Viscous fingering in periodically heterogeneous porous media. II. Numerical simulations

Stable and unstable miscible displacements in layered porous media

The effect of permeability heterogeneities and viscosity variations on miscible displacement processes in porous media is examined using high-resolution numerical simulations and reduced theoretical modelling. The planar injection of one fluid into a fluid-saturated, two-dimensional porous medium with a permeability that varies perpendicular to the flow direction is studied. Three cases are considered, in which the injected fluid is equally viscous, more viscous or less viscous than the ambient fluid. In general it is found that the flow in each case evolves through three regimes. At early times, the flow exhibits the concentration evolves diffusively, independent of both the permeability structure and the viscosity ratio. At intermediate times, the flow exhibits different dynamics including channelling and fingering, depending on whether the injected fluid is more or less viscous than the ambient fluid, and depending on the relative magnitude of the viscosity and permeability variations. Finally, at late times, the flow becomes independent of the viscosity ratio and dominated by shear-enhanced (Taylor) dispersion. For each of the regimes identified above, we develop reduced-order models for the evolution of the transversely averaged concentration and compare them to the full numerical simulations.

Fingering dynamics driven by a precipitation reaction: Nonlinear simulations

A fingering instability can develop at the interface between two fluids when the more mobile fluid is injected into the less-mobile one. For example, viscous fingering appears when a less viscous (i.e., more mobile) fluid displaces a more viscous (and hence less mobile) one in a porous medium. Fingering can also be due to a local change in mobility arising when a precipitation reaction locally decreases the permeability. We numerically analyze the properties of the related precipitation fingering patterns occurring when an A+B→C chemical reaction takes place, where A and B are reactants in solution and C is a solid product. We show that, similarly to reactive viscous fingering patterns, the precipitation fingering structures differ depending on whether A invades B or vice versa. This asymmetry can be related to underlying asymmetric concentration profiles developing when diffusion coefficients or initial concentrations of the reactants differ. In contrast to reactive viscous fingering, however, precipitation fingering patterns appear at shorter time scales than viscous fingers because the solid product C has a diffusivity tending to zero which destabilizes the displacement. Moreover, contrary to reactive viscous fingering, the system is more unstable with regard to precipitation fingering when the high-concentrated solution is injected into the low-concentrated one or when the faster diffusing reactant displaces the slower diffusing one.

Interface evolution during radial miscible viscous fingering

We study experimentally the miscible radial displacement of a more viscous fluid by a less viscous one in a horizontal Hele-Shaw cell. For the range of tested injection rates and viscosity ratios we observe two regimes for the evolution of the fluid-fluid interface. At early times the interface length increases linearly with time, which is typical of the Saffman-Taylor instability for this radial configuration. However, as time increases, the interface growth slows down and scales as ∼t(1/2), as one expects in a stable displacement, indicating that the overall flow instability has shut down. Surprisingly, the crossover time between these two regimes decreases with increasing injection rate. We propose a theoretical model that is consistent with our experimental results, explains the origin of this second regime, and predicts the scaling of the crossover time with injection rate and the mobility ratio. The key determinant of the observed scalings is the competition between advection and diffusion time scales at the displacement front, suggesting that our analysis can be applied to other interfacial-evolution problems such as the Rayleigh-Bénard-Darcy instability.

Scaling and unified characterization of flow instabilities in layered heterogeneous porous media

The physics of miscible flow displacements with unfavorable mobility ratios through horizontal layered heterogeneous media is investigated. The flow model is solved numerically, and the effects of various physical parameters such as the injection velocity, diffusion, viscosity, and the heterogeneity length scale and variance are examined. The flow instability is characterized qualitatively through concentration contours as well as quantitatively through the mixing zone length and the breakthrough time. This characterization allowed us to identify four distinct regimes that govern the flow displacement. Furthermore, a scaling of the model resulted in generalized curves of the mixing zone length for any flow scenario in which the first three regimes of diffusion, channeling, and lateral dispersion superpose into a single unifying curve and allowed us to clearly identify the onset of the fourth regime. A critical effective Péclet number w_{c} based on the layers' width is proposed to identify flows where heterogeneity effects are expected to be important and those where the flow can be safely treated as homogeneous. A similar scaling of the breakthrough time was obtained and allowed us to identify two optimal effective Péclet numbers w_{opt} that result in the longest and shortest breakthrough times for any flow displacement.

