second order expressions for velocity moments in two and three dimensional statistically anisotropic media second order expressions are derived for the mean and covariance of steady state seepage velocity under mean uniform flows in infinite two and three dimensional domains the order of approximation is defined in terms of the variance 2 of a statistically homogeneous and anisotropic natural log hydraulic conductivity field y with a gaussian spatial autocorrelation function results show that second order mean velocity either exceeds or is close to its first order counterpart depending on anisotropy head fluctuations of order larger than affect second order velocity moments to the same extent as do head fluctuations of order in virtually all cases hence neglecting the former renders the results nonasymptotic velocity variances are generally larger when approximated consistently to second than to first order the ratio between second and first order variance approximations is larger in three than in two dimensions larger for transverse than for longitudinal velocity and increases with 2 anisotropy has a significant effect on second order velocity variance second order effects have the greatest influence on longitudinal velocity variance at extreme anisotropy ratios and on transverse velocity variance in isotropic domains

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

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

higher order effects on flow and transport in randomly heterogeneous porous media advective solute transport in nonuniform geologic media is generally nonlocal and non fickian neuman 1993 cushman and ginn 1993 in statistically homogeneous log conductivity fields under uniform mean flow the transport is expected to become asymptotically local and fickian at late time during the earlier preasymptotic regime macrodispersivity a measure of the rate at which a plume spreads is expected to vary with solute residence time a first order linear in the natural log hydraulic conductivity variance 2 analysis of this variation has been performed by dagan 1984 1987 1988 he found that when local dispersion is neglected the longitudinal macrodispersivity increases monotonically from zero toward a constant asymptote however the transverse macrodispersivity first increases from zero to a peak value then decreases monotonically toward zero the first order asymptotic analyses by winter 1982 gelhar and axness 1983 and winter et al 1984 also yield zero transverse macrodispersivity when local dispersion is disregarded

a new insight into serre s reduction problem abstract the purpose of this paper is to study the connections existing between serre s reduction of linear functional systems which aims at finding an equivalent system defined by fewer equations and fewer unknowns and the decomposition problem which aims at finding an equivalent system having a diagonal block structure in which one of the diagonal blocks is assumed to be the identity matrix in order to do that we further develop results on serre s reduction problem and on the decomposition problem obtained in 2 3 finally these techniques are used to analyze the decomposability of linear systems of partial differential equations studied in hydrodynamics such as the oseen equations

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

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

on generating conductivity fields with known fractal dimension and nonstationary increments 1 fractional brownian motion fbm is a stochastic process that has stationary increments with long range correlations and known fractal dimension we study a multiple dimensional extension of fbm with nonstationary increments that allows for trends in the statistical structure while maintaining the gaussian nature and fractal dimension of fbm two methods for simulating this extension are employed and described in detail one approach combines cholesky decomposition with a generalization of random midpoint displacement the other makes repeated use of the cholesky decomposition the resulting fields can be employed in various geophysical settings e g as log conductivity fields in hydrology and topographic elevation in geomorphology

generalized scale invariance in the atmosphere and fractal models of rain in recent years there has been considerable interest in stochastic rain models by developing new ideas about scale invariance and intermittency we argue that the scope of such models can be greatly extended the notions of scale invariance intermittency and the associated idea of fractal dimension have lately gained considerable ground particularly in the context of extremely variable phenomena such as those found in the mesoscale and in hydrodynamic turbulence we review some relevant work and argue that the atmosphere respects a symmetry principle that we call generalized scale invariance in which the statistical properties of the small and large scale are related to each other by a magnification coupled with a differential stratification due to gravity and differential rotation due to the coriolis force we further argue that the extremely erratic intermittent nature of the atmosphere is characterized by scale invariant fractal measures leading to hyperbolic fat tailed probability distributions of the fluctuations the standard statistical methods based on exponential fall offs in both correlations and probability distributions are inappropriate when the variability is of this type we show how both the scaling and intermittency may be exploited to develop extremely variable stochastic models of rain although the models examined here are the simplest of a family of extremely variable processes depending on only two radar determined parameters they lead to realizations possessing many realistic features of the rainfield including complexity of form clustering and bands at all scales as well as differential stratification and rotation finally we point to weaknesses in the models in particular their monodimensional nature and suggest possible improvements

on higher order corrections to the flow velocity covariance tensor second order log fluctuating conductivity variance 2 corrections to the head and velocity covariance functions are derived for a lognormal stationary hydraulic conductivity field the fourier transform method proposed by deng et al 1993 is used extensively to obtain numerical estimates of these functions for an exponential log fluctuating conductivity covariance it is shown that the velocity covariance is insensitive to second order corrections in the head field the velocity covariance on the other hand is highly sensitive to second order corrections in the velocity when the log fluctuating conductivity variance approaches unity a closed expression is derived for a second order correction to the velocity variance when there is no second order correction to the head field the longitudinal second order correction to the velocity variance is 0 4 2 different from the first order approximation in isotropic media 1 5 2 different in a highly stratified formation and no different when the ratio of vertical to horizontal integral scales approaches infinity the second order corrections to the horizontal and vertical transverse velocity variances are 2 2 different from the first order approximations for both isotropic and anisotropic systems

