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

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

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

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

a birkhoff lewis type theorem for the nonlinear wave equation we give an extension of the celebrated birkhoff lewis theorem to the nonlinear wave equation accordingly we find infinitely many periodic orbits with longer and longer minimal periods accumulating at the origin which is an elliptic equilibrium of the associated infinite dimensional hamiltonian system

some new results on stability of ekeland s ϵ variational principle abstract in this paper we give an example and point out that solutions of ekeland s variational principle are not always lower semicontinuous in infinite dimensional banach spaces even with respect to the uniform metric further the example shows that the solutions need not be almost lower semicontinuous when the convergence of sequence of functions is weakened to painleve kuratowski epigraphical convergence to provide some results of stability we prove the almost lower semicontinuity of solutions in a general framework

a fractal based stochastic interpolation scheme in subsurface hydrology real porosity and hydraulic conductivity data do not vary smoothly over space so an interpolation scheme that preserves irregularity is desirable such a scheme based on the properties of fractional brownian motion fbm and fractional gaussian noise fgn is presented following the methodology of hewett 1986 the authors test for the presence of fgn in a set of 459 hydraulic conductivity k measurements the use of rescaled range analysis strongly indicated the presence of fgn when applied to the natural logs of the k data and the resulting hurst coefficient h was determined to be 0 82 this h value was then used along with the methodology for successive random additions to generate a fbm k interpolation realization in the vertical cross section between two wells the results appeared realistic and the overall methodology presented herein may serve as an improved basis for a conditional simulation approach to the study of various transport processes in porous media copyright c 1993 american geophysical union

periodic solutions for completely resonant nonlinear wave equations with dirichlet boundary conditions we consider the nonlinear string equation with dirichlet boundary conditions utt uxx u with u u3 o u5 odd and analytic 0 and we construct small amplitude periodic solutions with frequency for a large lebesgue measure set of close to 1 this extends previous results where only a zero measure set of frequencies could be treated the ones for which no small divisors appear the proof is based on combining the lyapunov schmidt decomposition which leads to two separate sets of equations dealing with the resonant and non resonant fourier components respectively the q and the p equations with resummation techniques of divergent powers series allowing us to control the small divisors problem the main difficulty with respect to the nonlinear wave equations utt uxx mu u m 0 is that not only the p equation but also the q equation is infinite dimensional

transport in multiscale log conductivity fields with truncated power variograms di federico and neuman this issue investigated mean uniform steady state groundwater flow in an unbounded domain with log hydraulic conductivity that forms a truncated multiscale hierarchy of statistically homogeneous and isotropic gaussian fields each associated with an exponential autocovariance here we present leading order expressions for the displacement covariance and dispersion coefficient of an ensemble of solute particles advected through such a flow field in two or three dimensions both quantities are functions of the mean travel distance s the hurst coefficient h and the low and high frequency cutoff integral scales l and u the latter two are related to the length scales of the sampling window region under investigation and sample volume data support respectively if one considers transport to be affected by a finite domain much larger than the mean travel distance so that s l then an early preasymptotic regime develops during which longitudinal and transverse dispersivities grow linearly with s if one considers transport to be affected by a domain which increases in proportion to s then l and s are of similar order and a preasymptotic regime never develops instead transport occurs under a regime that is perpetually close to asymptotic under the control of an evolving scale l s we show that if additionally u l then the corresponding longitudinal dispersivity grows in proportion to l1 2h or equivalently s1 2h both these preasymptotic and asymptotic theoretical growth rates are shown to be consistent with the observed variation of apparent longitudinal fickian dispersivities with scale we conclude by investigating the effect of variable separations between cutoff scales on dispersion

comparison of fickian and temporally nonlocal transport theories over many scales in an exhaustively sampled sandstone slab 1 it is not a compelling argument solely on the basis of a better fit to solute breakthrough curve btc data that a temporally nonlocal model is necessary to simulate transport in an advection dominated system one may counter that the classical advection dispersion equation ade is a valid model at some small scale and that the detailed hydraulic conductivity k data must be well represented then the nonlocality is only a result of upscaling and loss of information but is the nonlocal model demonstrably necessary at all scales we examine the experiment conducted by klise et al 2008 in which a 30 5 30 5 cm slab of relatively homogeneous cross bedded sandstone was exhaustively sampled for k the slab was sealed saturated with potassium iodide and x rayed 10 times while being flushed with fresh water the 8 649 air permeameter measurements were down and upscaled to make finer and coarser grids on which the velocity field was solved and the ade applied the optimized parameters in the ade were found to scale predictably most notably the longitudinal dispersivity which grew linearly with upscaling but at all levels of up and downscaling including the original k measurement scale of 0 33 cm the ade did not adequately represent the late time tails the temporally nonlocal time fractional ade t fade was applied and the optimized parameters and the immobile capacity did not depend on scale the better fit provided by the t fade in the late btc tails did not bring about a sacrificed fit elsewhere in the btc furthermore the optimized ade and t fade solutions do not converge at the smallest scale directly implying that the temporal nonlocality is a necessary model component we conclude that the logical inference if the ade is valid in heterogeneous material then there is tailing in the btc is not a proof that the reverse is true we provide a clear counterexample a corollary is that a mismatch between data and a discretized solution to the ade does not imply that more data will improve fits or predictive ability

