Dispersion In Heterogeneous Porous Media Research Articles

Smoothed Particle Hydrodynamics for anisotropic dispersion in heterogeneous porous media

In the context of modeling solute transport through heterogeneous porous media, particle methods possess inherent advantages with respect to mesh-based (Eulerian) methods. In Smoothed Particle Hydrodynamics (SPH), particles represent fluid volumes exchanging concentrations with their neighbors to emulate hydrodynamic dispersion, and advection is simulated by the particles’ displacement. This crucially prevents problems that are otherwise typically associated with Eulerian advection schemes (especially at high grid-Péclet numbers), such as numerical diffusion. Despite the advantages of SPH, modeling dispersion with anisotropic coefficients remains a challenge for the approach, with studies reporting unphysical negative concentrations in conservative problems. This has likely hindered its practical use because dispersion is intrinsically anisotropic in porous media. This article provides a review and numerical evaluation of SPH for simulating dispersion, focusing on three formulations compatible with anisotropic dispersion coefficients. The analysis includes a scheme for which negative concentrations have been formerly reported, plus two more recently developed methods which are applied here for the first time to the problem of anisotropic dispersion in heterogeneous porous media. The SPH schemes are tested under different degrees of dispersion anisotropy for both homogeneous and heterogeneous velocity fields. The results indicate that the newer SPH schemes can produce accurate results without negative concentrations while considering anisotropic dispersion, providing a valid alternative to simulate solute transport through heterogeneous domains.

Linear Stochastic Analysis of the Partial Reversibility of Ensemble and Effective Dispersion in Heterogeneous Porous Media

AbstractMacrodispersion in heterogeneous formations is caused by spatial variability of the velocity field, making different parts of a plume experience different advective displacements. Differential advection interacts with diffusion, which hardly affects longitudinal ensemble dispersion (i.e., the spread of the ensemble‐averaged concentration) but determines effective dispersion, that is, the expected spread of individual plumes for point injections. The latter has been suggested as metric of solute mixing. Pure advection is fully reversible, whereas diffusion is completely irreversible. We quantify the partial reversibility of macrodispersion by analyzing the second central ensemble and effective spatial moments for advective‐diffusive transport in heterogeneous domains with flow reversal, applying linear stochastic theory to approximate the corresponding moments and comparing them to particle‐tracking random‐walk simulations in periodic media. Diffusion causes solute particles to deviate from their forward trajectories when flow is reversed. As long as advective memory dominates, both types of second central moments decrease during backward motion, then reach a minimum, and increase again. The reversibility is considerably bigger for ensemble than effective dispersion but the latter also shows partial reversibility, challenging its use as metric of mixing. The stronger diffusion is in comparison to advection, the less reversible dispersion becomes. After equally long times of forward and backward motion, the two types of second central moments differ, but to a much smaller extent than in pure forward motion. In realistic settings, the advective memory affects dispersion so strongly that the asymptotic regime is not reached before the plume center has returned to its origin.

Mechanisms, Upscaling, and Prediction of Anomalous Dispersion in Heterogeneous Porous Media

AbstractWe study the upscaling and prediction of large‐scale solute dispersion in heterogeneous porous media with focus on preasymptotic or anomalous features such as tailing in breakthrough curves and spatial concentration profiles as well as nonlinear evolution of the spatial variance of the concentration distribution. Spatial heterogeneity in the hydraulic medium properties is represented in a stochastic modeling approach. Direct numerical Monte Carlo simulations of flow and advective particle motion combined with a Markov model for streamwise particle velocities give insight in the mechanisms of preasymptotic and asymptotic solute transport in terms of the statistical signatures of the medium and flow heterogeneity. Based on the representation of equidistantly sampled particle velocities as a Markov process, we derive an upscaled continuous time random walk approach that can be conditioned on the flow velocities and thus hydraulic conductivity in the injection region. In this modeling framework, we identify the Eulerian velocity distribution, advective tortuosity, and the correlation length of particle velocities as the key quantities for large‐scale transport prediction. Thus, the upscaled model predicts the spatial concentration profiles, their first and second centered moments, and the breakthrough curves obtained from direct numerical Monte Carlo simulations in spatially heterogeneous conductivity fields. The presented approach allows to relate the medium and flow properties to large‐scale preasymptotic and asymptotic solute dispersion.

