fractional brownian motion and fractional gaussian noise in subsurface hydrology a review presentation of fundamental properties and extensions recent studies have shown that fractional brownian motion fbm and fractional gaussian noise fgn are useful in characterizing subsurface heterogeneities in addition to geophysical time series although these studies have led to a fairly good understanding of some aspects of fbm fgn a comprehensive introduction to these stochastic fractal functions is still lacking in the subsurface hydrology literature in this paper efforts have been made to define fbm fgn and present a development of their mathematical properties in a direct yet rigorous manner use of the spectral representation theorem allows one to derive spectral representations for fbm fgn even though these functions do not have classical fourier transforms the discrete and truncated forms of these representations have served as a basis for synthetic generation of fbm fgn the discrete spectral representations are developed and various implications discussed in particular it is shown that a discrete form of the fbm spectral representation is equivalent to the well known weierstrass mandelbrot random fractal function although the full implications are beyond the scope of the present paper it is observed that discrete spectral representations of fbm constitute stationary processes even though fbm is nonstationary a new and general spectral density function is introduced for construction of complicated anisotropic 3 d fractals including those characterized by vertical fgn and horizontal fbm such fractals are useful for modeling anisotropic subsurface heterogeneities but cannot be generated with existing schemes finally some basic properties of fractional levy motion and concepts of universal multifractals which can be considered as generalizations of fbm fgn are reviewed briefly

on the estimation of parameters for frequency domain models the problem of estimating parameters for frequency domain models is considered past approaches have most commonly based estimation criteria upon modulus and phase transformations of the model and sample frequency response functions as an alternative a complex vector estimation criterion is proposed and is implemented in a nonlinear gauss marquardt optimization algorithm when compared to the previous methods of fitting modulus and phase transformations the complex vector estimation methodology has less bias and variance and is more robust

multifractal anisotropic scaling of the hydraulic conductivity 1 we analyzed the scaling properties of the hydraulic conductivity k at three sites in northern america made borden and cape cod we found that k at all sites exhibits multifractality fractal and multiscaling in both the vertical and horizontal directions though the multiscaling was within a range smaller than that of the maximum distance between measurements in the vertical direction the k data for made borden and cape cod were multiscaling from 0 15 to 1 35 m 0 05 to 0 5 m and 0 15 to 0 9 m respectively they were multiscaling in the horizontal direction from 9 to 45 m 1 to 10 m and 1 to 17 m respectively the multiscaling was also anisotropic evidence of scaling was poorest for the horizontal direction of the made site and it spanned half an order of magnitude such results compel one to treat multifractality in the horizontal direction of made as supported by heuristic arguments rather than by pure statistical evaluation we fitted a multifractal model to the data and estimated its parameters we found the underlying statistics of all data to be non gaussian and the model capable of reproducing the probability distribution of k data especially the negative skewness of log k we also generated two dimensional isotropic multifractal fields illustrating the role of the parameters of the selected multifractal model

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

optimization of the porous material described by the biot model abstract the paper is devoted to the shape optimization of microstructures generating porous locally periodic materials saturated by viscous fluids at the macroscopic level the porous material is described by the biot model defined in terms of the effective medium coefficients involving the drained skeleton elasticity the biot stress coupling the biot compressibility coefficients and by the hydraulic permeability of the darcy flow model by virtue of the homogenization these coefficients are computed using characteristic responses of the representative unit cell consisting of an elastic solid skeleton and a viscous pore fluid for the purpose of optimization the sensitivity analysis on the continuous level of the problem is derived we provide sensitivities of objective functions constituted by the biot model coefficients with respect to the underlying pore shape described by a b spline box which embeds the whole representative cell we consider material design problems in the framework of which the layout of a single representative cell is optimized then we propose a sequential linearization approach to the two scale problem in which local microstructures are optimized with respect to macroscopic design criteria numerical experiments are reported which include stiffness maximization with constraints guaranteeing minimum required permeability and vice versa issues of the design and anisotropy and the spline box parametrization are discussed in order to avoid remeshing a geometric regularization technique based on injectivity constraints is applied

