Nonlinear simulation of transverse flow interactions with chemically driven convective mixing in porous media

Authors

  • S. H. Hejazi,

    Corresponding author
    1. Department of Chemical and Petroleum Engineering, University of Calgary, Schulich School of Engineering, Calgary, Alberta, Canada
    • Corresponding author: S. H. Hejazi, Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada. (shhejazi@ucalgary.ca)

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  • J. Azaiez

    1. Department of Chemical and Petroleum Engineering, University of Calgary, Schulich School of Engineering, Calgary, Alberta, Canada
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Abstract

[1] Buoyancy-driven hydrodynamic instabilities of a miscible reactive interface in a homogeneous porous medium is examined. A bimolecular chemical reaction (A+B→C) is triggered at the interface between two reactant solutions A and B resulting in a chemical product solution C with different density and the viscosity from those of the reactants. The effects of the chemical reaction and a transverse flow parallel to the initial interface between the reactants are numerically analyzed. It was found that as a result of the transverse flow, fingers with sharp concentration gradients tend to develop and advance fast downward leading to higher rates of chemical production. Furthermore, a detailed analysis of the finger growth and the effects of buoyancy, transverse flow and chemical reaction allowed to reach a physical interpretation of the trends observed. Finally, a special tuning of the transverse velocity is proposed to ensure maximum or minimum chemical production applicable to subsurface flows.

1. Introduction

[2] Interfacial instabilities in miscible or immiscible porous media displacement flows have been the focus of numerous studies in the past [Hill, 1952; Slobod and Caudle, 1952; Saffman and Taylor, 1958; Bachmat and Elrick, 1970; Homsy, 1987; McCloud and Maher, 1995]. Also known as Saffman-Taylor or Rayleigh-Taylor instabilities, this frontal instability occurs when a less viscous or more dens solution displaces a more viscous or less dense solution, respectively. In some practical cases, both viscosities and densities of the solutions vary simultaneously [Abbas and Rose, 2010], and the first mathematical expression for the growth of instabilities in such a system was presented by Bacri et al. [1992]. In an interesting study, Manickam and Homsy [1995] showed the break of symmetry in the growth of fingers of buoyancy-driven instabilities when there are nonmonotonic viscosity and density profiles in vertical miscible displacements.

[3] More recently, experimental studies have shown the evidence of interfacial instabilities when chemical reactions could modify the physical properties of solutions flowing in porous media, such as viscosity or density. Flow instabilities can be purely chemically driven when a bimolecular chemical reaction (BCR), A+B→C, results in the production of a viscoelastic/elastic component C with a different viscosity form those of the reactants A and B [Podgorski et al., 2007]. Also, chemical reactions can change the fate of fluid interface developments in initially unstable systems [Nagatsu et al., 2007, 2009]. The properties of unstable miscible reactive interfaces in horizontal displacements have also been analyzed in a number of studies. Gérard and De Wit [2009] numerically examined the differences between the displacement of A by B or of B by A, with the assumption that the viscosity of the solution depends only on the concentration of the product C, and that the chemical species have different rates of diffusion. In a series of numerical studies, Hejazi and Azaiez [2010a, 2010b] investigated horizontal displacements of a single vertical BCR interface (between reactants A and B) and two BCR interfaces (B-reactant layer sandwiched and transported by the reactant A), respectively. It was assumed that the solution's viscosity was a function of all chemical components, and a parametric study in terms of the species' viscosity ratios and Péclet number was conducted. For the former configuration, a detailed linear stability analysis [Hejazi et al., 2010] allowed to classify the different possible viscous fingering scenarios. It was reported that chemical reactions can actually lead to frontal instabilities even in the case of a more viscous fluid displacing a less viscous one. This interesting phenomenon was later observed in a subsequent experimental study by [Riolfo et al., 2012a]. The influence of gravity on a vertical BCR interface in the absence of displacement flows was analyzed by Rongy et al. [2008]. These authors showed that even in the case of equal diffusion coefficients and equal initial reactants concentrations, a fluid motion can be triggered by buoyancy forces as soon as the chemical reaction changes the densities of the species.