Modelling solute dispersion in periodic heterogeneous porous media: Model benchmarking against intermediate scale experiments

This paper is devoted to theoretical and experimental investigations of solute dispersion in heterogeneous porous media. Dispersion in heterogenous porous media has been reported to be scale-dependent, a likely indication that the proposed dispersion models are incompletely formulated. A high quality experimental data set of breakthrough curves in periodic model heterogeneous porous media is presented. In contrast with most previously published experiments, the present experiments involve numerous replicates. This allows the statistical variability of experimental data to be accounted for. Several models are benchmarked against the data set: the Fickian-based advection-dispersion, mobile-immobile, multirate, multiple region advection dispersion models, and a newly proposed transport model based on pure advection. A salient property of the latter model is that its solutions exhibit a ballistic behaviour for small times, while tending to the Fickian behaviour for large time scales. Model performance is assessed using a novel objective function accounting for the statistical variability of the experimental data set, while putting equal emphasis on both small and large time scale behaviours. Besides being as accurate as the other models, the new purely advective model has the advantages that (i) it does not exhibit the undesirable effects associated with the usual Fickian operator (namely the infinite solute front propagation speed), and (ii) it allows dispersive transport to be simulated on every heterogeneity scale using scale-independent parameters.

Taylor dispersion in heterogeneous porous media: Extended method of moments, theory, and modelling with two-relaxation-times lattice Boltzmann scheme

This article describes a generalization of the method of moments, called extended method of moments (EMM), for dispersion in periodic structures composed of impermeable or permeable porous inclusions. Prescribing pre-computed steady state velocity field in a single periodic cell, the EMM sequentially solves specific linear stationary advection-diffusion equations and restores any-order moments of the resident time distribution or the averaged concentration distribution. Like the pioneering Brenner's method, the EMM recovers mean seepage velocity and Taylor dispersion coefficient as the first two terms of the perturbative expansion. We consider two types of dispersion: spatial dispersion, i.e., spread of initially narrow pulse of concentration, and temporal dispersion, where different portions of the solute have different residence times inside the system. While the first (mean velocity) and the second (Taylor dispersion coefficient) moments coincide for both problems, the higher moments are different. Our perturbative approach allows to link them through simple analytical expressions. Although the relative importance of the higher moments decays downstream, they manifest the non-Gaussian behaviour of the breakthrough curves, especially if the solute can diffuse into less porous phase. The EMM quantifies two principal effects of bi-modality, as the appearance of sharp peaks and elongated tails of the distributions. In addition, the moments can be used for the numerical reconstruction of the corresponding distribution, avoiding time-consuming computations of solute transition through heterogeneous media. As illustration, solutions for Taylor dispersion, skewness, and kurtosis in Poiseuille flow and open/impermeable stratified systems, both in rectangular and cylindrical channels, power-law duct flows, shallow channels, and Darcy flow in parallel porous layers are obtained in closed analytical form for the entire range of Péclet numbers. The high-order moments and reconstructed profiles are compared to their predictions from the advection-diffusion equation for averaged concentration, based on the same averaged seepage velocity and Taylor dispersion coefficient. In parallel, we construct Lattice-Boltzmann equation (LBE) two-relaxation-times scheme to simulate transport of a passive scalar directly in heterogeneous media specified by discontinuous porosity distribution. We focus our numerical analysis and assessment on (i) truncation corrections, because of their impact on the moments, (ii) stability, since we show that stable Darcy velocity amplitude reduces with the porosity, and (iii) interface accuracy which is found to play the crucial role. The task is twofold: the LBE supports the EMM predictions, while the EMM provides non-trivial benchmarks for the numerical schemes.

Dispersion variance for transport in heterogeneous porous media

[1] We study dispersion in heterogeneous porous media for solutes evolving from point-like and extended source distributions in and spatial dimensions. The impact of heterogeneity on the dispersion behavior is captured by a stochastic modeling approach that represents the spatially fluctuating flow velocity as a spatial random field. We focus here on the sample-to-sample fluctuations of the dispersion coefficients about their ensemble mean. For finite source sizes, the definition of dispersion coefficients in single realizations is not unique. We consider dispersion measures that describe the extension of the solute distribution, as well as dispersion coefficients that quantify the solute spreading relative to injection points of the partial plumes that constitute the solute distribution. While the ensemble averages of these dispersion quantities may be identical, their fluctuation behavior is found to be different. Using a perturbation approach in the fluctuations of the random flow field, we derive explicit expressions for the temporal evolution of the variances of the dispersion coefficients between realizations. Their evolution is governed by the typical dispersion time over the characteristic heterogeneity scale and the dimensions of the source distribution. We find that the dispersion variance decreases toward zero with time in spatial dimensions, while in it converges toward a finite long time value that is independent of the source dimensions.

