convergent radial dispersion a note on evaluation of the laplace transform solution a numerical inversion algorithm for laplace transforms that is capable of handling rapid changes in the computed function is applied to the laplace transform solution to the problem of convergent radial dispersion in a homogeneous aquifer prior attempts by the author to invert this solution were unsuccessful for highly advective systems where the peclet number was relatively large the algorithm used in this note allows for rapid and accurate inversion of the solution for all peclet numbers of practical interest and beyond dimensionless breakthrough curves are illustrated for tracer input in the form of a step function a dirac impulse or a rectangular input

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

a new class of truly consistent splitting schemes for incompressible flows a new class of splitting schemes for incompressible flows is introduced the new schemes are based on a weak form of the pressure poisson equation and at each time step only require to solve a set of helmholtz type equations for the velocity and a poisson equation in the weak form for the pressure just as pressure correction and velocity correction schemes however unlike pressure correction and velocity correction schemes the new splitting schemes are free of splitting errors and deliver full accuracy on the vorticity and the pressure

solute dispersion in multidimensional periodic saturated porous media suppose in a convection dispersion equation governing solute movement in a saturated porous medium of infinite extent the convection velocity components are periodic functions of spatial coordinates then it follows from a general mathematical result that the solute concentration can be asymptotically approximated by a gaussian density two theoretical examples with and without a constant vertical velocity are given to illustrate an application of this mathematical result to solute dispersion in a parallel bedded three dimensional aquifer of infinite extent analytical expressions are obtained for dispersion coefficients in the asymptotic gaussian approximations in the two examples the dispersion coefficients functionally depend on a velocity scaling parameter u0 and a spatial scaling parameter period a via their product au0 and this functional dependence on au0 is different in the two examples the expressions for dispersion coefficients are formally contrasted and compared with those obtained previously by other authors in a parallel bedded infinite aquifer in which the convection velocity is a random field some physical implications of dependence of dispersion coefficients on a spatial scaling parameter i e the scale effect are discussed in the context of modeling in an aquifer with evolving heterogeneities this interpretation of scale effect in dispersion coefficients is contrasted with the preasymptotic interpretation in the current literature

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

nonasymptotic behavior of a common eulerian approximation for transport in random velocity fields lagrangian methods of analyzing solute transport in random velocity fields deal with the motion of indivisible solute particles through such fields whereas eulerian methods treat the solute concentration c as the dependent variable in a stochastic mass balance equation both approaches lead to nonlinear stochastic differential equations which are difficult to solve analytically without approximation a particular eulerian approximation which has become popular in recent years derives from the premise that in mildly fluctuating velocity fields terms involving the products of fluctuating quantities can be disregarded in comparison to terms which involve only one such quantity this leads to a non fickian low order approximation for the ensemble mean concentration c where the dispersive flux is not proportional to c but is given instead by a convolution integral the spatial moments of c based on this approximation are shown to be in conflict with those obtained from a first order lagrangian analysis we explore this apparent contradiction and conclude that the reason lies in the nonasymptotic nature of the eulerian approximation whereby terms neglected are of the same order as terms retained

assessment of discretized errors and adaptive refinement with quadrilateral finite elements this paper studies discretized errors and their estimation in conjunction with quadrilateral finite element meshes which are generated by the intelligent mesh generator xformq 1 the exact energy error is used to evaluate the distortion effect of the quadrilateral mesh the zienkiewicz zhu2 error estimate and actaptive procedure are applied to the short cantilever and the square plate problems using the quadrilateral mesh generator xformq it is shown that the multistage quadrilateral element refinement produces results superior to the triangular element refinement in the test cases

steady state groundwater flow simulation with imprecise parameters a methodology based on fuzzy set theory is developed to incorporate imprecise parameters into steady state groundwater flow models in this case fuzzy numbers are used to represent parameter imprecision as such they are also used as a measure for the uncertainty associated with the hydraulic heads due to the imprecision in the input parameters the imprecise input parameters may come from indirect measurements subjective interpretation and expert judgment of available information in the methodology a finite difference method is combined with level set operations to formulate the fuzzy groundwater flow model this fuzzy modeling technique can handle imprecise parameters in a direct way without generating a large number of realizations two numerical solution methods are used to solve the fuzzy groundwater flow model the groundwater model operator method proposed in this methodology and the iterative algorithm based on conventional interval arithmetics the iterative method is simple but may overestimate the uncertainty of hydraulic heads the groundwater model operator method not only provides the hull of the solution set for the hydraulic heads but also considers the dependence of hydraulic head coefficients which are functions of imprecise parameters sensitivity analysis shows that the dependence of hydraulic head coefficients has a critical impact on the model results and neglecting this dependence may result in significant overestimation of the uncertainty of hydraulic heads a numerical model based on the methodology is tested by comparing it with the analytical solution for a homogeneous radial flow problem it is also applied to a simplified two dimensional heterogeneous flow case to demonstrate the methodology

