effective pore scale dispersion upscaling with a correlated continuous time random walk approach 1 we investigate the upscaling of dispersion from a pore scale analysis of lagrangian velocities a key challenge in the upscaling procedure is to relate the temporal evolution of spreading to the pore scale velocity field properties we test the hypothesis that one can represent lagrangian velocities at the pore scale as a markov process in space the resulting effective transport model is a continuous time random walk ctrw characterized by a correlated random time increment here denoted as correlated ctrw we consider a simplified sinusoidal wavy channel model as well as a more complex heterogeneous pore space for both systems the predictions of the correlated ctrw model with parameters defined from the velocity field properties both distribution and correlation are found to be in good agreement with results from direct pore scale simulations over preasymptotic and asymptotic times in this framework the nontrivial dependence of dispersion on the pore boundary fluctuations is shown to be related to the competition between distribution and correlation effects in particular explicit inclusion of spatial velocity correlation in the effective ctrw model is found to be important to represent incomplete mixing in the pore throats

modeling flow and transport in highly heterogeneous three dimensional aquifers ergodicity gaussianity and anomalous behavior 2 approximate semianalytical solution 1 flow and transport take place in a heterogeneous medium of lognormal distribution of the conductivity k flow is uniform in the mean and the system is defined by u mean velocity y2 log conductivity variance and integral scale i transport is analyzed in terms of the breakthrough curve of the solute identical to the traveltime distribution at control planes at distance x from the source the self consistent approximation is used where the traveltime is approximated by the sum of pertinent to the different separate inclusions and the neglected interaction between inclusions is accounted for by using the effective conductivity the pdf f x where x is the control plane distance is derived by a simple convolution it is found that f x has an early arrival time portion that captures most of the mass and a long tail which is related to the slow solute particles that are trapped in blocks of low k the macrodispersivity is very large and is independent of x the tail f x is highly skewed and only for extremely large x i depending on y2 the plume becomes gaussian comparison with numerical simulations shows very good agreement in spite of the absence of parameter fitting it is found that finite plumes are not ergodic and a cutoff of f x is needed in order to fit the mass flux of a finite plume depending on y2 and x i the bulk of f x can be approximated by a gaussian shape with fitted equivalent parameters the issue of anomalous behavior is examined with the aid of the model

a continuous variant for grunwald letnikov fractional derivatives abstract the names of grunwald and letnikov are associated with discrete convolutions of mesh h multiplied by h when h tends to zero the result tends to a marchaud s derivative of the order of of the function to which the convolution is applied the weights w k of such discrete convolutions form well defined sequences proportional to k 1 near infinity and all moments of integer order r are equal to zero provided is not an integer we present a continuous variant of grunwald letnikov formulas with integrals instead of series it involves a convolution kernel which mimics the above mentioned features of grunwald letnikov weights a first application consists in computing the flux of particles spreading according to random walks with heavy tailed jump distributions possibly involving boundary conditions

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

langevin equations for continuous time levy flights we consider the combined effects of a power law l evy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker the main point of our analysis is a formulation in terms of coupled langevin equations which allows in a natural way for the inclusion of external force fields in the anomalous case for f2 and g1 the dynamic exponent z locks onto the ratio f g drawing on recent results on l evy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder for d below the critical dimension mathit d mathit c 2f 2 the disorder is relevant corresponding to a nontrivial fixed point for the force correlation function

a mass balance based numerical method for the fractional advection dispersion equation theory and application 1 the inherent heterogeneity of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement and one approach to model solute transport through such media is the continuous time random walk ctrw one special asymptotic case of the ctrw is the fractional advection dispersion equation fade which has proven to be a promising alternative to model anomalous dispersion and has been increasingly used in hydrology to model chemical transport in both surface and subsurface water most practical problems in hydrology have complicated initial and boundary conditions and need to be solved numerically but the numerical solution of the fade is not trivial in this paper we present a finite volume approach to solve the fade where the spatial derivative of the dispersion term is fractional we also give methods to solve different boundary conditions often encountered in practical applications the linear system resulting from the temporal spatial discretization is solved using a semi implicit scheme the numerical method is derived on the basis of mass balance and its accuracy is tested against analytical solutions the method is then applied to simulate tracer movement in a stream and a near saturated hillslope in a naturally structured upland podzol field in northeast scotland