New stochastic model for dispersion in heterogeneous porous media: 2. Gaussian plume transmission across stepwise velocity fluctuations

The stochastic solute dispersion model studied in the previous article, can be applied to more realistic velocity variations by approximating them as piecewise constant. This requires treatment by a boundary value formulation, which raises problems connected with entropy considerations. A method is developed to deal with these by the introduction of a specially designed compensator function into the boundary value probability integral for calculating solute concentration. Applying this even for a single velocity step yields an intractable integration, but a suitable approximation is constructed that allows it to be evaluated in analytical form. The result is that a Gaussian solute plume impinging on a velocity step is transmitted as a modulated and compressed or dilated quasi-Gaussian. Plume dispersion is encapsulated in an enhancement factor F that multiplies the diffusive, linear time, dispersion. F is also time dependent; at the time of step penetration it equals kinematical dilation, but anneals away non-linearly so that a length scale can be established over which downstream effects of a velocity step on the dispersion extends. With multiple steps, the cumulative effect is expressed by a product of enhancement factors, each related to a single step. While kinematical compression and dilation effects cancel out across each stepped fluctuation, stochastic dispersion exceeds diffusive values for any plausible combination of fluctuation amplitude and length. A simple formula is obtained for the effective enhancement factor of a fluctuation in terms of its length and amplitude. Moreover, the algebra leads to the definition of a natural length scale Λ related to the Peclet number of the flow. Algebraic evaluation of the cumulative dispersion enhancement by a sequence of equally spaced identical fluctuations, leads to an expression for dispersivity as a function of the distance traversed by a solute plume. Key features of the model in agreement with observations are that the dispersivity behaves differently for traversal lengths above and below Λ, and that above this transition it is proportional to a fractional power of the traversal length. Simple stepwise models do not produce the full extent of the observed exponential increase below the transition, but modifying F by introduction of an adjustable parameter addresses this and leads to good agreement with experimental values over a range of 5 orders of magnitude in both traversal length and dispersivity.

A new mobile‐immobile model for reactive solute transport with scale‐dependent dispersion

This study proposed a new mobile‐immobile model (MIM) to describe reactive solute transport with scale‐dependent dispersion in heterogeneous porous media. The model was derived from the conventional MIM but assumed the dispersivity to be a linear or exponential function of travel distance. The linear adsorption and the first‐order degradation of solute were also considered in the model. The Laplace transform technique and the de Hoog numerical Laplace inversion method were applied to solve the developed model. Solute breakthrough curves (BTCs) obtained from MIM with scale‐dependent and constant dispersions were compared, and a constant effective dispersivity was provided to reflect the lumped scale‐dependent dispersion effect. The effective dispersivity was calculated by arithmetically averaging the distance‐dependent dispersivity. With this effective dispersivity, MIM could produce similar BTC as that from MIM with scale‐dependent dispersion in porous media with moderate heterogeneity. The applicability of the proposed new model was tested with concentration data from a 1,250‐cm long and highly heterogeneous soil column. The simulation results indicated that MIM with constant and linear distance‐dependent dispersivities were unable to adequately describe the measured BTCs in the column, while MIM with exponential distance‐dependent dispersivity satisfactorily captured the evolution of BTCs.

