Random Porous Media Research Articles

The critical scaling property of random percolation porous media

A percolation model of random porous media， with its nonzero bonds obeying the uniform distribution， is researched in this article- The scaling exponents of the permeability are calculated at the percolation critical point- The simulation suggests that the exponents are not universal，but depending on the parameter of the uniform distribution- This indicates that even if the first negative moment of the probability density function of the nonzero bonds exists， the corresponding scaling exponent at the critical point will not be universal- Our simulation result is different from the conclusion reached by Sahimi-

Stochastic Krylov subspace methods for flow in random porous media

AbstractThe present work establishes a comparative analysis of two different Krylov‐based methods for assessing uncertainty in porous media flow applications. They are: (1) the stochastic reduced basis method (SRBM) and the (2) Krylov‐Karhunen‐Loeve moment equation (KKLME). The former relies on the construction of an orthogonal basis from which projectors may be realized to perform the uncertainty analysis on a lower‐dimensional space. The second approach relies on the idea of expressing input and corresponding outputs in terms of stochastic and perturbative polynomial expansions. By grouping terms and moments of the same order, the original stochastic equation is replaced by successive (and parallel) solutions of a set of deterministic equations. We provide a set of numerical experiments illustrating the capabilities of SRBM and KKLME against Monte Carlo simulations (MCS) on stationary permeability cases. We show that KKLME is superior to SRBM. Furthermore, KKLME appears to be also a more efficient alternative than MCS for non‐stationary permeability field distributions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Comment on “Nonstationary flow and nonergodic transport in random porous media” by G. Darvini and P. Salandin

Comment on “Nonstationary flow and nonergodic transport in random porous media” by G. Darvini and P. Salandin

Reply to comment by N. Suciu and C. Vamoş on “Nonstationary flow and nonergodic transport in random porous media”

Reply to comment by N. Suciu and C. Vamoş on “Nonstationary flow and nonergodic transport in random porous media”

Shape factor characterization of fibrous media with a temperature gradient

Recent theoretical work generalizes thermoacoustic theory to random porous media [H. S. Roh et. al., J. Acoust. Soc. Am. 121(3), 1413–1422 (2007)]. Characteristics of the porous media, such as the tortuosity and dynamic shape factors for viscous and thermal effects, are introduced into the thermoacoustic wave equation and may be determined by suitable impedance measurements at zero temperature gradient. This theoretical approach may also be used to model fibrous media, such as fiberglass or steel wool. A new technique to determine the scaling factors for fibrous media utilizing a single-step finite difference inversion of the thermoacoustic wave equation [R. Raspet et. al., J. Acoust. Soc. Am. 103(5), 2395–2402 (1998)] with zero temperature gradient will be presented. Roh et. al. predict that acoustic gain with a nonzero temperature gradient may be written in terms of these scaled cylindrical dissipation functions. The acoustic gain term may also be determined by suitable application of the single-step finite difference technique. [This material is based upon work supported by the U. S. Army Space and Missile Defense Command under Contract No. W9113M-06-C-0029.]

Fluid flow simulation at open–porous medium interface using the lattice Boltzmann method

AbstractThe present study investigates, using the lattice Boltzmann method, the influence on the flow of the boundary condition at the open and porous medium interface for a two‐dimensional channel with half of the channel filled with a random porous medium. By simulating the fluid flow at pore level, the proposed macroscopic hydrodynamic boundary conditions are evaluated; correlations for the empirical constant in one of the most widely used models are also presented. The predicted values for the flow rate are in overall agreement with the previously published values. Copyright © 2007 John Wiley & Sons, Ltd.

Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods

An efficient method for uncertainty analysis of flow in random porous media is explored in this study, on the basis of combination of Karhunen‐Loeve expansion and probabilistic collocation method (PCM). The random log transformed hydraulic conductivity field is represented by the Karhunen‐Loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion. Probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points. The procedure is straightforward and analogous to the Monte Carlo method, but the number of simulations required in PCM is significantly reduced. Steady state flows in saturated random porous media are simulated with the probabilistic collocation method, and comparisons are made with other stochastic methods: Monte Carlo method, the traditional polynomial chaos expansion (PCE) approach based on Galerkin scheme, and the moment‐equation approach based on Karhunen‐Loeve expansion (KLME). This study reveals that PCM and KLME are more efficient than the Galerkin PCE approach. While the computational efforts are greatly reduced compared to the direct sampling Monte Carlo method, the PCM and KLME approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head.

