scholarly journals An integral equation for immiscible fluid isplacement in a two-dimensional porous medium or Hele-Shaw cell

1984 ◽  
Vol 26 (1) ◽  
pp. 14-30 ◽  
Author(s):  
M. R. Davidson
Keyword(s):  
Integral Equation ◽  
Porous Medium ◽  
Interfacial Tension ◽  
Immiscible Fluids ◽  
Physical Parameters ◽  
Immiscible Fluid ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Flat Interface ◽  
Hele Shaw Cell

AbstractAn integral equation for the normal velocity of the interface between two immiscible fluids flowing in a two-dimensional porous medium or Hele-Shaw cell (one fluid displaces the other) is derived in terms of the physical parameters (including interfacial tension), a Green's function and the given interface. When the displacement is unstable, ‘fingering’ of the interface occurs. The Saffman-Taylor interface solutions for the steady advance of a single parallel-sided finger in the absence of interfacial tension are seen to satisfy the integral equation, and the error incurred in that equation by the corresponding Pitts approximating profile, when interfacial tension is included, is shown. In addition, the numerical solution of the integral equation is illustrated for a sinusoidal and a semicircular interface and, in each case, the amplitude behaviour inferred from the velocity distribution is consistent with conclusions based on the stability of an initially flat interface.

scholarly journals Numerical calculation of unstable immiscible fluid displacement in a two-dimensional porous medium or Hele-Shaw cell

1985 ◽  
Vol 26 (4) ◽  
pp. 452-469 ◽  
Author(s):  
M. R. Davidson
Keyword(s):  
Porous Medium ◽  
Numerical Procedure ◽  
Immiscible Fluids ◽  
Boundary Integral ◽  
Immiscible Fluid ◽  
Two Dimensional ◽  
Time Step ◽  
Fluid Displacement ◽  
Numerical Errors ◽  
Hele Shaw Cell

AbstractA numerical procedure for calculating the evolution of a periodic interface between two immiscible fluids flowing in a two-dimensional porous medium or Hele-Shaw cell is described. The motion of the interface is determined in a stepwise manner with its new velocity at exach time step being derived as a numerical solution of a boundary integral equation. Attention is focused on the case of unstable displacement charaterised physically by the “fingering” of the interface and computationally by the growth of numerical errors regardless of the numerical method employed. Here the growth of such error is reduced and the usable part of the calculation extended to finite amplitudes. Numerical results are compared with an exact “finger” solution and the calculated behaviour of an initial sinusoidal displacement, as a function of interfacial tension, initial amplitude and wavelength, is discussed.

scholarly journals Lattice Boltzmann Simulation of Immiscible Two-Phase Displacement in Two-Dimensional Berea Sandstone

2018 ◽  
Vol 8 (9) ◽  
pp. 1497 ◽  
Author(s):  
Qingqing Gu ◽  
Haihu Liu ◽  
Yonghao Zhang
Keyword(s):  
Interfacial Tension ◽  
Lattice Boltzmann ◽  
Viscosity Ratio ◽  
Immiscible Fluids ◽  
Lattice Boltzmann Model ◽  
Immiscible Fluid ◽  
Two Dimensional ◽  
Berea Sandstone ◽  
Fluid Saturation ◽  
Wetting Fluid

Understanding the dynamic displacement of immiscible fluids in porous media is important for carbon dioxide injection and storage, enhanced oil recovery, and non-aqueous phase liquid contamination of groundwater. However, the process is not well understood at the pore scale. This work therefore focuses on the effects of interfacial tension, wettability, and the viscosity ratio on displacement of one fluid by another immiscible fluid in a two-dimensional (2D) Berea sandstone using the colour gradient lattice Boltzmann model with a modified implementation of the wetting boundary condition. Through invasion of the wetting phase into the porous matrix, it is observed that the viscosity ratio plays an important role in the non-wetting phase recovery. At the viscosity ratio ( λ ) of unity, the saturation of the wetting fluid is highest, and it linearly increases with time. The displacing fluid saturation reduces drastically when λ increases to 20; however, when λ is beyond 20, the reduction becomes less significant for both imbibition and drainage. The front of the bottom fingers is finally halted at a position near the inlet as the viscosity ratio increases to 10. Increasing the interfacial tension generally results in higher saturation of the wetting fluid. Finally, the contact angle is found to have a limited effect on the efficiency of displacement in the 2D Berea sandstone.

