Author: R Hilfer

Stochastic Multiscale Model for Carbonate Rocks

Post author By R Hilfer
Post date January 9, 2007

B. Biswal, P.E. Øren, R. Held, S. Bakke, R. Hilfer

Physical Review E 75, 061303 (2007)
https://doi.org/10.1103/PhysRevE.75.061303

submitted on
Tuesday, January 9, 2007

A multiscale model for the diagenesis of carbonate rocks is proposed. It captures important pore scale characteristics of carbonate rocks: wide range of length scales in the pore diameters; large variability in the permeability; and strong dependence of the geometrical and transport parameters on the resolution. A pore scale microstructure of an oolithic dolostone with generic diagenetic features is successfully generated. The continuum representation of a reconstructed cubic sample of sidelength 2 mm contains roughly 42⫻ 106 crystallites and pore diameters varying over many decades. Petrophysical parameters are computed on discretized samples of sizes up to 10003. The model can be easily adapted to represent the multiscale microstructure of a wide variety of carbonate rocks.

For more information see

url

pdf

diffusion Fractional Calculus Porous Media Two-Phase Flow

Modeling Infiltration by Means of a Nonlinear Fractional Diffusion Model

Post author By R Hilfer
Post date May 18, 2006

E. Gerolymatou, I. Vardoulakis, R. Hilfer

Journal of Physics D: Applied Physics 39, 4104 (2006)

submitted on
Thursday, May 18, 2006

The classical Richards equation describes infiltration into porous soil as a nonlinear diffusion process. Recent experiments have suggested that this process exhibits anomalous scaling behaviour. These observations suggest generalizing the classical Richards equation by introducing fractional time derivatives. The resulting fractional Richards equation with appropriate initial and boundary values is solved numerically in this paper. The numerical code is tested against analytical solutions in the linear case. Saturation profiles are calculated for the fully nonlinear fractional Richards equation. Isochrones and isosaturation curves are given. The cumulative moisture intake is found as a function of the order of the fractional derivative. These results are compared against experiment.

For more information see

url

pdf

Porous Media Two-Phase Flow

Macroscopic capillarity without a constitutive capillary pressure function

Post author By R Hilfer
Post date January 20, 2006

R. Hilfer

Physica A 371, 209 (2006)
https://doi.org/10.1016/j.physa.2006.04.051

submitted on
Friday, January 20, 2006

This paper challenges the foundations of the macroscopic capillary pressure concept. The capillary pressure function, as it is traditionally assumed in the constitutive theory of two-phase immiscible displacement in porous media, relates the pressure difference between nonwetting and wetting fluid to the saturation of the wetting fluid. The traditional capillary pressure function neglects the fundamental difference between percolating and nonpercolating fluid regions as first emphasized in R. Hilfer [Macroscopic equations of motion for two phase flow in porous media, Phys. Rev. E 58 (1998) 2090]. The theoretical approach proposed here starts from residual saturations as the volume fractions of nonpercolating phases. The resulting equations of motion open the possibility to describe flow processes where drainage and imbibition occur simultaneously. The theory predicts hysteresis and process dependence of capillary phenomena. The traditional theory is recovered as a special case in the residual decoupling approximation. Explicit calculations are presented for quasistatic equilibrium profiles near hydrostatic equilibrium. The results are found to agree with experiment. r 2006 Elsevier B.V. All rights reserved.

For more information see

url

pdf

diffusion Fractional Calculus Porous Media Two-Phase Flow

Simulating the Saturation Front Using a Fractional Diffusion Model

Post author By R Hilfer
Post date June 30, 2005

E. Gerolymatou, I. Vardoulakis, R. Hilfer

in: Proceedings of the GRACM05 International Congress on Computational Mechanics, Limassol 2005
edited by: G. Georgiou, P. Papanastasiou, M. Papadrakakis
GRACM, Athens, 653 (2005)

submitted on
Thursday, June 30, 2005

In this paper the possibility of making use of fractional derivatives for the simulation of the flow of water through porous media and in particular through soils is considered. The Richards equation, which is a non-linear diffusion equation, will be taken as a basis and is used for the comparison of results. Fractional derivatives differ from derivatives of integer order in that they entail the whole history of the function in a weighted form and not only its local behavior, meaning that a different numerical approach is required. Previous work on the topic will be examined and a consistent approach based on fractional time evolutions will be presented.

For more information see

url

pdf

Porous Media Two-Phase Flow

Macroscopic Capillarity and Hysteresis for Flow in Porous Media

Post author By R Hilfer
Post date May 27, 2005

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.

For more information see

url

pdf

html

Porous Media Two-Phase Flow

Capillary Pressure, Hysteresis and Residual Saturation in Porous Media

Post author By R Hilfer
Post date May 20, 2005

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.