Seismic attenuation in porous rocks with random patchy saturation

ABSTRACTSaturation of porous rocks with a mixture of two fluids has a substantial effect on seismic‐wave propagation. In particular, partial saturation causes significant attenuation and dispersion of the propagating waves due to the mechanism of wave‐induced fluid‐flow. Such flow arises when a passing wave induces different fluid pressures in regions of rock saturated by different fluids. Most models of attenuation and dispersion due to mesoscopic heterogeneities imply that fluid heterogeneities are distributed in a regular way. However, recent experimental studies show that mesoscopic heterogeneities have less idealized distributions and that the distribution itself affects attenuation and dispersion. Based on an approximation for the coherent wavefield in random porous media, we develop a model which assumes a continuous distribution of fluid heterogeneities. As this continuous random media approach assumes that there will be a distribution of different patch sizes, it is expected to be better suited to modelling experimental data. We also show how to relate the random functions to experimentally measurable parameters.

Exact Averaging of Stochastic Equations for Flow in Porous Media

It is well-known that at present, exact averaging of the equations for flow and transport in random porous media have been proposed for limited special fields. Moreover, approximate averaging methods—for example, the convergence behavior and the accuracy of truncated perturbation series—are not well-studied, and in addition, calculation of high-order perturbations is very complicated. These problems have for a long time stimulated attempts to find the answer to the question: Are there in existence some, exact, and sufficiently general forms of averaged equations? Here, we present an approach for finding the general exactly averaged system of basic equations for steady flow with sources in unbounded stochastically homogeneous fields. We do this by using (1) the existence and some general properties of Green’s functions for the appropriate stochastic problem, and (2) some information about the random field of conductivity. This approach enables us to find the form of the averaged equations without directly solving the stochastic equations or using the usual assumption regarding any small parameters. In the common case of a stochastically homogeneous conductivity field we present the exactly averaged new basic non-local equation with a unique kernel-vector. We show that in the case of some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for three-dimensional and two-dimensional flow in the same way derive the exact averaged non-local equations with a unique kernel-tensor. When global symmetry does not exist, the non-local equation with a kernel-tensor involves complications and leads to an ill-posed problem.

Electroosmosis in homogeneously charged micro- and nanoscale random porous media

Electroosmosis in homogeneously charged micro- and nanoscale random porous media has been numerically investigated using mesoscopic simulation methods which involve a random generation-growth method for reproducing three-dimensional random microstructures of porous media and a high-efficiency lattice Poisson–Boltzmann algorithm for solving the strongly nonlinear governing equations of electroosmosis in three-dimensional porous media. The numerical modeling and predictions of EOF in micro- and nanoscale random porous media indicate that the electroosmotic permeability increases monotonically with the porosity of porous media and the increasing rate rises with the porosity as well; the electroosmotic permeability increases with the average solid particle size for a given porosity and with the bulk ionic concentration also; the proportionally linear relationship between the electroosmotic permeability and the zeta potential on solid surfaces breaks down for high zeta potentials. The present predictions agree well with the available experimental data while some results deviate from the predictions based on the macroscopic theories.

Determination of the effective diffusion coefficient in porous media including Knudsen effects

A 3-D bond pore network model is presented and used to evaluate the effect of pore size and connectivity on the effective diffusion coefficient in random porous media. The control equations of the system are set up and the simulation method is discussed. The simulation results show that when the average pore size dm < 1 μm, the effective diffusion coefficient is strongly dependent on pore size. The analysis shows that Knudsen and bulk diffusion effects can be decoupled, and for any given diffusion conditions, the effect coefficient accounting for Knudsen diffusion can be obtained. Thus, the effect of pore size can been readily accounted for by correcting the bulk effective diffusivity with the Knudsen effect coefficient. The simulations also show that the percolation threshold in random porous materials decreases with increased pore network connectivity.

Prediction of the effective diffusion coefficient in random porous media using the finite element method

A finite-element-based method is presented for evaluating the effective gas diffusion coefficient of porous solids. Using this method, the 3-D micro-scale geometries of the porous solids are constructed under the ANSYS platform by the parametric code; the relation between effective gas diffusivity and micro-scale features of random-distributed porous solids is established. The results show that in random-distributed pore media, there is a percolation threshold e p, and this percolation threshold decreases with increasing coordination number of the pore network. The relationship between the effective diffusivity and porosity is strongly nonlinear when the porosity, e, is less than a certain value e L ; for e > e L , the relationship becomes quasi-linear. This dividing point e L decreases with increasing coordination number. The larger the coordination number of the pore network, the higher the effective gas diffusivity. Based on the simulation results and observations, a formula relating the effective diffusion coefficient with porosity is proposed.

