Harvey Scher

Modeling non‐Fickian transport in geological formations as a continuous time random walk

[1] Non-Fickian (or anomalous) transport of contaminants has been observed at field and laboratory scales in a wide variety of porous and fractured geological formations. Over many years a basic challenge to the hydrology community has been to develop a theoretical framework that quantitatively accounts for this widespread phenomenon. Recently, continuous time random walk (CTRW) formulations have been demonstrated to provide general and effective means to quantify non-Fickian transport. We introduce and develop the CTRW framework from its conceptual picture of transport through its mathematical development to applications relevant to laboratoryand field-scale systems. The CTRW approach contrasts with ones used extensively on the basis of the advectiondispersion equation and use of upscaling, volume averaging, and homogenization. We examine the underlying assumptions, scope, and differences of these approaches, as well as stochastic formulations, relative to CTRW. We argue why these methods have not been successful in fitting actual measurements. The CTRW has now been developed within the framework of partial differential equations and has been generalized to apply to nonstationary domains and interactions with immobile states (matrix effects). We survey models based on multirate mass transfer (mobile-immobile) and fractional derivatives and show their connection as subsets within the CTRW framework.

On Characterization of Anomalous Dispersion in Porous and Fractured Media

A key characterization of dispersion in aquifers and other porous media has been to map the effects of inhomogeneous velocity fields onto a Fickian dispersion term (D) within the context of the conventional advection-dispersion equation (ADE). Recent compilations of data have revealed, however, that the effective D coefficient is not constant but varies systematically with the length or timescale over which transport occurs. A natural strategy to encompass this “anomalous” behavior into the context of the conventional ADE is to make D time dependent. This approach, to use D(t) to handle the same anomalous dispersion phenomena, has also been common in the field of electronic transport in disordered materials. In this paper we discuss the intrinsic inadequacy of considering a time-dependent dispersivity in the conventional ADE context, and show that the D = D(t) generalization leads to quantifiably incorrect solutions. In the course of proving this result we discuss the nature of anomalous dispersion and provide physical insight into this important problem in hydrogeology via analysis of a class of kinetic approaches. Particular emphasis is placed on the effects of a distribution of solute “delay times” with a diverging mean time, which we relate to configurations of preferential pathways in heterogeneous media.

The Role of Probabilistic Approaches to Transport Theory in Heterogeneous Media

A physical picture of contaminant transport in highly heterogeneous porous media is presented. In any specific formation the associated governing transport equation is valid at any time and space scale. Furthermore, the advective and dispersive contributions are inextricably combined. The ensemble average of the basic transport equation is equivalent to a continuous time random walk (CTRW). The connection between the CTRW transport equation, in a limiting case and the familiar advection-dispersion equation (ADE) is derived. The CTRW theory is applied to the results of laboratory experiments, field observations, and simulations of random fracture networks. All of these results manifest dominant non-Gaussian features in the transport, over different scales, which are accounted for quantitatively by the theory. The key parameter β controlling the entire shape of the contaminant plume evolution and breakthrough curves is advanced as a more useful characterization of the transport than the dispersion tensor, which is based on moments of the plume. The role of probabilistic approaches, such as CTRW, is appraised in the context of the interplay of spatial scales and levels of uncertainty. We then discuss a hybrid approach, which uses knowledge of non-stationary aspects of a field site on a larger spatial scale (trends) with a probabilistic treatment of unresolved structure on a smaller scale (residues).

Analysis of field observations of tracer transport in a fractured till.

We analyze a set of observations from a recently published, field-scale tracer test in a fractured till. These observations demonstrate a dominant, underlying non-Fickian behavior, which cannot be quantified using traditional modeling approaches. We use a continuous time random walk (CTRW) approach which thoroughly accounts for the measurements, and which is based on a physical picture of contaminant motion that is consistent with the geometric and hydraulic characterization of the fractured formation. We also incorporate convolution techniques in the CTRW theory, to consider transport between different regions containing distinct heterogeneity patterns. These results enhance the possibility that limitations in predicting non-Fickian modes of contaminant migration can be overcome.

