Effective Diffusion Coefficient: From Homogenization to Experiment

In this paper, an example of the application of the homogenization approach (asymptotic expansion technique) to predict the effective diffusion coefficient for an equivalent continuum, together with the experimental verification of the theoretical results is presented. The experimental setup was constructed for the measurements of diffusion in a model periodic porous medium made of Plexiglas. The computer program using the FEM was elaborated to solve the local boundary value problem for a period and to calculate the effective diffusion coefficient. The comparison between the theory and the experiment indicates good agreement between the numerical and experimental values of the effective diffusion coefficient. Interpretation of the test data from the point of view of the homogenization theory is also incorporated.     

Determination of Salt Diffusion Coefficient in Brick: Analytical Methods

This paper presents an investigation into salt diffusion in new, fully saturated brick under isothermal conditions. A commonly used experiment methodology, diffusion cell method, is adopted. The analytical and numerical solutions are obtained. The analytical solution is simple and straightforward, which determines temporally salt concentrations in the monitored chamber. It enables us to estimate salt diffusion coefficients in a fast and accurate way.     

General Anisotropic Effective Medium Theory for the Effective Permeability of Heterogeneous Reservoirs

One of the techniques to calculate the effective property of a heterogeneous medium is the effective medium theory. The present paper presents a general mathematical formulation for the effective medium approximation using a self-consistent choice of the effective permeability, to apply it to the case of a general anisotropic 2D medium and to the case of a 3D isotropic medium with randomly oriented ellipsoidal inclusions. The 2D results are compared with analytical results and with a homogenization technique with good result. The 3D correlations are used to derive percolation thresholds in two-phase systems with a large permeability contrast, which are compared to numerical results from the literature, also with good results.     

Effective-medium treatment of flow through anisotropic fracture system — Improved permeability estimates using a new lattice mapping

The permeability tensor of a fractured reservoir, which will typically be anisotropic because of the presence of stress, is an important parameter to be taken into account when formulating a production strategy for the reservoir. Extensive computational effort is involved in calculating the permeabilities of model fracture systems by solving the fluid flow equations through finite realisations of the systems, and this renders a search for alternative techniques worthwhile. An attractive approach is to perform a rough mapping of the fracture system onto a lattice so that effective medium theory can be applied. For isotropic systems that are well-connected, this technique works well, but it gives increasingly poor results as the degree of anisotropy increases. In this contribution, a refinement of the lattice mapping is presented that incorporates an important aspect of the randomness present in the original system. This greatly increases the applicability of the technique.     

均匀化方法在粘弹性多层复合材料中的应用

主要研究了由各向同性线弹性加强体和各向同性线粘弹性基体组成的多层复合材料的问题,在已有的线弹性多层材料的均匀化方法的基础上,应用弹性一粘弹性对应原理,在Carson域中求解粘弹性多层材料的问题。通过Burgers模型表示线粘弹性基体材料,反演得到了多层材料的有效松弛模量和有效泊松比在时间域中的表达式,并且与实验结果和其他结果进行了比较。     

Transient moisture transport in a cracked porous medium

Condensation under high relative humidity, imbibition under zero capillary pressure, and drying in a cracked mesoporous slab is numerically simulated. The porous medium is homogeneous, the crack lattice is periodic and has uniform aperture and transport properties. It is found that the crack lattice density and the crack hydraulic conductivity has minor influence on the global kinetics of condensation and drying, and a strong influence on the imbibition kinetics. The transient effects of the heterogeneity of the medium are examined from three view-points: the study of the spatial distribution of pressure head, the tentative definition of an effective diffusivity, and the comparison between the quasi-static and transient transport properties. The equivalent homogeneous medium approach is found to be relatively satisfactory to describe the global kinetics in the three processes. The transient effects appear in secondary features of the processes.     

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe

A model is presented describing the changes that occur in the diffusion boundary layer upon injection of a macromolecular solution (PEO) into a cylindrical pipe under turbulent flow conditions (Re 40,000). A shape parameter was introduced to describe the shape of the turbulent plume. The value of this parameter was found to be the same for water and various dilute PEO solutions. The proposed model gives a good approximation at low homogeneous concentrations. x downstream distance from the slot - y normal distance from the wall - R radius of the pipe - C concentration - C _w wall concentration - Q _i flow rate injection - Q _t flow rate - C _j =C _i *Q _i /Q _t equivalent homogeneous polymer concentration - L _tf characteristic length of the diffusion plume - characteristic height of the diffusion plume, i.e. the value ofy at whichC/C _w = 0.5 - thickness of the diffusion boundary layer - x ₀ characteristic distance from the slot, i.e. the value ofx at which/R = 1/2 - ⁺ shape parameter of the diffusion boundary layer - ⁺ —/R nondimensionalized variables - x ⁺ —x/L _tf nondimensionalized variables     

Upscaling of conductivity of heterogeneous formations: General approach and application to isotropic media