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

on diffusion in fractal porous media generalized hydrodynamics and irreversible thermodynamics are used to derive the relation between a general wave vector and frequency dependent diffusion tensor and the power spectral density the results are valid for porous media with evolving heterogeneities an integrodifferential transport equation valid in porous media with evolving heterogeneities is presented

existence of coexistence states a generic property for cyclic 3 dimensional competitive systems abstract in this paper we consider the class c t of all dissipative 3 dimensional t periodic kolmogorov competitive and cyclic systems such that the trivial solution is a source and we prove that almost every such system possesses a coexistence state more precisely we characterize an open and dense subset u of c t with respect to the topology of the uniform convergence in compact sets such that each member of u has a coexistence state

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

hopf lemma for the fractional diffusion operator and its application to a fractional free boundary problem abstract we consider a one dimensional moving boundary problem for the time fractional diffusion equation where the time fractional derivative of order 0 1 is taken in the caputo sense a generalization of the hopf lemma is proved and then used to prove a monotonicity property for the free boundary when a fractional free boundary stefan problem is investigated

multiscaling fractional advection dispersion equations and their solutions 1 the multiscaling fractional advection dispersion equation ade is a multidimensional model of solute transport that encompasses linear advection fickian dispersion and super fickian dispersion the super fickian term in these equations has a fractional derivative of matrix order that describes unique plume scaling rates in different directions the directions need not be orthogonal so the model can be applied to irregular noncontinuum fracture networks the statistical model underlying multiscaling fractional dispersion is a continuous time random walk ctrw in which particles have arbitrary jump length distributions and finite mean waiting time distributions the meaning of the parameters in a compound poisson process a subset of ctrws is used to develop a physical interpretation of the equation variables the green s function solutions are the densities of operator stable probability distributions the limit distributions of normalized sums of independent and identically distributed random vectors these densities can be skewed heavy tailed and scale nonlinearly resembling solute plumes in granular aquifers they can also have fingers in any direction resembling transport along discrete pathways such as fractures

comments on the description and initialization of fractional partial differential equations using riemann liouville s and caputo s definitions abstract in this paper it is shown that neither riemann liouville s nor caputo s definitions can be used to properly take into account initial conditions in the description of a fractional partial differential equation this demonstration is done on a counterexample some suggestions are made to address this problem

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

reprint of boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

parameter estimation for fractional transport a particle tracking approach 1 space fractional advection dispersion models provide attractive alternatives to the classical advection dispersion equation for model applications that exhibit early arrivals and plume skewness this paper develops a flexible method for estimating the parameters of the fractional transport model on the basis of spatial plume snapshots or temporal breakthrough curve data a particle tracking approach provides error bars for the parameter estimates and a general method for model fitting and comparison via optimal weighted least squares a simple model of concentration variance based on the particle tracking approach identifies the optimal weights

modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit lagrangian scheme this study develops an explicit two step lagrangian scheme based on the renewal reward process to capture transient anomalous diffusion with mixed retention and early arrivals in multidimensional media the resulting 3 d anomalous transport simulator provides a flexible platform for modeling transport the first step explicitly models retention due to mass exchange between one mobile zone and any number of parallel immobile zones the mobile component of the renewal process can be calculated as either an exponential random variable or a preassigned time step and the subsequent random immobile time follows a hyper exponential distribution for finite immobile zones or a tempered stable distribution for infinite immobile zones with an exponentially tempered power law memory function the second step describes well documented early arrivals which can follow streamlines due to mechanical dispersion using the method of subordination to regional flow applicability and implementation of the lagrangian solver are further checked against transport observed in various media results show that although the time nonlocal model parameters are predictable for transport with retention in alluvial settings the standard time nonlocal model cannot capture early arrivals retention and early arrivals observed in porous and fractured media can be efficiently modeled by our lagrangian solver allowing anomalous transport to be incorporated into 2 d 3 d models with irregular flow fields extensions of the particle tracking approach are also discussed for transport with parameters conditioned on local aquifer properties as required by transient flow and nonstationary media