solute transport in heterogeneous porous media with long range correlations transport of finite plumes in three dimensional heterogeneous and statistically isotropic aquifers is investigated where the log hydraulic conductivity is characterized by a fractional gaussian noise fgn covariance structure of hurst coefficient h leading order analytical expressions for velocity autocovariance functions uij one particle displacement covariance xij and macrodispersivity tensor ij are derived under ergodic conditions and mean uniform steady state flow nonergodic transport is then discussed for a line source of finite length either normal or parallel to the mean flow by evaluating time dependent ensemble averages of the second spatial moments zij aij aij 0 xij rij and the effective dispersivity tensor ij defined as 0 5 d aij dt where aij 0 is the initial value of the second spatial moments of a plume aij and rij is the plume centroid covariance the main finding is that in a fgn log k field the spreading of a solute plume is never ergodic as h increases effective dispersivity results differ more from their ergodic counterparts since a larger h implies the medium is more correlated the most interesting results are as follows for a source parallel to flow 22 does not decrease below zero at large time but remains strictly positive in variance with exponential or gaussian covariance for a source normal to flow 11 reaches a large time asymptote whose value depends on h as follows it decreases with h for a small source it increases reaches a peak and then decreases as h goes from 1 2 to 1 for intermediate and large sources for h 1 11 is zero irrespective of the source size

topology optimization of fluid structure interaction problems in poroelasticity abstract this paper presents a method for applying topology optimization to fluid structure interaction problems in saturated poroelastic media the method relies on a multiple scale method applied to periodic media the resulting model couples the stokes flow in the pores of the structure with the deformation of the elastic skeleton through a macroscopic darcy type flow law the method allows to impose pressure loads for static problems through a one way coupling while transient problems are fully coupled modeling the interaction between fluid and solid the material distribution is determined by topology optimization in order to optimize the performance of a shock absorber and test the pressure loading capabilities and optimization of an internally pressurized lid

multifractal scaling of the intrinsic permeability existing fractal studies dealing with subsurface heterogeneity treat the logarithm of the permeability k as the variable of concern we treat k as a multifractal and investigate its scaling and fractality using measured horizontal k data from two locations in the united states the first data set was from a shoreline sandstone near coalinga california and the second was from an eolian sandstone goggin 1988 by applying spectral analyses and computing the scaling of moments of various orders using the double trace moment method lavallee 1991 lavallee et al 1992 we found that k is multiscaling i e scaling and multifractal we also found that the so called universal multifractal um schertzer and lovejoy 1987 model essentially a log levy multifractal was able to reproduce the multiscaling behavior reasonably well the um model has three parameters a s and h representing the multifractality index the codimension of the mean field and the distance to stationary multifractal respectively we found a 5 1 7 s 5 0 23 h 5 0 22 and a 5 1 6 s 5 0 11 h 5 0 075 for the shoreline and eolian data sets respectively the fact that a values were less than 2 indicates that the underlying statistics are non gaussian we generated stationary and nonstationary multifractals and illustrated the role of the um parameters on simulated fields studies that treated log k as the variable of concern have pointed out the necessity for large data records especially when the underlying distribution is levy stable our investigation revealed that even larger data records are required when treating k as a multifractal because log k is less intermittent or irregular than k

approximation of probability density functions by the multilevel monte carlo maximum entropy method we develop a complete convergence theory for the maximum entropy method based on moment matching for a sequence of approximate statistical moments estimated by the multilevel monte carlo method under appropriate regularity assumptions on the target probability density function the proposed method is superior to the maximum entropy method with moments estimated by the monte carlo method new theoretical results are illustrated in numerical examples

exact pdf equations and closure approximations for advective reactive transport mathematical models of advection reaction phenomena rely on advective flow velocity and bio chemical reaction rates that are notoriously random by using functional integral methods we derive exact evolution equations for the probability density function pdf of the state variables of the advection reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions these pdf equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterog eneous and strongly correlated random field our analytical solution is then used to investigate the accuracy and robustness of the recently proposed large eddy diffusivity led closure approximation 1 we find that the solution to the led based pdf equation which is exact for uncorrelated reaction rates is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

further analysis of multilevel monte carlo methods for elliptic pdes with random coefficients we consider the application of multilevel monte carlo methods to elliptic pdes with random coefficients we focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter and that only have limited spatial regularity we extend the finite element error analysis for this type of equation carried out in charrier et al siam j numer anal 2013 to more difficult problems posed on non smooth domains and with discontinuities in the coefficient for this wider class of model problem we prove convergence of the multilevel monte carlo algorithm for estimating any bounded linear functional and any continuously frechet differentiable non linear functional of the solution we further improve the performance of the multilevel estimator by introducing level dependent truncations of the karhunen loeve expansion of the random coefficient numerical results complete the paper