[4] In a gravity field and for a horizontal reactive interface, experimental evidence of chemically driven convective mixing was initially reported by Almarcha et al. [2010b, 2011] and Kuster et al. [2011]. In these experiments, the neutralization of a strong acid (HCl) by a strong base (NaOH) in aqueous miscible solutions allowed to change the solution density profiles. As a result, hydrodynamic instabilities developed at the horizontal reactive interface. Moreover, in another flow configuration, systems with heterogeneous viscosity and density profiles were generated through a chemical reaction involving secondary amine and an acrylate [Riolfo et al., 2012b]. In a series of related theoretical studies for a horizontal BCR, Almarcha et al. [2010a] and Lemaigre et al. [2013] showed that nonmonotonic density profiles or even different diffusion rates of solutions do lead to an asymmetric development of the flow with respect to the initial reactant interface. In fact, understanding the subtle interactions between chemical reactions and natural convection are at the heart of numerous applications and particularly in subsurface flows [Almarcha et al., 2010b, 2011]. For instance, one may refer to the geological storage of carbon dioxide, i.e., CO2 sequestration [Riaz et al., 2006; Hassanzadeh et al., 2007; Ennis-King and Paterson, 2007]. In such a process, CO2 is dissolved into the brine of saline aquifer[Adamczyk et al., 2009; Iglauer, 2011], forming a denser CO2 saturated zone on top of the less dense brine which naturally induces buoyancy-driven convective mixing. Such frontal instabilities are favorable as they facilitate the storage process and minimize the risk of CO2 leaking to the upper formations [Johnson et al., 2004; Neufeld et al., 2010; Ghesmat et al., 2011; Emami Meybodi and Hassanzadeh, 2011].

[5] Another important aspect influencing buoyancy-driven flows is the presence of a discharge flow as encountered in geological systems. The influence of through flows on buoyancy-driven motions has been extensively investigated in cases of thermally induced density variations [Prats, 1966; Nield and Barletta, 2010] as well as in situation where the physical properties are concentration dependent rather than temperature dependent [Rogerson and Meiburg, 1993a, 1993b]. In these studies, the fluids had different densities and viscosities and the inclined geometry introduced two velocity components, parallel and normal to the fluid interface. In particular, Rogerson and Meiburg [1993a] reported that the shear velocity component has a stabilizing effect on the frontal instabilities where diagonal fingering and secondary side-fingers were observed [Rogerson and Meiburg 1993b].

[6] In spite of a very rich literature on the interaction between through flows and natural convection for nonreactive systems, studies on reactive interfaces are limited to the linear stability analysis of Hejazi and Azaiez [2012]. This study presented a general classification of density fingering instability scenarios along with the influence of transverse velocities and component viscosities. It was found that the chemical reaction can actually destabilize a buoyancy-stable initial interface by generating a nonmonotonic density profile. It was also reported that the system was unstable and the transverse flow will not advect the perturbations resulting in the development of locally unstable regions. Furthermore, the study revealed that a chemical product which is denser but less viscous than the reactants would result in a dramatic shift of the most unstable wave number toward the longer wavelengths. However, such conclusions are limited to the early stages of the instability and do not apply to the later nonlinear development of the flow nor do they allow to determine the evolution of the finger structures and the subsequent roles of hydrodynamics and chemistry. Hence, the need to conduct full nonlinear simulations of the flow. Still, the conclusions from the linear stability analysis were essential in the choice of the flow parameters that result in the most interesting dynamics when running the numerical simulations.

[7] In order to understand the nonlinear evolution of the system, a general flow displacement where the reactants and the chemical product have different densities and viscosities will be examined. Furthermore, a transverse flow will be introduced parallel to the reactant interface. This aspect will allow addressing the important practical question of how the coupling of natural horizontal flows in underground reservoirs and gravity forces affect the development of reactive flows and chemical production. The results will be enlightening for a variety of subsurface flow processes. For example in the geological storage of carbon dioxide, it was recently reported that the mean flow discharge could delay the onset of convective mixing when coupled with dispersion [Hassanzadeh et al., 2009] and also could enhance the storage capacity when coupled with capillary and solubility trapping mechanisms [MacMinn et al., 2010, 2011].

[8] The outline of the article is as follows. In section 'Mathematical Model', the mathematical model is introduced. Results of nonlinear simulations and quantitative analyses are presented in section 'Results and Discussion' followed by the conclusion in section 'Conclusions'.

2. Mathematical Model

2.1. Problem Description

[9] A schematic of a two-dimensional homogenous porous medium is shown in Figure 1. The direction of gravity is along the x axis while the y axis is parallel to the initial plane of the interface. The length, width, and thickness of the medium are Lx, Ly, and b, respectively [Hejazi and Azaiez, 2012].

Figure 1.

Schematic of a horizontal reactive interface.

[10] Initially the porous medium is saturated with the fully miscible solutions of scalars (A) and (B) in a solvent. The solution of (A) with density and viscosity math formula is on top of a second solution (B) of density and viscosity math formula. A product (C) is generated as a result of chemical reaction:

display math(1)

[11] The chemical product solution (C) has a viscosity math formula and density math formula that in general will be assumed to be different from that of either reactants. All three species are fully miscible solutions. A transverse velocity, V1 is introduced along the positive y axis. Conceptually, we consider that a layer of chemical product (C) is formed between the reactants (A) and (B). As a consequence, three layers are identified with two interfaces in the reaction zone, namely an math formula interface at the top and a math formula at the bottom, which will be referred to as the trailing and leading reaction zones. Depending on the species' densities and viscosities, these interfaces may become unstable [Hejazi and Azaiez, 2012].

