Quantifying mixing in viscously unstable porous media flows

Abstract

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.

Investigation of concentration‐dependent diffusion on frontal instabilities and mass transfer in homogeneous porous media

“…It is worth noting that the NCG is similar to the typical mean scalar dissipation rate e in a laminar flow, which is used to characterize the mixing rate of two miscible fluids. [34] In contrast, for the CDDC case, it is more appropriate to use D(c) instead of a constant diffusion coefficient: e ¼ h D c ð Þ Á rc j j 2 i. Figure 14 depicts the NCG versus time for different log mobility ratios at Pe ¼ 900.…”

Nonlinear simulations of miscible viscous fingering with gradient stresses in porous media

“…Extensive studies have been conducted for various industrial and environmental processes such as secondary and tertiary oil recovery in oil industries, hydrology, filtration, chromatographic column (De Wit et al, 2005;Mishra et al, 2008;Rousseaux et al, 2007;Shalliker et al, 2007), fixed bed regeneration, aquifers (De Wit et al, 2005) and geodynamics (Morra and Yuen, 2008). VF has also been used as an alternative source to enhance mixing of two fluids in micro-channel in low Reynolds number regime (Jha et al, 2011), where turbulence does not appear. Although the miscible and immiscible fluids are of two completely different categories, they encounter qualitatively similar VF instability except the stabilizing phenomena in the respective cases.…”

Dynamics of a Highly Viscous Circular Blob in Homogeneous Porous Media

“…Its wide applicabilities, e.g., enhanced oil recovery [1], CO 2 sequestration [2], chromatographic separation [3,4], contaminant transport in aquifers [5], mixing in low-Reynolds number flow [6], intrinsic characteristics of nonlinear dynamics and pattern formation have fascinated active theoretical, numerical [6][7][8][9][10][11][12][13], and experimental [14] researchers for more than half a decade. Both rectilinear and radial displacements have been investigated rigorously and have their own significance that can be investigated independently as well as comparatively.…”

“…When the underlying fluids are miscible to each other, the finger-like patterns that develop at the fluid-fluid interface depend, to a great extent, on the shape of the interface [13], viscosity contrast [1], and diffusion rate [16]. As a consequence, mixing, spreading, and breakthrough time alter greatly [6,13,15]. Pramanik et al used the Fourier pseudospectral method to capture the dynamics of a more viscous blob in a homogeneous porous medium [13], and the results are summarized in the R-Pe parameter plane.…”

“…We exploit this mathematical analogy between a two-phase porous media flow with capillary diffusion and a single phase species transport in porous media. In the absence of the capillary pressure, the momentum conservation of both the phases is governed by single Darcy's law, constrained with the incompressibility condition (divergence-free velocity for constant density), described by Equations (5) and (6). This is coupled with a convection-diffusion equation, Equation (7), which describes the mass conservation of the concentration of phase-1.…”