Pore-throat size correlation from capillary pressure curves

Void spaces in porous media can be considered as three-dimensional networks consisting of bulges (pores) connected by constrictions (throats). Computer simulations of drainage-imbibition processes show that the critical end points of wetting-phase and nonwetting-phase saturation, in drainage and imbibition respectively, and the form of simulated relative permeability curves all were significantly different for uncorrelated and correlated pore-throat models. Since these models were identical except for the arrangement of throats in relation to pores, the degree of pore-throat size correlation appears to be an important property influencing flow and fluid displacement. Examples of uncorrelated and correlated pore-throat structures in rocks are presented and it is shown that this property, although difficult to quantify by direct observation, can be evaluated from capillary pressure curves.     

Stability of convective viscous fluid flow in porous and free vertical layers

The linear stability of convection in a system consisting of a vertical layer of fluid and an adjacent layer of porous medium saturated with that fluid is investigated. The fluid and the porous medium are bounded by isothermal surfaces heated to different temperatures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–9, September–October, 1990.The author wishes to thank G. Z. Gershuni for supervising the work and D. V. Lyubimov for useful discussions.     

High temperature and pressure reactive flows through porous media

Large heat load are encountered in hypersonic and space flight applications due to the high vehicle speed (over Mach 5, i.e. 5000 km h⁻¹) and to the combustion heat release. If passive and ablative protections are a way to ensure the thermal management, the active cooling is probably the most efficient way to enable the structures withstanding of such large heat load. In some conditions, transpiration cooling will be used. In this paper, the permeation of fuels and other fluids through porous media is studied up to 1150 K and 60 bars. A dedicated experimental bench has been established to ensure the monitoring of temperature, pressure, mass flow rate and chemical composition (Gas Chromatograph, Mass Spectrometer, Infra Red spectrometer) in stationary and transient conditions. The tests on metallic and composite samples have been conducted with N₂, CH₄, H₂ + CH₄ mixtures and synthetic fuels (n-C₁₂H₂₆). The pressure losses comparison with the mass flow rate has enabled the determination depending on the temperature of the Darcian permeability, K_D the linear contribution, and of the Forchheimer’s term, K_F the quadratic one. The fuel pyrolysis in such low Reynolds flow has been investigated. The blockage effect due to coking activity has been estimated.     

The effect of transient viscoelastic properties on interfacial instabilities in superposed pressure driven channel flows

In a recent study, Ganpule and Khomami (submitted to J. Non-Newtonian Fluid Mech.) have shown that in order to accurately describe the experimentally observed interfacial instability phenomenon in superposed channel flow of viscoelastic fluids, a constitutive equation that can accurately depict not only the steady viscometric properties of the experimental test fluids, but also their transient viscoelastic properties must be used in the analysis. In the present study, the effect of differences in transient viscoelastic properties which can arise either due to the differences in the predictive capabilities of various constitutive models or from the presence of multiple modes of relaxation on the interfacial instabilities of the superposed pressure driven channel flows has been investigated. Specifically, a linear stability analysis is performed using nonlinear constitutive equations which predict identical steady viscometric properties but different transient viscoelastic properties. It is shown that different nonlinear constitutive equations give rise to the same mechanism of interfacial instability, but the boundaries of the neutral stability contours and the magnitudes of the growth/decay rates, especially at intermediate and shortwaves, are shifted due to the overshoots in the transient viscoelastic responses predicted by the constitutive equations. In addition, the effect of the presence of multiple modes of relaxation on interfacial stability is studied using single and multiple mode upper convected Maxwell (UCM) fluids and it is shown that pronounced differences in the intermediate and shortwave linear stability predictions arise due to the fact that the increase in the number of modes gives rise to additional fast as well as slow relaxation modes and the presence of these additional relaxation modes gives rise to differences in the transient viscoelastic response of the fluids in the absence of any overshoots. The effect of fluid inertia on the interfacial stability of viscoelastic liquids is examined and it is shown that at longwaves, inertia has a pronounced effect on the stability of the interface, whereas at shortwaves, elastic and viscous effects dominate. Furthermore, the mechanism of viscoelastic interfacial instabilities is studied by a careful examination of disturbance eigenfunctions as well as performing a disturbance energy analysis. The results indicate that the mechanism of viscoelastic interfacial instabilities can be described in terms of interaction of mechanisms of purely viscous and purely elastic instabilities. However, since more than one mechanism for the instability is at work, the disturbance energy analysis can not clearly distinguish between them due to the fact that the eigenfunctions used in the energy analysis contain the information regarding both viscous and elastic effects. Hence, the mechanism of the instability must be determined by a careful examination of disturbance eigenfunctions.     

