Matrix Diffusion as a Mechanism Contributing to Fractal Stream Chemistry and Long-Tailed Transit Time Distributions

Solute transit or travel time distributions (TTDs) in catchments are relevant to both hydrochemical response and inference of hydrologic mechanisms. Long-tailed TTDs and fractal scaling behavior of stream concentration power spectra (~1/frequency, or 1/frequency to a power < 2) are widely observed in catchment studies. In several catchments, a significant fraction of streamflow is derived from groundwater flowing through shallow fractured bedrock. It is well established that matrix diffusion, a phenomenon first invoked to explain anomalous tracer ages, significantly influences the travel time of tracers in fractured rock. However, few models of catchment-scale solute TTDs explicitly incorporate the influence of matrix diffusion, and its contribution to fractal scaling and long-tailed TTDs has not received sufficient attention. I present time and frequency domain theoretical analyses of the combined influence of matrix diffusion and variable advective travel times along streamlines on solute transport in a catchment hillslope flow system. The theoretical concentration power spectra are shown to exhibit fractal scaling. The corresponding TTDs decline steeply at early time and also exhibit long tails. For finite matrix block widths, the TTDs impacted by matrix diffusion resemble a gamma distribution with shape parameter =0.5, which has been proposed previously as a suitable TTD model for representing fractal scaling behavior. The tails of the TTDs are influenced by the accessible matrix width, exhibiting a sustained power-law (rather than exponential) decline for large matrix widths. The theoretical framework enables superposition of the influence of matrix diffusion on any general advective travel time distribution, derived from either analytical or numerical subsurface flow models. Application to the Lower Hafren catchment, a well-studied experimental catchment at Plynlimon, Wales; shows that theoretical spectra match previously reported power spectral estimates derived from high-resolution concentration measurements, collected over a 28 year period.