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

TL;DR

This work establishes the mathematical well-posedness of a coupled $A+B\rightarrow C$ reactive transport problem in porous media modeled by a time-dependent Brinkman equation, incorporating density via the Oberbeck–Boussinesq framework and permeability $K(c)=e^{-\alpha c}$. It proves global existence and uniqueness of weak solutions using a Galerkin approach, along with a maximum principle that bounds concentrations and preserves nonnegativity, and it derives a continuum dependence estimate to guarantee stability with respect to initial data. The authors formulate a semi-discrete finite-element scheme and perform 2D numerical simulations (and 3D extensions) using COMSOL to study fingering instabilities under density stratification and permeability variation, validating scaling laws from classical reaction–diffusion theory and demonstrating continuous data dependence across initial geometries (flat and elliptic interfaces). The numerical experiments reveal how reaction-induced density changes and mobility contrasts shape finger wavelengths and front propagation, offering insights applicable to CO$_2$ sequestration, petroleum migration, and karst-reservoir processes. Overall, the paper provides a solid theoretical foundation for reactive fingering patterns and a versatile computational framework for exploring complex porous-media flows.

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.

Figures (8)

• Figure 1: (a) Spatio-temporal evolution of $x$-averaged reaction-diffusion profiles $\bar{a}(y,t)$, $\bar{b}(y,t)$, and $\bar{c}(y,t)$. Darker gradients indicate increasing time from $t = 0$ to $t = 1000$, with intervals of 100 units. The diffusion ratio is fixed at $d = 1$. (b) Temporal evolution of the reaction front position $y_f(t)$, fitted with a constant of 400. Temporal evolution of (c) reaction front width $w_f(t)$, (d) total reaction rate $R(t)$, and (e) reaction rate at the front $R(y_f(t))$, each with corresponding fitted power laws and proportionality constants of 2.73, 322, and 0.147, respectively.
• Figure 2: The maximum relative errors $E_{A,h}$, $E_{B,h}$, and $E_{ml,h}$ as functions of the maximum element size $h$.
• Figure 3: Typical density profiles $\rho$ for each case I to VI as a function of the vertical coordinate $y$, given by $\rho = 1 + R_{A} \bar{a} + R_{B} \bar{b} + R_{C} \bar{c}$, where $\bar{a}(y)$, $\bar{b}(y)$, and $\bar{c}(y)$ are reaction-diffusion averaged concentrations at $t = 100$, shown in Fig. \ref{['fig:galfi']}(a).
• Figure 4: Spatio-temporal evolution of concentration profiles for case-I, with $R_A = 2$, $R_C = 0$, and $\alpha = 0$. The top panel shows concentration $a$, the middle panel shows $b$, and the bottom panel shows $c$. In this configuration, only the more dense fluid, reactant $A$ (the top fluid), is present, leading to Rayleigh-Taylor fingering as it falls. The reaction occurs at the interface between reactants $A$ and $B$.
• Figure 5: Spatio-temporal evolution of the reaction product concentration profile $c$ for cases II to VI, shown from the top to bottom panels, respectively. The axis labels are same as Fig. \ref{['fig:abc']}.

Theorems & Definitions (21)

• Theorem 2.1: Gagliardo–Nirenberg, cf. migorski2019nonmonotone, garcke2019
• Lemma 2.1: Theorem 7, bresch2007effect
• Definition 2.1
• Definition 2.2
• Definition 2.3: Weak solution to the problem \ref{['model']}--\ref{['boundry and initial data']}
• Theorem 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof