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.     

On the flow of fluids through inhomogeneous porous media due to high pressure gradients

Most porous solids are inhomogeneous and anisotropic, and the flows of fluids taking place through such porous solids may show features very different from that of flow through a porous medium with constant porosity and permeability. In this short paper we allow for the possibility that the medium is inhomogeneous and that the viscosity and drag are dependent on the pressure (there is considerable experimental evidence to support the fact that the viscosity of a fluid depends on the pressure). We then investigate the flow through a rectangular slab for two different permeability distributions, considering both the generalized Darcy and Brinkman models. We observe that the solutions using the Darcy and Brinkman models could be drastically different or practically identical, depending on the inhomogeneity, that is, the permeability and hence the Darcy number.     

Mixing zone of filtering gases in various porous media

The article gives the results of an experimental investigation of the mixing zone of consecutively filtering gases in saturated inhomogeneous porous media of different permeability and in inhomogeneous laminar strata.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 179–181, January–February, 1977.     

论岩体的渗透特性

从岩体结构的控渗作用出发, 可将岩体的渗流介质划分为多孔介质、准多孔介质、面状流不连续介质及脉状流不连续介质四类；论述了岩体复杂的渗透特性, 包括渗流的不均匀性、各向异性、非饱和性及渗流与变形的耦合等问题。文中还对岩体渗流分析的技术思路进行了讨论。     

各向异性粘弹性多孔截止中平面波的传播

This study discusses wave propagation in perhaps the most general model of a poroelastic medium. The medium is considered as a viscoelastic, anisotropic and porous solid frame such that its pores of anisotropic permeability are filled with a viscous fluid. The anisotropy considered is of general type, and the attenuating waves in the medium are treated as the inhomogeneous waves. The complex slowness vector is resolved to define the phase velocity, homogeneous attenuation, inhomogeneous attenuation, and angle of attenuation for each of the four attenuating waves in the medium. A non-dimensional parameter measures the deviation of an inhomogeneous wave from its homogeneous version. An numerical model of a North-Sea sandstone is used to analyze the effects of the propagation direction, inhomogeneity parameter, frequency regime, anisotropy symmetry, anelasticity of the frame, and viscosity of the pore-fluid on the propagation characteristics of waves in such a medium.     

Numerical simulation of formation of a concentrated brine lens subject to magma chamber degassing

The mathematical model of flow of a binary salt-water mixture through a porous medium in a wide range of pressure and temperature is developed taking different multiphase thermodynamic equilibria of the mixture into account. Formation of concentrated brine lenses above a degassing magma chamber is investigated within the framework of the model. The lenses are assumed to be coupled with generation of ore deposits. It is shown that the lens formation is caused by phase transitions of two different types undergoing at different depths in the magmatic fluid rising towards the surface. In the shallow zones salt precipitation on the skeleton of the porous medium in the form of a solid phase leads to clogging of pore space and reduction of the permeability. As a result, the magmatic fluid flow towards the surface is blocked and this facilitates the concentrated brine accumulation in a local zone.     

Filtration model of longitudinal flow in a finned cylindrical channel

The problem of laminar flow of a viscous incompressible fluid in a finned circular tube is considered. A solution is obtained in the form of series in eigenfunctions of the Laplace operator; the coefficients in the series are found numerically. For the same problem, a simpler filtration approximation is proposed in which the system of fins is modeled by a radially inhomogeneous porous layer, and fluid flow in it is described by the Brinkman equation. A formula for the effective permeability of the porous medium is obtained by varying the number and height of fins. The formula provides an accurate evaluation of the mean flow velocity and viscous drag coefficient in finned channels.     

Measurement of Void Size Correlation in Inhomogeneous Porous Media

In previous works, we have described a void space reconstruction method based on non-wetting fluid intrusion, wetting fluid drainage, and image analysis data. The method has been applied to a wide range of substances, including sandstone, compressed and sintered powders, paper substrates and coatings, soil and fibrous mats. We have also demonstrated in a previous work that the spatial correlation of similarly sized voids within inhomogeneous porous media has a huge effect on permeability. We therefore describe a method of measuring such correlation, suitable for use in our void space reconstructions. The method involves a cubic spline smoothing of a variogram of the void sizes in a binary image of the porous medium. It has been successfully tested on an artificially correlated void network, comprising two sintered glass discs of different void size ranges. Stereological effects, caused by the off-centre sectioning of voids, can interfere with the variogram features. Our method is sh own to be insensitive to artificially generated stereological interference. The method is also applied to sandstone samples.     

