Non-monotonic Viscosity Profiles Research Articles

Stabilization of periodically forced Hele-Shaw flows by means of a nonmonotonic viscosity profile.

The onset of viscous fingering in the presence of a viscosity profile is investigated theoretically for two immiscible fluids undergoing a time-dependent injection. Here, we show that the presence of a positive viscosity gradient at the interface between both fluids stabilizes the interface facilitating the spread of the perturbation. This effect is much more pronounced in the case of sinusoidal injection flows. The influence of the viscosity gradient on the dispersion relation is analyzed. Numerical simulations of the Navier-Stokes equationconfirm the linear stability analysis.

A numerical study on reaction-induced radial fingering instability

The dynamics of $A+B\rightarrow C$ fronts is analysed numerically in a radial geometry. We are interested to understand miscible fingering instabilities when the simple chemical reaction changes the viscosity of the fluid locally and a non-monotonic viscosity profile with a global maximum or minimum is formed. We consider viscosity-matched reactants $A$ and $B$ generating a product $C$ having different viscosity than the reactants. Depending on the effect of $C$ on the viscosity relative to the reactants, different viscous fingering (VF) patterns are captured which are in good qualitative agreement with the existing radial experiments. We have found that, for a given chemical reaction rate, an unfavourable viscosity contrast is not always sufficient to trigger the instability. For every fixed Péclet number ($Pe$), these effects of chemical reaction on VF are summarized in the Damköhler number ($Da$) $-$ the log-mobility ratio ($R_{c}$) parameter space that exhibits a stable region separating two unstable regions corresponding to the cases of more and less viscous product. Fixing $Pe$, we determine $Da$-dependent critical log-mobility ratios $R_{c}^{+}$ and $R_{c}^{-}$ such that no VF is observable whenever $R_{c}^{-}\leqslant R_{c}\leqslant R_{c}^{+}$. The effect of geometry is observable on the onset of instability, where we obtain significant differences from existing results in the rectilinear geometry.

Non-modal stability analysis of miscible viscous fingering with non-monotonic viscosity profiles

A non-modal linear stability analysis (NMA) of the miscible viscous fingering in a porous medium is studied for a toy model of non-monotonic viscosity variation. The onset of instability and its physical mechanism are captured in terms of the singular values of the propagator matrix corresponding to the non-autonomous linear equations. We discuss two types of non-monotonic viscosity profiles, namely, with unfavourable (when a less viscous fluid displaces a high viscous fluid) and with favourable (when a more viscous fluid displaces a less viscous fluid) endpoint viscosities. A linear stability analysis yields instabilities for such viscosity variations. Using the optimal perturbation structure, we are able to show that an initially unconditional stable state becomes unstable corresponding to the most unstable initial disturbance. In addition, we also show that to understand the spatio-temporal evolution of the perturbations it is necessary to analyse the viscosity gradient with respect to the concentration and the location of the maximum concentration $c_{m}$. For the favourable endpoint viscosities, a weak transient instability is observed when the viscosity maximum moves close to the pure invading or defending fluid. This instability is attributed to an interplay between the sharp viscosity gradient and the favourable endpoint viscosity contrast. Further, the usefulness of the non-modal analysis demonstrating the physical mechanism of the quadruple structure of the perturbations from the optimal concentration disturbances is discussed. We demonstrate the dissimilarity between the quasi-steady-state approach and NMA in finding the correct perturbation structure and the onset, for both the favourable and unfavourable viscosity profiles. The correctness of the linear perturbation structure obtained from the non-modal stability analysis is validated through nonlinear simulations. We have found that the nonlinear simulations and NMA results are in good agreement. In summary, a non-monotonic variation of the viscosity of a miscible fluid pair is seen to have a larger influence on the onset of fingering instabilities than the corresponding Arrhenius type relationship.

Stabilization of miscible viscous fingering by a step growth polymerization reaction

Fingering is a hydrodynamic instability that occurs when a more mobile fluid displaces a fluid of lower mobility. When the primary source of the mobility difference is viscosity, the instability is termed viscous fingering. Viscous fingering is often, though not always, undesirable in industrial processes, particularly secondary petroleum recovery. Linear stability analysis by Hejazi et al. has indicated that the production of a non-monotonic viscosity profile can stabilize the interface. Herein, we use step-growth polymerization at the interface between two miscible monomers as a model system. In particular, a dithiol monomer displaced a diacrylate that reacted to form a linear polymer that behaves as a Newtonian fluid. Viscous fingering was imaged in a horizontal Hele-Shaw cell via Schlieren, which is sensitive to changes in index of refraction, and therefore polymer conversion. By varying reaction rate via initiator concentration along with flow rate via a syringe pump, we were able to demonstrate increasing stabilization of the flow with increasing Damkohler number.