The numerical simulation of flow through heterogeneous formations requires the assignment of the conductivity value to each numerical block. The conductivity is subjected to uncertainty and is modeled as a stationary random space function. In this study a methodology is proposed to relate the statistical moments of the block conductivity to the given moments of the continuously distributed conductivity and to the size of the numerical blocks. After formulating the necessary conditions to be satisfied by the flow in the upscaled medium, it is found that they are obeyed if the mean and the two-point covariance of the space averaged energy disspation function over numerical elements in the two media, of point value and of upscaled conductivity, are identical. This general approach leads to a systematic upscaling procedure for uniform average flow in an unbounded domain. It yields the statistical moments of upscaled logconductivity that depend only on those of the original one and on the size and shape of the numerical elements.The approach is applied to formations of isotropic heterogeneity and to isotropic partition elements. After a general discussion based on dimensional analysis, the procedure is illustrated by using a first-order approximation in the logconductivity variance. The upscaled logconductivity moments (mean, two-point covariance) are computed for two and three dimensional flows, isotropic heterogeneous media and elements of circular or spherical shape. The asymptotic cases of elements of small size, which preserve the point value conductivity structure on one hand, and of large blocks for which the medium can be replaced by one of deterministic effective properties, on the other hand, are analyzed in detail. The results can be used in order to generate the conductivity of numerical elements in Monte Carlo simulations.Nomenclature C covariance - e rate of dissipation of mechanical energy per unit weight of fluid - E total rate of energy dissipation in the flow domain - H overlap function - K hydraulic conductivity - K _G geometrical mean of conductivity - I integral scale - J=P mean head gradient - L characteristic size of - l characteristic size of also diameter of circle and sphere - n number of dimensions - P pressure head - Q total fluid discharge - S _A ,S _B inlet and outlet boundaries of flow domain - v velocity - Y logconductivity - characteristic scale of flow nonuniformity - autocorrelation function - ² variance - flow domain - partition element Overlining space averaged over - Ã upscaled quantity - â Fourier transform ofa     

Application of generalised effective-medium theory to transport in porous media

The use of effective-medium treatments to estimate bulk properties pertaining to transport (of, for example, fluids, heat, particles or electricity) through random composite media (such as reservoir rocks), is widespread. This is because they are relatively simple, often reasonably accurate (on occasion, remarkably so) and in many cases yield closed-form expressions for the properties concerned. However, the single-bond effective-medium treatment (EMT) of random resistor networks that has been used to determine transport coefficients for various transport problems in pore networks is limited to some special isotropic networks with nearest-neighbour connections. We demonstrate here that transport through two different fracture system models, with stress-induced anisotropy, can be treated using an EMT originally applied to anisotropic resistor networks. The main purpose of the present contribution, however, is to present a new, more general effective medium formalism applicable to networks of arbitrary topology. This new generalised EMT is used to obtain a new criterion for percolation of an arbitrary conducting network under random dilution. A specific application to unsaturated flow through a pore network with nearest- and next-nearest-neighbour connections is also given.     

Diffusion stability of bubbles in a cluster

The diffusion stability of gas bubbles in one-fraction and two-fraction clusters subjected to an acoustic field is studied. For a one-fraction cluster, numerical values were obtained for the initial gas concentrations in the liquid at which the bubble tends to one of two equilibrium states because of diffusion processes between the bubble and the ambient liquid. It is found that a two-fraction cluster tends to become a one-fraction cluster. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 40–48, July–August, 2007.     

Application of the finite Hankel transform to a diffusion problem without azimuthal symmetry

The problem treated in this paper concerns calculating the evolution of the pressure in a single-phase, slightly compressible fluid in a porous medium consisting of communicating layers. The fluid is produced through a point sink located on the side of an otherwise sealed cylindrical wellbore. This location of the sink causes the flow around the wellbore to be azimuthally asymmetric.The problem is solved through successive application of Laplace, finite Fourier and finite Hankel transforms. Although apparently straightforward, this approach leads to serious numerical difficulties. The published form of the inversion formula for the finite Hankel transform leads to inaccurate computation for the higher azimuthal modes even with 128 bit arithmetic. An alternative form is developed which enables accurate evaluation of the solution with the more practical 64 bit arithmetic. The technique for two-layer solution presented here can be directly extended to a problem with a larger number of communicating layers. This is the first instance of successful application of the finite Hankel transform to an azimuthally asymmetric diffusion problem.     

幂律流体在多孔介质中径向渗流的分形模型

基于分形理论及毛细管模型,本文研究了非牛顿幂律流体在各向同性多孔介质中径向流动问题,推导了幂律流体径向有效渗透率的分形解析表达式．研究结果表明,幂律流体径向有效无量纲渗透率模型和Chang and Yortsos’s模型吻合很好;同时还得出幂律流体径向有效渗透率随孔隙率、幂指数的增加而增加,随迂曲度分形维数的增加而减少．     

Seepage-Diffusion Model of Brine Migration in Inhomogeneous Aquifers

A new phenomenological mathematical model of the propagation of high- and low-salinity solutions through inhomogeneous aquifers is proposed. The model consists of interrelated equations of two-phase flow and diffusive mass transfer through a porous medium in a region with a traveling boundary. Features of different contamination scenarios of are analyzed with reference to particular examples.     

