[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.

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We re‐examine—in light of recent theoretical developments—classical experiments on dispersion of a passive tracer in fully and partially saturated porous columns. We find that the dispersion breakthrough curves (BTCs) exhibit anomalous (non‐Fickian) early arrival times and late time tailing, which can be explained by the Continuous Time Random Walk (CTRW) theory. The CTRW framework includes as a special case the classical advection‐dispersion equation (ADE) for Fickian transport. We argue that existing measurements and interpretations of dispersion should be carefully reconsidered in the framework of these advances in conceptual understanding and quantification.

[1] The transport of solutes in rivers is influenced by the exchange of water between the river and the underlying hyporheic zone. The residence times of solutes in the hyporheic zone are typically much longer than traveltimes in the stream, resulting in a significant delay in the downstream propagation of solutes. A new model for this process is proposed here on the basis of the continuous time random walk (CTRW) approach. The CTRW is a generalization of the classic random walk that can include arbitrary distributions of waiting times, and it is particularly suited to deal with the long residence times arising from hyporheic exchange. Inclusion of suitable hyporheic residence time distributions in the CTRW leads to a generalized advection-dispersion equation for instream concentration breakthrough curves that includes the effects of specific hyporheic exchange processes. Here examples are presented for advective hyporheic exchange resulting from regular and irregular series of bedforms. A second major advantage of the CTRW approach is that the combined effects of different processes affecting overall downstream transport can be incorporated in the model by convolving separate waiting time distributions for each relevant process. The utility of this approach is illustrated by analyzing the effects of local-scale sediment heterogeneity on bedform-induced hyporheic exchange. The ability to handle arbitrarily wide residence time distributions and the ability to assess the combined effects of multiple transport processes makes the CTRW model framework very useful for the study of solute transport problems in rivers. The model presented here can be easily extended to represent different types of surfacesubsurface exchange processes and the transport of both conservative and nonconservative substances in rivers.

[1] We develop a numerical method to model contaminant transport in heterogeneous geological formations. The method is based on a unified framework that takes into account the different levels of uncertainty often associated with characterizing heterogeneities at different spatial scales. It treats the unresolved, small-scale heterogeneities (residues) probabilistically using a continuous time random walk (CTRW) formalism and the largescale heterogeneity variations (trends) deterministically. This formulation leads to a Fokker-Planck equation with a memory term (FPME) and a generalized concentration flux term. The former term captures the non-Fickian behavior arising from the residues, and the latter term accounts for the trends, which are included with explicit treatment at the heterogeneity interfaces. The memory term allows a transition between non-Fickian and Fickian transport, while the coupling of these dynamics with the trends quantifies the unique nature of the transport over a broad range of temporal and spatial scales. The advection-dispersion equation (ADE) is shown to be a special case of our unified framework. The solution of the ADE is used as a reference for the FPME solutions for the same formation structure. Numerical treatment of the equations involves solution for the Laplace transformed concentration by means of classical finite element methods, and subsequent inversion in the time domain. We use the numerical method to quantify transport in a two-dimensional domain for different expressions for the memory term. The parameters defining these expressions are measurable quantities. The calculations demonstrate long tailing arising (principally) from the memory term and the effects on arrival times that are controlled largely by the generalized concentration flux term.