Lattice Boltzmann Pore Scale Simulation of Natural Convection in a Differentially Heated Enclosure Filled with a Detached or Attached Bidisperse Porous Medium

In this research, pore scale simulation of natural convection in a differentially heated enclosure filled with a conducting bidisperse porous medium is investigated using the thermal lattice Boltzmann method. For the first time, the effect of connection of the bidisperse porous medium to the enclosure walls is studied by considering the attached geometry in addition to the detached one. Effect of most relevant parameters on the streamlines and isotherms as well as hot wall average Nusselt number is studied for two of the bidisperse porous medium configurations. It is observed that effect of geometrical and thermo-physical parameters of the bidisperse porous medium on the heat transfer characteristics is more complicated for the attached configuration. To assess the validity of the local thermal equilibrium condition in the micro-porous media, the pore scale results are used to compute the percentage of the local thermal non-equilibrium for two of the bidisperse porous medium configurations. It is concluded that for the detached configuration, the local thermal equilibrium condition is confirmed in the entire micro-porous media for the ranges of the parameters studied here. However, for the attached geometry, it is shown that departure from the local thermal equilibrium condition is observed for the higher values of the Rayleigh number, micro-porous porosity, solid–fluid thermal conductivity ratio, and the smaller values of the macro-pores volume fraction.     

Instability of a liquid–vapor phase transition front in inhomogeneous wettable porous media

The stability of vertical flows through a horizontally extended two-dimensional region of a porous medium is considered in the case of presence of a phase transition front. It is shown that the plane steady-state phase transition front may have several steady-state positions in the wettable porous medium and the necessary condition of their existence is obtained. The spectral stability of the plane phase transition interface is investigated. It is found that in the presence of capillary forces exerted on the phase transition front in the wettable medium the plane front can be destabilized on the mode with both infinite and zero wavenumbers (short- and long-wave instabilities); the short-wave instability can then exist even in the case of the sole steady-state position of the front.     

The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach

After recalling the constitutive equations of finite strain poroelasticity formulated at the macroscopic level, we adopt a microscopic point of view which consists of describing the fluid-saturated porous medium at a space scale on which the fluid and solid phases are geometrically distinct. The constitutive equations of poroelasticity are recovered from the analysis conducted on a representative elementary volume of porous material open to fluid mass exchange. The procedure relies upon the solution of a boundary value problem defined on the solid domain of the representative volume undergoing large elastic strains. The macroscopic potential, computed as the integral of the free energy density over the solid domain, is shown to depend on the macroscopic deformation gradient and the porous space volume as relevant variables. The corresponding stress-type variables obtained through the differentiation of this potential turn out to be the macroscopic Boussinesq stress tensor and the pore pressure. Furthermore, such a procedure makes it possible to establish the necessary and sufficient conditions to ensure the validity of an ‘effective stress’ formulation of the constitutive equations of finite strain poroelasticity. Such conditions are notably satisfied in the important case of an incompressible solid matrix.     

Seismic Reflection from an Interface Between an Elastic Solid and a Fractured Porous Medium with Partial Saturation

The theory of Tuncay and Corapcioglu (Transp Porous Media 23:237–258, 1996a) has been employed to investigate the possibility of plane wave propagation in a fractured porous medium containing two immiscible fluids. Solid phase of the porous medium is assumed to be linearly elastic, isotropic and the fractures are assumed to be distributed isotropically throughout the medium. It has been shown that there can exist four compressional waves and one rotational wave. The phase speeds of these waves are found to be affected by the presence of fractures, in general. Of the four compressional waves, one arises due to the presence of fractures in the medium and the remaining three are those encountered by Tuncay and Corapcioglu (J Appl Mech 64:313–319, 1997). Reflection and transmission phenomena at a plane interface between a uniform elastic half-space and a fractured porous half-space containing two immiscible fluids, are analyzed due to incidence of plane longitudinal/transverse wave from uniform elastic half-space. Variation of modulus of amplitude and energy ratios with the angle of incidence are computed numerically by taking the elastic half-space as granite and the fractured porous half-space as sandstone material containing non-viscous wetting and non-wetting fluid phases. The results obtained in case of porous half-space with fractures, are compared graphically with those in case of porous half-space without fractures. It is found that the presence of fractures in the porous half-space do affect the reflection/transmission of waves, which is responsible for raising the reflection and lowering the transmission coefficients.     