Exploring the nature of non-Fickian transport in laboratory experiments

Observations of non-Fickian transport in sandbox experiments [Levy M, Berkowitz B. Measurement and analysis of non-Fickian dispersion in heterogeneous porous media. J Contam Hydrol 2003;64:203–26] were analyzed previously using a power law tail ψ( t) ∼ t −1− β with 0 < β < 2 for the spectrum of transition times comprising a tracer plume migration. For each sandbox medium a choice of β resulted in an excellent fit to the breakthrough curve (BTC) data, and the value of β decreased slowly with increasing flow velocity. Here, the data are reanalyzed with the full spectrum of ψ( t) gleaned from analytical calculations [Cortis A, Chen Y, Scher H, Berkowitz B. Quantitative characterization of pore-scale disorder effects on transport in “homogeneous” granular media. Phys Rev E 2004;10(70):041108. doi: 10.1103/PhysRevE.70.041108], numerical simulations [Bijeljic B, Blunt MJ. Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour Res 2006;42:W01202. doi: 10.1029/2005WR004578] and permeability fields [Di Donato G, Obi E-O, Blunt MJ. Anomalous transport in heterogeneous media demonstrated by streamline-based simulation. Geophys Res Lett 2003;30:1608–12s. doi: 10.1029/2003GL017196]. We represent the main features of the full spectrum of transition times with a truncated power law (TPL), ψ( t) ∼ ( t 1 + t) −1− β exp(− t/ t 2), where t 1 and t 2 are the limits of the power law spectrum. An excellent fit to the entire BTC data set, including the changes in flow velocity, for each sandbox medium is obtained with a single set of values of t 1, β, t 2. The influence of the cutoff time t 2 is apparent even in the regime t < t 2. Significantly, we demonstrate that the previous apparent velocity dependence of β is a result of choosing a pure power law tail for ψ( t). The key is the change in the log–log slope of the TPL form of ψ( t) with a shifting observational time window caused by the change in the mean velocity. Hence, the use of the full spectrum of ψ( t) is not only necessary for the transition to Fickian behavior, but also to account for the dynamics of these laboratory observations of non-Fickian transport.

Relaxation and reversibility of extended Taylor dispersion from a Markovian–Lagrangian point of view

Taylor dispersion in a two-dimensional (2D) stratified velocity field describes a transition, called relaxation, from convective behaviour for short times, towards Fickian behaviour for large times and is partially reversible upon reversal of the flow direction. In 2D the physics are assumed to be governed by the unidirectional convection diffusion equation (2D uCDE). The approximate height-averaged 1D Generalised Telegraph Equation (GTE) catches an essential part of the longitudinal spreading. Contrary to the 1D Fickian approach, it explicitly accounts for the transient reversible nature [Camacho J. Purely global model for Taylor dispersion. Phys Rev E 1993/2;48(1); Berentsen CWJ, Verlaan ML, van Kruijsdijk CPJW. Upscaling and reversibility of Taylor dispersion in heterogeneous porous media. Phys Rev E 2005;71:046308]. Here we approach Taylor dispersion from a Lagrangian–Markovian point of view. In the redistribution model (RM) that we present, the Euler forward method accounts for convection and a probability redistribution matrix generates the transverse movement by diffusion over time. We consider two redistribution matrices. The first results from the direct discretisation of the Gaussian distribution function associated with the transverse mixing of the 2D uCDE. The resulting Gaussian redistribution model (GRM) is able to capture the multi-scale relaxation and reversibility behaviour of the full 2D uCDE. The correlated redistribution model (CRM) is the RM model with a redistribution matrix based on auto-correlation. The CRM is a generalisation of the correlated random walk model of [Scheidegger AE. The random walk model with auto-correlation of flow through porous media, Can J Phys 1958;36]. For uniform auto-correlation, the CRM model approximates the multi-scale relaxation nature of the 2D uCDE as a single scale relaxation process similar to the GTE. Moreover, it has the same variance as the GTE in the limit of the time step over relaxation time ratio to zero. For specific conditions the equality of the CRM model and the GTE is proven up to Δ t 2 order accuracy.

New stochastic model for dispersion in heterogeneous porous media: 1. Application to unbounded domains

A new model of solute dispersion in porous media that avoids Fickian assumptions and that can be applied to variable drift velocities as in non-homogeneous or geometrically constricted aquifers, is presented. A key feature is the recognition that because drift velocity acts as a driving coefficient in the kinematical equation that describes random fluid displacements at the pore scale, the use of Ito calculus and related tools from stochastic differential equation theory (SPDE) is required to properly model interaction between pore scale randomness and macroscopic change of the drift velocity. Solute transport is described by formulating an integral version of the solute mass conservation equations, using a probability density. By appropriate linking of this to the related but distinct probability density arising from the kinematical SPDE, it is shown that the evolution of a Gaussian solute plume can be calculated, and in particular its time-dependent variance and hence dispersivity. Exact analytical solutions of the differential and integral equations that this procedure involves, are presented for the case of a constant drift velocity, as well as for a constant velocity gradient. In the former case, diffusive dispersion as familiar from the advection–dispersion equation is recovered. However, in the latter case, it is shown that there are not only reversible kinematical dispersion effects, but also irreversible, intrinsically stochastic contributions in excess of that predicted by diffusive dispersion. Moreover, this intrinsic contribution has a non-linear time dependence and hence opens up the way for an explanation of the strong observed scale dependence of dispersivity.

