Tag: local porosity theory

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Heterogeneous Materials Mathematical Physics Percolation Porous Media

Multiscale local porosity theory, weak limits, and dielectric response in composite and porous media

R. Hilfer

Journal of Mathematical Physics 59, 103511 (2018)
https://doi.org/10.1063/1.5063466

submitted on
Thursday, December 22, 2016

A mathematical scaling approach to macroscopic heterogeneity of composite and porous media is introduced. It is based on weak limits of uniformly bounded measurable functions. The limiting local porosity distributions, that were introduced in Advances in Chemical Physics, vol XCII, p. 299-424 (1996), are found to be related to Young measures of a weakly convergent sequence of local volume fractions. The Young measures determine frequency dependent complex dielectric functions of multiscale media within a generalized selfconsistent effective medium approximation. The approach separates scales by scale factor functions of regular variation. It renders upscaled results independent of the shape of averaging windows upon reaching the scaling limit.



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Porous Media review article Transport Processes

Local Porosity Theory and Stochastic Reconstruction for Porous Media

R. Hilfer

in: Räumliche Statistik und Statistische Physik
edited by: D. Stoyan and K. Mecke
Springer, Berlin, 203 (2000)
10.1007/3-540-45043-2
ISBN: 978-3-642-08725-7

submitted on
Tuesday, February 23, 1999

The paper reviews recent developments in local porosity theory, and discusses its application to the analysis of stochastic reconstruction models for sedimentary rocks. Special emphasis is placed on the geometric observables in local porosity theory and their relation with the Hadwiger theorem from stochastic geometry. In addition recent results for the exact calculation of effective physical transport properties are given for a Fontainebleau sandstone. The calculations pertain to potential type problems such as electrical conduction, dielectric relaxation, diffusion or Darcy flow. The exact results are compared to the approximate parameterfree predictions from local porosity, and are found to be in good agreement.



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dielectric relaxation diffusion electrical conductivity fluid flow Heterogeneous Materials Porous Media

Quantitative Prediction of Effective Material Properties of Heterogeneous Media

J. Widjajakusuma, B. Biswal, R. Hilfer

Computational Materials Science 16, 70 (1999)
https://doi.org/10.1016/S0927-0256(99)00047-6

submitted on
Thursday, October 8, 1998

Effective electrical conductivity and electrical permittivity of water-saturated natural sandstones are evaluated on the basis of local porosity theory (LPT). In contrast to earlier methods, which characterize the underlying microstructure only through the volume fraction, LPT incorporates geometric information about the stochastic microstructure in terms of local porosity distribution and local percolation probabilities. We compare the prediction of LPT and of traditional effective medium theory with the exact results. The exact results for the conductivity and permittivity are obtained by solving the microscopic mixed boundary value problem for the Maxwell equations in the quasistatic approximation. Contrary to the predictions from effective medium theory, the predictions of LPT are in better quantitative agreement with the exact results.



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image analysis Porous Media Simulations

Threedimensional Local Porosity Analysis of Porous Media

B. Biswal, C. Manwart, R. Hilfer

Physica A 255, 221 (1998)
https://doi.org/10.1016/S0378-4371(98)00111-3

submitted on
Thursday, February 5, 1998

A quantitative comparison of the pore space geometry for three natural sandstones is presented. The comparison is based on local porosity theory which provides a geometric characterization of stochastic microstructures. The characterization focusses on porosity and connectivity fluctuations. Porosity fluctuations are measured using local porosity distributions while connectivity fluctuations are measured using local percolation probabilities. We report the first measurement of local percolation probability functions for experimentally obtained three-dimensional pore space reconstructions. Our results suggest the use of local porosity distributions and percolation probabilities as a quantitative method to compare microstructures of models and experiment.



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Heterogeneous Materials Percolation Porous Media

Local Entropy Characterization of Correlated Random Microstructures

C. Andraud, A. Beghdadi, E. Haslund, R. Hilfer, J. Lafait, B. Virgin

Physica A 235, 307 (1997)
https://doi.org/10.1016/S0378-4371(96)00354-8

submitted on
Tuesday, August 13, 1996

A rigorous connection is established between the local porosity entropy introduced by Boger et al. (Physica A 187 (1992) 55) and the configurational entropy of Andraud et al. (Physica A 207 (1994) 208). These entropies were introduced as morphological descriptors derived from local volume fluctuations in arbitrary correlated microstructures occurring in porous media, composites or other heterogeneous systems. It is found that the entropy lengths at which the entropies assume an extremum become identical for high enough resolution of the underlying configurations. Several examples of porous and heterogeneous media are given which demonstrate the usefulness and importance of this morphological local entropy concept.