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

multifractal analysis of flow velocity simulated with the lattice model approach in idealized three dimensional porous media 1 the strange attractor formalism has been used to check whether the flow velocity in idealized three dimensional porous media could be described by the multifractal approach the flow velocity magnitude was simulated with a lattice model in six media with different porosities the results obtained from the analysis of the mass exponent functions and the multifractal and generalized dimensions spectra show that the velocity of the simulated flows has a multifractal type of scaling that becomes stronger as the porosity decreases a correlation analysis has been performed showing the relationships between the statistical properties and the multifractal parameters of the velocity magnitude distribution

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

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

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

a note on higher order corrections of the head covariances in steady aquifer flow average uniform flow takes place in a heterogeneous aquifer of infinite extent the input to the problem is the hydraulic conductivity which is regarded as a random space function that is lognormal stationary and statistically isotropic the output is the water head field which is a random function satisfying the equation of steady flow the first order approximation of the head in an asymptotic expansion for small log conductivity variance is a normal function characterized completely by the head log conductivity cross covariance and the head covariance or variogram these covariances are proportional to the log conductivity variance by using spectral methods second order corrections of the head covariances proportional to the log conductivity variance squared are derived explicitly detailed calculations are carried out for an exponential log conductivity covariance the main finding of the note is that the first order approximation is very robust and even for a log conductivity variance equal to unity the second order correction of the head variances is smaller than 10 of the first order approximation

information geometry and phase transitions abstract the introduction of a metric onto the space of parameters in models in statistical mechanics and beyond gives an alternative perspective on their phase structure in such a geometrisation the scalar curvature r plays a central role a non interacting model has a flat geometry r 0 while r diverges at the critical point of an interacting one here the information geometry is studied for a number of solvable statistical mechanical models

the information geometry of the one dimensional potts model in various statistical mechanical models the introduction of a metric into the space of parameters e g the temperature variable and the external field variable h in the case of spin models gives an alternative perspective on the phase structure for the one dimensional ising model the scalar curvature of this metric can be calculated explicitly in the thermodynamic limit and is found to be 1 cosh h sinh2 h exp 4 this is positive definite and for physical fields and temperatures diverges only at the zero temperature zero field critical point of the model in this paper we calculate for the one dimensional q state potts model finding an expression of the form a q h b q h q h where q h is the potts analogue of sinh2 h exp 4 this is no longer positive definite but once again it diverges only at the critical point in the space of real parameters we remark however that a naive analytic continuation to complex field reveals a further divergence in the ising and potts curvatures at the lee yang edge

higher order corrections to the flow velocity covariance tensor revisited recently the authors presented an accurate in the variance of fluctuating log conductivity second order solution to the steady saturated flow problem the longitudinal second spatial moment was evaluated numerically on the basis of the perturbation solution and it was found that second order corrections to the head field had little effect on this moment here we use the same theory and evaluate the second transverse spatial moment we find in agreement with others that our theory suggests that the second order head corrections as well as the nonlinear velocity corrections are important

effective saturated hydraulic conductivity of two dimensional random multifractal fields 0 991 p 0 0001 thisrelationship indicates that hkidecreases with increasing long range spatial correlation among the k values in the field furthermore as hki decreases it becomes increasingly dominated by flow channeling this is because high values of k become more and more clustered as p decreases this approach may prove useful for the prediction of hki from generalized dimensions estimated by multifractal analysis of field measurements of k the results may also be applicable to the design of sampling strategies for multiple small scale slug tests at a given resolution citation koirala s r e perfect r w gentry and j w kim 2008 effective saturated hydraulic conductivity of two dimensional random multifractal fields water resour res 44 w08410 doi 10 1029 2007wr006199

flow through porous media with multifractal hydraulic conductivity 1 we make a nonlinear analysis of flow through saturated porous media when the hydraulic conductivity k is an isotropic lognormal field with multifractal scale invariance in this case logk is isotropic gaussian with spectral density where d is the space dimension our main result is that the hydraulic gradient h and specific flow q are also multifractal fields whose renormalization under space contraction involves random rotation of the field and random scaling of its amplitude the scaling properties and marginal distributions of h and q are obtained analytically as functions of the space dimension d and a multifractal parameter of k the codimension ck the fields h and q are anisotropic at large scales but approach isotropy at very small scales using scaling arguments we obtain the effective conductivity of the medium keff as an explicit function of d ck and the scaling range of k