[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 laboratory-and 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.

Abstract. Scaling in fracture systems has become an active field of research in the last 25 years motivated by practical applications in hazardous waste disposal, hydrocarbon reservoir management, and earthquake hazard assessment. Relevant publications are therefore spread widely through the literature. Although it is recognized that some fracture systems are best described by scale-limited laws (lognormal, exponential), it is now recognized that power laws and fractal geometry provide widely applicable descriptive tools for fracture system characterization. A key argument for power law and fractal scaling is the absence of characteristic length scales in the fracture growth process. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studies emphasize the importance of layering on all scales in limiting the scaling characteristics of natural fracture systems. The determination of power law exponents and fractal dimensions from observations, although outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these techniques and suggest guidelines for the accurate and objective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after critical appraisal of published studies, to show a wide variation, frequently spanning the theoretically possible range. Extrapolations from one dimension to two and from two dimensions to three are found to be nontrivial, and simple laws must be used with caution. Directions for future research include improved techniques for gathering data sets over great scale ranges and more rigorous application of existing analysis methods. More data are needed on joints and veins to illuminate the differences between different fracture modes. The physical causes of power law scaling and variation in exponents and fractal dimensions are still poorly understood. INTRODUCTIONThe study of fracture systems (terms in italic are defined in the glossary, after the main text) has been an active area of research for the last 25 years motivated to a large extent by the siting of hazardous waste disposal sites in crystalline rocks, by the problems of multiphase flow in fractured hydrocarbon reservoirs, and by earthquake hazards and the possibility of prediction. Here we define a fracture as any discontinuity within a rock mass that developed as a response to stress. This comprises primarily mode I and mode II fractures. In mode I fracturing, fractures are in tensile or opening mode in which displacements are normal to the discontinuity walls (joints and many veins). Faults correspond to mode II fractures, i.e., an in-plane shear mode, in which the displacements are in the plane of the discontinuity. Fractures exist on a wide range of scales from microns to hundreds of kilometers, and it is known that throughout this scale range they have a sign...

No abstract

No abstract

No abstract

We show that dominant aspects of chemical ͑particle͒ transport in fracture networks-non-Gaussian propagation-result from subtle features of the steady flow-field distribution through the network. This is an outcome of a theory, based on a continuous time random walk formalism, structured to retain the key spacetime correlations of particles as they are advected across each fracture segment. The approach is designed to treat the complex geometries of a large variety of fracture networks and multiscale interactions. Monte Carlo simulations of steady flow in these networks are used to determine the distribution of velocities in individual fractures as a function of their orientation. The geometry and velocity distributions are used, in conjunction with particle mixing rules, to map the particle movement between fracture intersections onto a joint probability density (r,t). The chemical concentration plume and breakthrough curves can then be calculated analytically. Particle tracking simulations on these networks exhibit the same non-Gaussian profiles, demonstrating quantitative agreement with the theory. The analytic plume shapes display the same basic behavior as extensive field observations at the Columbus Air Force Base, Mississippi. The quantitative correlation between the time dependence of the mean and standard deviation of the field plumes, and their shape, is predicted by the theory.

[1] We consider the transport behavior of a passive solute in a heterogeneous medium, modeled by continuous time random walks (CTRW) and linear multirate mass transfer (MRMT). Within the CTRW framework, we formulate a transport model which is formally equivalent to MRMT. In both approaches the total concentration is divided into mobile and immobile parts. The immobile concentration is given by the convolution in time of the mobile concentration and a memory function. The memory function is a functional of the distribution of transition and trapping times for the CTRW and the MRMT approach, respectively, and determines the transport behavior of the solute. Based on different expressions for the memory function in the two frameworks, we derive conditions for which both approaches describe the same transport behavior. We focus on anomalous transport behavior that can arise if the transition and trapping time distributions behave algebraically in a given time regime. Using an expansion of the Laplace transform of the memory function, we develop explicit expressions for the time behavior of the flux concentration as well as for the center of mass velocity and the (macro) dispersion coefficients of the solute distribution. We observe (anomalous) power law as well as (normal) Fickian transport behavior, depending on the exponents that dominate the trapping and transition time distributions, respectively. The results show that the character of the anomalous transport does not depend on the details of the transport model but only on the exponents dominating the transition or trapping time distributions. The unified transport framework presented here comprising CTRW and MRMT shows new aspects and opens new perspectives for the modeling of transport in heterogeneous media.INDEX TERMS: 1832 Hydrology: Groundwater transport; 1869 Hydrology: Stochastic processes; 3210 Mathematical Geophysics: Modeling; 1829 Hydrology: Groundwater hydrology; KEYWORDS: Groundwater hydrology, groundwater transport, stochastic processes, modeling Citation: Dentz, M., and B. Berkowitz, Transport behavior of a passive solute in continuous time random walks and multirate mass transfer, Water Resour.

No abstract