Real Porous Materials Research Articles

Gas Sorption Characterization of Porous Materials Employing a Statistical Theory for Bethe Lattices.

In the present work, a recently developed statistical theory for adsorption and desorption processes in mesoporous solids, modeled by random Bethe lattices, has been applied to obtain pore size distributions and interpore connectivity from sorption isotherms in real random porous materials, employing a robust and validated methodology. Using the experimental adsorption-desorption N2 isotherms at 77.4 K on Vycor glass, a porous material with random pore structure, we demonstrate the solution of the inverse problem resulting in extracted pore size distribution and interpore connectivity, notably different from the predictions of earlier theories. The results presented are corroborated by the analysis of 3D digital images of reconstructed Vycor porous glass, showing excellent agreement between the predictions of geometric analysis and the new statistical theory.

A Pore‐Throat Segmentation Method Based on Local Hydraulic Resistance Equivalence for Pore‐Network Modeling

AbstractThe pore network is an approximate representation of the void space of porous materials, such as rocks and soil, via pores (corresponding to large cavities) and throats (narrow constrictions). During extraction of networks from real void space, ambiguous definitions or determinations of pores and throats may cause significant errors in the prediction of single/multi‐phase transport properties. Meanwhile, the pore‐throat segmentation needs to exclude non‐physical parameters as much as possible. In this work, we propose a pore‐throat segmentation method based on local hydraulic resistance equivalence between the real void space and the simplified pore‐throat geometry. Each pore‐throat interface is carefully determined at the position where the simplified tubes preserve the local hydraulic resistance of the real space best. This local segmentation method ensures equivalency between extracted pore network and real pore space without any empirical or non‐physical parameters. After validations of accuracy and reliability by benchmarks, this method is applied to real porous materials including spherical pack, sand pack, sandstone, limestone, and carbonate. The single/two‐phase transport properties predicted by the new method agree well with the experimental data and the direct simulation results. The proposed method improves the accuracy of pore network model (PNM) predictions significantly with a slight increase in computational cost. This local pore‐throat segmentation method may enhance the capability of PNM for more complicated applications.

A Multi-Scale Homogenization Procedure for the Estimation of Young’s Modulus of Porous Materials by a Multi-Void Shape Model

Transversal cross-section pores of lotus-type porous materials are generally considered circular; however, they exhibit various pore geometries, which affect their effective properties. The main objective of this work is to develop a generalized model which allows estimating the effective Young’s modulus of multi-void shape porous microstructures by exploiting a relationship developed to evaluate the effective Young’s modulus of porous materials with single-void shape. A procedure based on free software is then proposed to allow the application of the proposed generalized model on real lotus-type porous material images to estimate the effective Young’s modulus. The free tool allows the processing of real porous materials images to obtain multi-void shape microstructures and their pores parameters data. The validation of the generalized model has been established by confronting the obtained results with experimental data taken from literature; an excellent agreement was observed. Therefore, it can be concluded that the proposed procedure associated with the generalized model can be used efficiently for predicting the effective Young’s modulus of the multi-void porous materials, particularly lotus-type porous materials.

Simulation of the Particle Transport Behaviors in Nanoporous Matter

The transport behaviors of proton into nanoporous materials were investigated using different Monte Carlo simulation codes such as GEANT4, Deeper and SRIM. The results indicated that porous structure could enhance the proton scattering effects due to a higher specific surface area and more boundaries. The existence of voids can deepen and widen the proton distribution in the targets due to relatively lower apparent density. Thus, the incident protons would transport deeper and form a wider Bragg peak in the end of the range, as the target materials are in a higher porosity state and/or have a larger pore size. The existence of voids also causes the local inhomogeneity of proton/energy distribution in micro/nano scales. As compared, the commonly used SRIM code can only be used to estimate roughly the incident proton range in nanoporous materials, based on a homogeneous apparent density equivalence rule. Moreover, the estimated errors of the proton range tend to increase with the porosity. The Deeper code (designed for evaluation of radiation effects of nuclear materials) can be used to simulate the transport behaviors of protons or heavy ions in a real porous material with porosity smaller than 52.3% due to its modeling difficulty, while the GEANT4 code has shown advantages in that it is suitable and has been proven to simulate proton transportation in nanoporous materials with porosity in its full range of 0~100%. The GEANT4 simulation results are proved consistent with the experimental data, implying compatibility to deal with ion transportation into homogeneously nanoporous materials.

