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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.

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 [

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. [

A simple chemical reaction,

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 [_{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 [

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

Schematic of a reactive front displacement process.

Fluid (

The flow is governed by the equations for conservation of mass, momentum (Darcy’s equation), and the transport of the three chemical species:

The above equations are expressed in a Lagrangian reference frame moving with constant velocity

Following a previous related study [

Following previous studies [

The problem can be solved by splitting the variables into a base-state part and a perturbation term, where the base-state is a numerical solution of the reactive-diffusive-convective equations (

Following previous studies [

The derivatives of a function in the Hartley transform space can be easily derived from the transfer of the function by using Hartley transform derivative theorems:

The numerical code was validated by comparing the time evolution and the related finger structures for the case where the chemical reaction is complete (

Concentration isosurfaces for

Furthermore, the numerical convergence of the solution was analysed by considering cases with different spatial resolutions that varied from

In what follows, concentration isosurfaces are presented to examine the development of the instability and analyze the role of the chemical reversibility. The flow evolution will depend on both viscosities and densities mismatch between the chemical species, in addition to

In this part, it is assumed that there is no viscosity mismatch between the chemical species (

In a vertical displacement process, the initial interface between the displacing and displaced fluids is unstable if the top fluid

The case where the density of the product

Concentration isosurfaces for

It is clear that, in this case, reversibility tends to attenuate the instability of the flow at the unstable leading front, while there was a noticeable increase in the instability at the trailing front. In this particular case (

It is worth mentioning that the effects of reversibility on the instability may vary based on the particular choice of density differences between the chemical species. To illustrate this, another simulation was carried out for the previous case of frontal instability, where instability still takes place at the leading front but uses different magnitudes of the densities (

Concentration isosurfaces for

When the density of the product

In a reactive displacement process, instability will not grow at the unstable front until a certain amount of product

For displacements where the density of the product

Concentration isosurfaces for

In all previous cases, reversibility tends to attenuate or slightly enhance the instability of the reactive system. However, the effect of reversibility on the instability where both the trailing and the leading fronts are unstable may vary as the densities gap between the reactants increases. This is well illustrated in Figure

Concentration isosurfaces for

The strong effect of reversibility in increasing the instability at the leading front can be explained by looking at the variation of the density coefficients at that interface. When the chemical reaction is complete, the growth of instability at the leading front depends on the densities gap between reactant

The initial interface between the two reactants is stable or neutrally stable if the density of the displacing fluid is either smaller or equal to that of the displaced fluid (

An unstable trailing front and a stable leading front are observed in the case where the density of chemical product is smaller than that of both reactants. On the other hand, an unstable leading front and a stable trailing front are observed when the density of the product

Concentration isosurfaces for

Concentration isosurfaces for

In a displacement process, the variations in the densities of the fluids involved are not the only source of instability. In fact, frontal instability can also be encountered as the viscosities of these fluids vary. In a displacement process, the interface between the displacing and displaced fluid is unstable or stable, if the viscosity of the displacing fluid is smaller or larger than that of the displaced one, respectively. In this part, the effect of viscosities mismatch on the instability of a reversible reactive displacement process is examined by including the variations of the viscosities of the species involved in the displacement in addition to their densities.

In a vertical displacement process, a stable or neutrally stable viscous interface (

Starting with the unstable initial viscous interface, mobility ratios of

Concentration isosurfaces for

The case where the viscous initial interface between the reactants is stable (

Concentration isosurfaces for

It is worth mentioning that, in the previous two cases, reversibility had an opposite influence on the instability where it enhanced/attenuated the instability instead of attenuating/enhancing the instability for the cases (

In this part, the effects of reversibility on the instability of the reactive system are illustrated by presenting the relative contact area (R.C.A.) of the system, which is defined as the area of contact between the species involved in the displacement process scaled by the cross-sectional area of the cell.

This contact area is determined by measuring the length of a contour that corresponds to a specific concentration’s value of one of the species, which is the product

Measuring the contact area between the species is a good criterion to quantify the development of the instability. Moreover, this contact area is expected to increase as the complexity of the fingering structures increases. The R.C.A. for the case where an increase in the instability of the reactive front was observed (

Variations of the relative contact area (R.C.A.) for

Furthermore, the relative contact areas were also determined for the cases where the instability is triggered by both density and viscosity mismatches and it was observed that the contact areas grow faster/slower than the cases with only density mismatch, as the viscosity mismatches help/attenuate the instability when the reaction reverses.

It is worth mentioning that the growth of the instability when the reaction reverses is always slower than that of the complete reactions, at both the trailing and the leading fronts at early stages. Again, this is a result of the reduction in the total amount of the chemical product

The main objective of this study was to determine the effects of chemical reversibility on the overall efficiency of a vertical reversible reactive displacement process. The frontal instability is triggered by the densities combined or not with viscosities’ mismatch between the reactants and the product. The nonlinear interactions between the fluids were captured for different scenarios of frontal instability at specific sets of parameters. It was found that the fate of the displacement process can be dramatically influenced when a reversible chemical reaction takes place between the displacing and the displaced fluids, which introduces a new fluid with a viscosity and density that might be different from those of both reactants. The dynamics of fingering were mainly controlled by the viscosities and densities mismatch at the initial front as well as the trailing and the leading fronts in addition to the magnitude of the reversibility of the reaction.

The development of the instability was much faster in cases where the initial front between the reactants was unstable (

In the absence of viscosity differences between the chemicals, the effect of chemical reversibility was found to be similar for cases where only one of the trailing or the leading fronts is unstable regardless of the nature of the instability at the initial front. For those cases, the attenuation of the fingering instability increases with the increase in the magnitude of reversibility. Moreover, a complete stabilization of the system was reported for cases with stable initial fronts (

Finally, it was observed that increasing the viscosities mismatch between the chemical species in order to achieve viscously unstable trailing and leading fronts attenuated the instability of the system by limiting the degree of mixing between the chemicals. On the other hand, the stable initial viscous interface (

The authors declare that there is no conflict of interests regarding the publication of this paper.

H. Alhumade acknowledges financial support from the Ministry of Education, Kingdom of Saudi Arabia. The use of computational resources from WestGrid is also acknowledged.