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

invariant measures on g γ for split simple lie groups g we study the left action of a cartan subgroup on the space x g where is a lattice in a simple split connected lie group g of rank n 1 let be an invariant measure on x we give several conditions using entropy and conditional measures each of which characterizes the haar measure on x furthermore we show that the conditional measure on the foliation of unstable manifolds has the structure of a product measure the main new element compared to the previous work on this subject is the use of noncommutativity of root foliations to establish rigidity of invariant measures 2003 wiley periodicals inc

inertial effects in dispersion in porous media 1 in this work we develop the macroscale transport equation for dispersion of a nonreactive chemical species with a particular focus on the influence of inertial contributions at moderate reynolds numbers our starting point is the continuum level description of transport written at the subpore scale volume averaging is used to upscale these equations to develop the macroscale solute balance that applies at the darcy scale we develop a fully transient version of the ancillary closure problem that predicts the total dispersion tensor and we solve the closure using finite fourier transforms the result of this effort is a nonlocal macroscale transport equation where the nonlocal dispersion depends upon the microscale geometry of the pore space and the physical characteristics of the fluid both the longitudinal and transverse components of the total dispersion tensor are computed for a simple three dimensional unit cell the computational results indicate that a simple three dimensional periodic unit cell is able to capture the correct behavior for the longitudinal dispersion in the range 101 pep 2 5 105 although the magnitude of the longitudinal dispersion coefficient is underpredicted by up to a factor of about 4 for the transverse dispersion coefficient the simple unit cell provides much less satisfactory results when compared with experimental data the inertial effects for the longitudinal dispersion coefficient were relatively small but for the transverse dispersion coefficient inertial effects were predicted to increase the transverse dispersion coefficient up to 40 times that which would be predicted for stokes flow

boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

fractional advection dispersion equation a classical mass balance with convolution fickian flux the classical form of the partial differential equation governing advective dispersive transport of a solute in idealized porous media is the advection dispersion equation ade this equation is built on a fickian constitutive theory that gives the local dispersive flux as the inner product of a constant dispersion tensor and the spatial gradient of the solute concentration over the past decade investigators in subsurface transport have been increasingly focused on anomalous i e non fickian dispersion in natural geologic formations it has been recognized that this phenomenon involves spatial and possibly temporal nonlocality in the constitutive theory describing the dispersive flux most recently several researchers have modeled anomalous dispersion using a constitutive theory that relies on fractional as opposed to integer derivatives of the concentration field the resulting ade itself is expressed in terms of fractional derivatives and they are described as fractional ades here we show that the fractional ade is obtained as a special case of the authors convolution fickian nonlocal ade cushman and ginn 1993 to obtain the fractional ade from the convolution fickian model requires only a judicial choice of the wave vector and frequency dependent dispersion tensor

non fickian dispersion of groundwater age 1 we expand the governing equation of groundwater age to account for non fickian dispersive fluxes using continuous random walks groundwater age is included as an additional fifth dimension on which the volumetric mass density of water is distributed and we follow the classical random walk derivation now in five dimensions the general solution of the random walk recovers the previous conventional model of age when the low order moments of the transition density functions remain finite at their limits and describes non fickian age distributions when the transition densities diverge previously published transition densities are then used to show how the added dimension in age affects the governing differential equations depending on which transition densities diverge the resulting models may be nonlocal in time space or age and can describe asymptotic or preasymptotic dispersion a joint distribution function of time and age transitions is developed as a conditional probability and a natural result of this is that time and age must always have identical transition densities this implies that a transition density defined for age can substitute for a density in time and this has implications for transport model parameter estimation we present examples of simulated age distributions from a geologically based heterogeneous domain that exhibit non fickian behavior and show that the non fickian model provides better descriptions of the distributions than the fickian model

