Solution of 1D Space Fractional Advection-Dispersion Equation with Nonlinear Source in Heterogeneous Medium

Abstract

Fractional derivatives, owing to their nonlocal behavior, are well suited for modeling the fate of contaminants in a heterogeneous medium. This study develops the mathematical formulation and the solution of a one-dimensional (1D) fractional flux advection-dispersion equation (FFADE) for an increasing or decreasing nonlinear source of contamination in the permeable region. In the proposed model, the first-order space derivative is replaced with the fractional derivative of order (α−1) (1<α<2) in a Riemann-Liouville sense. The model assumes a spatiotemporally varying dispersion and seepage velocity, constant dispersion and uniform seepage velocity, linear dispersion and linear seepage velocity, and quadratic dispersion and linear seepage velocity, exhibiting various levels of medium heterogeneity. Initially, the medium is considered to be polluted with solute concentration c0 and spatially varying specific concentration ci. The approximate solution to the model is obtained by the modified Adomian decomposition method. It is remarked that change in the medium’s heterogeneity alters the peak of solute concentration, and the presence or absence of advection in the solute transport process significantly affects the concentration distribution. The obtained results are validated with field data available in the literature. The obtained solutions may be used to predict solute concentration in the permeable region from penetrating contaminant sources situated at any place along a vertical plane perpendicular to the direction of, for example, groundwater flow.