Three-Dimensional Simulations for Convection Problem in Anisotropic Porous Media with Nonhomogeneous Porosity,Thermal Diffusivity,and Variable Gravity Effects

A convection problem in anisotropic and inhomogeneous porous media has been analyzed. In particular, the effect of variable permeability, thermal diffusivity, and variable gravity with respect to the vertical direction, has been studied. A linear and nonlinear stability analysis of the conduction solution has been performed. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three- dimensional simulation. Our results show that the linear threshold accurately predicts on the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.     

Hydrodynamic averaging of highly inhomogeneous porous media with linear flow

A method is proposed for constructing homogeneous anisotropic models of new classes of highly inhomogeneous nondeformable porous media consisting of arbitrarily oriented systems of layers representing fractures and impermeable barriers (screens) embedded in each other with an arbitrary depth of embedment. It is assumed that the permeability functions of the elementary cells of the media can be represented in some Cartesian coordinate system (proper to each cell) in the form of a product of three integrable functions that depend on the corresponding coordinates. As distinct from known methods of averaging differential operators, the method in question is based on porous media flow considerations and reduces to replacing the highly inhomogeneous soils with homogeneous anistropic soils, so that on the boundaries of the domain considered the basic flow parameters remain the same.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika, Zhidkosti i Gaza, No. 5, pp. 190–192, September–October, 1993.     

A Fractal Model for Oil Transport in Tight Porous Media

Tight porous media are mainly composed of micro/nano-pores and throats, which leads to obvious microscale effect and nonlinear seepage characteristics. Based on the capillary bundle model and the fractal theory, a new nonlinear seepage equation was deduced, and a further fractal permeability model was obtained for oil transport in tight porous media by considering the effect of the boundary layer. The predictions of the model were then compared with experimental data to demonstrate that the model is valid. This model clarifies the oil transport mechanisms in tight porous media: the effective permeability is no longer a constant value and is governed by properties of tight porous media and oil. Furthermore, parameters influencing effective permeability were analyzed. The model can accurately present the seepage characteristics of the oil in tight porous media and provide a reliable basis for the development of unconventional reservoirs.     

Effect of yield stress on the flow of a Casson fluid in a homogeneous porous medium bounded by a circular tube

The effect of yield stress on the flow characteristics of a Casson fluid in a homogeneous porous medium bounded by a circular tube is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. The non-linear coupled implicit system of differential equations governing the flow is first transformed into suitable integral equations and are solved numerically. Analytical solution is obtained for a Newtonian fluid in the case of constant permeability, and the numerical solution is verified with that of the analytic solution. The effect of yield stress of the fluid and permeability of the porous medium on shear stress and velocity distributions, plug flow radius and flow rate are examined. The minimum pressure gradient required to start the flow is found to be independent of the permeability of the porous medium and is equal to the yield stress of the fluid.     

Motion through spherical droplet with non-homogenous porous layer in spherical container

The problem of the creeping flow through a spherical droplet with a nonhomogenous porous layer in a spherical container has been studied analytically. Darcy’s model for the flow inside the porous annular region and the Stokes equation for the flow inside the spherical cavity and container are used to analyze the flow. The drag force is exerted on the porous spherical particles enclosing a cavity, and the hydrodynamic permeability of the spherical droplet with a non-homogeneous porous layer is ca...     

Flows through Porous Media Containing a Shielded Spherical Inclusion

Compact formulas are obtained for constructing flow potentials in media containing a spherical inclusion shielded by a high- or low-permeability film (fracture or barrier) using the known potentials of steady-state incompressible fluid flows through homogeneous porous media. The formulas obtained can be extended both to inhomogeneous porous media in which the permeability functions have different constant factors inside and outside the inclusion and to bounded zones. The type of singular points of the potentials (sources, sinks, etc.) and the boundary conditions assigned in media without an inclusion are conserved for media containing a shielded inclusion. As an example, translational flow past a spherical shielded polluted zone is studied. This is of interest in connection with environmental problems.     

