Modeling Anomalous Groundwater Flow with a Fractional Laplacian in Heterogeneous Porous Media

Abstract

The study is an extensive computational analysis of groundwater flow in ‎heterogeneous porous media using classical and fractional partial differential ‎equations (PDEs). The Finite Difference Method (FDM) is used to solve the ‎classical Laplace equation, and in order to model memory and non-local ‎phenomena, a fractional Laplacian formulation, discretized through Grünwald-‎Letnikov approximation, is used. These numerical methods are used to model ‎steady-state piezometric head distribution with slope angles, flow coefficients, ‎and domain heterogeneity. Simulations done with MATLAB show that fractional ‎models capture anomalous diffusion patterns more accurately than classical ‎ones, especially in systems with spatial intricacy and long-range interactions. ‎Surface and contour plots expose the more diffuse, smoother behavior ‎indicative of fractional diffusion. This is a singular demonstration of how ‎composite geological environments with by fractional PDEs of groundwater ‎dynamics can be modeled within far more flexible and descriptive frameworks. ‎It also serves as foundation for the integration of hydrogeological simulation ‎tools and fractional calculus.