Mesoscopic predictions of the effective thermal conductivity for microscale random porous media

A mesoscopic numerical tool has been developed in this study for predictions of the effective thermal conductivities for microscale random porous media. To solve the energy transport equation with complex multiphase porous geometries, a lattice Boltzmann algorithm has been introduced to tackle the conjugate heat transfer among different phases. With boundary conditions correctly chosen, the algorithm has been initially validated by comparison with theoretical solutions for simpler cases and with the existing experimental data. Furthermore, to reflect the stochastic phase distribution characteristics of most porous media, a random internal morphology and structure generation-growth method, termed the quartet structure generation set (QSGS), has been proposed based on the stochastic cluster growth theory for generating more realistic microstructures of porous media. Thus by using the present lattice Boltzmann algorithm along with the structure generating tool QSGS, we can predict the effective thermal conductivities of porous media with multiphase structure and stochastic complex geometries, without resorting to any empirical parameters determined case by case. The methodology has been applied in this contribution to several two- and three-phase systems, and the results agree well with published experimental data, thus demonstrating that the present method is rigorous, general, and robust. Besides conventional porous media, the present approach is applicable in dealing with other multiphase mixtures, alloys, and multicomponent composites as well.

Parallel capillary-tube-based extension of thermoacoustic theory for random porous media

Thermoacoustic theory is extended to stacks made of random bulk media. Characteristics of the porous stack such as the tortuosity and dynamic shape factors are introduced into the thermoacoustic wave equation in the low reduced frequency approximation. Basic thermoacoustic equations for a bulk porous medium are formulated analogously to the equations for a single pore. Use of different dynamic shape factors for the viscous and thermal effects is adopted and scaling using the dynamic shape factors and tortuosity is demonstrated. Comparisons of the calculated and experimentally derived thermoacoustic properties of reticulated vitreous carbon and aluminum foam show good agreement. A consistent mathematical model of sound propagation in a random porous medium with an imposed temperature is developed. This treatment leads to an expression for the coefficient of the temperature gradient in terms of scaled cylindrical thermoviscous functions.

Fluid dynamics in capillary and chip electrochromatography

This review is concerned with the phenomenological fluid dynamics in capillary and chip electrochromatography (EC) using high-surface-area random porous media as stationary phases. Specifically, the pore space morphology of packed beds and monoliths is analyzed with respect to the nonuniformity of local and macroscopic EOF, as well as the achievable separation efficiency. It is first pointed out that the pore-level velocity profile of EOF through packed beds and monoliths is generally nonuniform. This contrasts with the plug-like EOF profile in a single homogeneous channel and is caused by a nonuniform distribution of the local electrical field strength in porous media due to the continuously converging and diverging pores. Wall effects of geometrical and electrokinetic nature form another origin for EOF nonuniformities in packed beds which are caused by packing hard particles against a hard wall with different zeta potential. The influence of the resulting, systematic porosity fluctuations close to the confining wall over a distance of a few particle diameters becomes aggravated at low column-to-particle diameter ratio. Due to the hierarchical structure of the pore space in packed beds and silica-based monoliths which are characterized by discrete intraparticle (intraskeleton) mesoporous and interparticle (interskeleton) macroporous spatial domains, charge-selective transport prevails within the porous particles and the monolith skeleton under most general conditions. It forms the basis for electrical field-induced concentration polarization (CP). Simultaneously, a finite and -- depending on morphology -- often significant perfusive EOF is realized in these hierarchically structured materials. The data collected in this review show that the existence of CP and its relative intensity compared to perfusive EOF form fundamental ingredients which tune the fluid dynamics in EC employing monoliths and packed beds as stationary phases. This addresses the (electro)hydrodynamics, associated hydrodynamic dispersion, as well as the migration and retention of charged analytes.

Fluids in porous media. I. A hard sponge model

The morphology of many porous materials is spongelike. Despite the abundance of such materials, simple models which allow for a theoretical description of these materials are still lacking. Here, we propose a hard sponge model which is made by digging spherical cavities in a solid continuum. We found an analytical expression for describing the interaction potential between fluid particles and the spongelike porous matrix. The diagrammatic expansions of different correlation functions are derived as well as that of grand potential. We derived also the Ornstein-Zernike (OZ) equations for this model. In contrast to Madden-Glandt model of random porous media [W. G. Madden and E. D. Glandt, J. Stat. Phys. 51, 537 (1988)], the OZ equations for a fluid confined in our hard sponge model have some similarity to the OZ equations of a three-component fluid mixture. We show also how the replica method can be extended to study our sponge model and that the same OZ equations can be derived also from the extended replica method.