On the stability of vertical constant throughflows for binary mixtures in porous layers

A system modeling fluid motions in horizontal porous layers, uniformly heated from below and salted from above by one salt, is analyzed. The definitely boundedness of solutions (existence of absorbing sets) is proved. Necessary and sufficient conditions ensuring the linear stability of a vertical constant throughflow have been obtained via a new approach. Moreover, conditions guaranteeing the global non-linear asymptotic stability are determined.     

Enhanced conduction in steady,vertical flows of water,steam and carbon dioxide in a porous medium

Transport in Porous Media - The steady, vertical transport of water, energy and carbon dioxide in a two-phase porous medium is analysed when capillarity can be ignored. Such flows are possible only...     

The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium: double diffusive case

The effects of both horizontal and vertical hydrodynamic, thermal and solutal heterogeneity, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are studied analytically using linear stability theory for the case of weak heterogeneity. The Brinkman model is employed. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent.     

Experimental investigation of capillary pressure effects on immiscible displacement in lensed and layered porous media

This paper reports the results of extensive experimental studies of the effects of well-defined heterogeneous porous media on immiscible flooding. The heterogeneities were layers and lenses, with some of the lenses being a wettability contrast. Drainage and imbibition displacements, with and without an initial residual fluid saturation, were carried out at a variety of flow rates on layered and lensed two-dimensional glass beads models of the size of a typical large core test (58×10×0.6 cm). These displacements were followed photographically and the effluent saturation profiles recorded. In most of the experiments the glass beads were water-wet, but in some the lens beads were coated with a water repellent chemical. In all experiments, the displacement fronts became highly irregular due to the different capillary pressures acting in the different areas of the models. In this paper, these displacements are fully reported and their implications for reservoir simulation and for interpretation of laboratory core tests, where the inner heterogeneities are not known, are discussed.     

Characterization of the crossover from capillary invasion to viscous fingering to fracturing during drainage in a vertical 2D porous medium

We experimentally studied the displacement of a viscous wetting fluid (water) by an inviscid non-wetting fluid (air) injected at the bottom of a vertical Hele-Shaw cell filled with glass microbeads. In order to cover a wide parameter space, the permeability of the porous medium was varied by using different bead size ranges and diverse air flow rates were generated by means of a syringe pump. A LED light table was used to back illuminate the experimental cell, allowing a high speed camera to capture images of the drainage process at equal time intervals. The invasion occurred in intermittent bursts. Image processing of the bursts and fractal analysis showed successive transitions from capillary invasion to viscous fingering to fracturing during the same experiment, dependent on the medium permeability, the air injection flow rate, and the vertical position in the cell. The interplay between the capillary, viscous and gravity forces determines the nature of the invasion pattern and the transitions, from capillary invasion to viscous fingering with decreasing fluid pressure on one hand and from viscous fingering to fracturing with decreasing effective overburden pressure on the other hand.     

The effect of leaching on mixing in porous media with a single heterogeneity

This paper presents an exploratory study of the effect of leaching on mixing in a porous medium containing a single heterogeneity to investigate the effect of the heterogeneity and time-dependent pore structure on dispersion. A percolation-convection simulation (PCS) model is used along with laboratory model experiments to study the mixing. The results show that mixing changes when the pores of the models are leached and that there is a change in regime influence during leaching. The simulation represents the mixing through a first leach for homogeneous media and for heterogeneous media with significant changes in permeability. If the pore structure is changing with time, prediction of mixing must include effects of heterogeneity and regime influence. Although the experimental results are representative of idealized laboratory sized systems they provide insight into the effects of leaching in heterogeneous media. Further the simulation may be useful on a field scale.Nomenclature b molecular weight, gm/mol - C concentration, mol/cm³ - C ₀ initial concentration, mol/cm³ - d _rms root-mean-squared distance, cm - d ₅₀ 50% grain size, cm - D channel depth, cm - f _n fraction of input tracer in effluent at time t _n - K ₁ permeability of flow field outside of heterogeneity, cm² - k ₂ permeability of heterogeneity, cm² - k _S reaction rate constant, cm/min - K _L microscopic dispersion coefficient in the longitudinal direction, cm²/sec - K _O overall dispersion coefficient in the longitudinal direction, cm²/sec - K _T microscopic dispersion coefficient in the transverse direction, cm²/sec - L length of channel, cm - n exponent for velocity - P pressure, N/M² - Pe Peclet number, Lv/K _O - P _ext local pressure outside heterogeneity, N/M² - P _int local pressure inside heterogeneity, N/M² - Q volumetric flow rate, cm³/sec - R channel half width, cm - t time, sec - W _c channel width, cm - W _c0 initial channel width, cm - v interstitial fluid velocity, cm/sec - v _k macroscopic velocity in transverse direction, cm/sec - v _y macroscopic velocity in longitudinal direction, cm/sec - v fluid velocity entering the medium, cm/sec - x _i transverse location of parcel at time t _i, cm - y _i longitudinal location of parcel at time t _i, cm - x microscopic movement in transverse direction, cm - y microscopic movement in longitudinal direction, cm Greek Letters t time increment, sec - ₀ overall dispersivity, cm - ² longitudinal variance of the distribution, cm² - porosity - _B bulk density, gm/cm³ - fractional grade of leachable material Currently with Center for Naval Analysis.     

