Theoretical Development of Fractional Calculus Models for Anomalous Diffusion in Nigerian Aquifer Systems

Classical Fickian diffusion models based on integer-order differential equations have been demonstrated empirically to fail in characterising solute transport phenomena in heterogeneous geological media, including the fractured crystalline basement aquifers and alluvial systems prevalent across Nigeria's hydrogeological provinces. This dissertation develops a theoretical framework for anomalous diffusion modelling using fractional calculus, specifically the space-time fractional advection-dispersion equation with Riesz space derivative and Caputo time derivative. The theoretical framework is developed from first principles using continuous time random walk theory, establishing the connection between microscopic particle jump statistics and macroscopic fractional transport equations. Analytical solutions for simplified boundary value problems are derived using the Laplace-Fourier transform method, and the asymptotic behaviour of plume spreading is characterised as a function of the fractional order parameters. A finite difference numerical scheme based on the shifted Grünwald-Letnikov approximation is developed and proved to be unconditionally stable and convergent using von Neumann analysis. The framework is calibrated and validated against tracer test data from borehole aquifer tests in Basement Complex terrain across Ekiti and Ondo States, South West Nigeria. The estimated anomalous diffusion exponent of 1.67 confirms superdiffusive transport inconsistent with Fickian assumptions. This work fills a theoretical gap in the application of fractional calculus to Nigerian hydrogeological modelling. Keywords: fractional calculus, anomalous diffusion, aquifer transport, fractional advection-dispersion, Caputo derivative.