Analysis of hydrodynamic and thermal dispersion in porous media by means of a local approach

A pore scale analysis is implemented in this numerical study to investigate the behavior of microscopic inertia and thermal dispersion in a porous medium with a periodic structure. The macroscopic characteristics of the transport phenomena are evaluated with an averaging technique of the controlling variables at a pore scale level in an elementary cell of the porous structure. The Darcy–Forchheimer model describes the fluid motion through the porous medium while the continuity and Navier–Stokes equations are applied within the unit cell. An average energy equation is employed for the thermal part of the porous medium. The macroscopic pressure loss is computed in order to evaluate the dominant microscopic inertial effects. Local fluctuations of velocity and temperature at the pore scale are instrumental in the quantification of the thermal dispersion through the total effective thermal diffusivity. The numerical results demonstrate that microscopic inertia contributes significantly to the magnitude of the macroscopic pressure loss, in some instances with as much as 70%. Depending on the nature of the porous medium, the thermal dispersion may have a marked bearing on the heat transfer, particularly in the streamwise direction for a highly conducting fluid and certain values of the Peclet number.     

Diffusion in isotropic and anisotropic porous systems: Three-dimensional calculations

Effective diffusion coefficients were calculated numerically for three-dimensional unit cells representative of different unconsolidated porous media. These numerical results were compared with the experimental results of Kim for packed beds of glass spheres, mica particles, and an artificial porous medium composed of mylar disks. These three-dimensional numerical results confirm that the porosity is the essential parameter for the determination of the effective diffusion coefficient in the case of unconsolidated isotropic systems. In the case of anisotropic systems, better agreement is obtained between numerical predictions and actual data when the unit cell is three-dimensional rather than twodimensional. This emphasizes the fact that three-dimensional unit cells feature more realistic geometrical properties which are needed to accurately describe anisotropic systems.     

Numerical Calculation of Effective Diffusion in Unsaturated Porous Media by the TRT Lattice Boltzmann Method

Numerical models that solve transport of pollutants at the macroscopic scale in unsaturated porous media need the effective diffusion dependence on saturation as an input. We conducted numerical computations at the pore scale in order to obtain the effective diffusion curve as a function of saturation for an academic sphere packing porous medium and for a real porous medium where pore structure knowledge was obtained through X-ray tomography. The computations were performed using a combination of lattice Boltzmann models based on two relaxation time (TRT) scheme. The first stage of the calculations consisted in recovering the water spatial distribution into the pore structure for several fixed saturations using a phase separation TRT lattice Boltzmann model. Then, we performed diffusion computation of a non-reactive solute in the connected water structure using a diffusion TRT lattice Boltzmann model. Finally, the effective diffusion for each selected saturation value was estimated through inversion of a macroscopic classical analytical solution.     

The Effect of Strong Heterogeneity on the Onset of Convection in a Porous Medium: 2D/3D Localization and Spatially Correlated Random Permeability Fields

The effect of strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. A computer package has been developed to implement an algorithm giving a criterion for instability, and this is now employed to investigate the case where there is two-dimensional variation in a horizontal plane and the case where the variation is generated by a log-normal distribution. In the latter case, spatially correlated fields with known stochastic properties are generated, and the results are analyzed in a statistical framework. We now test cases that are representative of natural, field-scale, geologic conditions—both in terms of the correlated structure and the much larger standard deviation of the permeability distribution.     

On the permeability of fibre-reinforced porous materials

Biological tissues can be considered as composite materials comprised of a porous matrix filled with interstitial fluid and reinforced by impermeable collagen fibres. Motivated by studies on fluid flow in articular cartilage, we would like to quantify the undeformed configuration permeability of fibre-reinforced composite materials. If there is a sufficient scale separation between the internal structure of the porous matrix and the arrangement of the fibres, the matrix can be taken as a porous continuum at the fibre scale. In this case, the fibres can be treated as inclusions in a porous continuum, and the overall permeability of the composite can be evaluated using homogenisation procedures. For an isotropic homogeneous matrix, the symmetry of the system is governed by the orientation of the fibres. Here, we propose to retrieve the overall permeability through geometrical considerations and directional averaging methods. The special case of transverse isotropy is discussed in detail, with particular attention to the sub-cases of aligned fibres and fibres lying on a plane.     

Homogenization Modeling and Parametric Study of Moisture Transfer in an Unsaturated Heterogeneous Porous Medium

The classical mass balance equation is usually used to model the transfer of humidity in unsaturated macroscopically homogeneous porous media. This equation is highly non-linear due to the pressure-dependence of the hydrodynamic characteristics. The formal homogenization method by asymptotic expansions is applied to derive the upscaled form of this equation in case of large-scale heterogeneities of periodic structure. The nature of such heterogeneities may be different, resulting in locally variable hydrodynamic parameters. The effective capillary capacity and the effective hydraulic conductivity are defined as functions of geometry and local characteristics of the porous medium. A study of a two-dimensional stone-mortar system is performed. The effect of the second medium (the mortar), on the global behavior of the system is investigated. Numerical results for the Brooks and Corey hydrodynamic model are provided. The sensitivity analysis of the parameters of the model in relation to the effective hydrodynamic parameters of the porous structure is presented.     