On the effective stress in unsaturated porous continua with double porosity

Using mixture theory we formulate the balance laws for unsaturated porous media composed of a double-porosity solid matrix infiltrated by liquid and gas. In this context, the term ‘double porosity’ pertains to the microstructural characteristic that allows the pore spaces in a continuum to be classified into two pore subspaces. We use the first law of thermodynamics to identify energy-conjugate variables and derive an expression for the ‘effective’, or constitutive, stress that is energy-conjugate to the rate of deformation of the solid matrix. The effective stress has the form , where σ is the total Cauchy stress tensor, B is the Biot coefficient, and is the mean fluid pressure weighted according to the local degrees of saturation and pore fractions. We identify other emerging energy-conjugate pairs relevant for constitutive modeling of double-porosity unsaturated continua, including the local suction versus degree of saturation pair and the pore volume fraction versus weighted pore pressure difference pair. Finally, we use the second law of thermodynamics to determine conditions for maximum plastic dissipation in the regime of inelastic deformation for the unsaturated two-porosity mixture.     

Heat and mass transfer in unsaturated porous media with solid–liquid change

 A model on heat and mass transfer in unsaturated porous media with solid/liquid phase change was developed with extending the three-variable model previously proposed. The movement of air phase and its effect on the motion of water is considered. The model has been checked with comparison of the experimental results of the temperature distribution for two dimensional freezing process. The evolution of air pressure, water and ice saturation were predicted by solving the governing equations. The ice segregation and moisture movement toward the front of freezing were numerically simulated. Received on 8 December 1999     

Electro-magnetic-field-induced flow and interfacial instabilities in confined stratified liquid layers

The electro-magneto-hydrodynamic (EMHD) flow and instabilities engendered by the Lorenz force arising from interaction between externally applied perpendicular electric and magnetic fields are investigated in layers of two immiscible liquids in a channel. A new finite wave-number EMHD instability mode is uncovered by the Orr–Sommerfeld analysis, in addition to the interfacial and shear modes which also arise in the pressure-driven flows. Thus, EMHD can be controlled for micro-channel transport, heat and mass transfer, mixing, micro-emulsion generation, etc.     

Electrode layers in flows of an inviscid plasma with good conductivity

The structure of the electromagnetic electrode layers that are produced in flows across a magnetic field by a completely ionized and inviscid plasma with good conductivity and a high magnetic Reynolds number is examined in a linear approximation. Flow past a corrugated wall and flow in a plane channel of slowly varying cross section with segmented electrodes are taken as specific examples. The possibility is demonstrated of the formation of nondissipative electrode layers with thicknesses on the order of the Debye distance or electron Larmor radius and of dissipative layers with thicknesses on the order of the skin thickness, as calculated from the diffusion rate in a magnetic field [2].In plasma flow in a transverse magnetic field, near the walls, along with the gasdynamie boundary layers, which owe their formation to viscosity, thermal conductivity, etc. (because of the presence of electromagnetic fields, their structures may vary considerably from that of ordinary gasdynamic layers), proper electromagnetic boundary layers may also be produced. An example of such layers is the Debye layer in which the quasi-neutrality of the plasma is upset. No less important, in a number of cases, is the quasi-neutral electromagnetic boundary layer, in which there is an abrupt change in the frozen-in parameter k=B/p (B is the magnetic field and p is the density of the medium). This layer plays a special role when we must explicitly allow for the Hall effect and the related formation of a longitudinal electric field (in the direction of the veloeiryv of the medium). We will call this the magnetic layer. The magnetic boundary layer can be dissipative as well as noudissipative (see below). The dissipative magnetic layer has been examined in a number of papers: for an incompressible medium with a given motion law in [1], for a compressible medium with good conductivity in [2], and with poor conductivity in [3]. In the present paper, particular attention will be devoted to nondissipative magnetic boundary layers.     