a rotational velocity correction projection method for unsteady incompressible magnetohydrodynamics equations abstract this paper provides a rotational velocity correction projection method for the unsteady incompressible magnetohydrodynamics mhd equations in this method the electrical field variable e was eliminated in this way the gauss s law can be conserved the velocity variable u and the pressure field p are solved by a rotational velocity correction projection method the optimal error estimates of the l 2 norm will be given finally we shall present some numerical results to confirm our analysis it shows that our method is effective and can conserve the gauss s law very well

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

stochastic modeling of macrodispersion in heterogeneous porous media a method for the prediction of dispersion processes of inert solutes occurring in heterogeneous porous media is presented its main features are as follows 1 the velocity field is assumed to be a space random function its moments are expressed through the physical parameters of the log transmissivity and head random fields by linearizing the flow equation the random velocity being a linear function of the head and log transmissivity which are assumed to be jointly multivariate normal mvn is thus also mvn its pdf is completely defined by its first two moments 2 the statistics of the dispersion process are obtained by the particle tracking method through monte carlo simulations which are based on gaussian conditioning the method is applied to the borden natural gradient tracer test freyberg 1986 showing favorable results

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

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

defect correction finite element method for the stationary incompressible magnetohydrodynamics equation in this study we give the defect correction finite element method for the stationary incompressible mhd equation firstly the nonlinear mhd equation is solved with an artificial viscosity term then the numerical solution is improved on the same grid by a linearized defect correction technique then we give the numerical analysis including stability analysis and error analysis the numerical analysis proves that our method is stable and has an optimal convergence rate then we give some numerical results from the numerical results we can see that our method is efficient for solving the mhd equations

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

impact of velocity correlation and distribution on transport in fractured media field evidence and theoretical model flow and transport through fractured geologic media often leads to anomalous non fickian transport behavior the origin of which remains a matter of debate whether it arises from variability in fracture permeability velocity distribution connectedness in the flow paths through fractures velocity correlation or interaction between fractures and matrix here we show that this uncertainty of distribution versus correlation controlled transport can be resolved by combining convergent and push pull tracer tests because flow reversibility is strongly dependent on velocity correlation whereas late time scaling of breakthrough curves is mainly controlled by velocity distribution we build on this insight and propose a lagrangian statistical model that takes the form of a continuous time random walk ctrw with correlated particle velocities in this framework velocity distribution and velocity correlation are quantified by a markov process of particle transition times that is characterized by a distribution function and a transition probability our transport model accurately captures the anomalous behavior in the breakthrough curves for both push pull and convergent flow geometries with the same set of parameters thus the proposed correlated ctrw modeling approach provides a simple yet powerful framework for characterizing the impact of velocity distribution and correlation on transport in fractured media

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

modeling of scale dependent dispersion in hydrogeologic systems an approach for handling scale dependent dispersion using a finite element linear triangular solute transport model was developed dispersivity is allowed to vary temporally as a function of the mean travel distance of the solute from an input source various scale dependent dispersivity relationships were presented including types which approach continuously to a maximum or asymptotic value the finite element model results were verified for a one dimensional case by comparison of the theoretical variances of the solute distribution for a given dispersivity travel distance function with the variances calculated from the finite element model concentration distributions the model was applied successfully to the results of several tracer tests reported in the literature that have exhibited a scale effect for systems that exhibit a constant asymptotic dispersivity at large times or mean travel distances the importance of scale dependent dispersion at early times or short travel distances was shown to be minimal in long term predictions of solute transport

