Fractional-derivative models for non-Fickian transport in a single fracture and its extension

Abstract

Fractures are ubiquitous in brittle geological media, providing critical paths for pollutant transport. Analytical solutions describing Fickian diffusion of pollutants in a single fracture or parallel fractures with diffusional mass exchange across the fracture-matrix interface are well known and have been used by hydrologists for more than three decades. Non-Fickian transport, however, has also been observed in natural fractures due to within-fracture flow channeling and the presence of stagnant zones. To address these observations, we develop fractional-derivative equations (FDEs) that consider more general cases of pollutant transport that include non-Fickian diffusive jumps for pollutant particles along the fracture and time nonlocal mass exchange across the fracture-matrix interface. General solutions are derived for the resultant FDEs for pollutant transport through a single fracture in a semi-infinite domain. Applications show that the new model is superior to the classical models in capturing heavy-tailed tracer breakthrough curves in single fracture systems observed in the laboratory. Further evaluation of the FDE solutions reveals that the three fractional derivatives defined in the FDEs capture depth- and time-dependent, subtle non-Fickian dynamics. By adjusting the FDE indices in the general solutions, we can quantify various non-Fickian transport behaviors. The single fracture-matrix system is then extended to discrete parallel fractures and large-scale, irregular fracture-matrix systems, where the corresponding FDE models and solutions are discussed. Therefore, as a logical extension of the classical solution for Fickian diffusion, this study provides physical models and solutions to quantify non-Fickian transport in a single fracture-matrix system and its extension.