Viscous boundary layers in flows through a domain with permeable boundary

The effect of small viscosity on nearly inviscid flows of an incompressible fluid through a given domain with permeable boundary is studied. The Vishik–Lyusternik method is applied to construct a boundary layer asymptotic at the outlet in the limit of vanishing viscosity. Mathematical problems with both consistent and inconsistent initial and boundary conditions at the outlet are considered. It is shown that in the former case, the viscosity leads to a boundary layer only at the outlet. In the latter case, in the leading term of the expansion there is a boundary layer at the outlet and there is no boundary layer at the inlet, but in higher order terms another boundary layer appears at the inlet. To verify the validity of the expansion, a number of simple examples are presented. The examples demonstrate that asymptotic solutions are in quite good agreement with exact or numerical solutions.     

The scattering of P-waves by a piezoelectric particle with FGPM interfacial layers in a polymer matrix

Propagation of P-wave in an unbounded elastic polymer medium which contains a set of nested concentric spherical piezoelectric inhomogeneities is formulated. The polymer matrix is made of Epoxy and is isotropic; each phase of the inhomogeneity is made of a different piezoelectric material and is radially polarized and has spherical isotropy. Note that the individual phases are homogeneous, and all interfaces are perfectly bonded. The scattered displacement and electric potentials in the matrix are expressed in terms of spherical wave vector functions and Legendre functions, respectively. The transmitted displacement and electric potentials within each phase of the piezoelectric particle are expressed in terms of Legendre functions. The equations of motion and electrostatics in each phase of the piezoelectric inhomogeneity lead to a system of coupled second order differential equations, which is solved using the generalized Frobenius series. The present theory is extended to the case where the core of the inhomogeneity is made of PZT-4 and its coating is made of functionally graded piezoelectric material (FGPM) whose microstructural composition varies smoothly from PZT-4 at the core–coating interface to Epoxy at the coating–matrix interface. The effects of different types of variation in the electro-mechanical properties of FGPM on scattering cross-section and other electro-mechanical fields are addressed. The present theory is valid for arbitrary coating thickness, and arbitrary frequencies.     

Liquid and gas flows in porous media with allowance for induced anisotropy

The changes in the permeability of a porous medium resulting from the reorientation of the solid matrix particles under the influence of the percolating flow are considered. The mathematical model also contains the angular momentum equation, including the moment of the viscous flow forces. The state of the elastic matrix is characterized not only by the repacking strains but also by the particle orientation vector. The latter determines the anisotropy of the permeability tensor. The effective stress, strain, pore pressure and orientation vector fields in the neighborhood of an operating well are constructed. The effect of the induced permeability anisotropy on well productivity is noted.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 96–103, May–June, 1992.The authors are grateful to G. A. Zotov and V. A. Chernykh for their interest and support.     

Mathematical modeling of flows in wind turbines with a vertical axis

Flow in the rotors of wind turbines with a vertical axis of the Darrieus-Savonius type and their combinations is considered within the framework of the Navier-Stokes equations, in the two-dimensional formulation. A parametric investigation of the Darrieus rotor characteristics is carried out. For all configurations the dependence of the moment and power coefficients on the specific speed of the rotor is calculated. The possibility of the Darrieus rotor self- starting from the state of rest is investigated. The effect of nonmonotonicmoment variation for the Darrieus-type rotor is studied for the specific speeds of 0.5 to 1. A combined design of the Darrieus-Savonius rotor is considered to solve the problems due to a small moment of the Darrieus rotor during the start.     

On pressure and thermal flux in gas-mixture flows and in two-phase flows

To begin with, two different definitions of pressure, thermal flux, etc. in the diffusion model and two-fluid model are given. Then the physical interpretations of the pressure and the thermal flux are provided by introducing the momentum and energy fluxes,M and ε, through a surface dS in the flow field. The quantities defined in the diffusion model are suggested when the motion of the mixture is studied as a whole, while the quantities defined in the two-fluid model are suggested when the motion of individual species is studied. The collision pressure and thermal flux in dense gas-mixtures are also discussed in detail, i.e. their origin, their expressions in the momentum and energy equations, and their distinctions from the normal partial pressure and thermal flux. A gas-particle flow can be treated as a flow of dense gas-mixtures. The long-standing controversy whether the “inertial coupling term” should exist in the momentum equation can be clarified by the two different definitions of pressure.