measurable quotients of unipotent translations on homogeneous spaces let u be a nilpotent unipotent subgroup of a lie group g and let f be a closed subgroup of g marina ratner showed that every ergodic u invariant probability measure on the homogeneous space f g is of a simple algebraic form we use this fundamental new result to show that every measurable quotient of the u action on j g is of a simple algebraic form roughly speaking any quotient is a double coset space a g k

iterative scheme for the observation of a competitive lotka volterra system abstract in this work in terms of the model parameters sufficient conditions are established to construct a sequence of approximate observers for a two species competitive lotka volterra system this iterative approach makes it possible to localize the solution of the system and reveal its long term behaviour the main results are also illustrated by numerical simulations

tempered fractional calculus fractional derivatives and integrals are convolutions with a power law multiplying by an exponential factor leads to tempered fractional derivatives and integrals tempered fractional diffusion equations where the usual second derivative in space is replaced by a tempered fractional derivative govern the limits of random walk models with an exponentially tempered power law jump distribution the limiting tempered stable probability densities exhibit semi heavy tails which are commonly observed in finance tempered power law waiting times lead to tempered fractional time derivatives which have proven useful in geophysics the tempered fractional derivative or integral of a brownian motion called a tempered fractional brownian motion can exhibit semi long range dependence the increments of this process called tempered fractional gaussian noise provide a useful new stochastic model for wind speed data a tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations and it also provides a useful new correlation model in time series

joinings of higher rank diagonalizable actions on locally homogeneous spaces we classify joinings between a fairly general class of higher rank diagonalizable actions on locally homogeneous spaces in particular we classify joinings of the action of a maximal r split torus on g with g a simple lie group of r rank 2 and g a lattice we deduce from this a classification of measurable factors of such actions as well as certain equidistribution properties

space fractional advection dispersion equations with variable parameters diverse formulas numerical solutions and application to the macrodispersion experiment site data 1 to model the observed local variation of transport speed an extension of the homogeneous space fractional advection dispersion equation fade to more general cases with space dependent coefficients drift velocity v and dispersion coefficient d has been suggested to provide a rigorous evaluation of this extension we explore the underlying physical meanings of two proposed and one other possible form of the fade by using the generalized mass balance law proposed by meerschaert et al 2006 when the classical fick s law is replaced by its generalized form in the first order mass conversation law the original fade with constant parameters extends to the advection dispersion equation ade with fractional flux denoted as ff ade when the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence we get the ade with fractional divergence fd ade when the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form the fade contains fully fractional divergence ffd ade these three fades with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by cushman et al 1994 while the space variable fades correspond to the conditional nonlocal theory proposed by neuman 1993 after specifying the general levy type functional form of the random distribution the corresponding langevin markov models can be found in many cases where the lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical measurable level and resolution the resulting lagrangian random walk particle tracking methods along with previous numerical solutions using implicit euler finite differences distinguish and elucidate the plume behavior described by these fades the fractional models are applied to fit the tritium plumes measured at the macrodispersion experiment test site when the local parameters gleaned from the hydraulic conductivity k distribution are varied even slightly i e a two zone model allowing the use of observed hydraulic conditions at the site the model fits are well within the variability of the data the extended fades describe the fast and space dependent leading edges of measured plumes in the regional scale alluvial system which was underestimated and could not be fully captured by the original fade with constant parameters applications also favor the fd ade model because of the ease of implementation and consistency with previous analysis of the k statistics