the multi level monte carlo finite element method for a stochastic brinkman problem we present the formulation and the numerical analysis of the brinkman problem derived in allaire arch rational mech anal 113 3 209 259 1990 doi 10 1007 bf00375065 arch rational mech anal 113 3 261 298 1990 doi 10 1007 bf00375066 with a lognormal random permeability specifically the permeability is assumed to be a lognormal random field taking values in the symmetric matrices of size d times d where d denotes the spatial dimension of the physical domain d we prove that the solutions admit bounded moments of any finite order with respect to the random input s gaussian measure we present a mixed finite element discretization in the physical domain d which is uniformly stable with respect to the realization of the lognormal permeability field based on the error analysis of this mixed finite element method mfem we develop a multi level monte carlo mlmc discretization of the stochastic brinkman problem and prove that the mlmc mfem allows the estimation of the statistical mean field with the same asymptotical accuracy versus work as the mfem for a single instance of the stochastic brinkman problem the robustness of the mfem implies in particular that the present analysis also covers the darcy diffusion limit numerical experiments confirm the theoretical results

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

impact of parametric uncertainty on estimation of the energy deposition into an irradiated brain tumor abstract predictions of the total energy deposited into a brain tumor through x ray irradiation are notoriously error prone we investigate how this predictive uncertainty is affected by uncertainty in both the location of the region occupied by a dose enhancing iodinated contrast agent and the agent s concentration this is done within the probabilistic framework in which these uncertain parameters are modeled as random variables we employ the stochastic collocation sc method to estimate statistical moments of the deposited energy in terms of statistical moments of the random inputs and the global sensitivity analysis gsa to quantify the relative importance of uncertainty in these parameters on the overall predictive uncertainty a nonlinear radiation diffusion equation dramatically magnifies the coefficient of variation of the uncertain parameters yielding a large coefficient of variation for the predicted energy deposition this demonstrates that accurate prediction of the energy deposition requires a proper treatment of even small parametric uncertainty our analysis also reveals that sc outperforms standard monte carlo but its relative efficiency decreases as the number of uncertain parameters increases from one to three a robust gsa ameliorates this problem by reducing this number

filters in topology optimization in this article a modified filtered version of the minimum compliance topology optimization problem is studied the direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator in this setting it is possible to establish the existence of solutions moreover convergence of an approximation by means of finite elements can be obtained this is illustrated through some numerical experiments the filtering technique is also shown to cope with two important numerical problems in topology optimization checkerboards and mesh dependent designs copyright 2001 john wiley sons ltd

a nonstationary spectral method for solving stochastic groundwater problems unconditional analysis stochastic analyses of groundwater flow and transport are frequently based on partial differential equations which have random coefficients or forcing terms analytical methods for solving these equations rely on restrictive assumptions which may not hold in some practical applications numerically oriented alternatives are computationally demanding and generally not able to deal with large three dimensional problems in this paper we describe a hybrid solution approach which combines classical fourier transform concepts with numerical solution techniques our approach is based on a nonstationary generalization of the spectral representation theorem commonly used in time series analysis the generalized spectral representation is expressed in terms of an unknown transfer function which depends on space time and wave number the transfer function is found by solving a linearized deterministic partial differential equation which has the same form as the original stochastic flow or transport equation this approach can accomodate boundary conditions spatially variable mean gradients measurement conditioning and other sources of nonstationarity which cannot be included in classical spectral methods here we introduce the nonstationary spectral method and show how it can be used to derive unconditional statistics of interest in groundwater flow and transport applications

temporal scaling of hydraulic head fluctuations nonstationary spectral analyses and numerical simulations 1 temporal scaling of the hydraulic head time series h t was found in a previous analysis of hourly measured head data this issue is further investigated in this paper with nonstationary spectral analyses and numerical simulations the results show that temporal scaling may indeed exist in h t which fluctuates like a fractional brownian motion in most aquifers on the basis of a linear reservoir model with a white noise recharge input we show that the variance and covariance of h t are functions of time the head variance increases with time and approaches a constant limit as time progresses while the covariance decreases with the separation time interval for a fixed time and approaches the typical exponential covariance as time increases the spectra of the simulated h t using a one dimensional transient groundwater flow model with a white noise recharge in both homogeneous and heterogeneous aquifers are shown to be proportional to f where f is frequency and 1 84 or h 0 42 heterogeneity in the hydraulic conductivity may affect the fractal dimension of h t in highly permeable aquifers but not in the low permeable aquifer simulated in this study