Lattice Boltzmann Simulations for Thermal Conductivity Estimation in Heterogeneous Materials

2009 ◽  
Vol 283-286 ◽  
pp. 364-369 ◽  
Author(s):  
M.R. Arab ◽  
Bernard Pateyron ◽  
Mohammed El Ganaoui ◽  
Nicolas Calvé
Keyword(s):  
Thermal Conductivity ◽  
Porous Medium ◽  
Lattice Boltzmann ◽  
Numerical Study ◽  
Theoretical Models ◽  
Short Description ◽  
Physical Parameters ◽  
Two Dimensional ◽  
Lattice Boltzmann Simulations ◽  
Boltzmann Method

For simulating flows in a porous medium, a numerical tool based on the Lattice Boltzmann Method (LBM) is developed with regards to the classical D2Q9 model. A short description of this model is presented. This technique, applied to two-dimensional configurations, indicates its ability to simulate phenomena of heat and mass transfer. The numerical study is extended to estimate physical parameters that characterize porous materials, like the so-called Effective Thermal Conductivity (ETC) which is of our interest in this paper. Obtained results are compared with those which could be found analytically and by theoretical models. Finally, a porous medium is considered to find its ETC.

Flow of Two-Immiscible Fluids in Porous and Nonporous Channels

1999 ◽  
Vol 122 (1) ◽  
pp. 117-124 ◽  
Author(s):  
Ali J. Chamkha
Keyword(s):  
Magnetic Field ◽  
Porous Medium ◽  
Oil Recovery ◽  
Flow Direction ◽  
Immiscible Fluids ◽  
Local Thermal Equilibrium ◽  
Physical Parameters ◽  
Electrical Insulator ◽  
Thermal Buoyancy ◽  
Different Temperatures

This study considers steady, laminar flow of two viscous, incompressible, electrically-conducting and heat-generating or absorbing immiscible fluids in an infinitely-long, impermeable parallel-plate channel filled with a uniform porous medium. A magnetic field of uniform strength is applied normal to the flow direction. The channel walls are assumed to be electrically nonconducting and are maintained at two different temperatures. When present, the porous medium is assumed to act as an electrical insulator and that it is in local thermal equilibrium with the fluid. The transport properties of both fluids are assumed to be constant. This study is expected to be useful in understanding the influence of the presence of slag layers on the flow and heat transfer aspects of coal-fired Magnetohydrodynamic (MHD) generators when the porous medium is absent and the effects of thermal buoyancy and a magnetic field on enhanced oil recovery and filtration systems where the porous medium is present. The problem is formulated by employing the balance laws of mass, linear momentum, and energy for both phases. Continuous conditions for the velocity and temperature as well as the shear stress and heat flux of both phases at the interface are employed. The resulting governing ordinary differential equations are solved numerically subject to the boundary and interface conditions for the velocity and temperature distributions of both fluids in the channel. Analytical solutions for a special case of the problem where the porous medium is absent or only its inertia effect is neglected are obtained. Comparisons with previously reported velocity profiles are performed and excellent agreements are obtained. A parametric study illustrating the influence of the physical parameters involved in the problem is conducted and the results are presented graphically and discussed. [S0098-2202(00)02101-5]

Using Dissipative Particle Dynamics to Model Binary Immiscible Fluids

1997 ◽  
Vol 08 (04) ◽  
pp. 909-918 ◽  
Author(s):  
Keir E. Novik ◽  
Peter V. Coveney
Keyword(s):  
Phase Separation ◽  
Dissipative Particle Dynamics ◽  
Particle Dynamics ◽  
Immiscible Fluids ◽  
Promising Method ◽  
Domain Growth ◽  
Immiscible Fluid ◽  
Two Dimensional ◽  
Fluid Systems

We investigate the domain growth and phase separation of two-dimensional binary immiscible fluid systems using dissipative particle dynamics. Our results are compared with similar simulations using other techniques, and we conclude that dissipative particle dynamics is a promising method for simulating these systems.

Lifting Surface Theory for the Problem of an Arbitrarily Yawed Sinusoidal Gust Incident on a Thin Aerofoil in Incompressible Flow

1970 ◽  
Vol 21 (2) ◽  
pp. 182-198 ◽  
Author(s):  
J. M. R. Graham
Keyword(s):  
Integral Equation ◽  
Kernel Function ◽  
Incompressible Flow ◽  
Velocity Component ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Surface Theory ◽  
Equation Solution ◽  
Chebyshev Expansion ◽  
Dimensional Integral

SummaryThe solution to the problem of the loading generated on a two-dimensional thin aerofoil by an incompressible flow whose normal velocity component is of the general form exp [i(λx+/μy — ωt)] is calculated. The method used extends the two-dimensional integral equation solution for the induced vorticity by means of a Chebyshev expansion of part of the kernel function. Thin aerofoil approximations are made throughout, but no collocation procedure, as such, is required.