Macroscopic simulation of relaxation mass transport in a porous medium

The problem of the macroscopic simulation of the motion of a viscous fluid and mass transport in a porous medium is considered under the assumption that the mass transport can locally be described by the Fick relaxation law. Several cases determined by the local inertia number of the mass flow and the Péclet number are investigated. The macroscopic transport models are analyzed and compared with well-known phenomenological models.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 21–30.Original Russian Text Copyright © 2004 by Khuzhayorov.     

Diffusion Characters of the Orbits in the Asteroid Motion

ＩｎｔｒｏｄｕｃｔｉｏｎＭｏｒｅａｎｄｍｏｒｅａｔｔｅｎｔｉｏｎｉｓｒｅｃｅｎｔｌｙｐａｉｄｔｏｔｈｅｄｉｆｆｕｓｉｏｎｏｒｃｈａｏｔｉｃｔｒａｎｓｐｏｒｔａｔｉｏｎｐｈｅｎｏｍｅｎａｉｎｄｙｎａｍｉｃａｌｓｙｓｔｅｍｓａｎｄｓｏｍｅｓｉｇｎｉｆｉｃａｎｔｒｅｓｕｌｔｓｈａｖｅｂｅｅｎｇａｉｎｅｄ[1].Ｉｎｃｅｌｅｓｔｉａｌｍｅｃｈａｎｉｃｓ,ｗｈｅｎａｐｐｌｉｅｄｔｏｔｈｅｉｎｖｅｓｔｉｇａｔｉｏｎｏｆｔｈｅａｓｔｅｒｏｉｄｂｅｌｔ,ｔｈｅｄｉｆｆｕｓｉｏｎｐｒｏｐｅｒｔｉｅｓｏｆｔｈｅｄｙｎａｍ…     

Dispersion Phenomena in Thermal Diffusion and Modelling of Thermogravitational Experiments in Porous Media

In a TIPM paper published in 1992, the authors presented a simple model of thermogravitational diffusion in packed columns (TPC). Though qualitatively in agreement with the experimental results, this model exhibited a systematic discrepancy with respect to the magnitude of the permeability of maximum separation in the TPC experiments. Here, the results of a re-examination of the classical phenomenology of irreversible thermodynamics in porous media, applied to TPC, are described. Through the interpretation of additional TPC experiments, we show that the effective thermal diffusion coefficient in TPC includes a dependency upon the fluid velocity. This dependency is consistent with a nonlinear extension of irreversible thermodynamics, and the model so amended accounts for a correct re-interpretation of the experiments.     

A multi-scale theory of swelling porous media: I. Application to one-dimensional consolidation

A theory is developed which describes flow in multi-scale, saturated swelling media. To upscale information, both the hybrid theory of mixtures and the homogenization technique are employed. In particular, a model is formulated in which vicinal water (water adsorbed to the solid phase) is treated as a separate phase from bulk (non-vicinal) water. A new form of Darcy's law governing the flow of both vicinal and bulk water is derived which involves an interaction potential to account for the swelling nature of the system. The theory is applied to the classical one-dimensional consolidation problem of Terzaghi and to verify Low's empirical, exponential, swelling result for clay at the macroscale.     

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe

This paper presents a model for the evolution of a transversal concentration profile of a macromolecular solution (PEO) injected into a cylindrical pipe at turbulent flow conditions (R 40000). This model, based on the diffusion of a scalar quantity emitted by two diametrically opposed point sinks, proves to be in good agreement with the experimental data. C concentration - C _w wall concentration - C _i initial concentration before injection - x downstream distance from the slot - y normal distance from the wall - characteristic height of diffusion, i.e. the value ofg at whichC/C _w = 0.5 - n characteristic exponent - R radius of pipe - D diameter of pipe - a, b constants - L _m mixing length, i.e. the value ofx at whichC _w /C _i =Q _i /Q _T - Q _i flow rate of injection - Q _T flow rate - f, g annex functions - n ₀ maximal value ofn     

On applications of the microlocal parameter method in modelling of temperature distributions in composite cylinders

Summary The problems of transient heat conduction in a periodically stratified medium consisting of a large number of alternating concentric cylinders of two homogeneous isotropic rigid materials and in a rotationally periodic cylinder consisting of a large number of circular homogeneous isotropic rigid sectors are considered. The equations of the homogenized models with microlocal parameters are derived by using the homogenization procedure given in [17]. The obtained models take into account certain microlocal effects connected with the microperiodic structure of the considered composites. Some examples of the application of the presented models to the problems of temperature distributions in composite cylinders are detailed. Received 10 March 1997; accepted for publication 23 October 1997