Nanoparticle Tracking to Probe Transport in Porous Media.

From the granular and fractured subsurface environment to highly engineered polymer membranes used in pharmaceutical purification, porous materials are ubiquitous in nature and industrial applications. In particular, porous media are used extensively in processes including water treatment, pharmaceutical sterilization, food/beverage processing, and heterogeneous catalysis, where hindered mass transport is either essential to the process or a necessary but undesirable limitation. Unfortunately, there are currently no universal models capable of predicting mass transport based on a description of the porous material because real porous materials are complex and because many coupled dynamic mechanisms (e.g., adsorption, steric effects, hydrodynamic effects, electrostatic interactions, etc.) give rise to the observed macroscopic transport phenomena.While classical techniques, like nuclear magnetic resonance and dynamic light scattering, provide useful information about mass transport in porous media at the ensemble level, they provide limited insight into the microscopic mechanisms that give rise to complex phenomena such as anomalous diffusion, hindered pore-space accessibility, and unexpected retention under flow, among many others. To address this issue, we have developed refractive index matching imaging systems, combined with single-particle tracking methods, allowing the direct visualization of single-particle motion within a variety of porous materials.In this Account, we summarize our recent efforts to advance the understanding of nanoparticle transport in porous media using single-particle tracking methods in both fundamental and applied scenarios. First, we describe the basic principles for two-dimensional and three-dimensional single-particle tracking in porous materials. Then, we provide concrete examples of nanoparticle transport in porous materials from two perspectives: (1) understanding fundamental elementary particle transport processes in porous media, including pore accessibility and cavity escape, which limit transport in porous media, and (2) facilitating applications in industrial processes, e.g., by understanding the mechanisms of particle fouling and remobilization in filtration membranes. Finally, we provide an outlook of opportunities associated with investigating other types of mass transport in confined environments using single-particle tracking methods, including electrophoretic and self-propelled motion.

Pore-scale modeling of nanoparticle transport and retention in real porous materials

Modern pore-scale modeling has the ability to simulate flow and particle (i.e., colloids, nanoparticles) transport at the pore- and particle-scale in realistic porous media. The pore structure of a material can be obtained through x-ray micro tomography (XCT) or a similar three-dimensional imaging technique. From these data sets, fluid and particle transport can be simulated by direct numerical simulation of the fundamental equations of motion. In this paper, we employ the finite element method to simulate Stokes flow and, after the flow field is resolved, Lagrangian particle tracking to track the fate and transport of nanoparticles (NPs). The presented methodology for direct numerical modeling allowed the simulation of NP transport in real materials (i.e., natural or engineered samples that experiments can be performed on rather than a computer-generated porous medium) in larger domains or with more particles than previous works. XCT images of a micromodel and a Berea sandstone were used as the computational domains to analyze the effect of particle diameter, attractive and repulsive surface forces, flow rate, surface capacity, and XCT image-based mineralogy on particle transport. In the micromodel simulations, in the presence of attractive surface forces, NP effluent recovery increased as particle diameter increased and as flow rate increased – findings qualitatively consistent with published experimental data. The use of XCT image-based mineralogy to spatially distribute attachment sites (i.e., clay) in the Berea showed no difference in particle retention when compared to a random distribution of attachment sites. These results are being used to help understand ongoing micromodel experiments and to design continuum-scale models of NP transport.

A personal view on pore network models in drying technology

After 15 years of activity in pore network modeling, the author recalls previously published individual research results and puts them in a broader context. The evolution from a merely academic drying model for fundamental investigations to recent studies of industrial relevance is tracked. The article also recalls the important role of sophisticated experiments with micromodels and real porous materials to validate the theoretical work. Altogether, strong emphasis is put on past and remaining challenges in pore network modeling; and insight is given in the decisions concerning development and use of the various model versions.

