Category: Disordered Systems

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Porous Media Two-Phase Flow

Macroscopic Capillarity and Hysteresis for Flow in Porous Media

R. Hilfer

Physical Review E 73, 016307 (2006)
https://doi.org/10.1103/PhysRevE.73.016307

submitted on
Friday, May 27, 2005

A macroscopic theory for capillarity in porous media is presented, challenging the established view that capillary pressure and relative permeability are constitutive parameter functions. The capillary pressure function in the present theory is not an input parameter but an outcome. The theoretical approach is based on introducing the residual saturations explicitly as state variables [as in Phys. Rev. E 58, 2090 (1998)]. Capillary pressure and relative permeability functions are predicted to exist for special cases. They exhibit hysteresis and process dependence as known from experiment.



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Porous Media Two-Phase Flow

Capillary Pressure, Hysteresis and Residual Saturation in Porous Media

R. Hilfer

Physica A 359, 119 (2006)
https://doi.org/10.1016/j.physa.2005.05.086

submitted on
Friday, May 20, 2005

A macroscopic theory for capillarity in porous media is presented. The capillary pressure function in this theory is not an input parameter but an outcome. The theory is based on introducing the trapped or residual saturations as state variables. It allows to predict spatiotemporal changes in residual saturation. The theory yields process dependence and hysteresis in capillary pressure as its main result.



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image analysis Porous Media

Characterization of Porous Media by Local Porosities, Minkowski- and Non-Minkowski Functionals

R. Hilfer

Microscopy and Microanalysis 10, 72 (2004)
DOI: 10.1017/S1431927604884472

submitted on
Friday, April 16, 2004

The presentation reviews local porosity theory. Recent results and new developments are discussed.



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Heterogeneous Materials Porous Media Two-Phase Flow

Dimensional analysis and upscaling of two-phase flow in porous media with piecewise constant heterogeneities

R. Hilfer, R. Helmig

Advances in Water Resources 27, 1033 (2004)
https://doi.org/10.1016/j.advwatres.2004.07.003

submitted on
Monday, March 15, 2004

Dimensional analysis of the traditional equations of motion for two-phase flow in porous media allows to quantify the influence of heterogeneities. The heterogeneities are represented by position dependent capillary entry pressures and position dependent permeabilities. Dimensionless groups quantifying the influence of random heterogeneities are identified. For the case of heterogeneities with piecewise constant constitutive parameters (e.g. permeabilities, capillary pressures) we find that the upscaling ratio defined as the ratio of system size and the scale at which the constitutive parameters are known has to be smaller than the fluctuation strength of the heterogeneities defined e.g. as the ratio of the standard deviation to the mean value of a fluctuating quantity.



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fluid flow Porous Media

Numerical Simulation of Creeping Fluid Flow in Reconstruction Models of Porous Media

C. Manwart, R. Hilfer

Physica A 314, 706 (2002)
https://doi.org/10.1016/S0378-4371(02)01193-7

submitted on
Sunday, March 30, 2003

In this paper we examine representative examples of realistic three-dimensional models for porous media by comparing their geometry and permeability with those of the original experimental specimen. The comparison is based on numerically exact evaluations of permeability, porosity, speci/c internal surface, mean curvature, Euler number and local percolation probabilities. The experimental specimen is a three-dimensional computer tomographic image of Fontainebleau sandstone. The three models are stochastic reconstructions for which many of the geometrical characteristics coincide with those of the experimental specimen. We /nd that in spite of the similarity in the geometrical properties the permeability and formation factor can differ greatly between models and the experiment.



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dielectric relaxation diffusion electrical conductivity Heterogeneous Materials Porous Media

Quantitative comparison of meanfield mixing laws for conductivity and dielectric constant of porous media

R. Hilfer, J. Widjajakusuma, B. Biswal

Physica A 318, 319 (2003)
https://doi.org/10.1016/S0378-4371(02)01197-4

submitted on
Tuesday, June 4, 2002

Exact numerical solution of the electrostatic disordered potential problem is carried out for four fully discretised threedimensional experimental reconstructions of sedimentary rocks. The measured effective macroscopic dielectric constants and electrical conductivities are compared with parameterfree predictions from several mean field type theories. All these theories give agreeable results for low contrast between the media. Predictions from Local porosity theory, however, match for the entire range of contrast.



