A continuous time random walk approach to the stream transport of solutes

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 in-stream 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 surface-subsurface exchange processes and the transport of both conservative and nonconservative substances in rivers