an explanation of scale dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry stochastic transport theories rely on the assumptions that the principles of stationarity and ergodicity are satisfied these assumptions imply that the pattern of heterogeneity can be viewed as being spatially periodic thus yielding a finite correlation scale we examine a type of heterogeneity which does not satisfy these principles self similar or fractal heterogeneity which exhibits a pattern over a large range of scales that is independent of the scale of observation and thus will possess a very large correlation scale we review the basic concepts of fractal geometry and develop scaling relationships for fractal travel distance versus scale of observation we then develop lagrangian models for dispersion in a single fractal streamtube and for a set or bundle of fractal streamtubes the results are compared to classical one dimensional advection dispersion theory existing fractal and stochastic theories and to a summary of field measured dispersivity data for the single streamtube model field measured dispersivity is proportional to the straight line travel distance raised to a power of d 1 where d is the fractal dimension for the set of fractal streamtubes field measured dispersivity is proportional to the straight line travel distance raised to a power of 2d 1 the analytical expressions are verified and elaborated by a fractal random walk computer model the computer model is used to show the self similar nature of heterogeneity that is independent of the scale of observation comparisons of the field dispersivity data with our fractal models indicate that most tracer tests have been performed in media that approximate a set of fractal streamtubes

an eulerian scheme for the second order approximation of subsurface transport moments the moments of a conservative tracer cloud migrating in a mean uniform flow field are estimated using an operator approximation scheme results are presented for the second third and fourth central moments in the mean flow direction it is assumed that the spatially variable flow field and therefore the tracer migration problem itself is amenable to a probabilistic description the effects of local dispersion on cloud migration are neglected in this study variation in the flow field is assumed to be the result of spatial variation in the hydraulic conductivity spatial variation in porosity is assumed negligible the operator approximation scheme as implemented in this study is second order correct which requires a second order correct approximation of the velocity field correlation structure because estimation of the velocity correlation structure is decidedly the most difficult aspect of second order analysis an ad hoc extension of the imperfectly stratified approximation developed earlier is implemented for this purpose the first order approximation resulting from the operator expansion scheme is equivalent to small perturbation eulerian results presented earlier naff 1990 1992 the infinite order approximation resulting from this scheme is equivalent to the exponential operator results obtained by van kampen 1976

analysis of the zienkiewicz zhu a posteriori error estimator in the finite element method an a posteriori error estimator for finite element analysis proposed by zienkiewicz and zhu is analysed and shown to be effective and convergent in addition we analyse wider classes of estimators of which the zienkiewicz zhu estimator is a special case it is shown that some of these estimators will be asymptotically exact numerical evidence is presented supporting the analysis

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

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

convergence analysis of three finite element iterative methods for the 2d 3d stationary incompressible magnetohydrodynamics abstract three finite element iterative methods are designed and analyzed for solving 2d 3d stationary incompressible magnetohydrodynamics mhd by a new technique strong uniqueness conditions for both stokes type iterative method iterative method i and newton iterative method iterative method ii are obtained which are weaker than the ones reported in open literature stability and optimal convergence rates for the above two methods are derived where the iterative method ii has an exponential convergent part with respect to iterative step m moreover oseen type iterative method iterative method iii is unconditionally stable and convergent under a uniqueness condition finally performance of the three proposed methods is investigated by numerical experiments

scale and directional dependence of macrodispersivities in colonnade networks macrodispersion in discrete colonnade network models is investigated using a series of monte carlo type numerical experiments the numerical simulations consider fluid flow and advective transport through a square flow region of a two dimensional network of hexagonal colonnades macrodispersivities are calculated as a function of the scale and orientation of the square flow region within the larger parent geometry particle tracking with flux weighted tracer injection and monitoring is used to generate experimental residence time distributions rtds the method of moments is used to characterize the longitudinal tracer spreading the simulated rtds are utilized to examine the macrodispersive behavior in colonnade networks column diameter 1 m with lognormally distributed fracture apertures b in millimeters the network models are assumed to consist of open fractures with ln b 1 945 and ln b 0 896 this translates to an equivalent continuum log conductivity variance ln k2 of 3 67 based on an ensemble average of 100 realizations the slope for spatial variance versus scale of observation ranges from 3 15 to 3 36 m for varying orientations of the hydraulic gradient for ensemble averaged data the varying orientations appear to have little effect on the macrodispersive behavior for single realization experiments the computed macrodispersivities are directionally dependent at a length scale as large as 30 times the column diameter and probably much beyond the computed asymptotic and preasymptotic macrodispersivities are compared with available stochastic solutions for two dimensional isotropic heterogeneity in the horizontal plane the ensemble based numerical data are in excellent agreement with the solutions of dagan 1984 1988 and gelhar and axness 1983 however for individual realizations nonergodic behavior is clearly apparent in the near source evolving region of transport and the numerical data are quite variable between realizations the study provides important insight on applicability of stochastic continuum theories to discrete colonnade network models having ln k2 much greater than 1

