Wave Equations of Porous Media Saturated With Two Immiscible Fluids Based on the Volume Averaging Method

Simulating and predicting wave propagation in porous media saturated with two fluids is an important issue in geophysical exploration studies. In this work, wave propagation in porous media with specified structures saturated with two immiscible fluids was studied, and the main objective was to establish a wave equation system with a relatively simple structure. The wave equations derived by Tuncay and Corapcioglu were analyzed first. It was found that the coefficient matrix of the equations tends to be singular due to the inclusion of a small parameter that characterizes the effect of capillary stiffening. Therefore, the previously established model consisting of three governing equations may be unstable under natural conditions. An improved model based on Tuncay and Corapcioglu’s work was proposed to ensure the nonsingularity of the coefficient matrix. By introducing an assumption in which one fluid was completely wrapped by the other, the governing equation of the wrapped fluid was degenerated. In this way, the coefficient matrix of wave equations became nonsingular. The dispersion and attenuation prediction resulting from the new model was compared with that of the original model. Numerical examples show that although the improved model consists of only two governing equations, it can obtain a result similar to that of the original model for the case of a porous medium containing gas and water, which simplifies the complexity of the calculations. However, in a porous medium with oil and water, the predictions of dispersion and attenuation produced by the original model obviously deviate from the normal trend. In contrast, the results of the improved model exhibit the correct trend with a smooth curve. This phenomenon shows the stability of the improved model and it could be used to describe wave propagation dispersions and attenuations of porous media containing two immiscible fluids in practical cases.

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Mathematical Formulation of the Global Coefficients of Transmission and Reflection of Ultrasonic Waves in Bi-Phase Porous Medium

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.1101.471 ◽

2015 ◽

Vol 1101 ◽

pp. 471-479

Author(s):

Georges Freiha ◽

Hiba Othman ◽

Michel Owayjan

Keyword(s):

Porous Medium ◽

Porous Media ◽

Wave Propagation ◽

Mathematical Formulation ◽

Ultrasonic Waves ◽

Ultrasonic Wave Propagation ◽

Local Coefficients ◽

Important Field ◽

New Formulation ◽

Transmission And Reflection

The study of signals propagation inside porous media is an important field especially in the biomedical research related to compact bones. The purpose of this paper is to determine a mathematical formulation of the global coefficients of transmission and reflection of nondestructive ultrasonic waves in any bi-phase porous medium. Local coefficients of transmission and reflection on the interface of the porous medium will be determined based on a study of boundary conditions. The behavior of different waves inside the porous medium will be developed so that we can derive a new formulation of global coefficients that takes interior phenomena into consideration. Results are found independently of the geometrical and physical characteristics of the medium. Note that this study is based on normal incident ultrasonic wave propagation.

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Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids

Transport in Porous Media ◽

10.1007/s11242-006-9059-2 ◽

2007 ◽

Vol 68 (1) ◽

pp. 91-105 ◽

Cited By ~ 20

Author(s):

Wei-Cheng Lo ◽

Garrison Sposito ◽

Ernest Majer

Keyword(s):

Porous Media ◽

Wave Propagation ◽

Low Frequency ◽

Immiscible Fluids ◽

Unsaturated Porous Media ◽

Dilatational Wave

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The Effect of Vertical Throughflow on Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: A Revised Model

Journal of Heat Transfer ◽

10.1115/1.4029773 ◽

2015 ◽

Vol 137 (5) ◽

Cited By ~ 7

Author(s):

D. A. Nield ◽

A. V. Kuznetsov

Keyword(s):

Boundary Condition ◽

Porous Medium ◽

Porous Media ◽

Thermal Instability ◽

Volume Fraction ◽

Critical Rayleigh Number ◽

Nanoparticle Volume Fraction ◽

Governing Equations ◽

Medium Layer ◽

Vertical Throughflow

The model developed in our previous paper (Nield and Kuznetsov, 2011, “The Effect of Vertical Throughflow on Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid,” Transp. Porous Media, 87(3), pp. 765–775) is now revised to accommodate a more realistic boundary condition on the nanoparticle volume fraction. The new boundary condition postulates zero nanoparticle flux through the boundaries. We established that in the new model, oscillatory instability is impossible. We also established that the critical Rayleigh number depends on three dimensionless parameters, and we derived these three parameters from the governing equations. We also briefly investigated the major trends.

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Features of wave propagation in the sand of different saturation

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2011.1.002 ◽

2011 ◽

Vol 8 (1) ◽

pp. 25-38

Author(s):

A.T. Akhmetov ◽

S.V. Lukin ◽

D.M. Balapanov ◽

S.F. Urmancheev ◽

N.M. Gumerov ◽

...

