Category: Percolation

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dielectric relaxation Disordered Systems electrical conductivity Heterogeneous Materials Percolation Transport Processes

Effective transport coefficients of anisotropic disordered materials

R. Hilfer, J. Hauskrecht

European Physical Journal B 95, 117 (2022)
https://doi.org/10.1140/epjb/s10051-022-00338-5

submitted on
Tuesday, January 4, 2022

A novel effective medium theory for homogenized transport coefficients of anisotropic mixtures of possibly anisotropic materials is developed. Existing theories for isotropic systems cannot be easily extended, because that would require geometric characterizations of anisotropic connectivity. In this work anisotropic connectivity is characterized by introducing a tensor that is constructed from a histogram of local percolating directions. The construction is inspired by local porosity theory. A large number of known and unknown generalized effective medium approximations for anisotropic media are obtained as limiting special cases from the new theory. Among these limiting cases the limit of strong cylindrical anisotropy is of particular interest. The parameter space of the generalized theory is explored, and the advanced results are applied to experiment.



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Disordered Systems Heterogeneous Materials Percolation Porous Media

Percolativity of Porous Media

R. Hilfer, J. Hauskrecht

Transport in Porous Media 145, 1-12 (2022)
https://doi.org/10.1007/s11242-021-01735-7

submitted on
Monday, April 19, 2021

Connectivity and connectedness are non-additive geometric functionals on the set of pore scale structures. They determine transport of mass, volume or momentum in porous media, because without connectivity there cannot be transport. Percolativity of porous media is introduced here as a geometric descriptor of connectivity, that can be computed from the pore scale and persists to the macroscale through a suitable upscaling limit. It is a measure that combines local percolation probabilities with a probability density of ratios of eigenvalues of the tensor of local percolating directions. Percolativity enters directly into generalized effective medium approximations. Predictions from these generalized effective medium approximations are found to be compatible with apparently anisotropic Archie correlations observed in experiment.



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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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Percolation Porous Media Two-Phase Flow

Percolation as a basic concept for macroscopic capillarity

R. Hilfer, F. Doster

Transport in Porous Media 82, 507 (2010)
https://doi.org/10.1007/s11242-009-9395-0

submitted on
Wednesday, November 12, 2008

The concepts of relative permeability and capillary pressure are crucial for the accepted traditional theory of two phase flow in porous media. Recently, a theoretical approach was introduced that does not require these concepts as input (Hilfer, Physica A, 359:119–128, 2006a; Phys. Rev. E, 73:016307, 2006b). Instead it was based on the concept of hydraulic percolation of fluid phases. This paper presents the first numerical solutions of the coupled nonlinear partial differential equations introduced in Hilfer (Phys. Rev. E, 73:016307, 2006b). Approximate numerical results for saturation profiles in one spatial dimension have been calculated. Long time limits of dynamic time-dependent profiles are compared to stationary solutions of the traditional theory. The initial and boundary conditions are chosen to model the displacement processes that occur when a closed porous column containing two immiscible fluids of different density is raised from a horizontal to a vertical position in a gravitational field. The nature of the displacement process may change locally in space and time between drainage and imbibition. The theory gives local saturations for nonpercolating trapped fluids near the endpoint of the displacement.



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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

Local Percolation Probabilities for a Natural Sandstone

R. Hilfer, T. Rage, B. Virgin

Physica A 241, 105 (1997)
https://doi.org/10.1016/S0378-4371(97)00067-8

submitted on
Thursday, July 25, 1996

Local percolation probabilities are used to characterize the connectivity in porous and heterogeneous media. Together with local porosity distributions they allow to predict transport properties. While local porosity distributions are readily obtained, measurements of the local percolation probabilities are more difficult and have not been attempted previously. First measurements of three-dimensional local porosity distributions and percolation probabilities from the pore space reconstruction of a natural sandstone show that theoretical expectations and experimental results are consistent.



