Well-posedness and Fingering Patterns in $A + B \rightarrow C$ Reactive Porous Media Flow

TL;DR

This paper establishes the mathematical well-posedness of reactive porous media flow systems involving density-driven convection and reaction, and explores fingering patterns through numerical simulations relevant to geological and environmental applications.

Contribution

It provides the first rigorous proof of existence and uniqueness for the coupled reaction-diffusion and flow system with density effects, along with numerical analysis of fingering instabilities.

Findings

Well-posedness of the reactive flow system is proven.

Numerical simulations reveal fingering patterns influenced by density stratification.

Explicit bounds on reactant and product concentrations are derived.

Abstract

The convection-diffusion-reaction system governing incompressible reactive fluids in porous media is studied, focusing on the \( A + B \to C \) reaction coupled with density-driven flow. The time-dependent Brinkman equation describes the velocity field, incorporating permeability variations modeled as an exponential function of the product concentration. Density variations are accounted for using the Oberbeck-Boussinesq approximation, with density as a function of reactants and product concentration. The existence and uniqueness of weak solutions are established via the Galerkin approach, proving the system's well-posedness. A maximum principle ensures reactant nonnegativity with nonnegative initial conditions, while the product concentration is shown to be bounded, with an explicit upper bound derived in a simplified setting. Numerical simulations are performed using the finite element method to explore reactive fingering instabilities and illustrate the effects of density stratification, differential product mobility, and two- or three-dimensionality. Two cases with initial flat and elliptic interfaces further demonstrate the theoretical result that solutions continuously depend on initial and boundary conditions. These theoretical and numerical findings provide a foundation for understanding chemically induced fingering patterns and their implications in applications such as carbon dioxide sequestration, petroleum migration, and rock dissolution in karst reservoirs.