Bimolecular reactive transport in a two-dimensional velocity field in disordered media

Abstract

The nonlocal-in-time, integro-partial differential equation (iPDE) describing the continuous time random walk (CTRW) is augmented by a nonlinear chemical-species term accounting for migrating bimolecular reactions in disordered media. This augmented form of the iPDE, previously shown to be in excellent agreement with a particle-tracking (CTRW-PT) version of reactive transport, is applied here to a reanalysis of recent experimental results to further establish its validity. The experimental set-up uses an injected solution of a pH indicator (Congo red) as both an inert tracer or a reactant, depending on the pH of the solution. A refraction index-matched, water-saturated porous medium allows tracking of the reacted concentration field, which is modeled by the solutions of the two-species coupled iPDE, whose nonlinear form precludes the use of a Laplace transform. The time-domain solutions are obtained with a finite element method (FEM) and a sum of exponentials representation of the kernel (memory function). The two-dimensional flow field subject to the macroscopic boundary conditions is calculated by solving the Darcy equation. The CTRW is well established for nonreactive transport in disordered media. The fit for the inert tracer in this case provides a reasonable match to these particular experimental data. The comparable, agreeable fit for the reactive case further establishes the validity of the iPDE augmented with a nonlinear, second-order reaction term.