a frequency domain approach to water quality modeling in groundwater theory a method for the analysis of complex temporal variations of environmental tracers or pollution time series in groundwater is examined using spectral analysis and linear filter theory for stationary stochastic processes the interpretation of solute fluctuations subject to a time varying source is accomplished via frequency domain solutions to stochastic differential equations for three widely applied transport models 1 a lumped parameter or linear reservoir model 2 convective advective transport in a curvilinear flow field and 3 convective dispersive transport in a uniform flow field frequency domain solutions are presented in terms of the theoretical transfer function and phase spectra which describe the amplitude attenuation and phase lag between frequencies in the input and output a comparison of the frequency response of the three models indicates that the unique filtering characteristics of each may provide a diagnostic tool for matching the appropriate theory to a sampled water quality signal a procedure is suggested for parameter estimation which involves comparison of the theoretical and field estimated transfer function and phase spectra

level set topology optimization of fluids in stokes flow we propose the level set method of topology optimization as a viable robust and efficient alternative to density based approaches in the setting of fluid flow the proposed algorithm maintains the discrete nature of the optimization problem throughout the optimization process leading to significant advantages over density based topology optimization algorithms specifically the no slip boundary condition is implemented directly this is accurate removes the need for interpolation schemes and continuation methods and gives significant computational savings by only requiring flow to be modeled in fluid regions topological sensitivity information is utilized to give a robust algorithm in two dimensions and familiar two dimensional power dissipation minimization problems are solved successfully computational efficiency of the algorithm is also clearly demonstrated on large scale three dimensional problems

universal scaling of hydraulic conductivities and dispersivities in geologic media an interpretation is offered for the observation that dispersivities increase with scale apparent longitudinal dispersivity data from a variety of hydrogeologic settings are assumed to represent a continuous hierarchy of log hydraulic conductivity fields with mutually uncorrelated increments each field having its own exponential autocovariance associated integral scale and variance that increases as a power of scale such a hierarchy is shown theoretically to form a self similar random field with homogeneous increments regardless of whether or not the underlying assumption is valid one can justify interpreting the apparent dispersivities in a manner consistent with a recent quasi linear theory of non fickian and fickian dispersion in homogeneous media which supports the notion of a self similar hierarchy a posteriori the hierarchy is revealed to possess a semivariogram s cs where c is a constant and a fractal dimension d e 0 75 where e is the topological dimension of interest this can be viewed as a universal scaling rule about which large deviations occur due to local influences including the existence of discrete natural scales at which log hydraulic conductivity is statistically homogeneous as such homogeneity is at best a local phenomenon occurring intermittently over narrow bands of the scale spectrum one must question the utility of associating medium properties with representative elementary volumes and relying on fickian models of dispersion over more than relatively narrow scale intervals porous and fractured media appear to follow the same idealized scaling rule for both flow and transport raising a question about the validity of many distinctions commonly drawn between such media finally the data suggest that conditioning transport models through calibration against hydraulic measurements has the effect of filtering out large scale modes from the hierarchy

scaling of random fields by means of truncated power variograms and associated spectra an interpretation is offered for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with domain size we first demonstrate that the power semi variogram and associated spectra of random fields having homogeneous isotropic increments can be constructed as weighted integrals from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated homogeneous isotropic fields modes we then analyze the effect of filtering out truncating high and low frequency modes from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous with an autocovariance function that varies monotonically with separation distance in a manner not too dissimilar than that of its constituent modes the integral scales of the lowest and highest frequency modes cutoffs are related respectively to the length scales of the sampling window domain and data support sample volume taking each relationship to be one of proportionality renders our expressions for the integral scale and variance of a truncated field dependent on window and support scales in a manner consistent with observations the traditional approach of truncating power spectral densities yields autocovariance functions that oscillate about zero with finite in one and two dimensions or vanishing in one dimension integral scales our hierarchical theory allows bridging across scales at a specific locale by calibrating a truncated variogram model to data observed on a given support in one domain and predicting the autocovariance structure of the corresponding multiscale field in domains that are either smaller or larger one may also venture we suspect with less predictive power to bridge across both domain scales and locales by adopting generalized variogram parameters derived on the basis of juxtaposed hydraulic and tracer data from many sites

two dimensional poroelastic acoustical foam shape design for absorption coefficient maximization by topology optimization method optimal shape design of a two dimensional poroelastic acoustical foam is formulated as a topology optimization problem for a poroelastic acoustical system consisting of an air region and a poroelastic foam region two different physical regions are continuously changed in an iterative design process to automatically account for the moving interfaces between two regions we propose a new unified model to analyze the whole poroelastic acoustical foam system with one set of governing equations biot s equations are modified with a material property interpolation from a topology optimization method with the unified analysis model we carry out two dimensional optimal shape design of a poroelastic acoustical foam by a gradient based topology optimization setting the specific objective is the maximization of the absorption coefficient in low and middle ranges of frequencies with different amounts of a poroelastic material the performances of the obtained shapes are compared with those of well known wedge s

