Radial source flows in porous media: Linear stability analysis of axial and helical perturbations in miscible displacements

2003 ◽  
Vol 15 (4) ◽  
pp. 938-946 ◽  
Author(s):  
Amir Riaz ◽  
Eckart Meiburg
Keyword(s):  
Porous Media ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Miscible Displacements ◽  
Flows In Porous Media

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.

Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 1. Linear stability analysis

2013 ◽  
Vol 721 ◽  
pp. 268-294 ◽  
Author(s):  
L. Talon ◽  
N. Goyal ◽  
E. Meiburg
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Stokes Equations ◽  
Variable Density ◽  
Local Concentration ◽  
Miscible Displacements ◽  
Instability Modes ◽  
Finger Tip

AbstractA computational investigation of variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells is presented. As a first step, two-dimensional base states are obtained by means of simulations of the Stokes equations, which are nonlinear due to the dependence of the viscosity on the local concentration. Here, the vertical position of the displacement front is seen to reach a quasisteady equilibrium value, reflecting a balance between viscous and gravitational forces. These base states allow for two instability modes: first, there is the familiar tip instability driven by the unfavourable viscosity contrast of the displacement, which is modulated by the presence of density variations in the gravitational field; second, a gravitational instability occurs at the unstably stratified horizontal interface along the side of the finger. Both of these instability modes are investigated by means of a linear stability analysis. The gravitational mode along the side of the finger is characterized by a wavelength of about one half to one full gap width. It becomes more unstable as the gravity parameter increases, even though the interface is shifted closer to the wall. The growth rate is largest far behind the finger tip, where the interface is both thicker, and located closer to the wall, than near the finger tip. The competing influences of interface thickness and wall proximity are clarified by means of a parametric stability analysis. The tip instability mode represents a gravity-modulated version of the neutrally buoyant mode. The analysis shows that in the presence of density stratification its growth rate increases, while the dominant wavelength decreases. This overall destabilizing effect of gravity is due to the additional terms appearing in the stability equations, which outweigh the stabilizing effects of gravity onto the base state.

Weak Turbulence in Small Prandtl Number Convection in Porous Media

2010 ◽  
Author(s):  
Peter Vadasz
Keyword(s):  
Porous Media ◽  
Stability Analysis ◽  
Prandtl Number ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Weak Turbulence ◽  
Numerical Computations ◽  
Prandtl Numbers ◽  
Convection In Porous Media

The dynamics of weak turbulence in small Prandtl number convection in porous media is substantially distinct than the corresponding dynamics for moderate and large Prandtl numbers. Linear stability analysis is performed and its results compared with numerical computations to reveal the underlying phenomena.

Variable-density miscible displacements in a vertical Hele-Shaw cell: linear stability

2007 ◽  
Vol 584 ◽  
pp. 357-372 ◽  
Author(s):  
N. GOYAL ◽  
H. PICHLER ◽  
E. MEIBURG
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Stokes Equations ◽  
Capillary Tube ◽  
Computational Study ◽  
Stability Limit ◽  
Dominant Wavelength ◽  
Miscible Displacements

A computational study based on the Stokes equations is conducted to investigate the effects of gravitational forces on miscible displacements in vertical Hele-Shaw cells. Nonlinear simulations provide the quasi-steady displacement fronts in the gap of the cell, whose stability to spanwise perturbations is subsequently examined by means of a linear stability analysis. The two-dimensional simulations indicate a marked thickening (thinning) and slowing down (speeding up) of the displacement front for flows stabilized (destabilized) by gravity. For the range investigated, the tip velocity is found to vary linearly with the gravity parameter. Strongly stable density stratifications lead to the emergence of flow patterns with spreading fronts, and to the emergence of a secondary needle-shaped finger, similar to earlier observations for capillary tube flows. In order to investigate the transition between viscously driven and purely gravitational instabilities, a comparison is presented between displacement flows and gravity-driven flows without net displacements.The linear stability analysis shows that both the growth rate and the dominant wavenumber depend only weakly on the Péclet number. The growth rate varies strongly with the gravity parameter, so that even a moderately stable density stratification can stabilize the displacement. Both the growth rate and the dominant wavelength increase with the viscosity ratio. For unstable density stratifications, the dominant wavelength is nearly independent of the gravity parameter, while it increases strongly for stable density stratifications. Finally, the kinematic wave theory of Lajeunesse et al. (J. Fluid Mech. vol. 398, 1999, p. 299) is seen to capture the stability limit quite accurately, while the Darcy analysis misses important aspects of the instability.

