Linear stability analysis and nonlinear simulation of the channeling effect on viscous fingering instability in miscible displacement

2018 ◽  
Vol 30 (3) ◽  
pp. 034106 ◽  
Author(s):  
M. R. Shahnazari ◽  
I. Maleka Ashtiani ◽  
A. Saberi
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Viscous Fingering ◽  
Miscible Displacement ◽  
Nonlinear Simulation ◽  
Fingering Instability ◽  
Channeling Effect

scholarly journals Non-modal stability analysis of miscible viscous fingering with non-monotonic viscosity profiles

2018 ◽  
Vol 856 ◽  
pp. 552-579
Author(s):  
Tapan Kumar Hota ◽  
Manoranjan Mishra
Keyword(s):  
Stability Analysis ◽  
Viscous Fluid ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Physical Mechanism ◽  
Stable State ◽  
Viscous Fingering ◽  
Linear Perturbation ◽  
Optimal Perturbation ◽  
Nonlinear Simulations

A non-modal linear stability analysis (NMA) of the miscible viscous fingering in a porous medium is studied for a toy model of non-monotonic viscosity variation. The onset of instability and its physical mechanism are captured in terms of the singular values of the propagator matrix corresponding to the non-autonomous linear equations. We discuss two types of non-monotonic viscosity profiles, namely, with unfavourable (when a less viscous fluid displaces a high viscous fluid) and with favourable (when a more viscous fluid displaces a less viscous fluid) endpoint viscosities. A linear stability analysis yields instabilities for such viscosity variations. Using the optimal perturbation structure, we are able to show that an initially unconditional stable state becomes unstable corresponding to the most unstable initial disturbance. In addition, we also show that to understand the spatio-temporal evolution of the perturbations it is necessary to analyse the viscosity gradient with respect to the concentration and the location of the maximum concentration $c_{m}$. For the favourable endpoint viscosities, a weak transient instability is observed when the viscosity maximum moves close to the pure invading or defending fluid. This instability is attributed to an interplay between the sharp viscosity gradient and the favourable endpoint viscosity contrast. Further, the usefulness of the non-modal analysis demonstrating the physical mechanism of the quadruple structure of the perturbations from the optimal concentration disturbances is discussed. We demonstrate the dissimilarity between the quasi-steady-state approach and NMA in finding the correct perturbation structure and the onset, for both the favourable and unfavourable viscosity profiles. The correctness of the linear perturbation structure obtained from the non-modal stability analysis is validated through nonlinear simulations. We have found that the nonlinear simulations and NMA results are in good agreement. In summary, a non-monotonic variation of the viscosity of a miscible fluid pair is seen to have a larger influence on the onset of fingering instabilities than the corresponding Arrhenius type relationship.

Fingering instabilities in vertical miscible displacement flows in porous media

1995 ◽  
Vol 288 ◽  
pp. 75-102 ◽  
Author(s):  
O. Manickam ◽  
G. M. Homsy
Keyword(s):  
Porous Media ◽  
Stability Analysis ◽  
Linear Stability ◽  
Critical Velocity ◽  
Linear Stability Analysis ◽  
Miscible Displacement ◽  
Stable Region ◽  
Miscible Displacements ◽  
Flows In Porous Media ◽  
Two Fluids

The fingering instabilities in vertical miscible displacement flows in porous media driven by both viscosity and density contrasts are studied using linear stability analysis and direct numerical simulations. The conditions under which vertical flows are different from horizontal flows are derived. A linear stability analysis of a sharp interface gives an expression for the critical velocity that determines the stability of the flow. It is shown that the critical velocity does not remain constant but changes as the two fluids disperse into each other. In a diffused profile, the flow can develop a potentially stable region followed downstream by a potentially unstable region or vice versa depending on the flow velocity, viscosity and density profiles, leading to the potential for ‘reverse’ fingering. As the flow evolves into the nonlinear regime, the strength and location of the stable region changes, which adds to the complexity and richness of finger propagation. The flow is numerically simulated using a Hartley-transform-based spectral method to study the nonlinear evolution of the instabilities. The simulations are validated by comparing to experiments. Miscible displacements with linear density and exponential viscosity dependencies on concentration are simulated to study the effects of stable zones on finger propagation. The growth rates of the mixing zone are parametrically obtained for various injection velocities and viscosity ratios.

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