topology optimization using a mixed formulation an alternative way to solve pressure load problems abstract a shortcoming of the traditional density based approach to topology optimization is the handling of design dependent loads that relate to boundary data such as for example pressure loads previous works have introduced spline and iso density curves or alternative parametrization schemes to determine the load surfaces in this work we suggest a new way to solve pressure load problems in topology optimization using a mixed displacement pressure formulation for the underlying finite element problem we define the void phase to be an incompressible hydrostatic fluid in this way we can transfer pressure loads through the fluid without any needs for special load surface parametrizations the method is easily implemented in the standard density approach and is demonstrated to work efficiently for both 2d and 3d problems by extending the method to a three phase solid fluid void design method we also demonstrate design of water containing dams

topology optimization for stationary fluid structure interaction problems using a new monolithic formulation this paper outlines a new procedure for topology optimization in the steady state fluid structure interaction fsi problem a review of current topology optimization methods highlights the difficulties in alternating between the two distinct sets of governing equations for fluid and structure dynamics hereafter the fluid and structural equations respectively and in imposing coupling boundary conditions between the separated fluid and solid domains to overcome these difficulties we propose an alternative monolithic procedure employing a unified domain rather than separated domains which is not computationally efficient in the proposed analysis procedure the spatial differential operator of the fluid and structural equations for a deformed configuration is transformed into that for an undeformed configuration with the help of the deformation gradient tensor for the coupling boundary conditions the divergence of the pressure and the darcy damping force are inserted into the solid and fluid equations respectively the proposed method is validated in several benchmark analysis problems topology optimization in the fsi problem is then made possible by interpolating young s modulus the fluid pressure of the modified solid equation and the inverse permeability from the damping force with respect to the design variables

identification of a constrained nonlinear hydrological system described by volterra functional series this paper addresses the problems of identification of a constrained nonlinear system cnls described by volterra functional series a workable and practical method in terms of a penalty function principle and orthogonal expansion was developed which involves an extension of the classical least squares solution to include a penalty function with arbitrarily defined weights which are adjusted to ensure that the constraints are satisfied a total of nine basins across a range of climates and catchment areas in china were selected for examination of both daily and hourly rainfall runoff forecasting it was found that the method described in this paper can provide a more reasonable and robust response function where hydrologic constraints are required the nonlinear model yields a better streamflow forecasting than the linear model particularly in peak flows

groundwater flow in heterogeneous composite aquifers 1 we introduce a stochastic model of flow through highly heterogeneous composite porous media that greatly improves estimates of pressure head statistics composite porous media consist of disjoint blocks of permeable materials each block comprising a single material type within a composite medium hydraulic conductivity can be represented through a pair of random processes 1 a boundary process that determines block arrangement and extent and 2 a stationary process that defines conductivity within a given block we obtain second order statistics for hydraulic conductivity in the composite model and then contrast them with statistics obtained from a standard univariate model that ignores the boundary process and treats a composite medium as if it were statistically homogeneous next we develop perturbation expansions for the first two moments of head and contrast them with expansions based on the homogeneous approximation in most cases the bivariate model leads to much sharper perturbation approximations than does the usual model of flow through an undifferentiated material when both are applied to highly heterogeneous media we make this statement precise we illustrate the composite model with examples of one dimensional flows which are interesting in their own right and which allow us to compare the accuracy of perturbation approximations of head statistics to exact analytical solutions we also show the boundary process of our bivariate model is equivalent to the indicator functions often used to represent composite media in monte carlo simulations

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

spectral and bispectral analysis for single and multiple input nonlinear phreatic aquifer systems in general stochastic partial differential equations and or boundary conditions describing flow in phreatic aquifers are nonlinear this paper uses second order perturbation to approximate the response of a phreatic aquifer system subject to multiple random inputs the approach is equivalent to assuming quadratic nonlinearities in the original nonlinear phreatic aquifer system using frequency domain representations the resulting model is decomposed into linear and nonlinear quadratic frequency response functions frf for gaussian input the response of a nonlinear phreatic aquifer system is non gaussian and is typically skewed thus the variance spectrum is insufficient to describe the response process the bispectrum defined as the fourier transform of the third order cumulant sequence represents the distribution of skewness with frequency and further serves as a measure of the nonlinearity of the system given input processes of aquifer recharge and stream stage fluctuations the linear and quadratic frf are used to construct the spectrum and bispectrum of the response process the role of nonlinearity in a stream aquifer system is examined