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

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

periodic solutions of some lienard and duffing equations on etudie certains aspects du probleme periodique u f u u g t u 0 u 0 u 2 u 0 u 2 ou f est une fonction continue sur r et g t u est une fonction de caratheodory soumise a certaines restrictions de croissances on donne deux resultats d existence

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

periodic solutions of nonlinear wave equations on obtient des conditions suffisantes pour l existence de solutions doublement periodiques de l equation d onde non lineaire u ts f t s u u t t s u t s u t s t ou f est une fonction continue donnee dans r 3 et f est t periodique en t et s

stability results for ekeland s e variational principle and cone extremal solutions given x a banach space and f x a a proper lower semicontinuous function which is bounded from below the ekeland s e variational principle asserts the existence of a point xi in x which we call e extremal with respect to f which satisfies fu fxi e u xi for all u x u xi by using set convergence notions kuratowski painleve mosco bounded hausdorff and their epigraphical versions we study the semi continuity properties of the mapping which to f associates e ext f the set of such e extremal points the key for the geometrical understanding of such properties is to consider the equivalent phelps extremization principle which given a closed set d in x and a partial ordering with respect to a pointed cone associates the set of elements of d maximal with respect to this order direct or potential applications are given in various fields multicriteria optimization numerical algorithmic calculus of variations

a semilinear elliptic problem which is not selfadjoint cesari 7 i in a preceding paper established conditions sufficient that a semilinear elliptic boundary value problem possibly not selfadjoint shall have a solution

hydraulic conductivity velocity and the order of the fractional dispersion derivative in a highly heterogeneous system 1 a one dimensional fractional order advection dispersion equation accurately models the movement of the core of the tritium plume at the highly heterogeneous made site an a priori estimate of the parameters in that equation including the order of the fractional dispersion derivative was based on the assumption that the observed power law heavy tail of the hydraulic conductivity k field would create a similarly distributed velocity field monte carlo simulations were performed to test this hypothesis results from the monte carlo analysis show that heavy tailed k fields do give rise to heavy tailed velocity fields however the exponent of the power law the tail parameter describing these two distributions is not necessarily the same the tail parameter that characterizes a velocity distribution is not solely dependent on the tail parameter that characterizes the k distribution the k field must also have long range dependence so that water may flow through relatively continuous high k channels

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

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

anisotropy lacunarity and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms it has been shown by di federico and neuman 1997 1998a b that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework which views hydraulic conductivity as a random fractal characterized by a power variogram such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments di federico and neuman 1997 have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated fields modes that are homogeneous and isotropic we show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic provided the ratios between principal integral spatial correlation scales are the same for all modes we then analyze the effect of filtering out truncating modes of low high and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous the integral scales of the lowest and highest frequency cutoff modes are related to length scales of the sampling window domain and data support sample volume respectively taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale in a manner previously shown to be consistent with observations in the isotropic case it also allows in principle bridging across scales at a specific locale as well as among locales by adopting either site specific or generalized variogram parameters the introduction of intermediate cutoffs allows us to account in a straightforward manner for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales for example where textural and structural features are associated with distinct ranges of scale such as fractures having discrete ranges of trace length and density which dissect the rock into matrix blocks having corresponding ranges of sizes we explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale variance covariance and spectra of a truncated fractal field we then develop an expression for the equivalent hydraulic conductivity of a box shaped porous block embedded within a multiscale log hydraulic conductivity field under mean uniform flow the block is larger than the support scale of the field but is smaller than a surrounding sampling window consequently its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function conditioned on a known mean value of support scale conductivity across the window are given explicitly by our multiscale theory

multifractal analyses of hydraulic conductivity distributions the concept of a universal multifractal a generalization of a monofractal is a recently developed scaling model for natural phenomena characterized by irregularity presented herein is an effort to use universal multifractal concepts to deal with spatial variations of hydraulic conductivity k which have a significant effect on contaminant transport in the subsurface structure function analyses of four k data sets show that some vertical variations of k display multifractal structures while others are consistent with monofractal behavior in order to make multifractal concepts more useful multifractal noise is introduced and defined as the increments of a multifractal it is concluded that the multifractal formalism of schertzer and lovejoy 1987 has provided a rather general approach for modeling in k variations in the vertical with the exception of horizontal variations of the borden data all results fell within the domain of universal multifractal behavior which includes the monofractal case parameters were well defined in an empirical sense and easy to calculate indicating a robust formalism results were consistent with the recent finding of liu and molz 1997 also non gaussian and scale variant behavior in hydraulic conductivity distributions submitted to water resources research 1997 hereinafter referred to as submitted paper that k variations display increasing heterogeneity at decreasing scales