The effect of a crosslinking chemical reaction on pattern formation in viscous fingering of miscible fluids in a Hele-Shaw cell.

Viscous fingering can occur in fluid motion whenever a high mobility fluid displaces a low mobility fluid in a Darcy type flow. When the mobility difference is primarily attributable to viscosity (e.g., flow between the two horizontal plates of a Hele-Shaw cell), viscous fingering (VF) occurs, which is sometimes termed the Saffman-Taylor instability. Alternatively, in the presence of differences in density in a gravity field, buoyancy-driven convection can occur. These instabilities have been studied for decades, in part because of their many applications in pollutant dispersal, ocean currents, enhanced petroleum recovery, and so on. More recent interest has emerged regarding the effects of chemical reactions on fingering instabilities. As chemical reactions change the key flow parameters (densities, viscosities, and concentrations), they may have either a destabilizing or stabilizing effect on the flow. Hence, new flow patterns can emerge; moreover, one can then hope to gain some control over flow instabilities through reaction rates, flow rates, and reaction products. We report effects of chemical reactions on VF in a Hele-Shaw cell for a reactive step-growth cross-linking polymerization system. The cross-linked reaction product results in a non-monotonic viscosity profile at the interface, which affects flow stability. Furthermore, three-dimensional internal flows influence the long-term pattern that results.

Influence of nonideal mixing properties on viscous fingering in micropillar array columns

A micropillar array column is used to study miscible viscous fingering in the presence of a nonmonotonic viscosity profile caused by nonideal mixing properties in a porous medium. Fingering is observed at both the frontal and rear miscible interfaces of a finite-size sample phase, with different spatial extents, however.

Cyclic time-dependent reactive flow displacements in porous media

Reactive miscible flow displacements in homogeneous porous media with time-dependent injection velocities are analyzed. The displacements consist of periodic cycles that involve alternating stages of injection and extraction. Nonlinear simulations are used and the dynamics of the flows are characterized qualitatively as well as quantitatively in terms of the amount of generated chemical product. The simulations revealed that, when compared with constant velocity flows with the same overall amount of injected fluid, time dependent displacements show drastically different flow structures and chemical productivity. These differences are strongly dependent on the period of the velocity cycles and the scenario of whether the displacement is initiated through an injection or an extraction process. In particular, for non-monotonic viscosity profiles, displacements with larger period result in stronger chemical production while the effects of the velocity scenarios depend on the nature of the instability at the initial front. Moreover, for monotonically decreasing viscosity profiles known to be stable under constant injection velocity, only displacements starting with extraction can trigger instabilities and increase the amount of chemical product. Finally for monotonically increasing viscosity profiles, displacements that start with injection tend to generate more chemical product than the constant velocity flow and its counterpart starting with an extraction instead.

Experimental evidence of reaction-driven miscible viscous fingering

An experimental demonstration of reaction-driven viscous fingering developing when a more viscous solution of a reactant A displaces a less viscous miscible solution of another reactant B is presented. In the absence of reaction, such a displacement of one fluid by another less mobile one is classically stable. However, a simple A+B→C reaction can destabilize this interface if the product C is either more or less viscous than both reactant solutions. Using the pH dependence of the viscosity of some polymer solutions, we provide experimental evidence of both scenarios. We demonstrate quantitatively that reactive viscous fingering results from the buildup in time of nonmonotonic viscosity profiles with patterns behind or ahead of the reaction zone, depending on whether the product is more or less viscous than the reactants. The experimental findings are backed up by numerical simulations.

The stability of miscible displacement in porous media: Nonmonotonic viscosity profiles

The stability of miscible displacement in porous media is analyzed theoretically. By considering the nonmonotonic viscosity variation effects, new stability equations are derived in a similar domain with and without the quasi-steady state approximations (QSSA). An analytical approach to solve the newly driven stability equations is proposed and its validity is confirmed by comparing its solutions with numerically obtained ones. Through the growth rate analysis without the QSSA, it is shown analytically that the system is unconditionally stable for the long-wave disturbance regardless of the viscosity profile. The present growth rate obtained for small time without the QSSA is quite different from the previous analyses based on the QSSA where the growth rate of the disturbance depends strongly on the viscosity profile. Through the stability characteristics for the finite time case, the validity of the QSSA is discussed. The present stability condition explains the system more reasonably than the previous results based on the conventional QSSA.