Keyword(s):

Mathematical Model ◽

Porous Medium ◽

Porous Media ◽

Wave Propagation ◽

Water Saturation ◽

Weak Shock ◽

Capillary Forces ◽

Theoretical Studies ◽

Longitudinal Waves ◽

Experimental And Theoretical Studies

There are the results of experimental and theoretical studies on the propagation of weak shock waves in the wet sand at different water saturation. There are mathematical model and numerical analysis of propagation of pressure pulses in porous media, taking into account capillary forces. Non-monotonic dependence of the amplitude of the wave resulting in a wet porous medium vs. the degree of water saturation is installed. The evolution of the fast, slow and filtration waves depends on the saturation of the system with water is analyzed. The influence of capillary forces on the propagation of longitudinal waves is evaluated.

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Statistics of Mathematical Two-Scale Closure of Momentum, Heat and Charge Transport Problems With Stochastic Orientation of Porous Medium Capillaries

10.1115/imece2001/htd-24157 ◽

2001 ◽

Author(s):

V. S. Travkin ◽

K. Hu ◽

I. Catton

Keyword(s):

Heat Transfer ◽

Porous Medium ◽

Porous Media ◽

Volume Averaging ◽

Governing Equations ◽

Rigorous Approach ◽

Flow And Heat Transfer ◽

Model Flow ◽

History Of ◽

Transport Problems

Abstract The history of stochastic capillary porous media transport problem treatments almost corresponds to the history of porous media transport developments. Volume Averaging Theory (VAT), shown to be an effective and rigorous approach for study of transport (laminar and turbulent) phenomena, is used to model flow and heat transfer in capillary porous media. VAT based modeling of pore level transport in stochastic capillaries results in two sets of scale governing equations. This work shows how the two scale equations could be solved and how the results could be presented using statistical analysis. We demonstrate that stochastic orientation and diameter of the pores are incorporated in the upper scale simulation procedures. We are treating this problem with conditions of Bi for each pore is in a range when Bi ≳ 0.1 which allows even greater distinction in assessing an each additional differential, integral, or integral-differential term in the VAT equations.

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On the Viscous Models for Wave Propagation in Solid Loaded with Viscous Liquid

Journal of Mechanics ◽

10.1017/s1727719100000381 ◽

1999 ◽

Vol 15 (3) ◽

pp. 103-108

Author(s):

M.-P. Chang ◽

T.T. Wu

Keyword(s):

Wave Propagation ◽

Viscous Liquid ◽

Wave Equations ◽

Layered Medium ◽

Viscous Liquids ◽

Surface Wave Propagation ◽

Stress Components ◽

Viscous Models ◽

Dispersion And Attenuation ◽

Elastic Coefficients

AbstractRecently, in the fields of biosensing and nondestructive of materials, there are increasing interests on the investigations of the surface wave propagation in fluid loaded layered medium. Several different models for the elastic coefficients of viscous liquids are usually adopted in the investigations. The purpose of this paper is to study the variations of choosing different viscous liquid models on the dispersion and attenuation of waves in liquid loaded solids. In the paper, a derivation of the elastic coefficients of a viscous liquid based on the Stokes' assumption is given first. Then, for the hypothetical solid assumption of a viscous liquid, the associated wave equations and expressions of the stress components for different viscous liquid models utilized in the literatures are given. Finally, dispersion and attenuation of waves in a viscous liquid loaded A1 half space and a SiC plate immersed in a viscous liquid are calculated and utilized to discuss the differences among these four different models.

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Modelling of Permeation Grouting Through Soils

Journal of Applied Engineering Sciences ◽

10.2478/jaes-2020-0003 ◽

2020 ◽

Vol 10 (1) ◽

pp. 11-16

Author(s):

S. B. Coskun ◽

T. Tokdemir

Keyword(s):

Mathematical Modeling ◽

Numerical Simulation ◽

Finite Element ◽

Porous Medium ◽

Capillary Pressure ◽

Immiscible Fluids ◽

Saturated Soils ◽

Governing Equations ◽

Permeation Grouting ◽

Element Technique

AbstractIn this study, mathematical modeling of permeation grouting through fully saturated soil is proposed based on immiscible multiphase flow theory. Grout flow in the medium is modeled together with the existing water as the simultaneous flow of two immiscible fluids. In the model, the porous medium is assumed as isotropic and rigid, fluids are assumed as incompressible and capillary pressure is assumed as negligible. Governing equations are discretized using upstream weighted finite element technique and results show that, proposed models give good results and may be used in the numerical simulation of grouting through fully saturated soils.

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