new stability results for ekeland s ϵ variational principles for vector valued and set valued maps in this paper we discuss the stability of ekeland s variational principles for vector valued and set valued maps when the dominating cone is a closed pointed convex cone whose interior may be empty we provide a new approach to the study of the stability of ekeland s variational principles for vector valued and set valued maps without scalarization

on the equivalence of some basic principles in variational analysis the primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization considering a broad class of banach spaces admitting smooth renorms with respect to some bornology we establish an equivalence between useful versions of a smooth variational principle for lower semicontinuous functions an extremal principle for nonconvex sets and an enhanced fuzzy sum rule formulated in terms of viscosity normals and subgradients with controlled ranks further refinements of the equivalence result are obtained in the case of a frechet differentiable norm based on the new enhanced sum rule we provide a simplified proof for the refined sequential description of approximate normals and subgradients in smooth spaces

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

monte carlo studies of flow and transport in fractal conductivity fields comparison with stochastic perturbation theory a monte carlo simulation of flow and transport is employed to study tracer migration in porous media with evolving scales of heterogeneity fractal media transport is studied with both conservative and reactive chemicals in media that possess physical as well as chemical heterogeneity linear kinetic equations are assumed to relate the sorbed phase and the aqueous phase concentrations the fluctuating log conductivity possesses the power law spectrum of a fractional brownian motion fbm chemical heterogeneity is represented as spatially varying reaction rates that also are assumed to obey fbm statistics and may be correlated to the conductivity field the model is based on a finite difference approximation to the flow problem and a random walk particle tracking approach for solving the solute transport equation the model is used to make comparisons with the nonlocal transport equations recently developed by deng et al 1993 and hu et al 1995 1997 the results presented herein support these nonlocal models for a wide range of heterogeneous systems however the infinite integral scale associated with the fractal conductivity has a significant effect on the prediction of the nonlocal theories this suggests that integral scale should play a role in stochastic eulerian perturbation theories the importance of the local scale dispersion depends to a great extent on the magnitude of the local dispersivities the effect of neglecting local dispersion decreases with the decrease in local dispersivity

random walk simulation of transport in heterogeneous porous media local mass conservation problem and implementation methods the random walk method for simulating solute transport in porous media is typically based on the assumption that the velocity and velocity dependent dispersion tensor vary smoothly in space however in cases where sharp interfaces separate materials with contrasting hydraulic properties these quantities may be discontinuous normally velocities are interpolated to arbitrary particle locations when finite difference or finite element methods are used to solve the flow equation the use of interpolation schemes that preserve discontinuities in velocity at material contacts can result in a random walk model that does not locally conserve mass unless a correction is applied at these contacts test simulations of random walk particle tracking with and without special treatment of material contacts demonstrate the problem techniques for resolving the problem including interpolation schemes and a reflection principle are reviewed and tested results from simulations of transport in porous media with discontinuities in the dispersion tensor show which methods satisfy continuity simulations of transport in two dimensional heterogeneous porous media demonstrate the potentially significant effect of using a nonconservative model to compute spatial moments and breakthrough of a solute plume

analyzing exact fractal time series evaluating dispersional analysis and rescaled range methods precise reference signals are required to evaluate methods for characterizing a fractal time series here we use fgp fractional gaussian process to generate exact fractional gaussian noise fgn reference signals for one dimensional time series the average autocorrelation of multiple realizations of fgn converges to the theoretically expected autocorrelation two methods commonly used to generate fractal time series an approximate spectral synthesis ssm method and the successive random addition sra method do not give the correct correlation structures and should be abandoned time series from fgp were used to test how well several versions of rescaled range analysis r s and dispersional analysis disp estimate the hurst coefficient 0 0 9 r s detrended overestimates h for time series with h 0 7 estimates of h from all versions of disp usually have lower bias and variance than those from r s all versions of dispersional analysis disp now tested on fgp are better than we previously thought and are recommended for evaluating time series as long memory processes

ensemble solute transport in two dimensional operator scaling random fields 1 motivated by field measurements of aquifer hydraulic conductivity k recent techniques were developed to construct anisotropic fractal random fields in which the scaling or self similarity parameter varies with direction and is defined by a matrix ensemble numerical results are analyzed for solute transport through these two dimensional operator scaling fractional brownian motion ln k fields both the longitudinal and transverse hurst coefficients as well as the radius of isotropy are important to both plume growth rates and the timing and duration of breakthrough it is possible to create operator scaling fractional brownian motion fields that have more continuity or stratification in the direction of transport the effects on a conservative solute plume are continually faster than fickian growth rates highly non gaussian shapes and a heavier tail early in the breakthrough curve contrary to some analytic stochastic theories for monofractal k fields the plume growth rates never exceed a mercado s 1967 purely stratified aquifer growth rate of plume apparent dispersivity proportional to mean distance apparent superstratified growth must be the result of other demonstrable factors such as initial plume size