Origins of anomalous transport in heterogeneous media: Structural and dynamic controls

Anomalous (or “non-Fickian”) transport is ubiquitous in the context of tracer migration in geological formations. We quantitatively identify the origin of anomalous transport in a representative model of a heterogeneous porous medium under uniform (in the mean) flow conditions; we focus on anomalous transport which arises in the complex flow patterns of lognormally distributed hydraulic conductivity (K) fields, with several decades of K values. Transport in the domains is determined by a particle tracking technique and characterized by breakthrough curves (BTCs). The BTC averaged over multiple realizations demonstrates anomalous transport in all cases, which is accounted for entirely by a power law distribution ∼t−1−β of local transition times. The latter is contained in the probability density function ψ(t) of transition times, embedded in the framework of a continuous time random walk (CTRW). A unique feature of our analysis is the derivation of ψ(t) as a function of parameters quantifying the heterogeneity of the domain. In this context, we first establish the dominance of preferential pathways across each domain, and characterize the statistics of these pathways by forming a particle-visitation weighted histogram, Hw(K), of the hydraulic conductivity. By converting the ln(K) dependence of Hw(K) into time, we demonstrate the equivalence of Hw(K) and ψ(t), and delineate the region of Hw(K) that forms the power law of ψ(t). This thus defines the origin of anomalous transport. Analysis of the preferential pathways clearly demonstrates the limitations of critical path analysis and percolation theory as a basis for determining the origin of anomalous transport. Furthermore, we derive an expression defining the power law exponent β in terms of the Hw(K) parameters. The equivalence between Hw(K) and ψ(t) is a remarkable result, particularly given the nature of the K heterogeneity, the complexity of the flow field within each realization, and the statistics of the particle transitions.

Structure, flow, and generalized conductivity scaling in fracture networks

We present a three-dimensional (3-D) model of fractures that within the same framework, allows a systematic study of the interplay and relative importance of the two key factors determining the character of flow in the system. The two factors of complexity are () the geometry of fracture plane structure and interconnections and (2) the aperture variability within these planes. Previous models have concentrated on each separately. We introduce anisotropic percolation to model a wide range of fracture structures and networks. The conclusion is that either of these elements, fracture geometry and aperture variability, can give rise to channeled flow and that the interplay between them is especially important for this type of flow. Significant outcomes of our study are (1) a functional relationship that quantifies the dependence of the effective hydraulic conductivity on aperture variability and on the network structure and fracture element density, (2) a relation between aperture variability and the Peclet number, and (3) a basis for a new explanation for the field-length dependence of permeability observed in fractured and heterogeneous porous formations.

Non‐Fickian transport and multiple‐rate mass transfer in porous media

[1] Non-Fickian behavior is due to a broad spectrum of rates limiting the solute transport. There are two generic mechanisms that can generate these spectra: the complex flow field of a highly heterogeneous medium and the mass exchange between a mobile phase and a distribution of immobile states. We have developed a physical model that incorporates both of these mechanisms into the continuous time random walk (CTRW) framework. We study their interacting dynamics as a function of the spectra of advective-diffusive transition times and exchange times and the relative separation of their respective time domains. Examples of interacting transport in a dispersive medium with immobile states include tracer migration in a random fracture network with matrix diffusion and transport in a porous medium with adsorption/desorption sites. To date, non-Fickian transport has been quantified effectively using the CTRW in a wide variety of porous and fractured geological formations. The basis of the CTRW framework is the portrayal of transport as a sequence of transition rates (e.g., between pore spaces, fracture intersections) and the incorporation of the full spectrum of these rates into the transport equations. The emphasis herein is on systems in which the time domains of the two different types of spectra are distinguishable, so that a more complete characterization of the transport can be obtained (i.e., rather than lumping all the rates together). Experimental data are analyzed from two of these systems: (1) tracer transport in a fractured shear zone and (2) sorbing species transported through a heterogeneous porous domain. The CTRW framework is found to produce excellent fits to and predictions from the experimental data.

