A distributed-order time fractional derivative model for simulating bimodal sub-diffusion in heterogeneous media

Bimodal transport due to dual-mobile advection or mass exchange between mobile and immobile zones has been widely observed for pollutants moving in heterogeneous media. Existing nonlocal transport models, however, cannot capture either heterogeneity in mobile domains or bi-peak concentration phenomena for breakthrough curves (BTCs) of solute transport in complex media. Therefore, this study proposed a dual heterogeneous domain model (DHDM) framework, conceptualized from the distributed-order time fractional derivative that incorporates a broader spectrum of particle movement in the dual-domain system than the classical single-domain model. A Lagrangian scheme was developed to solve the DHDM using the Bernoulli trial with the transition probability simulating mass exchange between different domains. Phase transition probabilities calculated based on the zeroth spatial moments of dual domain transport equations were used to simulate the particle partitioning between different domains. Applications to transport experiments in silt-clay columns conducted in our laboratory showed that the DHDM yielded significant improvement for simulating solute transport behavior compared to the single-domain anomalous transport model. Parameter analysis of the DHDM further revealed that the velocity disparity of the two domains controlled the BTC rising time and the shape of the second peak in the BTC. When the velocity of the slow domain is extremely small, the second BTC peak develops a 'shoulder' characteristic during the late-time tail of the BTC. This study improved our understanding for bimodal transport in a dual domain system and provides a feasible tool for capturing a wide range of bimodal transport in complex media.