upscaling transport with mass transfer models mean behavior and propagation of uncertainty 1 the choice of an adequate large scale conceptual transport model constitutes a major challenge associated with the upscaling of solute transport among the different alternatives to the classical advection dispersion model the multirate mass transfer model has been proposed as a valuable and convenient alternative to model the large scale behavior of solute transport this paper evaluates the use of mass transfer models as a constitutive equation for upscaling solute transport to achieve this we compare monte carlo simulations of solute transport at two different support scales transport simulations performed at the smallest scale represent a set of reference transport solutions described at a high resolution which are contrasted against transport simulations obtained using an upscaled model low resolution several formulations of the multirate mass transfer model which differ in the type of memory function single rate double rate and truncated power law are used as a constitutive transport equation the large scale scenario represents a simplified model obtained by partially homogenizing the reference solution results show that the double rate and the truncated power law mass transfer models are capable of properly describing the ensemble average behavior of the main features associated with the integrated breakthrough curves however the uncertainty associated with the upscaled mass transfer models was substantially smaller than that attributed to the reference solution importantly the cumulative distribution function of concentrations associated with the upscaled model follows a distribution similar to the reference solution but with smaller statistical dispersion the reason is that while appropriate memory functions can be used to preserve the residence time distribution of mass particles during upscaling the lack of memory in space prevents the model from reproducing mass fluxes in all directions specifically the reproduction of mass fluxes taking place at the interface between two homogenized blocks of the upscaled model is not satisfied thus providing a poor description of the spatial distribution of mass particles in a given realization

moment analysis for spatiotemporal fractional dispersion 1 the evolution of the first five nonnegative integer order spatial moments corresponding to the mass mean variance skewness and kurtosis are investigated systematically for spatiotemporal nonlocal fractional dispersion three commonly used fractional order transport equations including the time fractional advection dispersion equation time fade the fractal mobile immobile mim equation and the fully fractional advection dispersion equation ffade are considered analytical solutions verify our numerical results and reveal the anomalous evolution of the moments following adams and gelhar s 1992 work on the classical ade we find that a simultaneous analysis of all moments is critical in discriminating between different nonlocal models the evolution of dispersion among the subdiffusive to superdiffusive rates is then further explored numerically by a non markovian random walk particle tracking method that can be used for any heterogeneous boundary or initial value problem in three dimensions both the analytical and the numerical results also show the similarity at the early time and the difference at the late time of moment growth for solutes in different phases mobile versus total described by the mim models further simulations of the 1 d bromide snapshots measured at the made experiments using all three models with parameters fitted by the observed zeroth to fourth moments indicate that 1 both the time and space nonlocality strongly affect the solute transport at the made site 2 all five spatial moments should be considered in transport model selection and calibration because those up to the variance cannot effectively discriminate between nonlocal models and 3 the log concentration should be used when evaluating the plume leading edge and the effects of space nonlocality

possible problems of scale dependency in applications of the three dimensional fractional advection dispersion equation to natural porous media 1 a three dimensional 3 d analysis of transport and macrodispersion at the macrodispersion experiment made site boggs et al 1993 using the fractional advection dispersion equation fade developed by meerschaert et al 1999 2001 shows that the levy dispersion process is scale dependent levy dispersion may be superior to gaussian dispersion on a sufficiently small scale on larger scales both theories are likely to suffer from the fact that because of depositional structures most flow fields display an evolving nonstationary structure motion in such fields is advection dominated displays a lot of memory and therefore is not modeled well by markov random processes which underlie the derivation of both the gaussian and levy advection dispersion equations berkowitz et al 2002 to improve plume simulation of an advection dominated transport process one would have to bring in more advective irregularity while simultaneously decreasing the levy dispersion coefficient therefore on a 3 d basis first order levy dispersion has limitations similar to gaussian dispersion however this and related theories such as the continuous time random walk ctrw formalism are in the early stages of development and thus may be fruitful areas for further research