Viscous fingering of a miscible reactive A+B→C interface for an infinitely fast chemical reaction: Nonlinear simulations

Nonlinear dynamics of miscible viscous fingering is analyzed numerically for a reactive system when a solution containing a reactant A is displacing another miscible solution containing another reactant B. A simple A+B→C reaction takes place upon contact of the solutions. The viscosity of the fluid depends on the concentration of the various chemicals. The nonlinear fingering dynamics is studied numerically for an infinite Damköhler number Da, i.e., for an infinitely fast reaction as a function of the log-mobility ratios Rb and Rc quantifying the viscosity ratios of the solutions of B and C, respectively, versus that of the solution of A. If Rb>0, i.e., if the system is genuinely viscously unstable because the displacing solution of A is less viscous than the displaced solution of B, we analyze the changes to classical nonreactive viscous fingering induced by the reaction. If on the contrary Rb<0, which corresponds to a hydrodynamically stable case in absence of reactions, we study chemically driven viscous fingering occurring when the chemical reaction triggers a nonmonotonic viscosity profile. Comparison between the present simulation results and corresponding linear stability analysis and experiments is also conducted.

Viscous fingering of a miscible reactive A + B → C interface: a linear stability analysis

When one solution of reactant A is displacing another miscible solution of reactant B, a miscible product C can be generated in the contact zone if a simple A + B → C chemical reaction takes place. Depending on the relative effect of A, B and C on the viscosity, different viscous fingering (VF) instabilities can be observed. In this context, a linear stability analysis of this reaction–diffusion–convection problem under the quasi-steady-state approximation is performed to classify the various possible instability scenarios. To do so, we determine the criteria for an instability, obtain dispersion curves both at initial contact time using an analytical implicit solution and at later times via numerical stability analysis. Along with recovering known results for non-reactive systems where the displacement of a more viscous fluid by a less viscous one leads to a VF instability, it is found that in the presence of a chemical reaction, injecting a more viscous fluid into a less viscous fluid can also lead to instabilities. This occurs when the chemical reaction leads to the build up of non-monotonic viscosity profiles. Various instability scenarios are classified in a parameter plane spanned by Rb and Rc representing the log-mobility ratios of the viscosities of the B and C solution respectively with respect to that of the injected solution of A. A parametric study of the influence on stability of the Damköhler number and of the time elapsed after contact of the two reactive solutions is also conducted.

Miscible displacements in Hele–Shaw cells: Nonmonotonic viscosity profiles

The influence of nonmonotonic viscosity–concentration relationships on viscous fingering of neutrally buoyant, miscible fluids in a Hele–Shaw cell has been investigated. In a first step, quasisteady base states are obtained by means of nonlinear Stokes simulations. The properties of these base states are analyzed as a function of the Péclet number, the viscosity ratio, and the profile parameters. Subsequently, the stability of these base states is investigated by means of a linear stability analysis. Overall, the nonmonotonicity of the viscosity–concentration relationship is seen to have a much smaller influence on Hele–Shaw displacements than on corresponding Darcy flows. The reason for this difference lies in the nature of the respective base states. For Darcy flows, the base state is characterized by constant velocity and a diffusively decaying concentration (and hence viscosity) profile. This base viscosity profile is strongly affected by the nonmonotonicity. On the other hand, for Hele–Shaw displacements the quasisteady base states are convectively dominated and characterized by sharp fronts, so that their shape depends only weakly on the details of the viscosity–concentration relationship. Hence, for Hele–Shaw displacements both the eigenfunctions and the associated growth rates are quite similar for monotonic and nonmonotonic profiles, in contrast to the findings by [O. Manickam, G.M. Homsy, Stability of miscible displacements in porous media with nonmonotonic viscosity profiles, Phys. Fluids A 5 (1993) 1356–1367] for Darcy flows.