Two-domain description of solute transport in heterogeneous porous media: Comparison between theoretical predictions and numerical experiments

This paper deals with two-equation models describing solute transport in highly heterogeneous porous systems and more particularly dual permeability structures composed of high- and low-permeability regions. A macroscopic two-equation model has been previously proposed in the literature based on the volume averaging technique [Ahmadi A, Quintard M, Whitaker S. Transport in chemically and mechanically heterogeneous porous media V: two-equation model for solute transport with adsorption, Adv Water Resour 1998;22:59–86; Cherblanc F, Ahmadi A, Quintard M. Two-medium description of dispersion in heterogeneous porous media: calculation of macroscopic properties. Water Resour Res 2003;39(6):1154–73]. Through this theoretical upscaling method, both convection and dispersion mechanisms are taken into account in both regions, allowing one to deal with a large range of heterogeneous systems. In this paper, the numerical tools associated with this model are developed in order to test the theory by comparing macroscopic concentration fields to those obtained by Darcy-scale numerical experiments. The heterogeneous structures considered are made up of low-permeability nodules embedded in a continuous high-permeability region. Several permeability ratios are used, leading to very different macroscopic behaviours. Taking advantage of the Darcy-scale simulations, the role of convection and dispersion in the mass exchange between the two regions is investigated. Large-scale averaged concentration fields and elution curves are extracted from the Darcy-scale numerical experiments and compared to the theoretical predictions given by the two-equation model. Very good agreement is found between experimental and theoretical results. A permeability ratio around 100 presents a behaviour characteristic of “mobile–mobile” systems emphasizing the relevance of this two-equation description. Eventually, the theory is used to set-up a criterion for the existence of local equilibrium conditions. The potential importance of local-scale dispersion in reducing large-scale dispersion is highlighted. The results also confirm that a non-equilibrium description may be necessary in such systems, even if local-equilibrium behaviour could be observed.

Upscaling and reversibility of Taylor dispersion in heterogeneous porous media

Tracer flow in stratified porous media is dominated by the interaction between convective transport and transverse diffusive mixing. By averaging the tracer concentration in the transverse direction, a one-dimensional non-Fickian dispersion model is derived. The model accounts for the relaxation process that reduces the convective transport to dispersive mixing. This process is (short-) time correlated and partially reversible upon reversal of flow direction. For multiscale velocity fields, the relaxation is a multiscale process. To date only single scale processes have been successfully upscaled. Our procedure extends this to multiscale processes, using scale separation. The model parameters can be calculated a priori based on the velocity profile. For periodic flow reversal, the results are essentially the same. Despite the non-Fickian behavior during a cycle, the net contribution of each cycle to the spreading relaxes to a Fickian process in a similar way as for unidirectional flow. The cycle time averaged dispersion coefficient is a monotonically increasing function of the reversal time. It asymptotically converges towards the effective dispersion coefficient in the absence of flow reversal.

Large‐scale sandbox experiment on longitudinal effective dispersion in heterogeneous porous media

In heterogeneous media the breakthrough curves observed at single points may differ significantly from those integrated over a control plane. While standard macrodispersion describes the spread of concentration averaged over the entire cross section, effective dispersion quantifies the average spread in the locally observed breakthrough curves. We perform a conservative tracer test in a 14‐m‐long sandbox filled heterogeneously with four types of silica sand. The filling resembles natural heterogeneities including the distribution of facies and micro‐structures within the various sand types. We use fluorescein as tracer, which we measure at 126 measurement points directly within the porous medium and at 19 levels in the outlet of the domain. We use fiber‐optic probes with a point‐like measurement tip for fluorescence intensity measurements. From the second central temporal moments of the locally obtained breakthrough curves we compute apparent effective dispersion coefficients, and from the average over all probes within a measurement plane we derive apparent macrodispersion coefficients. The effective dispersion coefficient is about two thirds of the macrodispersion coefficient. Considering the typical length scales of the heterogeneities, the experimental findings are qualitatively in agreement with linear stochastic theory.

