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

Abstract

We study nonlinear viscous fingering in heterogeneous media through direct numerical simulation. A pseudospectral method is developed and applied to our spatially periodic model introduced in Paper I ͓J. Chem. Phys. 107, 9609 ͑1997͔͒. The problem involves several parameters, including the Peclet number, Pe, the magnitude and wave numbers of the heterogeneity, , n x , n y , respectively, and the log of the viscosity ratio R. Progress is made by fixing R at 3.0 and then working first with layered systems n x ϭ0 and finally with ''checkerboard'' systems in which both wave numbers are nonzero. Strongly nonlinear finger dynamics are compared and contrasted with those occurring in the homogeneous case. For layered systems, it is found that very low levels of heterogeneity leads to an enhancement of the growth rate of the fingered zones, and that both harmonic and subharmonic resonances between the intrinsic scale of nonlinear fingering and those of the heterogeneity occur. We also find that the fingering regime of layered systems can be completely disrupted by modest levels of heterogeneity, leading to a ''channeling'' regime and dispersive behavior which is identified as a Taylor dispersion mechanism. The effective axial dispersion coefficient in this regime is found to be strongly dependent on the viscosity ratio. The situation becomes more complex for the checkerboard case. The channeling regime can in turn be disrupted by the axial dependence of the heterogeneity, which stimulates tip splitting and a return to complex nonlinear finger dynamics in regions of parameter space, including very large , that would otherwise be strongly dispersive. The effectiveness of the axial variation in stimulating tip splitting is studied by a short parametric study in n x , and is found to be maximized for certain axial frequencies in a manner similar to that found in Paper I. All our results are found to be in general qualitative agreement with available ͑but limited͒ experimental visualizations.

Viscous fingering in reaction-diffusion systems

“…As has been shown in many previous studies, see for example Refs. 8,26, and the references therein, use of such boundary conditions does not influence the dynamics of an infinite domain front propagation until the unstable propagating front encounters its ͑stable͒ periodic extension, limiting the time over which the simulation can proceed, but not its accuracy. When using a step function as initial condition for the concentration, we thus work with the full cosine expansion of the displacement front.…”

Interface evolution during radial miscible viscous fingering

“…Much work has focused on characterizing miscible viscous fingering, including laboratory experiments [25][26][27], numerical simulations [28][29][30][31][32], and linear stability analyses to model the onset and growth of instabilities for rectilinear [33] and radial [34,35] geometries. Other studies have also focused on the effects of anisotropic dispersion [31,33,36], medium heterogeneity [32,[37][38][39][40], gravity [41][42][43][44][45][46], chemical reactions [3,[47][48][49], absorption [50], and flow configuration [51][52][53][54][55] on the viscous fingering instability. Despite the extensive work done, the effect of viscous fingering on mixing has only recently been investigated numerically for a rectilinear geometry [14,56].…”

Nonlinear interactions of chemical reactions and viscous fingering in porous media

“…If R Ͼ0 the front is viscously unstable and develops fingers even when f (c)ϭ0 as already studied in detail in numerous works. [5][6][7][8][9][10] We refer to such fingering in absence of chemical reactions as ''standard fingering.'' It is known that the nonlinear dynamics of standard fingers involves fading, shielding, and tip splitting.…”

“…In this sense, the enhanced splitting in the presence of chemistry mimics that induced by permeability heterogeneities. 6,10 In summary, the characteristics of viscous fingering in miscible systems are profoundly modified when chemical reactions leading to bistability come into play. FIG.…”

“…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.…”