Towards a unified framework for anomalous transport in heterogeneous media

Abstract We develop a unified framework to model the anomalous transport of tracers in highly heterogeneous media. While the framework is general, our working media in this study are geological formations. The basis of our approach takes into account the different levels of uncertainty, often associated with spatial scale, in characterizing these formations. The effects on the transport of smaller spatial scale heterogeneities are treated probabilistically with a model based on a continuous time random walk (CTRW), while the larger scale variations are included deterministically. The CTRW formulation derives from the ensemble average of a disordered system, in which the transport in each realization is described by a Master Equation. A generic example of such a system – a 3D discrete fracture network (DFN) – is treated in detail with the CTRW formalism. The key step in our approach is the derivation of a physically based ψ( s ,t) , the joint probability density for a displacement s with an event-time t . We relate the ψ( s ,t) to the velocity spectrum Φ(ξ) (|ξ|=1/v, ξ = v ) of the steady flow-field in a fluid-saturated DFN. Heterogeneous porous media are often characterized by a log-normal permeability distribution; the Φ ( ξ ) we use in this case is an analytic form approximating the velocity spectrum derived from this distribution. The common approximation of ψ( s ,t)∼p( s )t −1−β with a constant β , is evaluated in these cases. For the former case it is necessary to include s −t coupling while the latter case points to the presence of an effective t -dependent β . The full range of these features can be included in the CTRW solution but, as is shown, not in the fractional-time derivative equation (FDE) formulation of CTRW. Finally, the methods used for the unified framework are critically examined.

Field observations of a capillary fringe before and after a rainy season

Abstract Field measurements of profiles of the water content of a 7-m deep, sandy, phreatic aquifer have been conducted at the same site before and after the rainy season. The field site is an uncultivated region located in a citrus orchard, north of the city of Netanya, on the Coastal Plain aquifer of Israel. In two single-day campaigns, 13 boreholes were drilled in relatively homogeneous sandy sediments in an area subjected to infiltration of rain only. Twelve of the thirteen boreholes were located within a radius of 5 m. Water content, as a function of depth, was obtained from continuous cores. We found that the distribution of water within the capillary fringe (CF) was compact — denoting that there was an abrupt change in water content with increasing height above the water table. The height of the CF was about 1.4 m at each point above the water table, and over a horizontal distance of about 4 m, this height varied by up to 33% and 50% before and after the rainy season, respectively. Saturated conditions were detected in some regions of the CF and unsaturated conditions were found up to 1.5 m below the water table. The distribution of water in the capillary fringe remained compact while being displaced vertically as the water table rose 35 cm after the winter rains. These observations are consistent with pore level models of the water distribution within the capillary fringe in a porous media.

On the Structure and Flow Processes in the Capillary Fringe of Phreatic Aquifers

The water distribution in the capillary fringe (CF) reflects the interaction of a strongly wetting fluid in a heterogeneous porous medium. Field profiles of gravimetric water content of the CF for a 30m deep, sandy, phreatic aquifer in Israel are critically analyzed in the context of the possible wetting and drainage processes in these sediments. A highly plausible explanation of the profiles is based on the spatial configuration of the CF surface determined from a model of the movement of water within the porous medium. The structural types of CF that can arise from a number of competing pore-scale displacement mechanisms, in the presence of gravity, are characterized by the model. We differentiate between two generic types of CF structures: a tenuous invasion-percolation type and a compact type. Flow, in response to a horizontal pressure gradient, associated with each structure is analyzed. Our interpretation of the field data supports the compact structure with a spatial variation in the height of the CF surface, above the water table, on the order of 1m. In this compact structure horizontal flow is characterized by stagnant regions in the CF above a critical height hc and flow only for regions below hc. The field water content (at hc) may be used to predict the onset of lateral water flow in the CF.