Fractal mobile/immobile solute transport

TLDR

A fractal mobile/immobile transport model assumes power‑law waiting times in the immobile zone, yielding fractional‑time derivatives that recover earlier mobile/immobile models and describe continuous‑time random walks with heavy‑tailed waiting times. The model solution is obtained by applying an integral transform to any boundary‑value solution without an immobile phase, allowing multidimensional numerical outputs to be modified to include the fractal immobile effect. The transformed solutions reproduce anomalous tracer plume behavior in heterogeneous aquifers, producing late‑time power‑law breakthrough curves and mobile‑mass decline that match observations at the MADE site (γ = 0.33) and the Haggerty et al.

Abstract

A fractal mobile/immobile model for solute transport assumes power law waiting times in the immobile zone, leading to a fractional time derivative in the model equations. The equations are equivalent to previous models of mobile/immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times. The solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase. In this regard, the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase. The solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers, including power law breakthrough curves at late time, and power law decline in the measured mobile mass. The MADE site mobile tritium mass decline is consistent with a fractional time derivative of order γ = 0.33, while Haggerty et al. 's [2002] stream tracer test is well modeled by a fractional time derivative of order γ = 0.3.