Fractional Differential Equations in Modelling Anomalous Diffusion in Oil-Contaminated Soils in Bayelsa State

This study applies fractional differential equations to model anomalous diffusion processes in oil-contaminated soils in the oil-producing communities of Bayelsa State, South South Nigeria, filling a research gap identified in the existing remediation and contaminant transport literature for the Niger Delta environment. Classical Fickian diffusion models assume Gaussian particle displacement distributions and linear mean squared displacement growth in time, assumptions that frequently fail to capture the complex, heterogeneous transport behaviour of petroleum hydrocarbons in the fractured and clay-rich soils characteristic of mangrove swamp environments in Bayelsa. Soil samples and field diffusion measurements are collected from contaminated plots in Ogbia and Yenagoa local government areas through collaboration with the Hydrocarbon Pollution Remediation Project, and time-domain diffusion experiments are conducted at controlled laboratory conditions. Fractional Fokker-Planck equations with subdiffusive and superdiffusive exponents are fitted to the observed mean squared displacement data, and Caputo and Riemann-Liouville fractional derivative formulations are compared in terms of fit to experimental data. Numerical solutions are obtained using the Grünwald-Letnikov finite difference scheme. Results confirm subdiffusive behaviour with anomalous exponent values between 0.6 and 0.82 in clay-dominant contaminated layers, indicating anomalous transport that standard models would misrepresent. The study recommends fractional calculus-based transport models for remediation planning in the Niger Delta. Keywords: fractional differential equations, anomalous diffusion, soil contamination, Niger Delta, Bayelsa State