Mathematical Problems in Engineering

Volume 2015, Article ID 920692, 13 pages

http://dx.doi.org/10.1155/2015/920692

## Numerical Simulations of Gravity Driven Reversible Reactive Flows in Homogeneous Porous Media

^{1}Department of Chemical Engineering, The University of Waterloo, Waterloo, ON, Canada N2L 3G1^{2}Department of Chemical and Petroleum Engineering, The University of Calgary, Calgary, AB, Canada T2N 1N4

Received 1 April 2015; Accepted 9 June 2015

Academic Editor: Junwu Wang

Copyright © 2015 H. Alhumade and J. Azaiez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The effect of reversibility on the instability of a miscible vertical reactive flow displacement is examined. A model, where densities and/or viscosities mismatches between the reactants and the chemical product trigger instability, is adopted. The problem is governed by the continuity equation, Darcy’s law, and the convection-diffusion-reaction equations. The problem is formulated and solved numerically using a combination of the highly accurate spectral methods based on Hartley’s transform and the finite-difference technique. Nonlinear simulations were carried out for a variety of parameters to analyse the effects of the reversibility of the chemical reaction on the development of the flow under different scenarios of the frontal instability. In general, faster attenuation in the development and growth of the instability is reported as the reversibility of the chemical reaction increases. However, it was observed that reversibility is capable of triggering instability for particular choices of the densities and viscosities mismatches. In addition, the effect of the reversibility in enhancing the instability was illustrated by presenting the total relative contact area between the reactants and the product.

#### 1. Introduction

Instability at the interface between flowing solutions in porous media can be triggered as a result of viscosities and/or densities mismatch between the fluids. This instability develops in the form of intruding fingers and is referred to as viscous fingering or Saffman-Taylor instability in the case of viscosities mismatch or as density fingering or Rayleigh-Taylor instability in the case of densities mismatch between the fluids [1–5].

The simultaneous variation in viscosities and densities is encountered in various applications. An analytical expression for the growth of instability in a nonreactive system with variation in viscosities and densities was derived by Bacri et al. [6]. In 1992, Rogerson and Meiburg carried out a linear stability analysis to investigate the interface of a nonreactive system with densities and viscosities mismatch in porous media where both normal and tangential velocities can be present [7]. It was reported that the growth rate of the instability was not affected by tangential velocity in immiscible displacements, unlike the miscible displacements where a stabilizing role was observed. In the same year, the same authors investigated the nonlinear evolution for the unstable modes of the same system [8], where remarkable features of instability such as diagonal fingering and secondary side-finger instabilities were observed. The effects of nonmonotonic viscosities and densities profiles on the instability of vertical nonreactive displacement processes were investigated by Manickam and Homsy by conducting nonlinear simulations of the system [9]. The authors found that a stable viscous interface between the fluids can break the symmetry of the buoyancy-driven instability by acting as a barrier against the upward growth of instability.

A simple chemical reaction, , can change the physical properties such as viscosity or density of the fluids, which might affect the fate of the displacement process. Reactive flows are encountered in many applications such as in situ oil recovery, carbon dioxide sequestration, in situ groundwater remediation and chromatographic columns, transport of contaminant, laminar combustive and exothermic reactive flows, natural convection heat and mass transport, and reactive porous media for biological applications [10–14]. It has been shown that a frontal instability can be purely driven by a chemical reaction, where a viscous product is generated at the interface of two reactants with smaller viscosities [15]. In a series of experiments, the effect of chemical reaction on the viscosity profile of an initially unstable system was investigated by Nagatsu et al. [16–18]. The properties of the unstable miscible reactive displacements were analyzed by a number of theoretical and numerical studies. For example, Rongy et al. [19] showed that the instability of the interface in a vertical displacement process can be influenced by buoyancy-driven convection once the densities of the species are changed by a chemical reaction even for equal diffusion coefficients and equal initial concentrations of the reactants. A related study examined the buoyancy chemically driven instability where the reaction introduces a heavier product at the initial interface and where a variation in the diffusion rates of the solutions exists [20]. For horizontal geometries, Alhumade and Azaiez carried out a linear stability analysis [21] followed by nonlinear simulations [22] of the reversible reactive flow displacements. The authors investigated quantitatively and qualitatively the effects of the chemical reversibility on the instability of the flows. Hejazi and Azaiez conducted a detailed linear stability analysis [23] followed by a nonlinear simulation [24] of miscible vertical reactive displacement processes with transverse velocity. More recently, the effects of time-dependent injections on the dynamics of reactive flows in homogeneous porous media have been investigated [25].

The instability of miscible reactive solutions under gravity force is encountered in many underground flows applications, such as in the geological storage of CO_{2} in addition to mixing of brine [26]. The former process involves the dissolution of carbon dioxide in the reservoir’s fluid and this can be modeled by a chemical reaction [27]. A heavier solution will be introduced on top of the reservoir’s lighter fluid as a result of the CO_{2} dissolution, which will establish instability at the fluids’ interface. The reversibility of the chemical reaction plays an important role in various fields such as in situ soil remediation [28] and liquid chromatographic columns [29]. This motivates the present study aimed at analyzing the effects of chemical reversibility on the frontal instability of miscible vertical reactive displacements. In this study, a system where the variations in viscosities and densities between the reactants and the product initiate a frontal instability in the absence of injection or transverse velocities is examined.

#### 2. Mathematical Model

##### 2.1. Physical Problem

Nonlinear simulations of a two-dimensional displacement process are carried out in a homogeneous reservoir, where the porosity and permeability are assumed to be constant. The displacement takes place in the vertical direction, which is referred to as the -axis, while the -axis is perpendicular to the direction of the flow. Furthermore, both the displacing and displaced fluids are assumed to be incompressible, Newtonian, and miscible. A schematic of the process is shown in Figure 1, where , , and are the length, width, and thickness of the medium, respectively. When the thickness of the medium is small compared to the other dimensions, the system corresponds to a Hele-Shaw cell, which is a common prototype for homogeneous porous media [4].