optimal model management for multifidelity monte carlo estimation this work presents an optimal model management strategy that exploits multifidelity surrogate models to accelerate the estimation of statistics of outputs of computationally expensive high fidelity models existing acceleration methods typically exploit a multilevel hierarchy of surrogate models that follow a known rate of error decay and computational costs however a general collection of surrogate models which may include projection based reduced models data fit models support vector machines and simplified physics models does not necessarily give rise to such a hierarchy our multifidelity approach provides a framework to combine an arbitrary number of surrogate models of any type instead of relying on error and cost rates an optimization problem balances the number of model evaluations across the high fidelity and surrogate models with respect to error and costs we show that a unique analytic solution of the model management optimization problem exists under mild conditions on the models ou

contaminant spreading in stratified soils with fractal permeability distribution stochastic analysis of flow and transport in the subsurface usually assumes that the soil permeability is a stationary homogeneous stochastic process with a finite variance some field data suggest however that the permeability distributions may have a fractal character with long range correlations it is of interest to investigate how the fractal character of permeability distribution influences the spreading and mixing process in porous media the results of our analysis of this problem for perfectly stratified media with fractal distribution of permeability along the vertical are presented results were obtained for the transient and asymptotic dispersivities in the longitudinal direction the results show that the macroscopic asymptotic dispersivity depends strongly on the fractal dimension of vertical permeability distribution specifically the higher the fractal dimension the lower the value of macroscopic dispersivity the macroscopic dispersivity was found to be problem scale dependent in transient development and asymptotic phases variance of fluctuation of concentration was also analyzed and found to be dependent on the fractal dimension in this case the higher fractal dimension results in more mixing of pore water and therefore smoother smaller 2 concentration distribution

nonlocal and localized analyses of conditional mean transient flow in bounded randomly heterogeneous porous media 1 we consider the numerical prediction of transient flow in bounded randomly heterogeneous porous media driven by random sources initial heads and boundary conditions without resorting to monte carlo simulation after applying the laplace transform to the governing stochastic flow equations we derive exact nonlocal integro differential equations for the mean and variance covariance of transformed head and flux conditioned on measured values of log conductivity y ln k approximating these conditional moment equations recursively to second order in the standard deviation y of y we solve them by finite elements for superimposed mean uniform and convergent flows in a two dimensional domain an alternative conditional mean solution is obtained through localization of the exact moment expressions the nonlocal and localized solutions are obtained using a highly efficient parallel algorithm and inverted numerically back into the time domain a comparison with monte carlo simulations demonstrates that the moment solutions are remarkably accurate for strongly heterogeneous media with y2 as large as 4 the nonlocal solution is only slightly more accurate than the much simpler localized solution but the latter does not yield information about predictive uncertainty the accuracy of each solution improves markedly with conditioning a preliminary comparison of computational efficiency suggests that both the nonlocal and localized solutions for mean head and its variance require significantly less computer time than is required for monte carlo statistics to stabilize when the same direct matrix solver is used for all three we do not presently know how using iterative solvers would have affected this conclusion this is true whether the laplace inversion and monte carlo simulations are conducted sequentially or in parallel on multiple processors and regardless of problem size the underlying exact and recursive moment equations as well as the proposed computational algorithm are valid in both two and three dimensions only the numerical implementation of our algorithm is two dimensional

using the nonstationary spectral method to analyze flow through heterogeneous trending media this paper describes a nonstationary spectral theory for analyzing flow in a heterogeneous porous medium with a systematic trend in log hydraulic conductivity this theory relies on a linearization of the groundwater flow equation but does not require the stationarity assumptions used in classical spectral theories the nonstationary theory is illustrated with a two dimensional analysis of a linear trend aligned with the mean flow direction in this case closed form solutions can be obtained for the effective hydraulic conductivity head covariance and log conductivity head cross covariance the effective hydraulic conductivity decreases from the geometrical mean as the mean slope of the log conductivity increases trending leads to a reduction of head variance and a structural change in the head covariance and the log conductivity head cross covariance such changes have important implications for measurement conditioning or cokriging methods which rely on the head covariance and log conductivity head covariance the nonstationary spectral analysis is also compared with classical spectral analysis this comparison indicates that the classical spectral method correctly predicts the normalized head covariance in a linear trending media the stationary spectral method fails to capture the qualitative influence of trends on the effective hydraulic conductivity and the log conductivity head cross covariance although the magnitude of the error is relatively small for realistic values of the mean log conductivity slope the stationary and nonstationary results are the same when there is no trend in log conductivity the trending conductivity example illustrates that the nonstationary spectral method has all the capabilities of the classical spectral approach while not requiring as many restrictive assumptions