2.2. Governing Equations and Numerical Technique

[12] Reaction-convection-diffusion (RCD) equations for the transport of the three equi-diffusive species are coupled to the equations of mass conservation and momentum conservation in the form of Darcy's law in the limit of Boussinesq approximation. The viscosity and density are scaled with math formula and math formula, the solution viscosity and a reference density difference, respectively. The velocity, length and time are scaled with math formula and math formula respectively, where math formula is the characteristic velocity. The concentration is scaled with that of the pure reactant math formula and the pressure p with math formula. In the previous expressions, g is the gravitational acceleration, D the diffusion coefficient, κ the medium permeability, and ϕ its porosity.

[13] The resulting dimensionless equations in a Lagrangian reference frame that moves with the uniform transverse velocity V1 in the y direction are [Hejazi and Azaiez, 2012]:

display math(2)
display math(3)
display math(4)
display math(5)
display math(6)

[14] In the above equations, math formula is the superficial velocity vector where math formula and math formula are the unit vectors along x and y, respectively. In the reaction term, the Damköhler number math formula represents the ratio between the characteristic hydrodynamic timescale, math formula, and the chemical timescale, math formula and k represents the reaction constant.

[15] Furthermore, an exponential and linear variations, with the chemical component concentrations are specified for the viscosity and density, respectively.

display math(7)
display math(8)

[16] Where math formula is the solutal expansion coefficient of species i. Rb and Rc are the log-mobility ratios between the viscosity of the pure chemical components B and C and that of A

display math(9)

[17] A detailed derivation of the above model has been presented by Hejazi and Azaiez [2012].

[18] The streamwise-direction boundary conditions in dimensionless form are:

display math(10)
display math(11)

where math formula represents the transverse velocity in a pure B solution. Also, the transverse-direction boundary conditions are:

display math(12)

[19] In the above equations, two other dimensionless groups, namely the Péclet number math formula and the cell aspect ratio math formula are introduced.

[20] In order to solve equations (2)-(8) numerically, the problem is formulated in terms of vorticity ω and streamfunction ψ. Taking the curl of Darcy's equation allows to eliminate the pressure term as:

display math(13)
display math(14)

[21] Equations (4)-(6), (13), and (14) form a closed set that can be solved for the concentration and the velocity field. The initial conditions used in this study consist of a base-state and a perturbation. The base-state is the numerical solution of equations (3)-(6) with math formula. The perturbation terms in the form of math formula consist of a random noise obtained from a random number generator, F(y), between −1 and 1. The perturbation of magnitude δ, is centered at the initial reactant interface math formula and decays rapidly far from the center at a rate dictated by σ.

[22] The boundary conditions for the streamfunction and vorticity in the steamwise- and transverse-directions are, respectively:

display math(15)

[23] Equations (4)-(6), (13), and (14) are solved numerically using the highly accurate Hartley-based pseudospectral method [Canuto et al., 1987; Bracewell, 1970; Tan and Homsy, 1988; Zimmerman and Homsy, 1992; Singh and Azaiez, 2001]. Moreover, a semiimplicit predictor-corrector scheme along with an operator-splitting technique is used for time stepping of the dynamic equations [Islam and Azaiez, 2005].

2.3. Validation of the Numerical Code

[24] The pseudospectral code was validated first by simulating the nonreactive density fingering system and comparing the results with the ones in the literature.

[25] Figure 2 depicts concentration contours for a buoyancy unstable but viscously stable nonreactive front. The flow parameters are the same as those reported by Manickam and Homsy [1995] (see Figure (7)b, math formula in Manickam and Homsy [1995]). The results show a very good qualitative agreement between the two cases in terms of the number of fingers and their advancements in both upstream and downstream directions. However, one should note that an exact reproduction of the concentration field is not possible unless exactly the same time steps, initial perturbation parameters and random number generators are used. The convergence of the numerical results has also been tested by simulating some reactive cases with spatial resolutions of 256×512 and 512×1024 while varying the time step accordingly. As similar finger structures were observed in all cases, a resolution of 256×512 was used throughout this study.

Figure 2.

The concentration fields at t = 800 reproduced by the numerical code of this study; to be compared with the one reported by Manickam and Homsy [1995], (see Figure (7)b, math formula in Manickam and Homsy [1995]) math formula.

[26] In what follows, results of the numerical simulations are presented for different values of the major dimensionless parameters that characterize the flow. We should however note that the numerical algorithm adopted in the present study requires periodicity of the system in the direction of the flow. Therefore, the validity of any results beyond those where the longest fingers reach the lower boundary is questionable and no valid argument can be built based on such results. Most results presented in this study have been run to the extent of validity of the numerical algorithm, and it is not possible to extend them to longer times without the risk of reporting nonreliable results.

