Existence and uniqueness of solutions of unsteady Darcy-Brinkman problem for modelling miscible reactive flows in porous media

TL;DR

This paper establishes the existence and uniqueness of solutions for a coupled unsteady Darcy-Brinkman model with reactive flows in porous media, analyzing long-term behavior and blow-up conditions.

Contribution

It proves existence and uniqueness of weak solutions for a broad class of initial data and characterizes long-term dynamics including blow-up scenarios.

Findings

Solutions exist globally if initial concentration is between 0 and 1.

Solutions blow up in finite time if initial concentration exceeds 1.

Numerical simulations confirm theoretical long-time decay and blow-up behaviors.

Abstract

In this work, we investigate a model describing flow through porous media with permeability heterogeneity, combining an advection-reaction-diffusion equation for solute concentration with an unsteady Darcy-Brinkman equation with Korteweg stresses in the presence of external body forces for the flow field. Such models are appropriate in describing flows in fractured karst reservoirs, mineral wool, industrial foam, coastal mud, etc. These equations are coupled with Neumann boundary conditions for the solute concentration and no-flow conditions for the fluid velocity. For a broad class of initial data, we proved the existence of weak solutions. In the presence of a second-order nonlinear reaction, we show that the long-time behaviour of the solution depends on the initial concentration \(C_0\). More precisely, the solution exists for all time if \(0\leq C_0\leq 1\), and blows up at finite time if $C_0>1$. Furthermore, the uniqueness of the solution is proved for a two-dimensional domain. Finally, numerical simulations based on the finite element method have been presented that illustrate non-negativity of the concentration, long-time decay, and finite-time blow-up in agreement with theoretical estimates.