unconditional stability and error estimates of the modified characteristics fems for the time dependent incompressible mhd equations abstract this paper focuses on the unconditional stability and convergence of characteristics type methods for the time dependent incompressible mhd equations for this purpose we introduce a new characteristics time discrete system the optimal error estimates in l 2 and h 1 norms for the typical modified characteristics finite element method unconditionally can be deduced while the whole previous works require certain time step restrictions some numerical experiments document performance of the characteristics type methods for the time dependent incompressible mhd equations

adaptive remeshing for compressible flow computations an adaptive mesh procedure for improving the quality of steady state solutions of the euler equations in two dimensions is described the procedure is implemented in conjunction with a finite element solution algorithm using linear triangular elements and an explicit time stepping scheme the mesh adaptation is accomplished by regeneration using information provided by the computed solution on the current mesh the meshes produced are characterised by the appearance of stretched elements in the vicinity of one dimensional flow features and can exhibit a large variation in element size this allows solutions of enhanced quality to be produced in a computationally efficient manner

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

a perturbation expansion for diffusion in a random velocity field we present a formal perturbation expansion for the large scale effective drift velocity and diffusion matrix of a medium with stationary random velocity v x and constant nonrandom diffusion matrix a on a small scale if mu denotes the mean of v x and we express v x as mu varepsilon u x then the expansion is in powers of varepsilon with mu u and a fixed we obtain explicit expressions up to second order in varepsilon which generalize the standard formulae to the case a ne 0

asymptotic dispersion for two dimensional highly heterogeneous permeability fields under temporally fluctuating flow 1 temporal fluctuations of water flux have been investigated as a mechanism that strongly enhances transverse dispersion in heterogeneous media unfortunately most results have been obtained by linear stochastic theories on permeability fields of limited variability worse results are inconsistent regarding the impact of fluctuations on longitudinal dispersion which motivates our work to find the effect of temporal velocity fluctuations on macrodispersion we perform numerical monte carlo simulations for highly variable permeability fields of up to 800 correlation lengths we find that fluctuations longitudinal to the main flow direction hardly modify macrodispersion because they do not alter the flow lines fluctuations transverse to the main flow direction not only increase transverse dispersion which is well known but also reduce the longitudinal macrodispersion in a significant and consistent way which contradicts previous findings the reduction of the longitudinal dispersion is comparable to the increase of transverse dispersion most surprisingly for high heterogeneity temporal fluctuations cause total longitudinal plus transverse macrodispersion to drop with respect to the steady state one enhancement of the transverse macrodispersion comes from both the increase of the transverse velocity variability and lagrangian correlation reduction of the longitudinal macrodispersion results from the reduction of the lagrangian correlation of the longitudinal velocity that is temporal fluctuations reduce longitudinal spreading both by breaking the fastest velocity paths on the plume front and by letting solute bypass the low permeability zones that tend to block or trap the solute in steady state flow conditions

a quasi linear theory of non fickian and fickian subsurface dispersion 1 theoretical analysis with application to isotropic media a theory is presented which accounts for nonlinearity caused by the deviation of plume particles from their mean trajectory in three dimensional statistically homogeneous but anisotropic porous media under an exponential covariance of log hydraulic conductivities existing linear theories predict that in the absence of local dispersion transverse dispersivities tend asymptotically to zero as fickian conditions are reached according to our new quasi linear theory these dispersivities ascend to peak values and then diminish gradually toward nonzero fickian asymptotes which are proportional to y4 when the log hydraulic conductivity variance y2 is much less than 1 all existing theories agree that in isotropic media the asymptotic longitudinal dispersivity is proportional to y2 when y2 1 and all are nominally restricted to mildly heterogeneous media in which this inequality is satisfied however the quasi linear theory appears to be less prone to error than linear theories when extended to strongly heterogeneous media because it deals with the above nonlinearity without formally limiting y2 it predicts that when y 1 in isotropic media both the longitudinal and transverse dispersivities ascend monotonically toward fickian asymptotes proportional to y

error estimates and convergence rates for various incompressible elements this note reports numerical experiments on the efficiency of simple error estimates derived earlier1 applied to incompressible mixed or related penalty type formulations the rate of convergence and performance of various mixed elements is compared numerical results from a driven cavity and an incompressible elastic problem demonstrate that the t6b1 3d and t6 3c elements give a faster rate of convergence than the t6 1d element however in the case of a plane extrusion analysis stronger singularity the rate of convergence for the t6b1 3d element drops and is inferior to that of the t6 1d while the t6 3c element still proves superior to the other two elements