Three-dimensional effect on the effective thermal conductivity of porous media

A three-dimensional mesoscopic method is developed for predicting the effective thermal conductivity of multiphase random porous media. The energy transport equations are solved by a lattice Boltzmann method for multiphase conjugate heat transfer through a porous structure whose morphology is characterized by a random generation-growth algorithm. Our numerical results show that the cell number in the third dimension influences the resulting effective thermal conductivity of three-dimensional porous media. The predicted effective thermal conductivity varies with the cell number in the third dimension following an exponential relationship, and it requires in the examples at least 10 cells along the third dimension before the predictions stabilize. Comparisons with the experimental data show that the effective thermal conductivities measured by the hot-probe and hot-wire techniques agree well with the predicted results by the two-dimensional model, whereas those measured by the transient comparative method agree more with the three-dimensional predictions.

Critical Behavior of Colloid-Polymer Mixtures in Random Porous Media

We show that the critical behavior of a colloid-polymer mixture inside a random porous matrix of quenched hard spheres belongs to the universality class of the random-field Ising model. We also demonstrate that random-field effects in colloid-polymer mixtures are surprisingly strong. This makes these systems attractive candidates to study random-field behavior experimentally.

Nonstationary flow and nonergodic transport in random porous media

Transport processes taking place in natural formations are often characterized by the spatial nonstationarity of flow field and the nonergodic condition of solute spreading. The former may originate from statistical inhomogeneity of porous media properties or from the influence of boundary conditions as well as from conditioning procedures on measured data, while the latter is generally due to the finite size of the solute source. In real‐world applications these aspects sometimes dominate the transport phenomena, and known literature results based on statistical homogeneity and ergodicity assumptions cannot be properly applied. In this paper a method is proposed to handle different and concurrent causes of the flow field nonstationarity and to describe nonergodic transport of inert solutes by spatial moments in a domain of finite size. The goal is reached by expanding the steady state flow equation in Taylor series limited to the first‐order and by the recursive application of finite element (FE) method. The unknowns are the piezometric head mean values and its derivatives in respect to the fluctuating porous media hydraulic conductivity. From the latter the velocity statistics are derived on the basis of Darcy's law according to the first‐order approximation. By neglecting the pore‐scale dispersion the spatial moments of a solute plume are a posteriori obtained by a consistent Lagrangian analysis starting from the knowledge of the velocity field covariance matrices without any restriction regarding the statistical homogeneity of flow and/or ergodicity conditions. The proposed method is here applied to investigate the influence of boundary conditions on the solute transport developing in a limited domain. The results obtained from different test cases give a velocity field behavior in agreement with literature findings and allow us to define the nonergodic transport in bounded domains. The comparison of the results with some ad hoc developed Monte Carlo simulations ensures the applicability of the proposed approach at least to the small heterogeneity cases here considered.

Seismic attenuation due to wave-induced fluid flow in a porous rock with spherical heterogeneities

SUMMARY Most natural porous rocks have heterogeneities at nearly all scales. Heterogeneities of mesoscopic scale—that is, much larger than the pore size but much smaller than wavelength—can cause significant attenuation and dispersion of elastic waves due to wave-induced flow between more compliant and less compliant areas. Analysis of this phenomenon for a saturated porous medium with a small volume concentration of randomly distributed spherical inclusions is performed using Waterman-Truell multiple scattering theorem, which relates attenuation and dispersion to the amplitude of the wavefield scattered by a single inclusion. This scattering amplitude is computed using recently published asymptotic analytical expressions and numerical results for elastic wave scattering by a single mesoscopic poroelastic sphere in a porous medium. This analysis reveals that attenuation and dispersion exhibit a typical relaxation-type behaviour with the maximum attenuation and dispersion corresponding to a frequency where fluid diffusion length (or Biot’s slow wavelength) is of the order of the inclusion diameter. In the limit of low volume concentration of inclusions the effective velocity is asymptotically consistent with the Gassmann theory in the low-frequency limit, and with the solution for an elastic medium with equivalent elastic inclusions (no-flow solution) in the low-frequency limit. Attenuation (expressed through inverse quality factor 1/Q) scales with frequency ω in the lowfrequency limit and with ω −1/2 in the high-frequency limit. These asymptotes are consistent with recent results on attenuation in a medium with a periodic distribution of poroelastic inclusions, and in continuous random porous media.