3. Results and Discussion

[27] In this section, concentration isosurfaces of the chemical product (C) are presented. These surfaces show qualitatively the development of instabilities in both leading and trailing reaction zones. For brevity, only the frames that reveal interesting finger structures and may help understand the instability development are shown. Moreover, the dynamics of the flow are characterized through the efficiency of the reaction expressed in terms of the amount of chemical product.

[28] The discussion will be limited to the case of an initially stable reactant front, i.e., identical reactant properties math formula and math formula with math formula. In such a situation, any flow instability will be the direct result of the chemical reaction. In fact, the chemical product can be more dense or less dense than the reactants which corresponds to stable/unstable trailing/leading reaction zones or unstable/stable trailing/leading reaction zones, respectively. Also, in order to keep the discussion brief, it will be assumed that the chemical product has a lower viscosity than the reactants, i.e., math formula. This situation represents a viscously unstable leading region and a stable trailing one when there is an injection velocity. Furthermore, unless stated otherwise, the Damköhler number, the cell aspect ratio and Péclet number will be fixed as math formula, math formula and math formula.

[29] It should be noted that based on experimental measurements [Nagatsu et al., 2009], a Damköhler math formula is a moderate and realistic value. The effects of varying the Damköhler number on unstable reactive fronts have been experimentally [Nagatsu et al., 2009] and numerically [Gérard and De Wit 2009; Nagatsu and De Wit, 2011] investigated. Furthermore, one may also define a Péclet number based on the Hele Shaw cell thickness; math formula, in order to evaluate the importance of inertial forces in this geometry. The ratio of math formula is equal to math formula where the medium thickness b is 2–3 orders of magnitude smaller than the medium length Lx.

3.1. Fingering in the Absence of Transverse Velocity

[30] We start the analysis by examining buoyancy chemically driven flows in the absence of transverse velocities. Figure 3 a depicts time sequence of concentration contours. The chemical reaction product is more dense but less viscous than the physically identical reactants, math formula and math formula and math formula. Since the reactants have the same physical properties, the initial front is stable. However, the instability develops in the system when the properties of the solution change as a result of chemical reactions. It is noticeable that fingers of the chemical product develop only in the downward direction and penetrate into the reactant B while the trailing zone math formula is virtually stable. The upward growth of fingers is clearly stalled because of the favorable density ratio in the trailing reaction zone math formula which acts as a barrier against the growth of fingers. This is to be contrasted with nonreactive density fingering, where a gravitationally unstable front results in symmetric finger growth in both upward and downward directions [Manickam and Homsy, 1995]. It should be noted that the same phenomenon was reported by Almarcha et al. [2010a] for a similar reactive interface when the chemical reaction could only change the density but there was no variations in the viscosity profile.

Figure 3.

Concentration isosurfaces of the chemical product for math formula and (a) math formula (b) math formula at time frames 500, 750, 1000, and 1250 respectively.

[31] Before closing this section, we need to comment on the movement of the interface between the reactants which is important in some practical applications. These include the process of underground carbon dioxide storage where the movement of the contact zone between the CO2 and the underground water may affect the mixing of dissolved CO2 in water and consequently the amount of carbon dioxide dissolution [Tsai et al., 2013]. For this purpose a black, broken line is embedded in all frames to indicate the initial position of the reactant front. The results in Figure 3a clearly show a trend toward an upward movement of the reactant interface in the course of time. Fingers provide channels for the downward flow of the less viscous chemical product. However, this downward flow is also accompanied with a reverse flow around the finger base. As a result, the downstream advancement of fingers is balanced by the upward movement of its underlaying solution (reactant B in this case). This allows for stronger mixing of reactants A and B and results in more chemical product, as it will be discussed in the next sections.

3.2. Effect of Transverse Velocity

[32] First, differences in fingering interactions are analyzed and compared in the two cases of transverse and no transverse flows. Afterward, the influence of variations in the transverse velocity will be examined. Before exploring the effect of V1, we need to stress that when all chemical species have the same viscosity, i.e., math formula, the transverse velocity will not have any effects on the development of the flow. All cases discussed in this section correspond to a situation where the chemical product is less viscous than the reactants.

[33] A time sequence of concentration of the chemical product for the same parameters used in Figure 3a but with a nonzero transverse velocity math formula is depicted in Figure 3b. Close inspection reveals that there is a delay in the development of instabilities in the presence of a transverse velocity. This time delay may be quantified by the changes in the transition time from a diffusion-dominated regime to a convection-dominated regime as introduced in section 'Total Amount of Chemical Product in Time'. Based on the analysis presented in section 'Total Amount of Chemical Product in Time', the time delay is the difference between the time for finger development in the case of a zero transverse velocity (t = 150) and that of the nonzero transverse velocity situation (t = 500) which is about 350. In spite of this initial delay, the fingers later grow faster and catch up with their counterparts with zero transverse velocity. Furthermore, the number of emerging fingers decreases when a transverse velocity is present. This is in agreement with the results of linear stability analysis where the most dangerous wave number, a wave number associated with a maximum growth rate in a spectrum of unstable wave numbers, was found to shift toward long wavelengths in the presence of shear flows [Hejazi and Azaiez, 2012]. It should be noted that in the case of non zero transverse velocity, the fingers are narrower than the wide and diffused fingers of the system with math formula. The mechanisms responsible for such a behavior will be analyzed in the upcoming sections.