Miscible porous media displacements in the quarter five-spot configuration. Part 3. Non-monotonic viscosity profiles

The influence of a non-monotonic viscosity–concentration relationship on miscible displacements in porous media is studied for radial source flows and the quarter five-spot configuration. Based on linear stability results, a parametric study is presented that demonstrates the dependence of the dispersion relations on both the Péclet number and the parameters of the viscosity profile. The stability analysis suggests that any displacement can become unstable provided only that the Péclet number is sufficiently high. In contrast to rectilinear flows, for a given end-point viscosity ratio an increase of the maximum viscosity generally has a destabilizing effect on the flow. The physical mechanisms behind this behaviour are examined by inspecting the eigensolutions to the linear stability problem. Nonlinear simulations of quarter five-spot displacements, which for small times correspond to radial source flows, confirm the linear stability results. Surprisingly, displacements characterized by the largest instability growth rates, and consequently by vigorous viscous fingering, lead to the highest breakthrough recoveries, which can even exceed that of a unit mobility ratio flow. It can be concluded that, for non-monotonic viscosity profiles, the interaction of viscous fingers with the base-flow vorticity can result in improved recovery rates.

Finite element simulation of nonlinear viscous fingering in miscible displacements with anisotropic dispersion and nonmonotonic viscosity profiles

In this work parallel finite element techniques for the simulation of nonlinear viscous fingering in miscible displacements are addressed. The governing equations are approximated in space by equal order elements. The resulting semi-discrete equations are approximated in time by a block-iterative predictor-multicorrector algorithm. Feedback control theory is used for time step selection. The linear systems of equations at each block-iteration are solved with parallel element-by-element iterative techniques. Numerical simulations in different physical situations, involving anisotropic dispersion and a nonmonotonic viscosity law are shown.

Simulation of viscous fingering in miscible displacements with nonmonotonic viscosity profiles

The nonlinear evolution of viscous fingering instabilities in miscible displacement flows in porous media with nonmonotonic viscosity profiles is investigated. The flow is accurately simulated using a Hartley transform based spectral method. A flow with nonmonotonic viscosity profile has an unstable region followed downstream by a stable region. Instabilities first begin in the unstable region and then grow and penetrate the stable region. The striking contrast between viscous fingering in flows with monotonic and nonmonotonic viscosity profiles is the direction of fluid penetration. The nonmonotonicity in the viscosity profile gives rise to a new phenomena of ‘‘reverse’’ fingering in which the displaced fluid fingers through the displacing fluid more readily than vice versa. A forward and a reverse mixing lengths are defined to characterize the growth of the mixing zone in the two directions. At large times, both the forward and reverse mixing lengths grow linearly in time. A model nonmonotonic viscosity profile is used to parametrically study the asymptotic mixing rates. The parametric study shows that for a given end point and maximum viscosities the growth rate of the mixing zone varies nonmonotonically with the length of the stable barrier in the viscosity profile. A physical mechanism is put forth to explain the observed phenomena of reverse fingering and its dependence on the parameters of the problem. Finally, a finger is isolated and its evolution is studied to understand the differences in the mechanisms that control the growth of a finger in flows with monotonic and nonmonotonic viscosity profiles.

Stability of miscible displacements in porous media with nonmonotonic viscosity profiles

The effect of a nonmonotonic viscosity profile on the stability of miscible displacements in porous medium is studied. A linear theory using the quasi-steady-state approximation is employed to find the growth rate of the perturbations in the flow. The resulting eigenvalue problem has both a discrete and continuous spectrum. It is possible to obtain an analytical solution for a step base state concentration profile. The step profile result indicates that a nonmonotonic viscosity profile can be stable even when the viscosity contrast, measured by the end-point viscosities, is unfavorable, and vice versa. Asymptotic expansions of the growth rate for short times and for small wave numbers of the disturbances are obtained. The short-time expansion shows that the diffusion of the base state does not always mitigate the instabilities and the small wave-number expansion gives a sufficient condition for the flow to be unstable. Finally, the eigenvalue problem is solved numerically for diffused concentration profiles using finite difference methods. A model viscosity profile is chosen to parametrically study the stability of nonmonotonic profiles. The parametric study shows that the diffusion of the base state can have a destabilizing effect, with the surprising result that the nonmonotonic profiles that are predicted to be stable by the step profile analysis eventually become unstable by the destabilizing effect of diffusion. The ‘‘critical’’ time at which a stable flow turns unstable is obtained as a function of the parameters of the problem. A physical mechanism is proposed to explain the effects of diffusion on the stability of the flow.