on the existence uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics the authors consider the equations of stationary incompressible magneto hydrodynamics posed in a bounded domain in three dimensions and treat the full coupled system of equations with inhomogeneous boundary conditions under certain conditions on the data they show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed they discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution

impact of concentration measurements upon estimation of flow and transport parameters the lagrangian approach transport of a conservative solute takes place in a heterogeneous formation of spatially variable conductivity the latter is modeled as a random space function of stationary lognormal distribution as a result the velocity field and the concentration are also random assuming that measurements of concentration of an existing plume are available the problem addressed here is to assess their effect upon identification of log conductivity and flow transport variables the solution is sought in a lagrangian framework in which transport is represented in terms of the random trajectories of particles originating from the initial plume a concentration measurement is equivalent to the passage of a trajectory through the measurement point at a given time the impact of measurements is achieved by conditioning any variable of interest on realizations for which at least one trajectory satisfies the requirement it is shown that cokriging of concentration and another flow or transport variable leads to the correct conditioned mean of the latter in contrast the conditional variance based on cokriging is erroneous the procedure is illustrated for two dimensional flow under a few simplifying assumptions the effect of a concentration measurement upon the expected value and variance of log transmissivity and plume centroid are examined in a few particular cases the procedure may improve the solution of the inverse problem and the prediction of transport of existing plumes

eulerian lagrangian analysis of transport conditioned on hydraulic data 1 analytical numerical approach recently a unified eulerian lagrangian theory has been developed by one of us for nonreactive solute transport in space time nonstationary velocity fields we describe a combined analytical numerical method of solution based on this theory for the special case of steady state flow in a mildly fluctuating statistically homogeneous lognormal hydraulic conductivity field we take the unconditional mean velocity to be uniform but allow conditioning on measurements of log hydraulic conductivity or transmissivity and or hydraulic head this renders the velocity field nonstationary we solve the conditional transport problem analytically at early time and express it in pseudo fickian form at later time the deterministic pseudo fickian equations involve a conditional space time dependent dispersion tensor which we evaluate numerically along mean particle trajectories these equations lend themselves to accurate solution by standard galerkin finite elements on a relatively coarse grid the final step is an explicit numerical computation of lower bounds on conditional concentration prediction variance covariance and coefficient of variation travel time distribution cumulative mass release across a compliance surface the associated error and plume spatial moments our method also allows quantification of the uncertainty in the original source location of any solute particle located anywhere in the field at any time this paper describes the methodology and presents some unconditional results conditioning and more advanced computations are presented in the subsequent papers

macrodispersivity tensor for nonreactive solute transport in isotropic and anisotropic fractal porous media analytical solutions using spectral stochastic theory macrodispersivity tensors are developed for one two and three dimensional isotropic and anisotropic fractal porous media since natural geologic aquifers are always bounded we introduce the concept of a maximum length scale lmax in the one dimensional case the asymptotic long time longitudinal macrodispersivity a is proportional to lmax2 and in the multidimensional cases a lmax in the multidimensional cases as the fractal dimension increases a decreases because a larger fractal dimension means the tortuosity of solute particles is larger leading to more mixing in the transverse direction which in turn reduces longitudinal spreading the transverse macrodispersivities are shown to be analogous to traditional spectral stochastic results in the multidimensional cases lmax is shown to be controlled by the aquifer thickness as a result ergodic conditions may be reached in relatively thin aquifers allowing the use of the asymptotic macrodispersivity expressions obtained here for single realizations

large time behavior of concentration variance and dilution in heterogeneous formations consider the advection and dispersion of a conservative nonsorbing solute in a spatially variable but statistically homogeneous velocity field a lagrangian approach leads to expressions for the large time mean and variance of concentration the expressions require only the macroscopic velocity vector um the macrodispersion tensor dm and an additional tensor the macroscopic velocities and macrodispersivities are well known from numerous previous studies but is introduced here for the first time the tensor is needed to describe the kinetics of dilution of a plume two cases with the same um anddm have different dilution characteristics depending on the characteristic times of dilution are given by tensor d 1m it is demonstrated for the first time through a lagrangian approach that the coefficient of variation of concentration at the center of the plume becomes proportionate to 1 t as was previously shown in the eulerian theory ofkapoor and gelhar 1994a b expressions for the geometric mean of the dilution index and the reactor ratio are derived numerical simulations support the validity of the approach

