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

Pankaj Roy, Satyajit Pramanik

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.

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

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 . More precisely, the solution exists for all time if , and blows up at finite time if . 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.

Paper Structure

This paper contains 16 sections, 13 theorems, 101 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Model
  3. Function spaces and preliminary results
  4. Weak formulation
  5. Assumptions
  6. Main results
  7. Proof of Theorem \ref{['th: existence']}
  8. A Priori Estimates
  9. Existence
  10. Proof of Theorem \ref{['th: bddness & blow_up']}
  11. Global existence of solution
  12. Finite time blow up
  13. Proof of Theorem \ref{['th: Uniqueness']}
  14. Some special cases
  15. Numerical validation
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $\Omega \subset \mathds R^N,\ N = 2,3,$ be a domain with $\mathcal{C}^{1}$ boundary. Then, for any $\boldsymbol{\phi} \in H^{1}(\Omega)$, there exists a constant $M>0$ depending on $\Omega$ such that the following inequality holds for $N=2$: and for $N=3$:

Figures (2)

  • Figure 1: Contour plots of solute concentration $C$ for different time levels corresponding to the initial condition \ref{['eq:numerical_IC1']}.
  • Figure 2: Temporal evolution of $\|C(t)\|_{L^1(\Omega)}$ corresponding to the initial conditions given in (a) equation \ref{['eq:numerical_IC1']}, and (b) equation \ref{['eq:numerical_IC2']}.

Theorems & Definitions (24)

  • Theorem 2.1: Gagliardo–Nirenberg (see migorski2019nonmonotone, ebenbeck2019cahn)
  • Theorem 2.2: Grönwall’s inequality (integral form) (see Appendix of evans2010partial)
  • Definition 2.1: Weak solution
  • Remark 1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 1
  • proof
  • Remark 2
  • ...and 14 more