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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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Disordered Systems electrical conductivity Percolation Transport Processes

Correlated Hopping in a Disordered Medium

R. Hilfer

Physical Review B 44, 628 (1991)
10.1103/PhysRevB.44.628

submitted on
Monday, March 6, 1989

This paper discusses random walks with memory on a percolating network as a model of correlated hopping transport through a disordered system. Correlations can arise from such sources as hard-core and Coulomb repulsions, correlated hops of groups of particles, or lattice-relaxation effects. In general these correlations will result in a difference between the hopping probability for return to the previously visited site and the probability to jump to another nearest neighbor of the currently occupied site. Thus the hopping process possesses a memory of its previous hop. Such a random walk is investigated in this paper for the case of bond percolation on a regular lattice. The frequency-dependent conductivity σ(ω) is calculated using a generalized effective-medium approximation. Results are presented for the linear chain and the hexagonal lattice. New features appear in both the real and the imaginary part of σ. These depend on the strength of the correlations and on the concentration of bonds. As an example, the possibility of a pronounced maximum in the real part of σ(ω) at finite frequencies is found, which is sometimes accompanied by a change of sign in the imaginary part. The results are found to agree qualitatively with experimental data on ionic transport in Na+ β-alumina, where both disorder and correlations are known to be important.



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Disordered Systems electrical conductivity Lattice Models Nonequilibrium Percolation Statistical Physics Transport Processes

Correlated Random Walks in Dynamically Disordered Systems

R. Hilfer, R. Orbach

in: Dynamical Processes in Condensed Molecular Systems
edited by: J. Klafter and J. Jortner and A. Blumen
World Scientific Publ.Co., Singapore, 175 (1989)
https://doi.org/10.1142/9789814434379_0009
ISBN: 978-981-4434-37-9

submitted on
Tuesday, November 22, 1988

We discuss correlated hopping motion in a dynamically disordered environment. Particles of type A with one hopping rate diffuse in a background of B-particles with a different hopping rate. Double occupancy of sites is forbidden. Without correlations the limit in which the ratio of hopping rates diverges corresponds to diffusion on a percolating network, while the case of equal hopping rates is that of self-diffusion in a lattice gas. We consider also the effect of correlations. In general these will change the transition rate of the A-particle to the previously occupied site as compared to the rate for transitions to all other neighbouring sites. We calculate the frequency dependent conductivity for this model with arbitrary ratio of hopping rates and correlation strength. Results are reported for the two dimensional hexagonal lattice and the three dimensional face centered cubic lattice. We obtain our results from a generalization of the effective medium approximation for frozen percolating networks. We predict the appearance of new features in real and imaginary part of the conductivity as a result of correlations. Crossover behaviour resulting from the combined effect of disorder and correlations leads to apparent power laws over roughly one to two decades in frequency. In addition we find a crossover between a low frequency regime where the response is governed by the rearrangements in the geometry and a high frequency regime where the geometry appears frozen. We calculate the correlation factor for the d.c. limit and check our results against Monte Carlo simulations on the hexagonal and face centered cubic lattices. In all cases we find good agreement.



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Disordered Systems electrical conductivity Percolation Random Walks Stochastic Processes Transport Processes

Continuous Time Random Walk Approach to Dynamic Percolation

R. Hilfer, R. Orbach

Chemical Physics 128, 275 (1988)
https://doi.org/10.1016/0301-0104(88)85076-6

submitted on
Friday, September 16, 1988

We present an approximate solution for time (frequency) dependent response under conditions of dynamic percolation which may be related to excitation transfer in some random structures. In particular, we investigate the dynamics of structures where one random component blocks a second (carrier) component. Finite concentrations of the former create a percolation network for the latter. When the blockers are allowed to move in time, the network seen by the carriers changes with time, allowing for long-range transport even if the instantaneous carrier site availability is less than pc, the critical percolation concentration. A specific example of this situation is electrical transport in sodium β”-alumina. The carriersare Na+ ions which can hop on a two-dimensional honeycomb lattice. The blockers are ions of much higher activation energy, such as Ba++. We study the frequency dependence of the conductivity for such a system. Given a fixed Ba++ hopping rate the Na+ ions experience a frozen site percolation environment for frequencies larger than the inverse hopping rate. At frequencies smaller than the inverse hopping rate, the Na+ ions experience a dynamic environment which allows long-rangetransport, even below the percoltion threshold. A continuous time random walk mode1 combined with an effective medium approximation allows us to arrive at a numerical solution for the frequency-dependent Na+ conductivity which clearly exhibits the crossover from frozen to dynamic environment.



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