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

development of the fokker planck equation and its solutions for modeling transport of conservative and reactive solutes in physically heterogeneous media when a scale dependent fractal dispersivity is introduced in the convective dispersive equation of transport in subsurface flow the fokker planck equation fpe also known as the second diffusion equation is derived similarity solutions of the one dimensional fpe subject to a dirac delta function input are also presented the similarity solution is shown to function as a kernel in the convolution integral to yield an output on a real timescale the input function is derived by a procedure known as the inverse problem with the aid of laplace transform the fpe and its solutions contain a fractal dimension d and cutoff limit two shape parameters and in the fpe and other parameters that appear in darcy s law a two dimensional fpe and its solutions have been presented elsewhere it is reasoned that the scale dependent dispersion in unsaturated flow is also an issue for investigation

a solute flux approach to transport in heterogeneous formations 1 the general framework it is common to represent solute tranport in heterogeneous formations in terms of the resident concentration c x t regarded as a random space function the present study investigates the alternative representation by q the solute mass flux at a point of a control plane normal to the mean flow this representation is appropriate for many field applications in which the variable of interest is the mass of solute discharged through a control surface a general framework to compute the statistical moments of q and of the associated total solute discharge q and mass m is established with x the direction of the mean flow a solute particle is crossing the control plane at y z and at the travel arrival time the associated expected solute flux value is proportional to the joint probability density function pdf g1 of and whereas the variance of q is shown to depend on the joint pdf g2 of the same variables for two particles in turn the statistical moments of and depend on those of the velocity components through a system of stochastic ordinary differential equations for a steady velocity field and neglecting the effect of pore scale dispersion a major simplification of the problem results in the independence of the random variables and as a consequence the pdf of and can be derived independently of a few approximate approaches to derive the statistical moments of and are outlined these methods will be explored in paper 2 in order to effectively derive the variances of the total solute discharge and mass while paper 3 will deal with the nonlinear effect of the velocity variance upon the moments of and

analytical solution of a convection dispersion model with time dependent transport coefficients mathematical studies of solute transport in porous media have often utilized equivalent models of the transport process to remove undesired variability in the transport coefficients at the space and time scales of direct interest both deterministic and stochastic approaches in this genre produce an effective convection dispersion equation with time dependent coefficients this type of equation in one spatial dimension is investigated mathematically in the present paper a closed form solution of the solute transport equation is derived for a semi infinite spatial domain with arbitrary initial and boundary flux conditions it is shown that the solution reduces to well known results for special forms of the time dependent coefficients in general however a volterra integral equation of the second kind must be solved to evaluate the analytical solution of the transport equation we present a stable and convergent numerical scheme utilizing a trapezoidal quadrature rule for the solution of the volterra equation the method of solution developed should be applicable to a broad variety of solute transport problems including particularly those in heterogeneous porous media

convergence of gauge method for incompressible flow a new formulation a gauge formulation of the incompressible navier stokes equations in terms of an auxiliary field a and a gauge variable u a was proposed recently by e and liu this paper provides a theoretical analysis of their formulation and verifies the computational advantages we discuss the implicit gauge method which uses backward euler or crank nicolson in time discretization however the boundary conditions for the auxiliary field a are implemented explicitly vertical extrapolation the resulting momentum equation is decoupled from the kinematic equation and the computational cost is reduced to solving a standard heat and poisson equation moreover such explicit boundary conditions for the auxiliary field a will be shown to be unconditionally stable for stokes equations for the full nonlinear navier stokes equations the time stepping constraint is reduced to the standard cfl constraint t x c we also prove first order convergence of the gauge method when we use mac grids as our spatial discretization the optimal error estimate for the velocity field is also obtained

an efficient high order perturbation approach for flow in random porous media via karhunen loeve and polynomial expansions in this study we attempt to obtain higher order solutions of the means and co variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of karhunen loeve decomposition polynomial expansion and perturbation methods we first decompose the log hydraulic conductivity y ln ks as an infinite series on the basis of a set of orthogonal gaussian standard random variables i the coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity we then write head as an infinite series whose terms h n represent head of nth order in terms of y the standard deviation of y and derive a set of recursive equations for h n we then decompose h n with polynomial expansions in terms of the products of n gaussian random variables the coefficients in these series are determined by substituting decompositions of y and h n into those recursive equations we solve the mean head up to fourth order in y and the head variances up to third order in y2 we conduct monte carlo mc simulation and compare mc results against approximations of different orders from the moment equation approach based on karhunen loeve decomposition klme we also explore the validity of the klme approach for different degrees of medium variability and various correlation scales it is evident that the klme approach with higher order corrections is superior to the conventional first order approximations and is computationally more efficient than the monte carlo simulation

