Physical mechanism analysis of anomalous diffusion characterized by scaling law

The scaling law of displacement variance serves as an important signature in identifying the characteristics of solute transport in heterogeneous media. However, classifying anomalous diffusion into sub-diffusion and super-diffusion based on scaling laws does not fully capture the underlying mechanisms of solute transport. We employ Continuous Time Random Walk (CTRW) and Spatial Markov Model (SMM) to capture various microscopic mechanisms and their corresponding displacement variance. A detailed analysis, illustrated by breakthrough curves and spatial snapshots, under fixed values of displacement variance scaling law, is performed to investigate the distinctions among given mechanisms. Analysis results show that the anomalous transport behavior driven by preferential flow, sorption and velocity correlation may have fundamental distinction under the same displacement variance scaling law. On the other hand, both sub-diffusion and super-diffusion can arise from the interplay of competing mechanisms, such as preferential flow, sorption, and velocity correlation. At last, some suggestions are provided for selecting the appropriate transport model, based on the physical mechanism analysis of real-world situations.