Formulation and Evaluation of CTRW Governing Equations for Irreversible, Bimolecular Reactions During Transport

Abstract:

The continuous time random walk (CTRW) formalism is a valuable tool for modeling both conservative and reversibly sorbing solute transport in heterogeneous porous media, underpinning both Eulerian (integrodifferential equation-based) and Lagrangian (particle tracking) approaches to transport modeling. More recently, there has been interest in modeling transport with irreversible reactions, and Lagrangian CTRW-based numerical models have been successfully applied to these problems. However, a corresponding Eulerian theory has been lacking. We recently developed Eulerian governing equations in the presence of irreversible bimolecular reactions, via the device of upscaling transport and treating reactions at a finer scale. The technique is generally valid, even in porous media that do not have an obvious division of length scales, subject to certain smoothness assumptions on the solution. We show that the governing equations we develop simplify, under appropriate circumstances, to both the generalized master equation for the unreactive CTRW and to the advection-dispersion-reaction equation. We also present a numerical corroboration of our development and its underlying smoothness assumptions obtained using a novel, indirect particle-tracking / partial differential equation hybrid technique. We discuss the implications of the new governing equations for practical reactive transport modeling and for conceptualization of subsurface processes, and highlight connections to other approaches. As applicable, we will also discuss aspects of numerical implementation.