The asymptotic homogenization elasticity tensor properties for composites with material discontinuities

The classical asymptotic homogenization approach for linear elastic composites with discontinuous material properties is considered as a starting point. The sharp length scale separation between the fine periodic structure and the whole material formally leads to anisotropic elastic-type balance equations on the coarse scale, where the arising fourth rank operator is to be computed solving single periodic cell problems on the fine scale. After revisiting the derivation of the problem, which here explicitly points out how the discontinuity in the individual constituents’ elastic coefficients translates into stress jump interface conditions for the cell problems, we prove that the gradient of the cell problem solution is minor symmetric and that its cell average is zero. This property holds for perfect interfaces only (i.e., when the elastic displacement is continuous across the composite’s interface) and can be used to assess the accuracy of the computed numerical solutions. These facts are further exploited, together with the individual constituents’ elastic coefficients and the specific form of the cell problems, to prove a theorem that characterizes the fourth rank operator appearing in the coarse-scale elastic-type balance equations as a composite material effective elasticity tensor. We both recover known facts, such as minor and major symmetries and positive definiteness, and establish new facts concerning the Voigt and Reuss bounds. The latter are shown for the first time without assuming any equivalence between coarse and fine-scale energies (Hill’s condition), which, in contrast to the case of representative volume elements, does not identically hold in the context of asymptotic homogenization. We conclude with instructive three-dimensional numerical simulations of a soft elastic matrix with an embedded cubic stiffer inclusion to show the profile of the physically relevant elastic moduli (Young’s and shear moduli) and Poisson’s ratio at increasing (up to 100 %) inclusion’s volume fraction, thus providing a proxy for the design of artificial elastic composites.     

Analyzing the deformation of a porous medium with account for the collapse of pores

The generalized rheological method is used to construct a mathematical model of small deformations of a porous media with open pores. Changes in the resistance of the material to external mechanical impact at the moment of collapse of the pores is described using the von Mises–Schleicher strength condition. The irreversible deformation is accounted for with the help of the classic versions of the von Mises–Tresca–Saint-Venant yield condition and the condition that simulates the plastic loss of stability of the porous skeleton. Within the framework of the constructed model, this paper describes the analysis of the propagation of plane longitudinal compression waves in a homogeneous medium accompanied with plastic strain of the skeleton and densification of the material. A parallel computational algorithm is developed for the study of the elastoplastic deformation of the porous medium under external dynamics loads. The algorithm and the program are tested by calculating the propagation of plane longitudinal compression shock waves and the extension of the cylindrical cavity in an infinite porous medium. The calculation results are compared with exact solutions, and it is shown that they are in good agreement.     

Wave regimes on power-law fluid film flowing down a porous plane

Non-linear waves on the surface of a falling film of power-law fluid on a vertical porous plane are investigated. The waves are described by evolution equations generalising equations previously derived in the case of solid plane. It is shown that the slip condition on the interface between pure liquid and the porous substrate drastically changes structure of the steady waves travelling in the film.     

Plane waves in a semi-infinite fluid saturated porous medium

The field equations governing the propagation of waves in an incompressible liquid-saturated porous medium are investigated and a general solution is presented. It has been revealed that coupled longitudinal and transverse waves propagate in the porous medium. The propagation of transverse waves in the fluid phase is completely due to the interaction between the solid and fluid phases. The dispersion relationship and attenuation features are discussed. Unlike other investigations, all explicit forms of the arguments are derived. The reflection of the plane harmonic waves at the plane, traction-free boundary, which shows the influence of the dissipation on the velocity, and the attenuation coefficients of the reflected waves is studied. It is of interest that pore pressure is produced in the process of reflection, even in the case of the incidence of transverse waves.     