Equivalent microstructure problem: Mathematical formulation and numerical solution

Analytical homogenization schemes, including the Mori–Tanaka (M–T) or Self-Consistent (S-C) schemes, are computationally attractive tools for estimating the homogenized properties of porous media. Utilizing these approaches, we evaluate the effective properties based on the solution of single inclusion problem and assuming a simplified morphology of microstructure (usually a finite number of inclusion families is postulated). The simplified microstructure is the main disadvantage of these methods since it does not conform in a geometrical sense to the microstructure of a real porous medium. In this work, we formulate the inverse problem of micromechanics in which we aim to identify a so-called equivalent microstructure for the real porous material. This microstructure has to preserve the overall response (thermal conductivity) that is analogous to that of real porous material, regardless of the conductivity of the fluid occupying the pore space. The equivalent microstructure (still simplified with respect to the real one) is a virtual one with morphology of oblate spheroids (pore space) embedded in a solid matrix (skeleton). The distribution of inclusions is described by the probability density function with a random variable being the semi-axis aspect ratio θ. The inverse problem is formulated as a linear Fredholm equation of the first kind supplemented with additional constraints. Stochastic optimization procedure is used to solve the inverse problem, i.e. identification of the probability density function. The methodology is verified against the theoretical results obtained via classical bounds on the effective thermal conductivity. Finally, the procedure is applied to real porous material, and an equivalent microstructure for sand, with respect to the overall thermal conductivity, is identified.

Freezing and Melting Transitions under Mesoscalic Confinement: Application of the Kossel–Stranski Crystal-Growth Model

The Kossel–Stranski model of crystal growth has been adopted to study freezing and melting of fluids in mesoporous materials. The model is found to exhibit the key features observed in the experiments, including shifted solid–liquid and liquid–solid transition temperatures, irreversibility between freezing and melting, and strong impact of the pore geometry. By first analyzing fluids confined to cylindrical pores, we obtain several important insights into the transition mechanisms. In particular, we establish the conditions for the occurrences of equilibrium and metastable transitions and derive exact analytical equations interrelating the transition temperatures, confinement size, and the interaction parameters of the Kossel crystal. Variation of the channel diameter along the channel axis, mimicking disorder in real porous materials, is shown to strongly affect both freezing and melting. The model predicts that the freezing transition in disordered materials is governed by the pore blocking mechanism. The melting transition is found to result as an interplay of two different transition mechanisms.

Diffusive Diffraction Phenomenon Observed by PGSE NMR Technique in a Sugar-Based Low-Molecular-Mass Gel

The paper presents the diffusive diffraction phenomenon observed by the single-pulse-gradient spin-echo (s-PGSE) NMR technique in a real porous material: a gel composed of low-molecular-mass gelator methyl-4,6-O-(p-nitrobenzylidene)-α-D-glucopyranoside and toluene. Thanks to this phenomenon, we can probe the true microstructure (not xerogel) in which the toluene diffuses. To analyze the measured diffusion-diffraction pattern, we employed a composite bicompartmental model that superimposes restricted diffusion in small cavities of the gel matrix within the bundles of crossing fibers, with free diffusion in large and unconfined compartments between the bundles of crossing fibers. For restricted diffusion a pore-hopping formalism was applied. The observation of the diffraction pattern and its analysis leads to the conclusion that the pores, in the slow diffusing compartment of studied gel are ordered, at least locally, and relatively monodisperse with a size of 64 μm. Moreover, the restricting walls formed by the crossing fibers are perpendicular to the direction of the diffusion gradient.

COMPUTATIONAL HOMOGENIZATION OF POLYCRYSTALLINE MATERIALS WITH PORES: A THREE-DIMENSIONAL GRAIN BOUNDARY FORMULATION

In this study, the influence of porosity on the elastic effective properties of polycrystalline materials is investigated using a 3D grain boundary micro mechanical model. The volume fraction of pores, their size and distribution can be varied to better simulate the response of real porous materials. The formulation is built on a boundary integral representation of the elastic problem for the grains, which are modeled as 3D linearly elastic orthotropic domains with arbitrary spatial orientation. The artificial polycrystalline morphology is represented using 3D Voronoi Tessellations. The formulation is expressed in terms of intergranular fields, namely displacements and tractions that play an important role in polycrystalline micromechanics. The continuity of the aggregate is enforced through suitable intergranular conditions. The effective material properties are obtained through material homogenization, computing the volume averages of micro-strains and stresses and taking the ensemble average over a certain number of microstructural samples. The obtained results show the capability of the model to assess the macroscopic effects of porosity.

