Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 2. Nonlinear simulations

Abstract

Direct numerical simulations of the variable density and viscosity Navier–Stokes equations are employed, in order to explore three-dimensional effects within miscible displacements in horizontal Hele-Shaw cells. These simulations identify a number of mechanisms concerning the interaction of viscous fingering with a spanwise Rayleigh–Taylor instability. The dominant wavelength of the Rayleigh–Taylor instability along the upper, gravitationally unstable side of the interface generally is shorter than that of the fingering instability. This results in the formation of plumes of the more viscous resident fluid not only in between neighbouring viscous fingers, but also along the centre of fingers, thereby destroying their shoulders and splitting them longitudinally. The streamwise vorticity dipoles forming as a result of the spanwise Rayleigh–Taylor instability place viscous resident fluid in between regions of less viscous, injected fluid, thereby resulting in the formation of gapwise vorticity via the traditional, gap-averaged viscous fingering mechanism. This leads to a strong spatial correlation of both vorticity components. For stronger density contrasts, the streamwise vorticity component increases, while the gapwise component is reduced, thus indicating a transition from viscously dominated to gravitationally dominated displacements. Gap-averaged, time-dependent concentration profiles show that variable density displacement fronts propagate more slowly than their constant density counterparts. This indicates that the gravitational mixing results in a more complete expulsion of the resident fluid from the Hele-Shaw cell. This observation may be of interest in the context of enhanced oil recovery or carbon sequestration applications.

Experimental study of a buoyancy-driven instability of a miscible horizontal displacement in a Hele-Shaw cell

“…Results of experiments in rectilinear displacements could also be easier to compare with theoretical stability analyses or full numerical solutions of the relevant equations. 6,7 Simulations could moreover test the influence of the different parameters of the problem on the evolution of the pattern in 3D. …”

“…1 In horizontal systems, the density jump is in the direction perpendicular to gravity and the stability of the interface is classically considered to depend only on the viscosity difference: viscous fingering can be observed when the less viscous fluid is injected into the more viscous one while the reverse displacement is viscously stable. The influence of density differences across the miscible interface on the viscous fingering pattern has been recently addressed by both linear stability analysis 6 and nonlinear simulations 7 of the full 3D Stokes problem. It has been shown that a buoyancy-driven instability can act within the gap of the cell on the upper side of the finger.…”

Schlieren imaging of viscous fingering in a horizontal Hele-Shaw cell

“…Fig. 8) (John et al 2013). The density difference between the displacing and displaced fluids gives rise to an additional dimensionless parameter F defined as…”

“…These stripes result from streamwise, counter-rotating vortices, such as can be seen in Figs. 10 and 20 in John et al (2013). Talon et al (2013) conducted a detailed linear stability analysis of this density-driven instability.…”

“…These Pihler-Puzović et al 2012). Finally, advances in computational modeling and hardware have begun to provide information on the three-dimensional flow structure within the gap (Heussler et al 2014;John et al 2013;Oliveira and Meiburg 2011), and specifically on novel types of instabilities.…”