[34] The effect of the strength of the transverse flow is examined in Figure 4a for the same parameters used in Figure 3 when math formula but for different velocity magnitudes; math formula, and 0.7 at time 1250. A counterpart case where the chemical reaction leads to a less dense product math formula is depicted in Figure 4b for math formula. When the chemical product is less dense than the reactants, only the trailing zone is gravitationally unstable. As a result, the reactant A fingers through the chemical product C or in other words, the chemical product fingers develop upward (Figure 4b). In fact, now the stable leading reaction zone acts as a barrier against the growth of the instability in the downward direction.

Figure 4.

Concentration isosurfaces of the chemical product for math formula, (a) math formula, (b) math formula for math formula, and math formula respectively.

[35] Figure 4 reveals that the transverse flow has the same effects on both systems with math formula and math formula. Increases in the transverse velocity from math formula to math formula systematically lead to more delays in the emergence of fingers as well as in less complex structures. Further increase to math formula results in an almost diffused but slightly wavy interface. It should be noted at this stage that the results of linear stability analysis revealed that the transverse flow, regardless of its magnitude, can not stabilize completely a buoyancy unstable reactive interface [Hejazi and Azaiez, 2012]. Therefore, it is expected that fingers will develop even for large V1 but at very large time scales. Figure 4 also reveals that the reactant front moves in an upward/downward direction from its initial position in the case of unstable leading math formula/trailing math formula reaction zone. Moreover, it is found that in spite of the initial delay due to the transverse velocity, fingers tend to later develop faster in the presence of the transverse velocity, and the larger V1 the faster the later growth of the fingers. As an example, Figure 5 depicts results for a large transverse velocity math formula where the fingers develop faster than the ones in Figure 3. The time sequences in Figure 5 for the case of math formula show that the finger appearance time and advancement time are math formula and math formula respectively, while the corresponding times for math formula are math formula and math formula, respectively (See Figure 3b). Note that these conclusions are based on the times that were explored, and may not extend beyond them. Moreover, comparing the last time frames (which is the end of simulations when the chemical product fingers get close to the bottom boundary) in Figures 3a, 3b, and 5 reveal that as V1 increases, larger pockets of the pure reactant B (the blue color at the down side of the reaction zone) are observed above the initial reactant interface identified by the dotted line.

Figure 5.

Concentration isosurfaces of the chemical product for math formula and math formula at time frames 1000, 1200, 1400, and 1600 respectively.

[36] In the following section, the effect of the chemical product's viscosity is analyzed.

3.3. Effect of Chemical Product Viscosity

[37] Figures 6a and 6b depict the chemical product concentration at t = 1250 for different values of the viscosity ratio; math formula, and −4, respectively with a more dense chemical product; math formula, and transverse velocities of (a) math formula and (b) math formula. Since the time of finger development varies significantly with Rc, the frames with the fully developed fingers are also shown in Figure 6c.

Figure 6.

Concentration isosurfaces of the chemical product for math formula and for math formula, and −4 respectively, (a) math formula and all frames are at t = 1250 (b) math formula and all frames are at t = 1250 (c) math formula frames at the times with the fully developed fingers.

[38] First, we would like to comment on the influence of the chemical product's viscosity on the buoyancy-driven reactive flows in the absence of transverse velocities. Based on the fact that when all species have the same viscosity, the transverse velocity has no effects on the finger structures, the frame for math formula math formula in Figures 6a and 6b actually represents concentration isosurfaces for both zero and nonzero transverse velocity scenarios. Also, in the case of math formula, since the finger advancement is slow, the simulation could be run for long times. In this situation, the more diffused fingers tended to merge together and were widely spread in the domain. Fingers develop earlier in the cases of math formula. In fact, in this situation, the leading reaction zone is both gravitationally and viscously unstable ( math formula and math formula). As a result, the unfavorable density and viscosity ratios have additive effects on the growth of the instabilities. Moreover, a comparison between the cases of math formula and math formula in Figures 6a and 6b reveals that fingers are more diffuse and widely spread in the domain when math formula.

[39] Figures 6a–6c show that increasing the viscosity ratio, math formula, systematically results in less diffuse and more advanced finger structures with sharp concentration gradients. Also, more amount of the pure reactant B is pushed up above the centerline as math formula increases.

[40] In order to gain some physical insight about the role of transverse velocities in the flow development of reactive interfaces, a single wave analysis is presented in the following section.