A mass-conservative switching algorithm for modeling fluid flow in variably saturated porous media

Numerical solutions of flow equation in fluid content-based form or in fluid pressure head-based form are often tradeoffs between speed, accuracy, and convenience. The fluid-content based form can be solved quite rapidly with low CPU time and perfect mass balance. However, it cannot be used in saturated regions (as diffusivity function becomes infinite) and strictly becomes invalid in composite, layered, and real heterogeneous porous materials, due to singularity and discontinuity in fluid content profile. This formulation also gives misleading impression that gradient in fluid content causes the flow of fluid in porous materials, where in reality gravity and fluid pressure potential gradient produce the motion. The pressure head-based form, on the other hand, is more flexible but due to its highly nonlinear nature is much more time-consuming and produces poor global mass balance for dry initial conditions. Very fine spatial and temporal discretizations are needed to maintain mass balance property for these scenarios. The mixed form of the flow equation partially solves these issues as it maintains acceptable mass balance and is applicable to layered, heterogeneous, and composite fractured foundations. However, it is only applicable in unsaturated zones. In this study, a switching algorithm was proposed and implemented in which the mass conservative mixed form and the pressure head-based form were, respectively, used in the unsaturated and saturated zones of an initial-boundary value flow problem involving a variably saturated porous medium. The algorithm showed excellent agreement with a reference solution, obtained on a very fine spatiotemporal mesh. The simulator was then calibrated with several real-world large-scale experimental datasets. In all cases, the proposed algorithm exhibited close agreements with the experimental time–space series. The algorithm poses excellent mass balance property and can easily be used in both saturated and unsaturated regions without special treatment of fluid content discontinuities in heterogeneous and layered porous media. The proposed algorithm can also be extended to simulate multiphase and multidimensional flow problems.

Restricted diffusion and exchange of water in porous media: Average structure determination and size distribution resolved from the effect of local field gradients on the proton NMR spectrum

NMR Pulsed field gradient measurements of the restrained diffusion of confined fluids constitute an efficient method to probe the local geometry in porous media. In most practical cases, the diffusion decay, when limited to its principal part, can be considered as Gaussian leading to an apparent diffusion coefficient. The evolution of the latter as a function of the diffusion interval yields average information on the surface/volume ratio of porosities and on the tortuosity of the network. In this paper, we investigate porous model systems of packed spheres (polystyrene and glass) with known mean diameter and polydispersity, and, in addition, a real porous polystyrene material. Applying an Inverse Laplace Transformation in the second dimension reveals an evolution of the apparent diffusion coefficient as a function of the resonance frequency. This evolution is related to a similar evolution of the transverse relaxation time T2. These results clearly show that each resonance frequency in the water proton spectrum corresponds to a particular magnetic environment produced by a given pore geometry in the porous media. This is due to the presence of local field gradients induced by magnetic susceptibility differences at the liquid/solid interface and to slow exchange rates between different pores as compared to the frequency differences in the spectrum. This interpretation is nicely confirmed by a series of two-dimensional exchange experiments.

Modelling diffusion from simulated porous structures

An algorithm has been developed for the simulation of liquid diffusion from a porous material in the presence of a concentration gradient. A network model, Pore-Cor, was used to create simulated porous networks. These structures comprised cubic pores connected by cylindrical throats each centred on the coordinates of a three-dimensional 10 × 10 × 10 Cartesian matrix of positions. The simulated structure was then saturated with a mixture of benzene and cyclohexane and immersed in static pure cyclohexane. The algorithm then simulated the diffusion of benzene from the porous structure into the surrounding cyclohexane. The effects on the diffusion process of the geometries of the simulated structures were investigated by generating predefined structures and isolating the effect of each geometric parameter. The method presented in this study should lead to a predictive tool for studying the diffusion processes within real samples, as the pore network model can generate void structures which have porosity and percolation properties that closely match experimental mercury intrusion porosimetry data of real porous materials.