an efficient eulerian lagrangian method for solving solute transport problems in steady and transient flow fields a computationally efficient yet relatively simple eulerian lagrangian method is proposed for solving the one dimensional convection dispersion solute transport equation assuming a steady or transient velocity field the method uses a modified single step reverse particle tracking msrpt technique to handle steep concentration fronts the scheme utilizes two weighting factors to control the movement of particles during a backward tracking step one weighting factor greater than unity is used in the upstream region of the convection front while another weighting factor less than unity is taken in the downstream region the two factors were related empirically to the grid peclet and courant numbers the msrpt technique is carried out only within the concentration plume at each time step for transient flow fields the weighting factors were determined using an automatically adjustable procedure based on mass balance errors the msrpt method maintains the advantages of the traditional single step reverse particle tracking srpt procedure i e producing efficient and oscillation free calculations but circumvents numerical dispersion introduced by srpt a large number of tests against analytical solutions for one dimensional transport in uniform flow fields indicate that the proposed method can handle the entire range of peclet numbers from zero to infinity numerical tests also show that the msrpt method is a relatively accurate efficient an d massconservative algorithm for solute transport in transient flow fields the courant number at present cannot exceed 1 the msrpt approach was found especially useful for convection dominated problems in fact an exact numerical solution may be obtained with msrpt for pure convection convection dispersion type equations are being widely used to model solute transport in soil and groundwater owing to the particular combination of hyperbolic and parabolic terms serious difficulties are often encountered in obtaining accurate numerical solutions of these equations a variety of numerical schemes have been developed to deal with these difficulties including an extensive number of

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

velocity correction projection methods for incompressible flows we introduce and study a new class of projection methods namely the velocity correction methods in standard form and in rotational form for solving the unsteady incompressible navier stokes equations we show that the rotational form provides improved error estimates in terms of the h1 norm for the velocity and of the l2 norm for the pressure we also show that the class of fractional step methods introduced in s a orsag m israeli and m deville j sci comput 1 1986 pp 75 111 and k e karniadakis m israeli and s a orsag j comput phys 97 1991 pp 414 443 can be interpreted as the rotational form of our velocity correction methods thus to the best of our knowledge our results provide the first rigorous proof of stability and convergence of the methods in those papers we also emphasize that contrary to those of the above groups our formulations are set in the standard l2 setting and consequently they can be easily implemented by means of any variational approximation techniques in particular the finite element methods

the significance of heterogeneity of evolving scales to transport in porous formations flow takes place in a heterogeneous formation of spatially variable conductivity which is modeled as a stationary space random function to model the variability at the regional scale the formation is viewed as one of a two dimensional horizontal structure a constant head gradient is applied on the formation boundary such that the flow is uniform in the mean a plume of inert solute is injected at t 0 in a volume v0 under ergodic conditions the plume centroid moves with the constant mean flow velocity u and a longitudinal macrodispersion coefficient dl may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid for a log conductivity covariance cy of finite integral scale i at first order in the variance y2 and for a travel distance l ut i dl y2ui and transport is coined as fickian ergodicity of the moments is ensured if l i where l is the initial plume scale some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with l without reaching a constant limit anomalous diffusion to model such a behavior previous studies have assumed that cy is stationary but of unbounded integral scale with cy ar 1 0 for large lag r under ergodic conditions it was found that asymptotically dl aul1 i e non fickian behavior and anomalous dispersion the present study claims that an ergodic behavior is not possible for a given finite plume of initial size l since the basic requirement that l i cannot be satisfied for cy of unbounded scale for instance the centroid does not move any more with u but is random figure 1 owing to the large scale heterogeneity in such a situation the actual effective dispersion coefficient dl is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization this is the accessible entity in a given experiment we show that in contrast with dl the behavior of dl is controlled by l and it has the fickian limit dl aul1 figure 3 we also discuss the case in which y is of stationary increments and is characterized by its variogram y then u and dl can be defined only if y is truncated equivalently an infrared cutoff is carried out in the spectrum of y however for a bounded u it is shown that dl depends only on y furthermore for y ar dl aul2l 1 i e dispersion is fickian for 0 1 whereas for 1 2 transport is non fickian since 2 dl cannot grow faster than l ut this is in contrast with a recently proposed model neuman 1990 in which the dispersion coefficient is independent of the plume size and it grows approximately like l1 5