Convection in a saturated porous medium at large Rayleigh number or Peclet number

1963 ◽  
Vol 15 (4) ◽  
pp. 527-544 ◽  
Author(s):  
R. A. Wooding
Keyword(s):  
Porous Medium ◽  
Saturated Porous Medium ◽  
Vertical Plane ◽  
Homogeneous Solution ◽  
Physical Parameters ◽  
Convective System ◽  
Two Dimensional ◽  
Source Strength ◽  
Special Cases ◽  
Plane Jet

When the dimensions of a convective system in a saturated porous medium are sufficiently great, diffusion effects can be neglected except in regions where the gradients of fluid properties are very large. A boundary-layer theory is developed for vertical plane flows in such regions. In special cases, the theory is equivalent to that for laminar incompressible flow in a two-dimensional half-jet, or in a plane jet or round jet, for which similarity solutions are well known.A number of experiments have been performed using a Hele-Shaw cell immersed in water, with a source of potassium permanganate solution located between the plates. At very small values of the source strength, a flow analogous to that of a plane jet from a slit is obtained. The distance advanced by the jet front, or cap, is measured as a function of time, and the velocity is found to be nearly proportional to the velocity of the fluid on the axis of the steady jet behind the cap, as given by the similarity law of Schlichting and Bickley. At large values of the source strength, a two-dimensional ‘broad jet’ of homogeneous solution descending under gravity is produced; the shape of the flow region can be calculated with little error from potential theory, neglecting the effect of the mixing layers.A possible example of a mixing layer observed in a geothermal region is examined. The theoretical form of the temperature distribution is calculated numerically, taking into account the large viscosity variation with temperature and also the possibility of a large permeability variation. These effects are found to have less influence upon the solution than might have been expected. Quantitative values obtained for the physical parameters are consistent with other geophysical observations.

Self-similar flows of multi-phase immiscible fluids

2000 ◽  
Vol 11 (6) ◽  
pp. 529-559
Author(s):  
A. OZTEKIN ◽  
B. R. SEYMOUR ◽  
E. VARLEY
Keyword(s):  
Viscous Fluid ◽  
Unsteady Flows ◽  
Immiscible Fluids ◽  
Newtonian Fluids ◽  
Two Dimensional ◽  
Constant Strength ◽  
Self Similar ◽  
Multi Phase ◽  
Similar Flows ◽  
Hele Shaw Cell

Exact analytical representations are obtained describing self-similar unsteady flows of multi-phase immiscible fluids in the vicinity of non-circular, but constant strength, fronts. It is assumed that Darcy's law holds for each phase and that the mobilities are known functions of the saturations. Equivalent representations are obtained for Hele-Shaw cell flows that are produced when a viscous fluid is injected into a region containing some other viscous fluid. The fluids may be Newtonian fluids or non-Newtonian fluids for which the coefficients of viscosity depend on the shear stress. Even though the flows are unsteady and two dimensional, the representations are obtained by using hodograph techniques.

The effect of a fixed vertical barrier on surface waves in deep water

1947 ◽  
Vol 43 (3) ◽  
pp. 374-382 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Integral Equation ◽  
Surface Waves ◽  
Deep Water ◽  
Vertical Plane ◽  
Standing Waves ◽  
Simple Type ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Vertical Barrier ◽  
The Mean

In this paper the two-dimensional reflection of surface waves from a vertical barrier in deep water is studied theoretically.It can be shown that when the normal velocity is prescribed at each point of an infinite vertical plane extending from the surface, the motion on each side of the plane is completely determined, apart from a motion consisting of simple standing waves. In the cases considered here the normal velocity is prescribed on a part of the vertical plane and is taken to be unknown elsewhere. From the condition of continuity of the motion above and below the barrier an integral equation for the normal velocity can be derived, which is of a simple type, in the case of deep water. We begin by considering in detail the reflection from a fixed vertical barrier extending from depth a to some point above the mean surface.

scholarly journals Two-phase MHD Flow through Porous Medium with Heat Transfer in a Horizontal Channel

2020 ◽  
pp. 79-90
Author(s):  
Devendra Kumar ◽  
B. Satyanarayana ◽  
Rajesh Kumar ◽  
Bholey Singh ◽  
R. K. Shrivastava
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Temperature Distribution ◽  
Differential Equations ◽  
Velocity Profile ◽  
Parallel Plate ◽  
Immiscible Fluids ◽  
Immiscible Fluid ◽  
Flow Through ◽  
Plate Channel

The present study deals with two layered MHD immiscible fluid flow through porous medium in presence of heat transfer through parallel plate channel. The fluids are incompressible, and flow is fully developed. The fluids are of different viscosities and thermal conductivities so flowing without mixing each other. Two different phases are accounted for study and are electrically conducting. Temperature of the walls of parallel plate channel is constant. Rheological properties of the immiscible fluids are constant in nature. The flow is governed by coupled partial differential equations which are converted to ordinary differential equations and exact solutions are obtained. The velocity profile and temperature distribution are evaluated and solved numerically for different heights and viscosity ratios for the two immiscible fluids. The effect of magnetic field parameter M and porosity parameter K is discussed for velocity profile and temperature distribution. Combined effects of porous medium and magnetic fields are accelerating the flow which, can be helpful in draining oil from oil wells.

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