3.4. Evolution of a Single Wave

[41] To better understand some of the nonlinear mechanisms that are difficult to observe in the presence of many fingers, the growth of an isolated finger is analyzed using the same parameters as in Figures 3a and 3b. The initial perturbation concentration is introduced in the form of a single wavelength cosine of amplitude 0.1 in the y direction weighted with a Gaussian distribution in the x direction:

display math(16)

[42] First we look at the case without transverse velocity depicted in Figure 7a. The concentration contours range from 0.05 (the most outside contours) to 0.45 (the most inside ones) with 0.05 increments. As the leading reaction zone is unstable, a single symmetric finger develops. The advancement of the finger is accompanied with its enlargement. However later on, the flow of the less viscous product through the finger channel decreases as a result of finger base contraction. In fact, a reverse flow around the lateral sides of the finger enhances mixing of the more viscous fluid B with the less viscous one C at the finger base. This results in more production at the base and the fading of the original finger which is almost disconnected from the base. Consequently, two fingerlets are initiated at the base around time 1000. To understand the flow direction and circulation, plots of the velocity field superposed on the concentration contours are shown in Figure 7a for t = 1000. Later on at t = 1150, these two fingerlets advance through the same path of the original finger. The mechanism of chocking of the original finger and fingerlet generations were also reported in horizontal displacements of a non reactive interface with nonmonotonic viscosity profile [Manickam and Homsy, 1993]. The flow circulation around the fingerlet sides further stretches the unperturbed parts of the math formula interface. Therefore, two new fingers start to develop independently on the left and right sides of the fingerlets.

Figure 7.

Single wave analysis for math formula and (a) math formula (b) math formula at math formula, and 1150, respectively.

[43] The effects of the transverse velocity are examined in Figure 7b for the same parameters used in Figure 7a but with a non zero transverse velocity; math formula. In this scenario, two competing driving forces are involved in the flow development, i.e., buoyancy-driven flows in the vertical direction and viscous driven flows in the horizontal direction. In the initial stages of the process when there is a limited amount of chemical product, the viscous forces are stronger than the buoyancy ones. As a result, the finger development is hindered. However, around t = 800 the single disturbance starts to grow diagonally. The nonsymmetric and diagonal growth of the finger and also the strong reverse flow illustrated by the velocity field at t = 1000, influence the system and constrict the chemical product layer. This leads to a nonuniform chemical production at the leading reaction zone. Consequently, it is interesting to see that around t = 1000 a second finger emerges right after the first one. At a later time, t = 1150, when there is enough chemical product to feed the fingers, gravity forces become dominant and the fingers are mainly dragged downward. Therefore, the product layer is not influenced anymore and no more fingers are generated. It should be mentioned that the generation of two fingers from a single wave disturbance is not observed when running the same simulation for a non reactive interface.

[44] For comparison purposes, the importance of the driving forces in each direction and their averages are qualitatively shown by green vectors for all time frames in Figure 7b and in the first frame of Figure 7a. The variation with time of these two forces are further quantified through the evaluation of the ratio of the maximum vertical perturbation velocity component to the maximum horizontal one. Figure 8 depicts the maximum perturbations velocity ratios versus time for the no-transverse flow case and the non zero velocity case with math formula.

Figure 8.

Variation with time of the maximum velocity component ratios for math formula and math formula. Circle: zero-velocity case, Triangle: nonzero-velocity case math formula.

[45] For the zero velocity system, initially the horizontal velocity component is larger than the vertical one. Mid way in the displacement process math formula the vertical and horizontal components of the velocity are of comparable magnitudes resulting in a ratio close to one. At later times, when there is enough heavy product to feed the growth, the finger advances fast downward and the vertical velocity component rises up to 3 times its horizontal counterpart. However, for math formula the ratio of the velocity components drops rapidly close to unity resulting in the slope discontinuity. This time is associated with the previously explained phenomenon of finger base constriction and no connection between the original finger and the heavy product layer. As a result, there is less or even no feeding of the original finger and consequently the fast advancement of the finger is stalled. For the nonzero velocity scenario math formula, initially the vertical component of the velocity is orders of magnitudes smaller than the horizontal one. But at later times, the ratio of the horizontal to vertical component evolves to one, implying that the two components become comparable. Hence, the disturbance starts to grow.

[46] Finally, before closing this section we need to make a last comment about Figure 7. A close inspection of the two scenarios in Figure 7 further confirms our previous observation where the transverse flow results in sharp concentration gradients. However, in the case of math formula, the contours are symmetrically diffused on both sides of the finger. The sharp concentration gradient may enhance the rate of reactants' mixing and consequently the rate of chemical production. We speculate that the fast advancement of fingers in the cases with non zero transverse flows reported in section 'Effect of Transverse Velocity' is due to the larger rates of chemical product. This idea is further explored in the following section through the amount of chemical product.

