Evolution of the balance equations in saturated thermoelastic porous media following abrupt simultaneous changes in pressure and temperature

Levy, Avi; Sorek, S.; Ben‐Dor, G.; Bear, Jacob

doi:10.1007/bf00617408

Cited by 49 publications

(40 citation statements)

References 8 publications

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“…At the REV (larger) spatial scale, on letting the α phase represent a Newtonian fluid (i.e., the fluid shear stress tensor τ is related to the rate of its strain tensor), then the macroscopic Navier-Stokes equation resulting from (11) can govern the propagation of shock waves through porous media (Levy et al 1995), conform to Forchheimer's law accounting for the transfer of fluid inertia through the microscopic solid-fluid interface or to Darcy's law when friction at that interface is dominant (Sorek et al 2005). At an adjacent spatial scale smaller than the REV, (12) describes an inertia momentum flux in the form of a wave equation that governs the concurrent propagation of the d M quantity.…”

Section: Momentum Balance Equation Of the α Phasementioning

confidence: 99%

“…These models, however, excluded the possibility of the exchange of microscopic inertia through the solid-fluid interface. Following Sorek et al (1992) and [4] Levy et al (1995), additional Forchheimer terms were introduced Levy et al (1999) and extended Sorek et al (2005) resulting in a variety of nonlinear wave equation forms. We prove Sorek et al (2005) that a phase balance equation is composed of a dominant form that refers to the REV (larger) scale and a secondary form valid at a scale smaller than that of the REV.…”

Section: Introductionmentioning

confidence: 99%

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Scale-dependent Macroscopic Balance Equations Governing Transport Through Porous Media: Theory and Observations

2009

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A theory is developed providing a rational framework for spatial scaledependent fluid's flow and heat transfer, and mass of a component migrating with it through porous media. Introducing the assumption of a non-Brownian type motion and referring to asymptotic expansion in powers of a small defined parameter, we develop a novel approach associated with macroscopic balance equations obtained by averaging over a Representative Elementary Volume (REV). We prove that these equations can be decomposed into a primary part that refers to the REV length scale and a secondary part valid at a length scale smaller than that of the corresponding REV length. Further to our previous development, we obtain two general forms of the primary and secondary macroscopic balance equations. One is based on the assumption that the advective flux of the extensive quantity is dominant over that of the dispersive flux, whereas the other disregards this assumption. Moreover, we also introduce the primary and secondary macroscopic forms for the fluid heattransfer equation. Considering a Newtonian fluid, the resulting primary Navier-Stokes equation can vary from a nonlinear wave equation to a drag-dominant equation at the fluid-solid interface (Darcy's law). The secondary momentum balance equation describes a wave equation governing the concurrent propagation of the intensive momentum and the dispersive momentum flux, deviating from their corresponding average terms. The primary macroscopic fluid heat-transfer equation accounts for advective and dispersive heat fluxes and the secondary macroscopic heat-transfer equation involves the simultaneous advection of heat 123 62 S. Sorek et al.deviating from its corresponding intensive average quantity. The primary macroscopic solute mass balance equation accounts for advection and hydrodynamic dispersion. The secondary macroscopic component mass balance equation is in the form of pure advection governing migration of the deviation from the average component concentration. At this stage, we focus on establishing the viability of the developed theory. We do this by arguing that field observations of motion at small spatial scales are coherent with the hyperbolic characteristics of the secondary balance equations. Field observations under natural gradient flow conditions show excessive high concentration (average of 50 mg/L) of colloids under land irrigated by sewage effluents. We argue that this displacement of condensed colloidal parcels manifests the theoretical findings for the smaller spatial scale. Further evidence show the accumulation of particles moving behind the front of an emitted shockwave. We consider this as an experimental proof reinforcing the argument that colloidal migration is subject to the action of a shockwave in the fluid and pure advection transport, governed by the respective suggested hyperbolic macroscopic balance equations of fluid momentum and component mass at the smaller spatial scale.Keywords Representative elementary volume · Deviation from spatial average · General mac...

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Section: Momentum Balance Equation Of the α Phasementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Scale-dependent Macroscopic Balance Equations Governing Transport Through Porous Media: Theory and Observations

2009

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“…The constitutive relations for effective stress tensors in the solid and fluid of an isotropic thermally conducting fluid-saturated porous medium are derived [2,20] as…”

Section: Basic Equationsmentioning

confidence: 99%

“…Green and Lindsay [11] proposed the second couple theory of thermoelasticity by introducing two parameters of relaxation time. Levy et al [20] developed a mathematical model for saturated flow of a Newtonian fluid in a thermoelastic, homogeneous, isotropic porous medium under non-isothermal conditions. Lord and Shulman [21] studied the theory of generalised thermoelasticity using a modified Fourier's law of heat conduction with a thermal relaxation time.…”

Section: Introductionmentioning

confidence: 99%

Elastic waves in thermoelastic saturated porous medium

Zorammuana

Singh

2015

Meccanica

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The problem of reflection of elastic waves at a plane free boundary of thermoelastic saturated porous half space has been investigated. There exist four types of plane waves in thermoelastic saturated porous medium which are three attenuating coupled longitudinal waves and one non-attenuating transverse wave. The amplitude and energy ratios corresponding to the reflected waves due to incident longitudinal as well as transverse waves are derived and computed numerically. Critical angles are observed for the incident transverse wave.

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“…This idea was advanced further by Levy et al [6], by performing a dimensional analysis on the resulting macroscopic balance equation developed by Sorek et al [4] and Bear et al [7]. This established a theoretical basis on which to model non-linear wave motion in a deformable porous media represented by a continuum of interacting solid and uid phases.…”

Section: Introductionmentioning

confidence: 98%

Weighted average flux method and flux limiters for the numerical simulation of shock waves in rigid porous media

Torrens¹,

Wrobel²

2002

Numerical Methods in Fluids

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SUMMARYThe one-dimensional ow ÿeld generated by the passage of a shock wave in a rigid, thermoelastic porous foam has been simulated using a two-phase mathematical model. The work presented here makes use of the weighted average ux method to solve the system of six equations that govern the problem. Spurious oscillations are eliminated through the application of total variation diminishing limiting methods. Four di erent limiters were tested: van Leer, SuperA, MinA and van Albada. Numerical tests were carried out to verify the performance of each ux limiter in terms of accuracy. The results were compared to analytical and previously obtained data to assess the performance of the mathematical model. Excellent agreement was obtained.

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Evolution of the balance equations in saturated thermoelastic porous media following abrupt simultaneous changes in pressure and temperature

Cited by 49 publications

References 8 publications

Scale-dependent Macroscopic Balance Equations Governing Transport Through Porous Media: Theory and Observations

Scale-dependent Macroscopic Balance Equations Governing Transport Through Porous Media: Theory and Observations

Elastic waves in thermoelastic saturated porous medium

Weighted average flux method and flux limiters for the numerical simulation of shock waves in rigid porous media

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