Co- and counter-current spontaneous imbibition into groups of capillary tubes with lateral connections permitting cross-flow

A model for co- and counter-current imbibition through independent capillaries has already been developed and experiments conducted to verify the theory [E. Unsal, G. Mason, N.R. Morrow, D.W. Ruth, J. Colloid Interface Sci. 306 (2007) 105]. In this paper, the work is extended to capillaries which are connected laterally and in which cross-flow can take place. The fundamental pore geometry is a rod in an angled round-bottomed slot with a gap between the rod and a capping glass plate. The surfaces of the slot, rod and plate form capillaries and interconnecting passages which have non-axisymmetric cross-sections. Depending on the gap size either (i) a large single meniscus, (ii) two menisci one on each side of the rod, or (iii) three menisci, one between the rod and the glass additional to the ones on each side can be formed. A viscous refined oil was applied to one end of the capillaries and co-current and counter-current spontaneous imbibition experiments were performed. The opposite end was left open to the atmosphere for co-current experiments. When the gap between the rod and the plate was large, the imbibing oil advanced into the tubes with the meniscus in the largest capillary always lagging behind the two menisci in the other two smaller capillaries. For counter-current imbibition experiments the open end was sealed and connected to a sensitive pressure transducer. In some experiments, the oil imbibed into the smaller capillaries and expelled air as a series of bubbles from the end of the largest capillary. In other experiments, the oil was allowed to imbibe part way into the tubes before counter-current imbibition was started. The meniscus curvatures of the capillaries have been calculated using the Mayer and Stowe–Princen method for different cell slot angles and gap sizes using a value of zero for the contact angle. These values have been compared with actual values by measuring the capillary rise in the tubes; agreement was very close. A model for co-current and counter-current imbibition has also been developed. The significance of this model is that some hydraulic/capillary properties are common for both co-current and counter-current imbibition. The experiments give an illustration of behavior expected in a real porous material and verify the importance of the ‘perfect cross-flow’ modification to the ‘bundle of parallel tubes’ model.

The effect of strain hardening in the extrusion of bimetallic tubes of porous internal layer

The effect of strain-hardening features of porous materials treated by cold extrusion through a conical die is studied theoretically in the present paper. The theoretical model developed by using the fundamentals of the Plasticity Theory of Real Porous Materials investigates all technological parameters affected in the case of extruding bimetallic tubes with a porous internal layer. Comparative analysis between strain-hardened and ideally plastic materials used in the fabrication of bimetallic tubular components, as far as their effect on various technological extrusion parameters is concerned, such as variation of porosity, applied load, level of total deformation, etc., has been conducted successfully, providing, in this manner, with useful concluding remarks from engineering point of view.

Mathematical modelling of plastic deformation processes of bimetallic tubes with porous-internal layer in conical dies

The present paper deals with the mathematical analysis of the extrusion of bimetallic tubes, made of porous materials. The Plasticity Theory of Real Porous Materials [G.L. Petrossian, Plastic Deformation of Powder Materials, Metallurgy, Moscow, 1988] is used for the establishment of real regularities of bimetal's plastic deformation and development of traditional technological schemes of its treatment. An algorithm and a computer program were composed to integrate the derived system of differential equations. The model created allows revealing the influence of initial porosity, applied loading, level of deformation, materials hardening and other technological parameters on the stress-strain state of the process. On the basis of numerical results obtained, graphs between components of stress-strain state of bimetal and porosity of internal layer are drawn, which allow solving important technological problems.

Wave-induced fluid flow in random porous media: Attenuation and dispersion of elastic waves

A detailed analysis of the relationship between elastic waves in inhomogeneous, porous media and the effect of wave-induced fluid flow is presented. Based on the results of the poroelastic first-order statistical smoothing approximation applied to Biot's equations of poroelasticity, a model for elastic wave attenuation and dispersion due to wave-induced fluid flow in 3-D randomly inhomogeneous poroelastic media is developed. Attenuation and dispersion depend on linear combinations of the spatial correlations of the fluctuating poroelastic parameters. The observed frequency dependence is typical for a relaxation phenomenon. Further, the analytic properties of attenuation and dispersion are analyzed. It is shown that the low-frequency asymptote of the attenuation coefficient of a plane compressional wave is proportional to the square of frequency. At high frequencies the attenuation coefficient becomes proportional to the square root of frequency. A comparison with the 1-D theory shows that attenuation is of the same order but slightly larger in 3-D random media. Several modeling choices of the approach including the effect of cross correlations between fluid and solid phase properties are demonstrated. The potential application of the results to real porous materials is discussed.