A linear stability analysis for miscible displacements

1986 ◽  
Vol 1 (2) ◽  
pp. 179-199 ◽  
Author(s):  
Shih-Hsien Chang ◽  
John C. Slattery
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Miscible Displacements

Stability of Piston-Like Displacements of Water by Steam and Nitrogen in Porous Media

1982 ◽  
Vol 22 (05) ◽  
pp. 625-634 ◽  
Author(s):  
David A. Krueger
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Gravitational Force ◽  
Absolute Permeability ◽  
Water Steam ◽  
The Stability ◽  
Condensation Front

Abstract Downhole steam generation leads to consideration of reservoir fluid displacement by a mixture of steam and nitrogen. The linear stability analysis of the steam condensation front has been generalized to include a noncondensing gas. Roughly speaking, the addition of nitrogen increases the likelihood of having fingers, but, compared with the no-nitrogen case, the fingers will grow more slowly. Introduction The theory of the stability of flows through porous media has been a subject of interest for more than 25 years, dating back to the pioneering work of Dietz, Chuoke et al., and Saffman and Taylor. They considered injecting one fluid (e.g., water) to force a second fluid (e.g., oil) out of a porous medium. The primary result was that instabilities (fingering) occurred when the driving fluid was more mobile than the driven fluid. Hagoort included multiple fluid phases. Miller generalized the original work to include steam driving water (liquid). He showed that the thermodynamic phase transition (steam to water) introduces two stabilizing effects. The first effect introduces a water/steam velocity ratio as a multiplier of the mobility ratio. This factor is less than one because of the volume change upon condensation. The second effect is the cooling of incipient steam fingers by the surrounding water, which retards their growth. Baker anticipated these effects in a qualitative way to explain his experiments, which showed a more stable displacement by steam than was expected on the basis of mobility ratios alone. Armento and Miller also have considered the stability of the in-situ combustion front in porous media. Their work deals with a region where steam is generated. This paper reformulates Miller's results for a condensation front in a more useful form including general numerical results and extends the theory to include injection of a noncondensing gas (e.g., nitrogen) together with the steam. Depending on the particular situation, the presence of nitrogen can be either stabilizing or destabilizing. The motivation for the generalization comes from enhanced oil recovery projects where the exhaust gases from the steam generator are injected into the reservoir along with the steam. This paper considers perturbations on a flat condensation front that is perpendicular to its velocity. The gravitational force along this velocity is included, but the component of the gravitational force perpendicular to the velocity is not. Thus we include the effect of gravity on fingering, but we do not discuss the gravity override problem. In Stability Analysis we present two steps:determination of the motion of a flat condensation front (details are in the Appendix) andevaluation of the characteristic time for growth or decay of a perturbation of that front. In Results wegive the results for a specific reservoir;discuss the sensitivity of these results to the important reservoir parameters (flow velocities and absolute permeabilities),show that, if surface tension and gravitation are unimportant, the stability condition is independent of the absolute permeability and absolute flow rates, anddiscuss the longest wavelength for a stable perturbation. In the final section we discuss the main conclusions. Stability Analysis We consider a homogeneous porous medium with fluids in two regions as illustrated in Fig. 1. A steam/nitrogen mixture is injected at the left, and water (liquid) and nitrogen are produced at the fight. The linear stability analysis proceeds in two main stages and follows the general methods as discussed by Chandrasekhar and the specific application of Miller. First, we assume that the condensation front is flat, moves with constant velocity, v, and has properties that vary with z alone. SPEJ P. 625^

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