Diffusion–Convection in a Porous Medium with Impervious Inclusions at Low Flow Rates

The aim of this paper is to develop a macroscopic model for the transport of a passive solute, by diffusion and convection, in a heterogeneous medium consisting of impervious solids periodically distributed in a porous matrix. In the porous part, the flow is described by Darcy's law. Attempt is made to derive the macroscopic equation governing the average concentration field in the equivalent macroscopic medium and the macroscopic transport parameters. The analysis is conducted in the case when convection and diffusion are of the same order of magnitude at the macroscopic level, that is, when the Péclet number is of order 1. The proposed macroscopic model is obtained using the homogenization method for periodic structures with a double scale asymptotic expansion, in which the small parameter is the ratio between the two characteristic lengths l (the period scale of the impervious bodies distribution) and L (the scale of the macroscopic sample). The macroscopic parameters, which characterize the multiporous medium, depend solely on the transport parameters in the porous matrix and on the geometry of the impervious inclusions without any phenomenological assumption. Numerical computations are performed using a finite element method for several geometries of the solid inclusions, in two- and three-dimensional cases.     

非饱和颗粒材料的多孔连续体有效压力与有效广义Biot应力

多孔连续体理论框架下的非饱和多孔介质广义有效压力定义和Bishop参数的定量表达式长期以来存在争议,这也影响了对与其直接相关联的非饱和多孔介质广义Biot有效应力的正确预测.基于随时间演变的离散固体颗粒-双联液桥-液膜体系描述的Voronoi胞元模型,利用由模型获得的非饱和颗粒材料表征元中水力-力学介观结构和响应信息,文章定义了低饱和度多孔介质局部材料点的有效内状态变量:非饱和多孔连续体的广义Biot有效应力和有效压力,导出了其表达式.所导出的有效压力公式表明,非饱和多孔连续体的有效压力张量为各向异性,它不仅对非饱和多孔连续体广义Biot有效应力张量的静水应力分量的影响呈各向异性,同时也对其剪切应力分量有影响.文章表明,非饱和多孔连续体中提出的广义Biot理论和双变量理论的基本缺陷在于它们均假定反映非混和两相孔隙流体对固相骨架水力-力学效应的有效压力张量为各向同性.此外,为定义各向同性有效压力张量和作为加权系数而引入的Bishop参数并不包含对非饱和多孔连续体中局部材料点水力-力学响应具有十分重要效应的基质吸力.所导出的非饱和多孔介质广义Biot有效应力和有效压力公式(包括反映有效压力...     

Use of a Downscaling Procedure for Macroscopic Heat Source Modeling in Porous Media

Many porous media such as rocks have mesoscale inhomogeneities. The characteristic sizes of such inhomogeneities are much larger than the pore size but much less than the characteristic scale of the problem, such as the length of the sample on which measurements are taken. In this paper, we have solved the one-particle problem for depolarization of an ellipsoidal particle located in a porous medium with electrokinetic effect. To calculate the effective physical properties of a porous medium with many ellipsoidal inclusions, we have applied the effective field method. The application of this method allows us to take into account the texture of an inhomogeneous medium. The analysis performed has shown that three effective properties of inhomogeneous media (permeability, electroosmotic coupling coefficient and electrical conductivity) are not completely independent variables. General theory is illustrated by calculations of the effective properties of media containing spherical and spheroidal inclusions.     

Hydrodynamic and Thermal Fields in a Porous Medium with Injection of Superheated Steam

The plane one-dimensional and radially symmetric problems of injection of superheated steam into a porous medium saturated with gas are considered. Self-similar solutions are constructed on the assumption that in this case four zones are formed in the porous medium, namely, a gas flow zone, superheated and wet steam zones, and a water slug zone formed due to steam condensation. On the basis of the solution obtained, both the effects of the boundary pressure, mass flow rate, and temperature of the injected superheated steam and the effect of the initial state of the porous medium on the propagation of the hydrodynamic and thermal fields in the porous medium are studied.     

Flow past a cylinder and a sphere in a porous medium within the framework of the Brinkman equation with the Navier boundary condition

Exact analytical solutions of the problem of flow past a sphere and a cylinder in a porous medium are derived within the framework of the Brinkman equationwith the Navier boundary condition. Attention is drawn to the fact that the no-slip condition imposed on the interface between the porous medium and a solid, used, in particular, in the case of the Brinkman equation, must be in the general case replaced by a condition that admits nonzero flow velocity at the boundary.     

Perfect gas convection in a porous medium between two coaxial horizontal cylinders of large length

Natural convection of a perfect gas in a porous medium between two coaxial horizontal cylinders of large length located in heat-conducting space is considered. The two-dimensional problem (thin porous ring) is investigated in a plane orthogonal of the axis of the cylinders. The dependence of the criterion of the onset of convection on the non-Boussinesq parameters is studied. In the steady-state case an analytic solution of the nonlinear problem is obtained and its asymptotic behavior is considered for large Rayleigh numbers and when the compressibility criterion tends to zero. The gas flow rate in the ring and other characteristics of convection are studied as functions of the gas compressibility criterion and a constant temperature gradient given far away from the contour.