From microstructure to memory: Basset-type fractional transport models in porous media

Fractional time-derivative operators have been increasingly used to model anomalous advection-diffusion-reaction phenomena in porous media, where classical models fail to capture non-locality and persistent memory effects observed in experimental data. Most existing fractional models are postulated phenomenologically, without direct links to microstructural dynamics. We derive a fractional transport model from first principles from a micro-macroscale framework for transport in a heterogeneous porous medium (i.e. a medium with microscale inclusions). The resulting equation is of the Basset type, with fractional order 1/3<α<1, and α=1/2 in the case of semi-infinite inclusions. We perform a qualitative analysis of the model, focusing on the early- and late-time asymptotic behaviour for a general fractional exponent 0<α<1. Our results highlight the loss of regularity in the solution, a characteristic phenomenon of fractional-order models and a source of significant numerical challenges. To address these challenges, we design a first-order product-integration scheme coupled with finite-difference spatial discretization and reformulate the problem into a vectorial Basset equation to handle boundary effects. Numerical simulations confirm the analytical predictions, showing early-time memory effects and late-time convergence to a self-similar profile governed by the principal eigenfunction of the transport operator. The work provides a rigorous, physically grounded framework for modelling anomalous transport in porous materials, bridging fractional calculus, asymptotic analysis and computational methods.