3.5. Total Amount of Chemical Product in Time

[47] The total amount of chemical product (C) which can be used to characterize the unstable RCD interface, is evaluated by integrating the concentration of the product through the whole domain scaled by the maximum amount of chemical product, Cmax:

display math(17)

[48] Variations in time of the total amount of chemical product (C) corresponding to the cases in Figure 3a for different V1 and in Figure 6 for different Rc are depicted in Figures 9a and 9b, respectively. The plots also include a purely stable reactive diffusive (RD) case with math formula and math formula (circle symbols). As expected, Figure 9 reveals that the amount of chemical product increases monotonically with time for all cases. In the RD stable case, the amount of production increases as math formula reflecting that the reaction is limited by the rate of diffusion between the reactants. In all other cases where the flow is unstable, there is a later deviation from the initial diffusion-limited regime. In these unstable situations, the fingers that develop at the trailing math formula and/or leading math formula reaction zones contribute to the mixing between the reactants and hence to the increase in the rate of chemical production. This results in a transition from the diffusion regime to a new flow regime mainly dominated by convection.

Figure 9.

Variation of the normalized total amount of chemical product with time for math formula, (a) math formula and various V1 (b) math formula and various Rc.

[49] Figure 9a reveals that the transition from the diffusion dominated regime to the convection dominated one is systematically delayed as math formula is increased. This implies that the transverse flow has an adverse effect on the rate of production, at least in the early stages of the flow. However, once in the second regime, the rate of chemical production is stronger for larger math formula resulting in larger amount of chemical production. The greater rate of chemical production is responsible for the fast finger advancements in the nonzero transverse velocity cases discussed earlier (Figures 3b and 5). The slopes of the curves shown in Figure 9a vary like tn where math formula for the RD front, math formula for the RCD front with zero transverse velocity and math formula for math formula, respectively.

[50] Total product curves for different viscosity ratios in Figure 9b reveal that increasing math formula systematically leads to a larger slope, i.e., a higher rate of production. However, the time for the onset of the convection regime is a nonmonotonic function of math formula in the presence of a transverse velocity. The transition time decreases as math formula increases from zero up to math formula. Larger viscosity ratios enhance the growth of unstable modes but at the same time are also associated with stronger transverse velocity effects. As a result, further increase of the viscosity ratio math formula increases the onset time of the convection regime. In other words, the initial delay in finger development is an increasing factor of both math formula and math formula and a decreasing factor of math formula. Therefore, there is an optimum value for Rc at which the increasing effect of math formula is more pronounced than the decreasing effect of math formula. It should be mentioned that results obtained for flows without transverse velocity show that the convection regime starts systematically earlier for larger viscosity ratio math formula.

[51] Here we would like to comment on the role of the sharp concentration gradients as a possible driving force for a higher rate of chemical production when math formula. In fact, the large concentration gradients in the reaction zones result in a strong mass transfer among the components, reactant A at the trailing reaction zone and reactant B at the leading one. Hence, more reactant mixing and consequently a higher rate of chemical production is expected.

[52] Figure 10 depicts the variation of the onset time of convection regime with V1 and Rc. It is clear that at a fixed math formula, the onset time increases monotonically with math formula and math formula, but through two regimes. In the first one, the increase is slow while it is faster in the second. The velocity magnitude at which there is a change of regime is math formula in Figure 10 was found to vary with Rc where larger values of math formula lead to smaller math formula. The two regimes indicate the significance of the transverse flow on suppressing the initial finger development as opposed to the viscosity and buoyancy destabilizing effect. For low-transverse velocity magnitudes, the later effect is more pronounced while for high-transverse velocity magnitudes the former effect is dominant.

Figure 10.

The variation of the time for the onset of the convection regime for math formula. Solid line: math formula and variable Rc. Broken line: math formula and variable V1.

[53] Moreover, for a fixed math formula, an optimum value for the viscosity ratio math formula is found where the onset time of convection regime is minimum. For larger values of math formula, i.e., math formula, the increasing effect of math formula on the onset time becomes stronger than the decreasing effect due to math formula. Hence, the onset time for math formula is higher than the case with math formula or lower values. However, as previously discussed, the higher value of math formula results in a stronger rate of chemical production in the second regime evidenced through the slope of the curve for math formula.

[54] The previous analysis indicates that the transverse velocity has an adverse effect on the instability and the rate of chemical production in the first flow regime (diffusion dominant) and an enhancing one in the second regime (convection dominant). In order to separate these two opposite effects, tests where the transverse flow is turned off or on in one of the two regimes, were conducted. In a first case, the flow initially proceeds without a transverse velocity. Then after the onset time for the convection regime, a transverse velocity is introduced. The transverse velocity for such a case will be referred to as math formula. An opposite case where the transverse velocity is turned off in the second regime; math formula. Figure 11 depicts the concentration iso surfaces for the cases with math formula at t = 1250 and for math formula and math formula. Clearly, turning on/off the transverse velocity in the middle of the simulation results in fast/slow finger advancement. Moreover, larger number of fingers develop when the flow is switched off, i.e., math formula. Therefore, for stronger finger growth and larger chemical production rate, it is better to impose the transverse flow later, while removing it after the diffusion dominated regime leads to weaker finger growth and smaller chemical production rate.

