Nonlinear interactions of chemical reactions and viscous fingering in porous media

Abstract

Nonlinear interactions of chemical reactions and viscous fingering are studied in porous media by direct numerical simulations of Darcy's law coupled to the evolution equation for the concentration of a chemically reacting solute controlling the viscosity of miscible solutions. Chemical kinetics introduce important topological changes in the fingering pattern: new robust pattern formation mechanisms such as droplet formation and enhanced tip splitting are evidenced and analyzed. © 1999 American Institute of Physics. ͓S1070-6631͑99͒01805-X͔The problem of fingering in flows with chemical reactions is of fundamental importance in several applications ranging from petroleum 1 or spill recovery, polymerization fronts 2 and chromatographic separations of concentrated mixtures 3 to deformation of chemical waves by hydrodynamical instabilities. 4 Fingering instabilities occur when a fluid with low mobility displaces another fluid with high mobility and take place in both miscible and immiscible systems. The difference in mobility can originate from a difference in viscosity and/or density. This hydrodynamical instability has been the subject of numerous studies. 1 Despite the wide number of examples where fingering occurs in systems where chemical reactions take place, very little theoretical work has been devoted to the study of nonlinear interactions of chemical reactions and fingering.In this letter, we find that viscous fingering of miscible fluids flowing in porous media is strongly influenced by chemical reactions, leading to new interactions and pattern formation mechanisms. In particular, the interplay between the hydrodynamical instability and chemical reactions leads to the formation of droplets of one solution disconnecting from the bulk and invading the other solution. Chemical reaction also catalyzes tip splitting phenomena and maintains sharp fronts between the miscible fluids.We consider a homogeneous two-dimensional porous medium of length L x and width L y with constant permeability K and a base uniform flow of speed U along the x direction. The viscosity ϭ (c) of the fluid is a function of the local concentration c(x,y,t) of the chemically reacting solute. Assuming that the fluid is neutrally buoyant and incompressible and that dispersion is isotropic, the equations governing the problem can be written as:where f (c) takes into account chemical reactions. Equations ͑1͒-͑3͒ can be rewritten by switching to a frame moving with the mean velocity of the fluid and nondimensionalizing with dispersive scales. 5,6 Introducing the stream function (x,y,t) such that uϭ␦ /␦y, wϭϪ␦ /␦x where u and w are the longitudinal and transverse velocity components, we then have:where is the vorticity and RϭϪd(ln )/dc. Important in what follows is the fact that vorticity is produced whenever the concentration gradient is not colinear with the velocity vector. We consider periodic boundary conditions and the initial condition of a step function between cϭ1 and 0 with noise added in the front and ϭ0 everywhere. We have here ass...

Viscous fingering of miscible slices

“…Unfortunately, dx = 1 is a resolution too high for fingering simulations especially if one wants to look at the dynamics at very long times. As an example, previous simulations on viscous fingering phenomena 15,19 were done with larger dx as typical dimensionless fingering wavelengths are around 100 for R = 3 for instance. Using typically dx = 4 gives roughly 25 points per wavelength which is numerically reasonable.…”

“…͑8͒, we get our final equations: 18 This model is numerically integrated using a pseudospectral code introduced by Tan and Homsy 15 and successfully implemented for various numerical studies of fingering. 19,20 The two-dimensional domain of integration is, in dimensionless units, of size Peϫ L where Pe= UL y / D x is the dimensionless width which is nothing else than the Péclet number of the problem, while L = UL x / D x . The dimensionless length of the sample is l = UW / D x .…”

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

“…The exact dependence of the viscosity on concentrations B and C still needs to be specified. Following earlier studies (Tan & Homsy 1986;De Wit & Homsy 1999a, 1999bAzaiez & Singh 2002;Gérard & De Wit 2009), an exponential dependence is adopted. The logarithm of the viscosity is thus taken to be a linear combination of the concentrations.…”

“…In the case of immiscible flows, this instability, known as the Saffman-Taylor instability, can also be modified by reactions that change the surface tension at the fluid interface (Jahoda & Hornof 2000;Fernandez & Homsy 2003). Several works have moreover addressed theoretically the coupling between VF and autocatalytic reactions (De Wit & Homsy 1999a, 1999bSwernath & Pushpavanam 2007Ghesmat & Azaiez 2009) without however corresponding experimental confirmations as autocatalytic reactions are more prone to change density rather than viscosity (De Wit 2001;De Wit et al 2003). In the context of chromatographic applications, adsorption-desorption phenomena have also been shown to influence VF patterns (Mishra, Martin & De Wit 2007).…”

Miscible density fingering of chemical fronts in porous media: Nonlinear simulations

“…Recently, reaction driven viscosity changes have been shown numerically to lead to new spatiotemporal dynamics of viscous fingering in chemical bistable autocatalytic systems providing traveling fronts between solutions of different viscosity. [6][7][8] Unfortunately the predicted nonlinear dynamics have not been observed to date mainly because autocatalytic reactions are usually performed in aqueous solutions for which the viscosity hardly changes with concentrations of solutes. The only striking evidence of chemically driven viscous fingering for miscible autocatalytic systems occurs in polymeric systems where the monomer solution and the polymer matrix can have quite strong differences in viscosity.…”