Strontium transport facilitated by bentonite and illite colloids with breakthrough tailing characterized by a fractional memory and kinetic exchange model

Colloid-facilitated transport plays a critical role in the long-distance transport of radionuclides in geological media, particularly in scenarios of nuclear waste disposal. Among various colloids, bentonite colloids formed from the swelling and dispersion of bentonite buffer material can significantly influence radionuclide transport due to their strong sorption capacity and high mobility. However, due to the media heterogeneity, sorption and desorption, traditional advection-dispersion models fail to capture the observed long-tailed desorption and retention behaviors of experimental breakthrough curves. To address this gap, we propose a fractional derivative-based co-transport model to capture the colloid-facilitated transport. In this framework, the retarded migration process of radioactive nuclides on the solid matrix is characterized by fractional derivatives and colloid-borne radionuclide transport is described by the classical advection-dispersion equation with first-order mass transfer between the free and colloid-associated radionuclide states. The specific contribution of the model lies in its phase-specific coupling of time-fractional memory and reversible kinetic exchange, which distinguishes the proposed framework from conventional single-equation fractional transport models and classical two-state kinetic models. The model was evaluated using column experiments of Sr transport facilitated by illite and bentonite colloids in saturated quartz sand. The results demonstrate that the proposed framework accurately characterizes both retention and fast-movement behaviors, providing solid accuracy and interpretability for modeling colloid-facilitated radionuclide transport in complex media. The proposed fractional-order framework provides a substantial improvement over classical integer-order models, enabling simultaneous representation of fast colloid-borne transport and long-tail retention. Quantitatively, the model reduces RMSE by approximately 51-77% across different colloid systems, demonstrating the effectiveness of fractional-order memory effects in describing heterogeneous radionuclide transport.