Eur. Phys. J. B (2018) 91: 15
https://doi.org/10.1140/epjb/e2017-80417-8

Regular Article

A continuous time random walk (CTRW) integro-differential equation with chemical interaction^★

Department of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot 7610001, Israel

^a e-mail: brian.berkowitz@weizmann.ac.il

Received: 11 July 2017
Received in final form: 23 October 2017
Published online: 17 January 2018

Abstract

A nonlocal-in-time integro-differential equation is introduced that accounts for close coupling between transport and chemical reaction terms. The structure of the equation contains these terms in a single convolution with a memory function M (t), which includes the source of non-Fickian (anomalous) behavior, within the framework of a continuous time random walk (CTRW). The interaction is non-linear and second-order, relevant for a bimolecular reaction A + B → C. The interaction term ΓP_A (s, t) P_B (s, t) is symmetric in the concentrations of A and B (i.e. P_A and P_B); thus the source terms in the equations for A, B and C are similar, but with a change in sign for that of C. Here, the chemical rate coefficient, Γ, is constant. The fully coupled equations are solved numerically using a finite element method (FEM) with a judicious representation of M (t) that eschews the need for the entire time history, instead using only values at the former time step. To begin to validate the equations, the FEM solution is compared, in lieu of experimental data, to a particle tracking method (CTRW-PT); the results from the two approaches, particularly for the C profiles, are in agreement. The FEM solution, for a range of initial and boundary conditions, can provide a good model for reactive transport in disordered media.