time periodic transport in heterogeneous porous media analytical solutions for the one dimensional convection diffusion equation in a semi infinite heterogeneous domain are obtained for periodic boundary conditions the kinematic dispersion of the porous medium is taken to be either a linear function of distance or an exponential function of distance that asymptotically becomes constant the analytic solutions are given in terms of kelvin functions and hypergeometric functions respectively which are computed using mathematica the solutions are compared with each other and with the constant dispersivity case and the amplitudes and phase shifts are studied with respect to the frequency of the input wave at the boundary all three models respond differently to different single frequency inputs showing different decay rates at lower frequencies and more pronounced phase shifts at higher frequencies the use of single frequency periodic inputs appears to give more distinguishable solutions than traditional pulse type data which contains multiple spectral frequencies and the resulting solutions can be used as a verification for numerical schemes for convection diffusion equations

error estimates and adaptive refinement for plate bending problems the zienkiewicz zhu error estimator is shown to be effective in problems of plate flexure when used in conjunction with triangular elements and an adaptive mesh generator allowing a prescribed size of elements to be developed very fast adaptive convergence for results of specified accuracy is achieved

a mixed finite element method with exactly divergence free velocities for incompressible magnetohydrodynamics abstract we introduce and analyze a mixed finite element method for the numerical discretization of a stationary incompressible magnetohydrodynamics problem in two and three dimensions the velocity field is discretized using divergence conforming brezzi douglas marini bdm elements and the magnetic field is approximated by curl conforming nedelec elements the h 1 continuity of the velocity field is enforced by a dg approach a central feature of the method is that it produces exactly divergence free velocity approximations and captures the strongest magnetic singularities we prove that the energy norm error is convergent in the mesh size in general lipschitz polyhedra under minimal regularity assumptions and derive nearly optimal a priori error estimates for the two dimensional case we present a comprehensive set of numerical experiments which indicate optimal convergence of the proposed method for two dimensional as well as three dimensional problems

stochastic theory of field scale fickian dispersion in anisotropic porous media a three dimensional theory is described for field scale fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities the study relies partly on earlier work by the authors the attributes of which are briefly reviewed it leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media we express the dispersion tensor d as the sum of a local component d and a field scale component the local component is assumed to be independent of velocity which is most appropriate if it represents molecular diffusion and its principal terms are taken to act parallel and normal to the mean velocity vector the field scale component is written as where is a dispersivity tensor and we show that at large peclet numbers p the dispersivity tensor reduces to a single principal component parallel to the mean velocity regardless of how is oriented this result valid for arbitrary covariance functions of log hydraulic conductivity differs from that of l w gelhar and c l axness 1983 according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value they calculate the direction of the largest principal dispersivity to be offset from the mean velocity toward the direction of least spatial correlation or away from the stratification in typical layered media we show that this principal dispersivity is offset in the opposite direction at small and intermediate peclet numbers but rotates toward the mean velocity as p increases the largest eigen value is constant and dominated by field scale velocity fluctuations at large p values the other two eigen values diminish asymptotically in proportion to p 1 and are controlled by d as well as by field scale differential convection the range of small peclet numbers has not been previously investigated under anisotropic conditions yet is of much importance for transport in low permeability rocks or soils we show that at low p values all three principal dispersivities are proportional to p and thus is proportional to 2 a phenomenon reminiscent of taylor diffusion when the mean velocity is inclined to the axes of anisotropy the eigen values of are neither parallel nor normal to however since d is dominated by d at small peclet numbers the principal dispersion coefficients are asymptotically as p 0 parallel and normal to the mean velocity just like when p is large their maximum deviation from these directions occurs at intermediate p values

a laplace transform power series solution for solute transport in a convergent flow field with scale dependent dispersion 1 this study presents a novel mathematical model to describe solute transport in a radially convergent flow field with scale dependent dispersion the scale dependent advection dispersion equation in cylindrical coordinates derived based on the dispersivity is assumed to increase linearly with the distance of the solute transported from its input source the laplace transformed power series technique is applied to solve the radially scale dependent advection dispersion equation with variable coefficients breakthrough curves obtained using the scale dependent dispersivity model are compared with those from the constant dispersivity model to illustrate the features of scale dependent dispersion in a radially convergent flow field the comparison results reveal that the constant dispersivity model can produce a type curve with the same shape as that from the proposed scale dependent dispersivity model this correspondence in type curves between the two models occurs when the product of the peclet number used in the constant dispersivity model and the dispersivity distance ratio used in the scale dependent dispersivity model equals 4 finally the scale dependent dispersivity model is applied to a set of previously reported field data to investigate the linearly scale dependent dispersion effect the analytical results reveal that the linearly scale dependent dispersion model is applicable to this test site

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