stochastic relationships for periodic responses in randomly heterogeneous aquifers 1 the aim of this work is to develop a theoretical framework for the analysis of groundwater head oscillations commonly observed in bores near boundaries of surface water bodies that are subject to periodic variations in stage height restricting attention to the linear groundwater flow equation the dynamics of head variations induced by periodic modes acting at boundaries are governed by a complex valued time independent equation parameterized by the modal frequency of interest for randomly heterogeneous aquifers the hydraulic conductivity field may be regarded as a spatial random variable stochastic relationships between the conductivity spectrum and the induced head oscillation spectrum are generated from a stochastic perturbation approach spatial correlative relationships are derived for several stochastic models incorporating up to three spatial dimensions explicit calculations of head oscillation autocovariances and spectral densities are parameterized by conductivity statistics including integral scale and variance and by modal frequency the results show that time domain head responses to periodic boundary forcing are strongly dependent on multidimensional effects and on spatial correlation structure computational simulations show that the stochastic variance estimators match simulated head fluctuation variances for a range of modal frequencies and aquifer diffusivities and that joint inversion of conductivity integral scale and variance is possible with moderate numbers of sampling points

topology optimization of non linear elastic structures and compliant mechanisms abstract a linear elastic response is assumed in most structural topology optimization problems while this assumption is valid for a wide variety of problems it is not valid for structures undergoing large displacements the elastic structural analysis used here accommodates geometric and material non linearities the material density field is filtered to enforce a length scale on the field variation and is penalized to remove less effective intermediate densities the filtering scheme is embedded in the structural analysis to resolve the non existent solution to the solid void topology problem in this way we know precisely what optimization problem is being solved the structural topology optimization formulation is also used to design compliant mechanisms

modeling nonlinear phenomena in deforming fluid saturated porous media using homogenization and sensitivity analysis concepts a new computationally efficient nonlinear model of fluid saturated porous media two scale modelling of the biot darcy continuum based on the homogenization deformation dependent coefficients are computed using the sensitivity analysis nonlinear effects captured without any need to update local microstructures nonlinearity is pronounced especially for complex geometries of pores homogenization of heterogeneous media with nonlinear effects leads to computationally complex problems requiring updating local microstructures and solving local auxiliary problems to compute characteristic responses in this paper we suggest how to circumvent such a computationally expensive updating procedure while using an efficient approximation scheme for the local homogenized coefficients so that the complexity of the whole two scale modeling is reduced substantially we consider the deforming porous fluid saturated media described by the biot model the proposed modeling approaches are based on the homogenization of the quasistatic fluid structure interaction whereby differentiation with respect to the microstructure deformation is used as a tool for linearization assuming the linear kinematics framework the physical nonlinearity in the biot continuum is introduced in terms of the deformation dependent material coefficients which are approximated as linear functions of the macroscopic response these functions are obtained by the sensitivity analysis of the homogenized coefficients computed for a given geometry of the porous structure which transforms due to the local deformation the deformation dependent material coefficients approximated in this way do not require any solving of local microscopic problems for updated configurations it appears that difference between the linear and nonlinear models depends significantly on the specific microstructure of the porous medium this observation is supported by numerical examples

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

on advective transport in fractal permeability and velocity fields we consider advective transport in a steady state random velocity field with homogeneous increments such a field is self affine with a power law dyadic semivariogram s proportional to d2 where d is distance and is a hurst coefficient it is characterized by a fractal dimension d e 1 where e is topological dimension as the mean and variance of such a field are undefined we condition them on measurement at some point x0 we then introduce a tracer at another point y0 and invoke elementary theoretical considerations to demonstrate that its conditional mean dispersion is local at all times its conditional mean concentration and variance are given explicitly by well established expressions which however have not been previously recognized as being valid in fractal fields once the conditional mean travel distance s of the tracer becomes large compared to the distance between y0 and x0 the corresponding dispersion and dispersivity tensors grow in proportion to s1 2 where 0 1 this supralinear rate of growth is consistent with that exhibited by apparent longitudinal dispersivities obtained by standard methods of interpretation from tracer behavior observed in a variety of geologic media under varied flow and transport regimes filtering out modes from the fractal velocity field with correlation scales larger than some s0 allows an asymptotic transport regime to develop when s s0 the corresponding asymptotic dispersivities grow in proportion to s02 when 0 this linear to sublinear rate of growth is consistent with that exhibited by apparent longitudinal dispersivities obtained from calibrated numerical models in a variety of media a self affine natural log permeability field gives rise to a self affine velocity field while s is sufficiently small to insure that the variance of the log permeabilities which grows as a power of s remains nominally less than one an analysis of published apparent longitudinal dispersivity data in light of the above theoretical results supports my earlier conclusion that when one juxtaposes data from a large number of generally dissimilar geologic media from a variety of locales one observes a tendency toward self affine behavior with a hurst coefficient 0 25 at any given locale such media may or may not exhibit fractal behavior if they do may or may not be close to 0 25

