matrix that satisfies a time varying dynamic equation whose coefficient matrix is the derivative of the mean lagrangian velocity field in the special cases of stationary velocity fields the transition matrix becomes the identical matrix and our model reduces to the well known model of dagan 1984 for nonstationary but unidirectional flow fields our model reduces to that of butera and tanda 1999 and sun and zhang 2000 the validity of the transport model is examined by comparisons with monte carlo simulations for the following three cases transport in a mean gravity dominated flow in an unsaturated flow with a water table boundary and in a saturated unsaturated flow an excellent agreement is found between our model results and those from monte carlo simulations

stochastic analysis of unsaturated flow with probabilistic collocation method 1 in this study we present an efficient approach called the probabilistic collocation method pcm for uncertainty analysis of flow in unsaturated zones in which the constitutive relationship between the pressure head and the unsaturated conductivity is assumed to follow the van genuchten mualem model spatial variability of soil parameters leads to uncertainty in predicting flow behaviors the aim is to quantify the uncertainty associated with flow quantities such as the pressure head and the effective saturation in the proposed approach input random fields i e the soil parameters are represented via the karhunen loeve expansion and the flow quantities are expressed by polynomial chaos expansions pces the coefficients in the pces are determined by solving the equations for a set of carefully selected collocation points in the probability space to illustrate this approach we use two dimensional examples with different input variances and correlation scales and under steady state and transient conditions we also demonstrate how to deal with multiple input random parameters to validate the pcm we compare the resulting mean and variance of the flow quantities with those from monte carlo mc simulations the comparison reveals that the pcm can accurately estimate the flow statistics with a much smaller computational effort than the mc

upscaling for unsaturated flow for non gaussian heterogeneous porous media 1 large scale models of transient flow processes in the unsaturated zone require in general upscaling of the flow problem in order to capture the impact of heterogeneities on a small scale which cannot be resolved by the model effective parameters for the upscaled models are often derived from second order stochastic properties of the parameter fields such properties are good quantifications for parameter fields which are multi gaussian however the structure of soil does rarely resemble these kinds of fields the non multi gaussian field properties can lead to strong discrepancies between predictions of upscaled models and the averaged real flow process in particular the connected paths of parameter ranges of the medium are important features which are usually not taken into account in stochastic approaches they are determined here by the euler number of one cut indicator fields methods to predict effective parameters are needed that incorporate this type of information we discuss different simple and fast approaches for estimating the effective parameter for upscaled models of slow transient flow processes in the unsaturated zone where connected paths of the material may be taken into account upscaled models are derived with the assumption of capillary equilibrium the effective parameters are calculated using effective media approaches we also discuss the limits of the applicability of these methods

stochastic models of subsurface flow infinite versus finite domains and stationarity stochastic solutions of the differential equation describing one dimensional flow through a porous medium with spatially variable hydraulic conductivity which is represented by a stationary statistically homogeneous random process are developed using several techniques the analytical approximations using first order analysis propagation of error covariance differential equations and variogram analysis all yield consistent results which demonstrate the important effects of boundary conditions and conditioning using the theory of intrinsic random functions stochastic solutions are developed for the case when the logarithm of the hydraulic conductivity is a three dimensional stationary random field in contrast to the one dimensional case it is found that the resulting three dimensional head perturbation will be locally stationary under very general conditions results from the one dimensional analytical solutions are found to be in agreement with previous monte carlo simulations for a flow system of finite length the solution based on linearization in the logarithm of hydraulic conductivity proved to be very robust showing reasonable agreement with monte carlo results even for the largest input standard deviation of lnk 4 6

numerical homogenization of the richards equation for unsaturated water flow through heterogeneous soils homogenized equations and the corresponding effective constitutive relations are generally necessary for numerically modeling large scale unsaturated flow processes in soils recently based on the kirchhoff transformation and the two scale convergence theory a homogenization method for the richards equation with the mualem van genuchten model has been proposed with a constant model parameter relating to the inverse of the air entry pressure and the soil pore size distribution the homogenized model is computationally efficient and convenient to use because of its explicit expression in this study we generalize this method allowing to be a spatially distributed random field and proposing a homogenized richards equation in the mixed form h under the condition that the effective hydraulic conductivity tensor is diagonal this generalization eliminates the limitation of a constant in practical applications the proposed homogenized model is meaningful in most situations because the flow problems are influenced mainly by the diagonal terms of conductivity and the off diagonal terms are often neglected two dimensional numerical tests are conducted in soil profiles with different degrees of spatial heterogeneity structure to illustrate that the homogenized model can capture the fine scale flow behaviors on coarse grids effectively homogenization for the richards equation with other two commonly used constitutive relations the brooks corey model and the gardner russo model is also illustrated in this study