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dielectric relaxation Fractional Calculus Fractional Time Glasses

Experimental Evidence for Fractional Time Evolution in Glass Forming Materials

R. Hilfer

Chem.Phys. 284, 399 (2002)
https://doi.org/10.1016/S0301-0104(02)00670-5

submitted on
Friday, December 7, 2001

The infinitesimal generator of time evolution in the standard equation for exponential (Debye) relaxation is replaced with the infinitesimal generator of composite fractional translations. Composite fractional translations are defined as a combination of translation and the fractional time evolution introduced in [Physica A, 221 (1995) 89]. The fractional differential equation for composite fractional relaxation is solved. The resulting dynamical susceptibility is used to fit broad band dielectric spectroscopy data of glycerol. The composite fractional susceptibility function can exhibit an asymmetric relaxation peak and an excess wing at high frequencies in the imaginary part. Nevertheless it contains only a single stretching exponent. Qualitative and quantitative agreement with dielectric data for glycerol is found that extends into the excess wing. The fits require fewer parameters than traditional fit functions and can extend over up to 13 decades in frequency.



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dielectric relaxation Fractional Calculus Fractional Time Glasses

Fitting the excess wing in the dielectric α-relaxation of propylene carbonate

R. Hilfer

Journal of Physics: Condensed Matter 14, 2297 (2002)
https://doi.org/10.1088/0953-8984/14/9/318

submitted on
Wednesday, November 28, 2001

A novel fitting function for the complex frequency-dependent dielectric susceptibility is introduced and compared against other fitting functions for experimental broadband dielectric loss spectra of propylene carbonate taken from Schneider et al (Schneider U, Lunkenheimer P, Brand R and Loidl A 1999 Phys. Rev. E 59 6924). The fitting function contains a single stretching exponent similar to the familiar Cole–Davidson or Kohlrausch stretched exponential fits. It is compared to these traditional fits as well as to the Havriliak–Negami susceptibility and a susceptibility for a two-step Debye relaxation. The results for the novel fit are found to give superior agreement.



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dielectric relaxation Glasses Special Functions

H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems

R. Hilfer

Physical Review E 65, 061510 (2002)
https://doi.org/10.1103/PhysRevE.65.061510

submitted on
Thursday, June 28, 2001

Analytical expressions in the time and frequency domains are derived for non-Debye relaxation processes. The complex frequency-dependent susceptibility function for the stretched exponential relaxation function is given for general values of the stretching exponent in terms of H-functions. The relaxation functions corresponding to the complex frequency-dependent Cole-Cole, Cole-Davidson, and Havriliak-Negami susceptibilities are given in the time domain in terms of H-functions. It is found that a commonly used correspondence between the stretching exponent of Kohlrausch functions and the stretching parameters of Havriliak-Negami susceptibilities are not generally valid.



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dielectric relaxation Glasses Nonequilibrium Special Functions

Analytical representations for relaxation functions of glasses

R. Hilfer

Journal of Non-Crystalline Solids 305, 122 (2002)
https://doi.org/10.1016/S0022-3093(02)01088-8

submitted on
Friday, April 13, 2001

Analytical representations in the time and frequency domains are derived for the most frequently used phenomenological fit functions for non-Debye relaxation processes. In the time domain the relaxation functions corresponding to the complex frequency dependent Cole–Cole, Cole–Davidson and Havriliak–Negami susceptibilities are also rep- resented in terms of H-functions. In the frequency domain the complex frequency dependent susceptibility function corresponding to the time dependent stretched exponential relaxation function is given in terms of H-functions. The new representations are useful for fitting to experiment.



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Fractional Calculus Glasses

On Fractional Relaxation

R. Hilfer

Fractals 11, 251 (2003)
https://doi.org/10.1142/S0218348X03001914

submitted on
Monday, April 2, 2001

Generalized fractional relaxation equations based on generalized Riemann-Liouville derivatives are combined with a simple short time regularization and solved exactly. The solution involves generalized Mittag-Leffler functions. The associated frequency dependent susceptibilities are related to symmetrically broadened Cole-Cole susceptibilities occurring as Johari Goldstein β -relaxation in many glass formers. The generalized susceptibilities exhibit a high frequency wing and strong minimum enhancement.