Effective dispersion in heterogeneous media under random transient flow conditions

Under steady state flow conditions, solute dispersion in heterogeneous porous media is much smaller in the transverse than in the longitudinal direction. This holds particularly for effective dispersion of a plume originating from a point‐like injection. The effective dispersion coefficient describes the actual dispersive mixing of solutes in the aquifer. The lack of dispersive transverse mixing may limit considerably natural attenuation of certain contaminants. Temporal fluctuations of the flow direction enhance horizontal transverse dispersion. This has been shown previously for uniform flow and for macrodispersion in stationary media. We present a linear stochastic theory for effective dispersion under quasi‐steady state flow conditions with random temporal fluctuations of the mean flow direction. As for macrodispersion, effective transverse dispersion proves to be dominated by transient flow effects. We compare semianalytical results derived from linear theory to those from particle‐tracking random‐walk simulations for a three‐dimensional test case. The parameters of the test case are similar to those obtained at the Borden site, where the mean transverse flow component fluctuated approximately by ±10°. Linear theory and particle simulations agree well.

Two‐medium description of dispersion in heterogeneous porous media: Calculation of macroscopic properties

In this paper, a numerical procedure is proposed to calculate effective properties associated with the generalized two‐equation model developed by Ahmadi et al. [1998]. A transformation of the original closure problems was found that allowed the introduction of a finite volume formulation. Results are presented for a heterogeneous porous medium made up of nodules embedded in a continuous matrix. The properties of the effective parameters are discussed in terms of the influence of the Péclet number, the permeability ratio, and the local‐scale dispersivity. In addition, the problem of the asymptotic behavior of the two‐equation model is discussed in some details. Finally, a comparison is made with results available from the literature in simplified cases.

Measurement and analysis of non-Fickian dispersion in heterogeneous porous media

Contaminant breakthrough behavior in a variety of heterogeneous porous media was measured in laboratory experiments, and evaluated in terms of both the classical advection–dispersion equation (ADE) and the continuous time random walk (CTRW) framework. Heterogeneity can give rise to non-Fickian transport patterns, which are distinguished by “anomalous” early arrival and late time tails in breakthrough curves. Experiments were conducted in two mid-scale laboratory flow cells packed with clean, sieved sand of specified grain sizes. Three sets of experiments were performed, using a “homogeneous” packing, a randomly heterogeneous packing using sand of two grain sizes, and an exponentially correlated structure using sand of three grain sizes. Concentrations of sodium chloride tracer were monitored at the inflow reservoir and measured at the outflow reservoir. Breakthrough curves were then analyzed by comparison to fitted solutions from the ADE and CTRW formulations. In all three systems, including the “homogeneous” one, subtle yet measurable differences between Fickian and non-Fickian transport were observed. Quantitative analysis demonstrated that the CTRW theory characterized the full shape of the breakthrough curves far more effectively than the ADE.

Dispersion in Heterogeneous Porous Media: One-Equation Non-equilibrium Model

In this paper, the method of large-scale averaging is used to develop two different one-equation models describing dispersion in heterogeneous porous media. The first model represents the case of large-scale mass equilibrium, while the second represents the asymptotic behavior of a two-equation model obtained by large-scale averaging. It is shown that a one-equation, non-equilibrium model can be developed even when the intrinsic large-scale averaged concentrations for each region are not equal. The solution of this non-equilibrium model is equivalent to the asymptotic behavior of the two-equation model.

Comment on ?fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media?, by Muhammad Sahimi

Two fundamental questions regarding the application of percolation theory to transport in porous media are addressed. First, when ‘critical path’ arguments (based on a sufficiently wide spread of microscopic transition rates) are invoked (in analogy to the case of transport in disordered semiconductors) to justify the application of percolation theory to the determination of relevant transport properties, then for long time scales (compared to the inverse of the ‘critical’ percolation rate), the fractal structure of the ‘critical’ path is relevant to transport, but not at short time scales. These results have been demonstrated concretely in the case of disordered semiconductors, and are in direct contradiction to the claims of the review. Second, the relevance of deterministic or stochastic methods to transport has been treated heretofore by most authors as a question of practicality. But, at least under some conditions, concrete criteria distinguish between the two types of transport. Percolative (deterministic) transport is temporally reproducible and spatially inhomogeneous while diffusive (stochastic) transport is temporally irreproducible, but homogeneous, and a cross-over from stochastic to percolative transport occurs when the spread of microscopic transition rates exceeds 4–5 orders of magnitude. It is likely that such conditions are frequently encountered in soil transport. Moreover, clear evidence for deterministic transport (although not necessarily percolative) exists in such phenomena as preferential flow. On the other hand, the physical limitation of transport to (fractally connected) pore spaces within soils (analogously to transport in metal-insulator composites) can make transport diffusive on a fractal structure, rather than percolative.

Reply to ?comment on fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media?

Reply to ?comment on fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media?