Stability of miscible displacements in porous media: Rectilinear flow

Abstract

A theoretical treatment of the stability of miscible displacement in a porous medium is presented. For a rectilinear displacement process, since the base state of uniform velocity and a dispersive concentration profile is time dependent, we make the quasi-steady-state approximation that the base state evolves slowly with respect to the growth of disturbances, leading to predictions of the growth rate. Comparison of results with initial value solutions of the partial differential equations shows that, excluding short times, there is good agreement between the two theories. Comparison of the theory with several experiments in the literature indicates that the theory gives a good prediction of the most dangerous wavelength of unstable fingers. An approximate analysis for transversely anisotropic media has elucidated the role of transverse dispersion in controlling the length scale of fingers.

Miscible displacements in Hele–Shaw cells: Nonmonotonic viscosity profiles

“…In a subsequent step, the stability of the quasisteady front to spanwise perturbations is examined, based on the three-dimensional Stokes equations. Generally poor agreement is observed between the Stokes results and corresponding findings for Darcy flows [4]. [5] extend this investigation to variable density displacements in vertical Hele-Shaw cells and find excellent agreement with the experimental data of [6] regarding the most amplified wavelength.…”

Double diffusive effects on pressure-driven miscible displacement flows in a channel

“…While there is no experimental evidence in support of (2.5) in a ternary system, it has been widely used to study viscosity-related instabilities (e.g. Chen & Meiburg 1996;Tan & Homsy 1986;Mishra et al 2010), and will be used here for mathematical convenience, and to allow for comparison with previous theoretical work using the same assumption. Since diffusivity and viscosity in a liquid are related approximately inversely through the Stokes-Einstein equation (Probstein 1994), one cannot conduct an experiment in which viscosity varies significantly and diffusivity does not.…”

“…The stability of this type of two-phase flow in a channel or pipe has been widely investigated both theoretically (Ranganathan & Govindarajan 2001;Selvam et al 2007;Sahu et al 2009a; and experimentally (Hickox 1971;Hu & Joseph 1989;Joseph & Renardy 1992;Joseph et al 1997). Linear stability analyses of displacement flows in porous media (Saffman & Taylor 1958;Chouke, Van Meurs & Van Der Pol 1959;Tan & Homsy 1986) explain that, if the displacing fluid is less viscous than the displaced one, the interface separating them becomes unstable and a fingering pattern develops at the interface. A review on such dynamics in porous media and Hele-Shaw cells is given by Homsy (1987).…”

Horizontal ethanol floods in clean, uniform, and layered sand packs under confined conditions

“…The times required for the experimentally measured angles to stabilize and the magnitudes of the angles were compared to the predictions determined using the method of Gardner et al The reason for the poor agreement appears to be related to the suitability of sharp interface models to describe interfacial phenomena during miscible flooding in liquid systems having density differences and nonmonotonic or exponential viscosity functions [Tan and Homsy, 1986;Homsy, 1993, 1994;Rogerson and Meiburg, 1993]. In particular, the estimation of the viscous forces and fingering (i.e., (9)) based on the difference in the end point viscosities has been questioned by Homsy [1993, 1994], who concluded that M < 1 is not necessarily a sufficient condition for stability in horizontal floods.…”