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Fractional Calculus Glasses

On Fractional Relaxation

R. Hilfer

in: Scaling and Disordered Systems
edited by: F. Family and M. Daoud and H. Herrmann and H.E. Stanley
World Scientific, Singapore, 251 (2002)
https://doi.org/10.1142/9789812778109_0026
ISBN: 978-981-02-4838-3

submitted on
Monday, April 2, 2001

Generalized fractional relaxation equations based on generalized Riemann-Liouville derivatives are combined with a simple short time regularization and solved exactly. The solution involves generalized Mittag-Leffler functions. The associated frequency dependent susceptibilities are related to symmetrically broadened Cole-Cole susceptibilities occurring as Johari Goldstein β-relaxation in many glass formers. The generalized susceptibilities exhibit a high frequency wing and strong minimum enhancement.



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electrical conductivity fluid flow Porous Media

Permeability and Conductivity for Reconstruction Models of Porous Media

R. Hilfer, C. Manwart

Physical Review E 64, 21304 (2001)
https://doi.org/10.1103/PhysRevE.64.021304

submitted on
Friday, October 27, 2000

The purpose of this paper is to examine representative examples of realistic three-dimensional models for porous media by comparing their geometrical and transport properties with those of the original experimental specimen. The comparison is based on numerically exact evaluations of permeability, formation factor, porosity, specific internal surface, mean curvature, Euler number, local porosity distributions, and local percolation probabilities. The experimental specimen is a three-dimensional computer tomographic image of Fontainebleau sandstone. The three models are examples of physical and stochastic reconstructions for which many of the geometrical characteristics coincide with those of the experimental specimen. We find that in spite of the similarity in the geometrical properties the permeability and formation factor can differ greatly between models and experiment. Our results seem to indicate that the truncation of correlations is responsible for some of these observed discrepancies. A physical reconstruction model by Bakke and Øren [SPEJ 2, 136 (1997)] based on sedimentation, compaction and diagenesis of sandstones yields surprisingly accurate predictions for permeability and conductivity. These findings imply that many of the presently used geometric descriptors of porous media are insufficient for the prediction of transport.



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image analysis Porous Media

On the Analysis of Spatial Binary Images

C. Lang, J. Ohser, R. Hilfer

Journal of Microscopy 203, 303 (2001)
https://doi.org/10.1046/j.1365-2818.2001.00899.x

submitted on
Tuesday, March 28, 2000

This paper deals with the analysis of spatial images taken from microscopically heterogeneous but macroscopically homogeneous microstructures. A new method is presented, which is strictly based on integral‐geometric formulae such as Crofton’s intersection formulae and Hadwiger’s recursive definition of the Euler number. By means of this approach the quermassdensities can be expressed as the inner products of two vectors where the first vector carries the ‘integrated local knowledge’ about the microstructure and the second vector depends on the lateral resolution of the image as well as the quadrature rules used in the discretization of the integral‐geometric formulae. As an example of application we consider the analysis of spatial microtomographic images obtained from natural sandstones.



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image analysis Lattice Models Porous Media

Erosion-Dilation Analysis for Experimental and Synthetic Microstructures of Sedimentary Rock

A. Tscheschel, D. Stoyan, R. Hilfer

Physica A 284, 46 (2000)
https://doi.org/10.1016/S0378-4371(00)00116-3

submitted on
Thursday, February 17, 2000

Microstructures such as rock samples or simulated structures can be described and characterized by means of ideas of spatial statistics and mathematical morphology. A powerful approach is to transform a given 3D structure by operations of mathematical morphology such as dilation and erosion. This leads to families of structures, for which various characteristics can be determined, for example, porosity, specific connectivity number or correlation and connectivity functions. An application of this idea leads to a clear discrimination between a sample of Fontainebleau sandstone and two simulated samples.