Dispersion of a filtration stream in media with random inhomogeneities

The transport of a dynamically neutral impurity by a stream into a porous medium with random inhomogeneities is considered. In contrast to [1, 2], in which a Markov random-walk process of the impurity particles was postulated substantially (taking a hypothesis about Markov random walk processes contradicts to a definite degree the representation of particle motion along a streamline, finiteness of the velocity, and smoothness of the trajectory), the complete system of equations for the filtration concentration and velocities is investigated here by a perturbation method, which results in a non-local equation for the mean concentrations after taking the average. It is shown that the local equation (parabolic or hyperbolic) is the limit case in the scheme considered. The effect of a regular drift of saturation, analogous to the effect of directed transport in the theory of inhomogeneous turbulent diffusion [3], is established. One-dimensional, plane, and three-dimensional flows are considered. The fundamental relationships contain moments of the random velocity field. The relationship between these moments and the characteristics of the random permeability and porosity fields has been established in [1, 2].     

Effective Correlation of Apparent Gas Permeability in Tight Porous Media

Gaseous flow regimes through tight porous media are described by rigorous application of a unified Hagen–Poiseuille-type equation. Proper implementation is accomplished based on the realization of the preferential flow paths in porous media as a bundle of tortuous capillary tubes. Improved formulations and methodology presented here are shown to provide accurate and meaningful correlations of data considering the effect of the characteristic parameters of porous media including intrinsic permeability, porosity, and tortuosity on the apparent gas permeability, rarefaction coefficient, and Klinkenberg gas slippage factor.     

Vortex structure of turbulence over permeable walls

In order to understand the effect of the wall permeability on the turbulent vortex structure near porous walls, based on PIV experimental data, a probability density analysis of fluctuating velocities, statistical quadrant and quadrant-hole analyses of the Reynolds shear stress are performed. The investigated flow fields are turbulent channel flows whose bottom walls are made of porous media. The porous media used are three kinds of foamed ceramics which have almost the same porosity (∼0.8) but different permeability. From the discussions on those analyses, a conceptual scenario of the development of the vortex structure near a permeable wall is proposed for a moderate permeability Reynolds number case. It explains the reason why the near-wall long streaky structure tends to vanish near a porous wall with increasing wall permeability.     

Network model evaluation of permeability and spatial correlation in a real random sphere packing

In principle, network models can replicate exactly the microstructure of porous media. In practice, however, network models have been constructed using various assumptions concerning pore structure. This paper presents a network model of a real, disordered porous medium that invokes no assumptions regarding pore structure. The calculated permeability of the model agrees well with measured permeabilities, providing a new and more rigorous confirmation of the validity of the network approach. Several assumptions commonly used in constructing network models are found to be invalid for a random packing of equal spheres. In addition, the model permits quantification of the effect of pore-scale correlation (departure from randomness) upon permeability. The effect is comparable to reported discrepancies between measured permeabilities and predictions of other network models. The implications of this finding are twofold. First, a key assumption of several theories of transport in porous media, namely that pore dimensions are randomly distributed upon a network, may be invalid for real porous systems. Second, efforts both to model and to measure pore-scale correlations could yield more accurate predictions of permeability.     

Effect of the microinhomogeneity of the medium on the flow law in porous media

A lattice percolation model of an inhomogeneous medium [6] introducing f(r), the probability density function of the microcapillaries with respect to their radius r, is employed. The calculation scheme described makes it possible to determine the dependence of the permeability K on the pressure gradient G for media with arbitrary f(r). It is shown that for inhomogeneous media the behavior of K(G) is mainly determined by the form of f(r). The question of the effect of the state of stress on the permeability of the medium is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 84–94, March–April, 1989.     

Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid

The body-force-driven motion of a homogeneous distribution of spherically symmetric porous shells in an incompressible Newtonian fluid and the fluid flow through a bed of these shell particles are investigated analytically. The effect of the hydrodynamic interaction among the porous shell particles is taken into account by employing a cell-model representation. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field around a single particle in a unit cell, and the drag force acting on the particle by the fluid is obtained in closed forms. For a suspension of porous spherical shells, the mobility of the particles decreases or the hydrodynamic interaction among the particles increases monotonically with a decrease in the permeability of the porous shells. The effect of particle interactions on the creeping motion of porous spherical shells relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solution describing the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres. The particle-interaction behavior for a suspension of porous spherical shells with a relatively low permeability may be approximated by that of permeable spheres when the porous shells are sufficiently thick.