A general stability analysis for immiscible viscous displacement in a porous medium

1981 ◽  
Vol 59 (5) ◽  
pp. 678-687 ◽  
Author(s):  
T. J. T. Spanos
Keyword(s):  
Porous Medium ◽  
Stability Analysis ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Immiscible Displacement ◽  
Stability Criteria ◽  
Macroscopic Scale ◽  
Two Fluids ◽  
General Stability ◽  
Inhomogeneous Porous Medium

A perturbation of an immiscible displacement process causes relative motion of the two fluids involved. At the macroscopic scale such relative motions are considered to propagate throughout the porous medium in the form of fluid waves. A description of these waves is given on surfaces of constant saturation in a similar fashion to the description of a surface wave propagating on the interface between two fluids. In the porous medium, however, the wave propagation is not restricted to the surface of constant saturation and as a result one obtains a wave equation that is both dissipative and diffusive.A stability analysis is also considered for the immiscible displacement process. Here, a characteristic time for instability to occur can be calculated when the inertial terms are included in the equations of motion. Also a generalization of the wave equations and stability criteria are considered for an inhomogeneous porous medium.

The equations of immiscible viscous displacement in a porous medium

1981 ◽  
Vol 59 (1) ◽  
pp. 45-56 ◽  
Author(s):  
T. J. T. Spanos
Keyword(s):  
Porous Medium ◽  
Statistical Theory ◽  
Boundary Conditions ◽  
Initial Conditions ◽  
Continuum Theory ◽  
Immiscible Displacement ◽  
Macroscopic Scale ◽  
Inertial Terms ◽  
Inhomogeneous Porous Medium ◽  
Relative Motions

A statistical theory for the construction of the equations of viscous displacement in a porous medium is considered. This yields a continuum theory for immiscible displacement which can be applied to either a homogeneous or inhomogeneous porous medium. The relative motions of the fluid are considered in terms of the motion of surfaces of constant saturation which are smoothed surfaces at the macroscopic scale considered. The boundary conditions and initial conditions at the injection boundary are considered as well as the boundary conditions and breakthrough conditions at the recovery boundary and the side boundary conditions. The inertial terms are included in the equations and shown to be of importance in describing these initial conditions and the breakthrough conditions.

scholarly journals Multi-scale flow patterns during immiscible displacement of oil by water in a layer-inhomogeneous porous media

2021 ◽  
Vol 2119 (1) ◽  
pp. 012048
Author(s):  
V V Kuznetsov ◽  
S A Safonov
Keyword(s):  
Porous Medium ◽  
Large Scale ◽  
Numerical Study ◽  
Immiscible Displacement ◽  
Multi Scale ◽  
Physical Mechanisms ◽  
Displacement Front ◽  
Viscous Instability ◽  
Induced Flow ◽  
Inhomogeneous Porous Medium

Abstract This paper presents the results of numerical study of the relationship between micro-and macroscale flows during immiscible displacement in a two-layer porous medium. A feature of the proposed approach is the allowance for large-scale capillarity induced flow due to curvature of the displacement front in macro-inhomogeneous porous medium. The physical mechanisms determining the development of viscous instability in a layer-inhomogeneous porous medium are considered, the methods for suppressing viscous fingers formation based on the stabilization of the displacement front due the action of capillary forces are proposed.

The surface conditions for viscous displacement in a homogeneous porous medium

1979 ◽  
Vol 57 (10) ◽  
pp. 1738-1745 ◽  
Author(s):  
T. J. T. Spanos
Keyword(s):  
Boundary Layer ◽  
Porous Medium ◽  
Continuum Theory ◽  
Immiscible Displacement ◽  
Surface Conditions ◽  
Fluid Displacement ◽  
Two Fluids ◽  
Mathematical Abstraction ◽  
Homogeneous Porous Medium ◽  
Macroscopic Shape

The viscous surface conditions between two fluids are considered for fluid displacement in a homogeneous porous medium. The term viscous surface is defined here as a mathematical abstraction used to approximate the macroscopic shape of the boundary layer between two fixed saturations of displacing fluid by continuum theory. In the limit as the saturations approach each other, one then obtains a convergence of the viscous surface to a constant saturation contour. This yields a mathematical description of immiscible displacement in porous media which contains a theory of viscous fingering.

Stability Analysis of a Supercritical Fluid in a Porous Medium Heated from Below

2016 ◽  
Author(s):  
Leonardo Santos de Brito Alves ◽  
Leandro Santos de Barros ◽  
Heitor Herculano de Barros
Keyword(s):  
Porous Medium ◽  
Stability Analysis ◽  
Supercritical Fluid

Fluid flow over and through a regular bundle of rigid fibres

2016 ◽  
Vol 792 ◽  
pp. 5-35 ◽  
Author(s):  
Giuseppe A. Zampogna ◽  
Alessandro Bottaro
Keyword(s):  
Porous Medium ◽  
Fluid Flow ◽  
Flow Field ◽  
Stokes Equations ◽  
Transversely Isotropic ◽  
Navier Stokes ◽  
Leading Order ◽  
Macroscopic Scale ◽  
Navier Stokes Equations ◽  
Homogenized Model

The interaction between a fluid flow and a transversely isotropic porous medium is described. A homogenized model is used to treat the flow field in the porous region, and different interface conditions, needed to match solutions at the boundary between the pure fluid and the porous regions, are evaluated. Two problems in different flow regimes (laminar and turbulent) are considered to validate the system, which includes inertia in the leading-order equations for the permeability tensor through a Oseen approximation. The components of the permeability, which characterize microscopically the porous medium and determine the flow field at the macroscopic scale, are reasonably well estimated by the theory, both in the laminar and the turbulent case. This is demonstrated by comparing the model’s results to both experimental measurements and direct numerical simulations of the Navier–Stokes equations which resolve the flow also through the pores of the medium.

Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell

1969 ◽  
Vol 39 (3) ◽  
pp. 477-495 ◽  
Author(s):  
R. A. Wooding
Keyword(s):  
Porous Medium ◽  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Saturated Porous Medium ◽  
Mean Amplitude ◽  
Wave Number ◽  
Two Fluids ◽  
Averaged Equations ◽  
The Mean ◽  
Hele Shaw Cell

Waves at an unstable horizontal interface between two fluids moving vertically through a saturated porous medium are observed to grow rapidly to become fingers (i.e. the amplitude greatly exceeds the wavelength). For a diffusing interface, in experiments using a Hele-Shaw cell, the mean amplitude taken over many fingers grows approximately as (time)2, followed by a transition to a growth proportional to time. Correspondingly, the mean wave-number decreases approximately as (time)−½. Because of the rapid increase in amplitude, longitudinal dispersion ultimately becomes negligible relative to wave growth. To represent the observed quantities at large time, the transport equation is suitably weighted and averaged over the horizontal plane. Hyperbolic equations result, and the ascending and descending zones containing the fronts of the fingers are replaced by discontinuities. These averaged equations form an unclosed set, but closure is achieved by assuming a law for the mean wave-number based on similarity. It is found that the mean amplitude is fairly insensitive to changes in wave-number. Numerical solutions of the averaged equations give more detailed information about the growth behaviour, in excellent agreement with the similarity results and with the Hele-Shaw experiments.

Duality and stability of MHD Darcy–Forchheimer porous medium flow of rotating nanofluid on a linear shrinking/stretching sheet: Buongiorno model

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Jawad Raza ◽  
Sumera Dero ◽  
Liaquat Ali Lund ◽  
Zurni Omar
Keyword(s):  
Porous Medium ◽  
Stability Analysis ◽  
Differential Equations ◽  
Shrinking Sheet ◽  
Dual Nature ◽  
Content Type ◽  
Great Achievement ◽  
Medium Flow ◽  
Buongiorno Model ◽  
Porous Medium Flow

Purpose The purpose of study is to examine the dual nature of the branches for the problem of Darcy–Forchheimer porous medium flow of rotating nanofluid on a linearly stretching/shrinking surface under the field of magnetic influence. The dual nature of the branches confronts the uniqueness and existence theorem, moreover, mathematically it is a great achievement. For engineering purposes, this study applied a linear stability test on the multiple branches to determine which solution is physically reliable (stable). Design/methodology/approach Nanofluid model has been developed with the help of Buongiorno model. The partial differential equations in space coordinates for the law of conservation of mass, momentum and energy have been transformed into ordinary differential equations by introducing the similarity variables. Two numerical techniques, namely, the shooting method in Maple software and the three-stage Lobatto IIIA method in Matlab software, have been used to find multiple branches and to accomplish stability analysis, respectively. Findings The parametric investigation has been executed to find the multiple branches and explore the effects on skin friction, Sherwood number, Nusselt number, concentration and temperature profiles. The findings exhibited the presence of dual branches only in the case of a shrinking sheet. Originality/value The originality of work is a determination of multiple branches and the performance of the stability analysis of the branches. It has also been confirmed that such a study has not yet been considered in the previous literature.

Darcy-Forchheimer porous medium effect on rotating hybrid nanofluid on a linear shrinking/stretching sheet

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Ilyas Khan
Keyword(s):  
Porous Medium ◽  
Stability Analysis ◽  
Differential Equations ◽  
Three Dimensional ◽  
Spherical Particles ◽  
Medium Effect ◽  
Hybrid Nanofluid ◽  
Three Dimensional Flow ◽  
Content Type ◽  
Dimensional Flow

Purpose The purpose of this study is to find the multiple branches of the three-dimensional flow of Cu-Al2 O3/water rotating hybrid nanofluid perfusing a porous medium over the stretching/shrinking surface. The extended model of Darcy due to Forchheimer and Brinkman has been considered to make the hybrid nanofluid model over the pores by considering the porosity and permeability effects. Design/methodology/approach The Tiwari and Das model with the thermophysical properties of spherical particles for efficient dynamic viscosity of the nanoparticle is used. The linear similarity transformations are applied to convert the partial differential equations into ordinary differential equations (ODEs). The system of governing ODEs is solved by using the three-stage Lobatto IIIa scheme in MATLAB for evolving parameters. Findings The system of governing ODEs produces dual branches. A unique stable branch is identified with help of stability analysis. The reduced heat transfer rate has been shown to increase with the reduced ϕ2 in both branches. Further, results revealed that the presence of multiple branches depends on the ranges of porosity, suction and stretching/shrinking parameters for the particular value of the rotating parameter. Originality/value Dual branches of the three-dimensional flow of Cu-Al2 O3/water rotating hybrid nanofluid have been found. Therefore, stability analysis of the branches is also conducted to know which branch is appropriate for the practical applications. To the best of the authors’ knowledge, this research is novel and there is no previously published work relevant to the present study.

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