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image analysis Porous Media review article

Review on Scale Dependent Characterization of the Microstructure of Porous Media

R. Hilfer

Transport in Porous Media 46, 373 (2002)
https://doi.org/10.1023/A:1015014302642

submitted on
Tuesday, February 8, 2000

The paper discusses local porosity theory and its relation with other geometric characterization methods for porous media such as correlation functions and contact distributions. Special emphasis is placed on the charcterization of geometric observables through Hadwigers theorem in stochastic geometry. The four basic Minkowski functionals are introduced into local porosity theory, and for the first time a relationship is established between the Euler characteristic and the local percolation probabilities. Local porosity distributions and local percolation probabilities provide a scale dependent characterization of the microstructure of porous media that can be used in an effective medium approach to predict transport.



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Porous Media

Stochastic Reconstruction of Sandstones

C. Manwart, S. Torquato, R. Hilfer

Physical Review E 62, 893 (2000)
https://doi.org/10.1103/PhysRevE.62.893

submitted on
Friday, February 4, 2000

A simulated annealing algorithm is employed to generate a stochastic model for a Berea sandstone and a Fontainebleau sandstone, with each a prescribed two-point probability function, lineal-path function, and “pore size” distribution function, respectively. We find that the temperature decrease of the annealing has to be rather quick to yield isotropic and percolating configurations. A comparison of simple morphological quantities indicates good agreement between the reconstructions and the original sandstones. Also, the mean survival time of a random walker in the pore space is reproduced with good accuracy. However, a more detailed investigation by means of local porosity theory shows that there may be significant differences of the geometrical connectivity between the reconstructed and the experimental samples.



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dielectric relaxation diffusion electrical conductivity fluid flow Porous Media

Effective Physical Properties of Sandstones

J. Widjajakusuma, R. Hilfer

in: IUTAM Symposium on Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials
edited by: W. Ehlers
Solid Mechanics and Its Applications, vol. 87,Kluwer, Dordrecht, 113 (2001)
10.1007/0-306-46953-7
ISBN: 978-0-7923-6766-6

submitted on
Wednesday, October 6, 1999

In this paper we continue the investigation of the effective transport parameters of a digitized sample of Fontainebleau sandstone and three reconstruction models discussed previously in Biswal et. al., Physica A 273, 452 (1999). The effective transport parameters are computed directly by solving the disordered Laplace equation via a finite-volume method. We find that the transport properties of two stochastic models differ significantly from the real sandstone. Moreover, the effective transport parameters are predicted by employing local porosity theory and various traditional mixing-laws (such as effective medium approximation or Maxwell-Garnet theory). The prediction of local porosity theory is in good agreement with the exact result.



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Porous Media Two-Phase Flow

Macroscopic Two Phase Flow in Porous Media

R. Hilfer, H. Besserer

Physica B 279, 125 (2000)
https://doi.org/10.1016/S0921-4526(99)00694-8

submitted on
Tuesday, July 6, 1999

A system of macroscopic equations for two-phase immiscible displacement in porous media is presented. The equations are based on continuum mixture theory. The pairwise character of interfacial energies is explicitly taken into account. The equations incorporate the spatiotemporal variation of interfacial energies and residual saturations. The connection between these equations and relative permeabilities is established, and found to be in qualitative agreement with experiment.



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dielectric relaxation electrical conductivity Heterogeneous Materials

Macroscopic Dielectric Constant for Microstructures of Sedimentary Rocks

R. Hilfer, J. Widjajakusuma, B. Biswal

Granular Matter 2, 137 (1999)
https://doi.org/10.1007/s100359900035

submitted on
Friday, May 21, 1999

An approximate method to calculate dielectric response and relaxation functions for water saturated sedimentary rocks is tested for realistic three-dimensional pore space images. The test is performed by comparing the prediction from the approximate method against the exact solution. The approximate method is based on image analysis and local porosity theory. An empirical rule for the specification of the length scale in local porosity theory is advanced. The results from the exact solution are compared to those obtained using local porosity theory and various other approximate mixing laws. The calculation based on local porosity theory is found to yield improved quantitative agreement with the exact result.



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