Grand Potential, Helmholtz Free Energy, and Entropy Calculation in Heterogeneous Cylindrical Pores by the Grand Canonical Monte Carlo Simulation Method

The adsorption of fluids in porous media is still an open area of research, since no model is able to explain all experimental features. The difficulties rise from the complexity of the real porous materials which present surface heterogeneities, large pore size distributions, and complex networks of interconnected pores. In parallel to experimental efforts trying to produce more ordered porous materials, theoreticians try to introduce more disorder in their models, with the help of molecular simulation for instance. This grand canonical Monte Carlo simulation study concentrates on the adsorption of a simple Lennard-Jones fluid in three porous substrates, to compare the effect of purely geometric heterogeneity (spatial deformation of the external potential) as opposed to purely chemical heterogeneity (amplitude variations of the external potential). This separation is unrealistic, since geometric fluctuations of a real pore diameter along its axis generally induce variations in the amplitude of the external potential created by the pore. However it enables one to compare both effects. In this paper, a thermodynamic integration scheme is applied to a complete set of adsorption/desorption isotherms. The grand potential, free energy, and entropy are calculated, which allows one to discuss the features of the phase diagrams. It is shown that a purely geometric deformation (undulation) of the external potential does not affect the thermodynamic characteristics of the confined fluid. On the other hand, amplitude modulation of the external potential (chemical heterogeneity) strongly distorts the phase diagram. This heterogeneity is actually able to stabilize a "bridgelike" phase which corresponds to an accumulation of molecules in the most attractive region of the pore.

Network Modelling of Apparent-Relative Permeability of Gas in Heavy Oils

Abstract Some of the solution-gas drive heavy oil reservoirs known as foamy oil reservoirs, in Canada, Venezuela, and other countries have demonstrated a high primary oil recovery factor (>10%), a low producing gas-oil ratio, a low reservoir pressure decline, and a high oil production rate. One of the hypotheses to explain these unusual behaviours is that the gas mobility in a foamy oil flow is much lower than that in conventional oil, leading to improved recovery performance. In this study, the immiscible two-phase micro-displacement in porous media is modelled by using a network of pores of converging-diverging geometry. The effect of viscosity of one phase (oil) on the mobility of another phase (gas) is included in the model. The developed model is used to simulate the motion of the dispersed bubbles in an initially oil-filled network, and to determine bubble mobility. The obtained results showed that bubble mobility decreased drastically by increasing the oil viscosity. The results also showed that dispersion of gas leads to lower mobility of bubbles. Dispersed gas flow and low bubble mobility are believed to lead to improved recovery in foamy oil reservoirs. Introduction Some of the solution-gas drive heavy oil reservoirs in Canada, Venezuela, China, and Oman have demonstrated unusually high primary production rates, high primary oil recovery factors (>10%), low producing gas-oil ratios, and low reservoir pressure declines(1, 2). To explain these unusual behaviours, three fundamental reasons have been suggested: geomechanical effects(3), special fluid properties(4), and unusual flow dependent properties of oil and gas(5). Most researchers now believe that the low mobility of gas is the main reason for low producing GOR and high recoveries obtained(1, 2). Gas mobility in a heavy oil system is investigated in this paper. In one study, solution-gas drive experiments were performed in an identical sand-pack, using light oil and heavy oil(5). The experimental studies clearly showed that gas mobility in the two experiments differed by about four orders of magnitude. It has been observed that matching of field(6) and laboratory depletion data(5) required assigning extremely low values of gas relative permeability. The relative permeability functions of these studies, however, were obtained through history matching. Of primary interest is, how can relative permeability functions be determined for a particular system a priori? Following this question, the first step is to find what parameters affect relative permeability functions; the second is to find how these factors are ranked in their importance. These steps are investigated in this paper. A microscopic scale (network) model is developed and used to investigate the effects of different parameters, in particular, oil viscosity on gas mobility in porous media. Network models are simplified mathematical representations of real porous material. The objective of a porous network model is to provide a reasonable idealization of the complex geometry of a real porous medium on a microscopic scale, so that the related fluid flow can be treated mathematically at a manageable level of complexity.