Statistical physical modeling of anomalous diffusion behavior in non-uniform media

Abstract

The non-uniformity of the medium and the flow field causes the diffusion process to deviate from Fick's law, resulting in anomalous diffusion. Anomalous diffusion is not only an important fundamental research topic in theoretical physics and statistical mechanics, but also a basic physical process of common concern in fields such as environment, hydrology, and engineering, with practical application backgrounds. The study addresses the issue that the classical convection-diffusion model cannot effectively describe the migration laws of gases from the perspective of flow modeling, and establishes a variable-order time fractional convection-diffusion equation model and numerically solves it using the finite difference method. It is used to simulate the migration laws and characteristics of gases, and to compare and analyze the experimental data with the simulation results. The results show that the fractional-order derivative model can well fit the anomalous diffusion phenomenon of the experimental data, and the time order characterizes the non-uniformity of the storage medium structure. The smaller the order, the stronger the non-uniformity. This verifies the accuracy and effectiveness of the model. This study solves the anomalous diffusion phenomenon that cannot be accurately described by traditional integer-order equations, providing a theoretical guidance for predicting the groundwater pollution risk during shale gas extraction under the dual-carbon goals.