SCHOOL OF MECHANICAL ENGINEERING SEMINAR Monday, January 14, 2013 at 15:00 Wolfson Building of Mechanical Engineering, Room 206

Application of a Particle Method to the Diffusion-Reaction Equation

Department of Civil & Environmental Engineering & Earth Sciences, University of Notre Dame

A chemical reaction between two species in porous media can only happen if molecules collide and react. Thus, the level of mixing of the species can become a limiting factor in the onset of reaction. The effect of incomplete mixing upon reaction has important implications on various processes in natural and engineered systems, ranging from mineral precipitation in geological formations to groundwater remediation in aquifers. For example, incomplete mixing that slows down the remediation of a contaminated site can delay site closure, and increase remediation costs.

Numerical models of flow and transport typically fail to describe incomplete mixing effects. Finite difference/element methods usually assume that each of the numerical cells is well-mixed. In order to take into account the incomplete mixing effects, the cells need to be extremely small, leading to impractical computational costs. Thus, we propose a different approach for the modeling of the ADRE (advectiondiffusionreaction equation) by means of a Monte-Carlo particle tracking approach. In this method, each numerical particle represents some reactant mass, and advection and diffusion are modeled by drift and a random walk of the particles (via Langevin equation). The novel part of the approach is the implementation of the reaction term. This is done by annihilating some of the particles in each time step. The probability of the annihilation is proportional to the reaction rate constant and the probability of the particles to become colocated .

To demonstrate the approach we study a relatively simple system with a bimolecular irreversible kinetic reaction A+B→0, where the initial concentrations are given in terms of an average and a perturbation. Such stochastic initial conditions are highly suitable for a particle approach, since noise is inherent to a representation of concentration by discrete particles. An approximate analytical solution for this system exists for one dimension; we extend it to d=2,3. We also derive the relationship between the initial number of particles in the system and the initial concentrations perturbations represented by that number. The numerical results of the particle-tracking simulations demonstrate the well-known phenomena of incomplete mixing (Ovchinnikov-Zeldovich segregation). We compare the results to the approximate analytical solution, and explain the late time discrepancy.

1. Paster A., Bolster D. and Benson D.A., Particle Tracking and the Diffusion-Reaction Equation. Water Resources Research (in press).
2. Ding D., Benson D. A., Paster A., and Bolster D. Modeling bimolecular reactions and transport in porous media via particle tracking. Advances in Water Resources, 53, pp. 56-65, 2013.

Application of a Particle Method to the Diffusion-Reaction Equation

Dr. Amir Paster