Quantifying mixing in viscously unstable porous media flows

Viscous fingering is a well-known hydrodynamic instability that sets in when a less viscous fluid displaces a more viscous fluid. When the two fluids are miscible, viscous fingering introduces disorder in the velocity field and exerts a fundamental control on the rate at which the fluids mix. Here we analyze the characteristic signature of the mixing process in viscously unstable flows, by means of high-resolution numerical simulations using a computational strategy that is stable for arbitrary viscosity ratios. We propose a reduced-order model of mixing, which, in the spirit of turbulence modeling and in contrast with previous approaches, recognizes the fundamental role played by the mechanical dissipation rate. The proposed model captures the nontrivial interplay between channeling and creation of interfacial area as a result of viscous fingering.

Influence of porosity on Rayleigh-Taylor instabilities in reaction-diffusion systems

We analyze the effect of porosity in a porous medium on hydrodynamic instabilities in reaction-diffusion fronts. We use an experimental device to create an effective two-dimensional porous medium which is vertically orientated. In this system the molecular diffusion coefficients and the acid autocatalysis of the chlorite-tetrathionate reaction satisfy the appropriate conditions to produce a chemical front that advances through the cell leading to the products overlaying the reactants. The reactants have a lower density than the products and therefore a buoyantly unstable front develops. To evaluate the influence of the porosity on the formation and propagation of such instabilities, media with different porosities were used in the experiments. The amplitude of the instability is found to reduce as the porosity of the medium is decreased. For sufficiently small porosity, the instability can almost disappear leading to a planar front.

Viscous fingering in packed chromatographic columns: Linear stability analysis

When a fluid is displaced by a less viscous one in a porous medium, a hydrodynamic instability appears leading to the formation of some kind of fingers of the upstream fluid invading the downstream one, hence the name "viscous fingering" (VF) given to this instability. In a LC column, such an instability is likely to appear at that of the two interfaces between the sample and the eluent which exhibits an unfavorable viscosity contrast. It leads to distorted peak shapes and contributes to peak broadening. This phenomenon has been observed for long in SEC and more recently in RPLC on elution peak shapes as well as with various methods of in-column visualization. A simplistic LC column model is described to explain the origin of the VF instability and its characteristics. The general principles for analyzing hydrodynamic instabilities are described and the results of the linear stability analysis performed by Tan and Homsy [C.T. Tan, G.M. Homsy, Phys. Fluids 29 (1986) 3549 [1]], at the onset of the VF phenomenon for a step interface between two fluids are here applied to typical operating conditions encountered in analytical LC. The most probable growth rate and wavelength (linked to the finger width) of the instability are given in terms of particle size and solute diffusion coefficient, with particular emphasis on the role of the carrier velocity. Previously published qualitative observations about VF in chromatography are examined and interpreted at the light of this theory. The role of the column geometry on the development of the instability, the possible sources of noise or fluctuations triggering the instability, and the various experimental situations in which a significant viscosity contrast is encountered in LC are discussed.

Periodic heterogeneity-driven resonance amplification in density fingering

Periodic heterogeneity is introduced in experiments with thin solution layers where downward propagating planar autocatalytic fronts are hydrodynamically unstable and cellular patterns develop. The evolution of fingers is greatly affected by the spatial heterogeneity when the wave number associated with it falls in the vicinity of the most unstable mode of the reference system with uniform thickness. The imposed heterogeneity will drive the instability by amplifying the modes with the matching wave numbers as indicated by the experimentally constructed dispersion curves.

Structural and dynamical characterization of Hele-Shaw viscous fingering

Viscous fingering occurs in the interfacial zone between two fluids confined between two plates with a narrow gap (Hele-Shaw geometry) when a highly viscous fluid is displaced by a fluid with relatively low viscosity. Using a mesoscopic approach--the lattice Boltzmann method--we investigate the dynamics of spatially extended Hele-Shaw flow under conditions corresponding to various experimental systems by tuning the 'surface tension' and the reactivity between the two fluids. We discuss the onset of the fingering instability (dispersion relation), analyse the structural properties (characterization of the interface) and the dynamical properties (growth of the mixing zone) of the Hele-Shaw systems, and show the effect of reactive processes on the structure of the interfacial zone.