Figure 11.

Concentration isosurfaces of the chemical product for math formula for math formula, and math formula respectively at t = 1250.

[55] The corresponding total amount of chemical product is depicted in Figure 12. The simple RD case is also included as a reference. Figure 12 reveals that the velocity configuration math formula initially decreases the rate of production. However, when sharp concentration gradients are formed, the rate of production jumps up above that of the no-transverse flow case. For the case math formula, one would have expected to see a slower rate of production when switching off the velocity in comparison with the case math formula. However, it is found that turning off the velocity results in an initial increase in the rate of production and then the expected slowdown in the production rate. This surprising result may be attributed to a delay in the diffusion of the sharp concentration gradients and also to differences in the number of growing fingers in these two systems: math formula and math formula. In fact, as depicted in Figure 11, after the velocity is turned off, a large number of fingers still continue their growth without merging in comparison to the lower number of fingers in math formula. Moreover, sharp concentration gradients are still observed for a period of time after the transverse velocity has been turned off. As a consequence, the rate of production initially increases followed later by a decrease that results in lower chemical production rate in the case math formula.

Figure 12.

Variation of the normalized total amount of chemical product with time for math formula, and RD case (empty circles), math formula (dashed line), math formula (dashed line with square), math formula (solid line), math formula (solid line with triangle).

4. Conclusions

[56] We have simulated flow instabilities of a horizontal miscible reactive interface A+B→C in porous media under a gravity field and a transverse velocity math formula parallel to the initial reactant interface math formula. It was assumed that the reactants are identical in terms of the physical properties. Therefore, the analyses were carried out based on the solutal expansion coefficient math formula representing the contribution of C to the density profile and the mobility ratio Rc as the log viscosity ratio of the chemical product C to that of the reactant A. It was shown that the chemical reaction could break the previously known symmetry of density fingering developments in nonreactive systems. In the reactive case, regardless of Rc values, the fingers of the chemical product developed only in one direction; downward when the chemical product was more dense and upward otherwise. In fact, the favorable density profile at the math formula region in the former and at the math formula region in the latter acts as a barrier against the growth of instabilities in the upward and downward directions, respectively. One should note that as soon as a chemical reaction is triggered, another solution is formed so that three layers of solutions exist as opposed to only two in the known symmetrical nonreactive cases. It was also found that the less viscous chemical product can result in early growth of the fingers.

[57] It was reported that introducing a transverse flow could affect the fate of the chemical reaction process and the later development of the flow when there was a viscosity variations in the system math formula. The coupling effects of a transverse velocities and a less viscous chemical product lead to a delay in the development of fingers. However, in spite of this initial delay, it was found that the fingers later grew faster toward the boundaries. Moreover, the flow instability was characterized through the total amount of chemical product. At early times, diffusive effects were predominant and as a result the rate of production was independent of the species physical properties and transverse flows. In the late time regime, the chemically buoyancy induced fluid motion played an important role and enhanced mixing between the reactants, which resulted in a larger rate of production. The rate of production was even larger in the second regime when math formula. Indeed, it was revealed that in the case of math formula, fingers with sharp concentration gradients were formed and as a result the stronger mixing of reactants and consequently a larger rate of chemical production and fast finger advancements were attained. When there is no viscosity contrast, all chemical components travel transversely with the same speed and this transverse flow has no effects on the finger structures.

[58] The study has also shown that the initial delay in finger development increased with the transverse velocity math formula and math formula and decreased with the viscosity ratio math formula. However, fast finger advancement was an increasing factor of math formula. The implication of these findings is that the rate of chemical production and finger advancement can be controlled by turning on and off the transverse velocity in one of the two flow regimes. An off (in the first flow regime) and on (in the second flow regime) V1 scenario should be preferred if one wants to attain higher rates of chemical production and fast finger advancements. In the opposite, an on and off V1 scenario results in slow finger developments.

[59] It should be stressed that these conclusions were based on the assumption that the porous medium is homogeneous. However, it is well known that heterogeneity can actually affect the flow dynamics, and depending of the scale of the heterogeneities and their correlation, regions of preferred flow path or channeling can be observed and may result in changes in the extent of mixing and reaction rates [Rolle et al., 2009; Bolster et al., 2011; Chiogna et al., 2012]. Still, the main conclusions of this study provide a good basis for the analysis of the general physics of the flow.

Acknowledgments

[60] The authors acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Alberta Innovation Fund (AIF). The use of the computing resources of the West-Grid cluster is also acknowledged. The authors also thank G. M. Homsy, K. Ghesmat, and H. Hassanzadeh for discussions.