conditional moment analysis of steady state unsaturated flow in bounded randomly heterogeneous soils 1 we consider steady state unsaturated flow in bounded randomly heterogeneous soils under the influence of random boundary and source terms our aim is to predict pressure heads and fluxes without resorting to monte carlo simulation upscaling or linearization of the constitutive relationship between unsaturated hydraulic conductivity and pressure head we represent this relationship through gardner s exponential model while treating its exponent as a random constant and saturated hydraulic conductivity ks as a spatially correlated random field we linearize the steady state unsaturated flow equations by means of the kirchhoff transformation and integrate them in probability space to obtain exact integro differential equations for the conditional mean and variance covariance of transformed pressure head and flux after approximating these equations recursively to second order in the standard deviation y of y ln ks we solve them by finite elements for superimposed mean uniform and divergent flows in the vertical plane with and without conditioning on measured y values comparison with monte carlo solutions demonstrates that whereas our nonlocal solution is nominally restricted to mildly nonuniform media with y2 1 it yields remarkably accurate results for strongly nonuniform media with y2 at least as large as 2 this accords well with a previous theoretical analysis which shows that the solution may remain asymptotic for values of y2 as large as 2

materials with prescribed constitutive parameters an inverse homogenization problem abstract this paper deals with the construction of materials with arbitrary prescribed positive semi definite constitutive tensors the construction problem can be called an inverse problem of finding a material with given homogenized coefficients the inverse problem is formulated as a topology optimization problem i e finding the interior topology of a base cell such that cost is minimized and the constraints are defined by the prescribed constitutive parameters numerical values of the constitutive parameters of a given material are found using a numerical homogenization method expressed in terms of element mutual energies numerical results show that arbitrary materials including materials with poisson s ratio 1 0 and other extreme materials can be obtained by modelling the base cell as a truss structure furthermore a wide spectrum of materials can be constructed from base cells modelled as continuous discs of varying thickness only the two dimensional case is considered in this paper but formulation and numerical procedures can easily be extended to the three dimensional case

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

stochastic subsurface hydrology from theory to applications research on stochastic analysis of subsurface flow has developed rapidly in the last decade but applications of this approach have been very limited the purpose of this paper is to illustrate how currently available techniques and results can be used to answer important questions about the large scale behavior of naturally heterogeneous aquifers perturbation based spectral theory which presumes local statistical homogeneity provides generic theoretical results for the head variance effective conductivity tensor and macrodispersivity tensor in a field these results emphasize the key role of the variance and spatial correlation scales of the log hydraulic conductivity field field information of variances and correlation scales of natural materials is summarized the validity of some of the generic stochastic results is evaluated through comparisons with monte carlo simulations and field observations a specific field application example is developed to illustrate how the stochastic results are used to estimate large scale parameters and determine the reliability of three dimensional numerical simulations using typical log conductivity covariance parameters the effective hydraulic conductivity tensor and the macrodispersivity tensor are estimated the calculated head variance based on the simulated mean hydraulic gradient is used as a measure of the adequacy of the calculation of the steady state flow model discussion emphasizes limitations and extensions of this approach and ongoing field evaluations of the results

discrimination of fractional brownian movement and fractional gaussian noise structures in permeability and related property distributions with range analyses recent studies have shown that a useful scheme to characterize subsurface heterogeneity is to use fractional brownian motion fbm and fractional gaussian noise fgn commonly used techniques to detect these fractal distributions include rescaled range spectral and semivariogram analyses although rescaled range analysis is considered to be superior to spectral or semivariogram techniques it cannot be used to clearly discriminate fgn and fbm from a given data set in some cases on the basis of the fact that both adjusted range and rescaled range analyses for a true fgn data set should yield the same slope value in the relevant log log plots a simple scheme is developed to discriminate fgn and fbm using a combination of the two analyses the usefulness of the proposed scheme is evaluated with one dimensional computational experiments and by an application to a data set of observed air permeability logs lack of clear discrimination may have contributed to inconsistent past results and it is recommended that old data sets be reanalyzed