Caputo-Fabrizio Fractional Advection-Dispersion Model for Contaminant Transport in Groundwater

Abstract

Groundwater is often the only reliable water supply in arid regions, so predicting how a dissolved contaminant spreads in a shallow aquifer matters for protecting it. The classical advection-dispersion equation (ADE) has a memoryless time derivative and does not reproduce the early arrival and long tails often seen in field breakthrough curves. Here the time derivative of one -dimensional ADE is replaced by the Caputo-Fabrizio (CF) fractional derivative, whose exponential kernel is non-singular and gives a fading memory controlled by the order α. For Semi-infinite column with a constant-concentration inlet, the problem is solved exactly in the Laplace domain and inverted numerically with the Talbot method. For α=1 the result matches the classical Ogata-Banks solution to machine precision. We show analytically that the mean breakthrough time is exactly 1/ α times the advective time, so α acts as a retardation factor, and that the model has a non-zero early-time response, given in closed form, and a heavier tail than the classical case. At a Péclet number of 10, lowering α from 1 to 0.5 delays the half-concentration arrival time by about 48%, from 0.91 to 1.35 advective time units. The model is analytically tractable, but it still has to be calibrated on tracer-test data before it can be used for risk screening.