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Percolation Porous Media

Rescaling Relations between Two- and Three Dimensional Local Porosity Distributions for Natural and Artificial Porous Media

B. Virgin, E. Haslund, R. Hilfer

Physica A 232, 1-10 (1996)
https://doi.org/10.1016/0378-4371(96)00131-8

submitted on
Friday, March 29, 1996

Local porosity distributions for a three-dimensional porous medium and local porosity distributions for a two-dimensional plane-section through the medium are generally different. However, for homogeneous and isotropic media having finite correlation lengths, a good degree of correspondence between the two sets of local porosity distributions can be obtained by rescaling lengths, and the mapping associating corresponding distributions can be found from two-dimensional observations alone. The agreement between associated distributions is good as long as the linear extent of the measurement cells involved is somewhat larger than the correlation length, and it improves as the linear extent increases. A simple application of the central limit theorem shows that there must be a correspondence in the limit of very large measurement cells, because the distributions from both sets approach normal distributions. A normal distribution has two independent parameters: the mean and the variance. If the sample is large enough, LPDs from both sets will have the same mean. Therefore corresponding distributions are found by matching variances of two and three dimensional local porosity distributions. The variance can be independently determined from correlation functions. Equating variances leads to a scaling relation for lengths in this limit. Three particular systems are examined in order to show that this scaling behavior persists at smaller length-scales.



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Percolation Porous Media

Local Porosity Theory for the Transition from Microscales to Macroscales in Porous Media

R. Hilfer, B. Virgin, T. Rage

ERCOFTAC Bull. 28, 6 (1996)

submitted on
Monday, February 12, 1996

A quantitative understanding of fluid flow and other transport processes in porous media remains a prerequisite for progress in many disciplines such as hydrology, petroleum technology, chemical engineering, environmental protection, nuclear waste storage, drug transport in biological tissues, catalysis, paleontology, filtration and separation technology to name but a few. While the microscopic equations governing flow and transport in porous media are often well known, the macroscopic laws are usually different and much less understood. Most approaches in computational fluid dynamics for porous media avoid to discuss or control the problems arising in the transition from a microscale (pores) to the macroscale (field or laboratory). As a consequence the upscaling of transport processes, particularly for immiscible fluid-fluid displacement, has remained difficult.



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dielectric relaxation Disordered Systems Porous Media

Measurement of Local Porosities and Dielectric Dispersion for a Water Saturated Porous Medium

E. Haslund, B.D. Hansen, R. Hilfer, B. Nøst

Journal of Applied Physics 76, 5473 (1994)
https://doi.org/10.1063/1.357205

submitted on
Monday, October 4, 1993

The frequency‐dependent conductivity and dielectric constant of a salt‐water‐saturated porous glass specimen have been measured. The measurements cover the full frequency range of the Maxwell–Wagner dispersion. The experimental results have been compared with the recently introduced local porosity theory and with previous theories. For the purpose of comparing with the local porosity theory experimental measurements of local porosity distributions from digitized pore space images are presented. These experimental porosity distributions are then used for a first experimental test of local porosity theory. The comparison with previous theoretical expressions for the frequency‐dependent effective dielectric function shows that local porosity theory constitutes a significant improvement in the quantitative agreement.



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Disordered Systems fluid flow Porous Media Transport Processes

Local Porosity Theory for Flow in Porous Media

R. Hilfer

Physical Review B 45, 7115 (1992)
https://doi.org/10.1103/PhysRevB.45.7115

submitted on
Thursday, March 28, 1991

A recently introduced geometric characterization of porous media based on local-porosity distributions and local-percolation probabilities is used to calculate dc permeabilities for porous media. The disorder in porous media is found to be intimately related to the percolation concept. The geometric characterization is shown to open a possibility for understanding experimentally observed scaling relations between permeability, formation factor, specific internal surface, and porosity. In particular, Kozeny’s equation relating effective permeability and bulk porosity and the power lawrelation between permeability and formation factor are analyzed. A simple and general consolidation model is introduced. It is based on the reduction of local porosities and emphasizes the general applicability and flexibility of the local-porosity concept. The theoretical predictions are compared with the experimentally observed range for the exponents, and are found to be in excellent agreement.



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