For more information see

url

pdf

Critical phenomena Lattice Models Statistical Physics

Multicanonical Simulations of the Tails of the Order-Parameter Distribution of the two-dimensional Ising Model

Post author By R Hilfer
Post date April 9, 2005

R. Hilfer, B. Biswal, H.G. Mattutis, W. Janke

Computer Physics Communications 169, 230 (2005)
https://doi.org/10.1016/j.cpc.2005.03.053

submitted on
Saturday, April 9, 2005

We report multicanonical Monte Carlo simulations of the tails of the order-parameter distribution of the two-dimensional Ising model for fixed boundary conditions. Clear numerical evidence for “fat” stretched exponential tails is found below the critical temperature, indicating the possible presence of fat tails at the critical temperature.

For more information see

url

pdf

Mathematics Special Functions

Computation of the Generalized Mittag-Leffler Function and its Inverse in the Complex Plane

Post author By R Hilfer
Post date March 21, 2005

R. Hilfer, H.J. Seybold

Integral Transforms and Special Functions 17, 637 (2006)
https://doi.org/10.1080/10652460600725341

submitted on
Monday, March 21, 2005

The generalized Mittag-Leffler function Eα,β (z) has been studied for arbitrary complex argument and real parameters. This function plays a fundamental role in the theory of fractional differential equations and numerous applications in physics. The Mittag-Leffler function interpolates smoothly between exponential and algebraic functional behaviour. A numerical algorithm for its evaluation has been developed. The algorithm is based on integral representations and exponential asymptotics. Results of extensive numerical calculations in the complex z-plane are reported here. We find that all complex zeros emerge from the point z = 1 for small alpha. They diverge towards negative infinity for alpha approaching unity. All the complex zeros collapse pairwise onto the negative real axis for α approaching 2. We introduce and study also the inverse generalized Mittag-Leffler function. We determine its principal branch numerically.

For more information see

url

pdf

html

image analysis Porous Media

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

Post author By R Hilfer
Post date April 16, 2004

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.

For more information see

url

pdf

Heterogeneous Materials Porous Media Two-Phase Flow

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

Post author By R Hilfer
Post date March 15, 2004

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.

For more information see

url

pdf

diffusion Fractional Calculus

On fractional diffusion and continuous time random walks

Post author By R Hilfer
Post date May 22, 2003

R. Hilfer

Physica A 329, 35 (2003)
https://doi.org/10.1016/S0378-4371(03)00583-1

submitted on
Thursday, May 22, 2003

A continuous time random walk model is presented with long-tailed waiting time density that approaches a Gaussian distribution in the continuum limit. This example shows that continuous time random walks with long time tails and di!usion equations with a fractional time derivative are in general not asymptotically equivalent.

For more information see

url

pdf

html

Mathematics Special Functions

Numerical Results for the Generalized Mittag-Leffler Function

Post author By R Hilfer
Post date May 4, 2003

H.J. Seybold, R. Hilfer

Fractional Calculus and Applied Analysis 8, 127 (2005)

submitted on
Wednesday, June 4, 2003

Results of extensive calculations for the generalized Mittag-Leffler function are presented in a region of the complex plane. This function is related to the eigenfunction of a fractional derivative.

For more information see

pdf

fluid flow Porous Media

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

Post author By R Hilfer
Post date March 30, 2003

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.

For more information see

url

pdf

Critical phenomena Lattice Models Statistical Physics

Multicanonical Monte-Carlo Study and Analysis of Tails for the Order-Parameter Distribution of the Two-Dimensional Ising Model

Post author By R Hilfer
Post date February 10, 2003

R. Hilfer, B. Biswal, H.G. Mattutis, W. Janke

Physical Review E 68, 046123 (2003)
https://doi.org/10.1103/PhysRevE.68.046123

submitted on
Monday, February 10, 2003

The tails of the critical order-parameter distribution of the two-dimensional Ising model are investigated through extensive multicanonical Monte Carlo simulations. Results for fixed boundary conditions are reported here, and compared with known results for periodic boundary conditions. Clear numerical evidence for ‘‘fat’’ stretched exponential tails exists below the critical temperature, indicating the possible presence of fat tails at the critical temperature. Our work suggests that the true order-parameter distribution at the critical temperature must be considered to be unknown at present.

For more information see

url

pdf

html

Critical phenomena Lattice Models Statistical Physics

Thermodynamic potentials for the infinite range Ising model with strong coupling

Post author By R Hilfer
Post date September 30, 2002

R. Hilfer

Physica A 320, 429 (2003)
https://doi.org/10.1016/S0378-4371(02)01585-6

submitted on
Monday, September 30, 2002

The specific Gibbs free energy has been calculated for the infinite range Ising model with fixed and finite interaction strength. The model shows a temperature driven first-order phase transition that differs from the infinite ranged Ising model with weak coupling. In the temperature-field phase diagram the strong coupling model shows a line of first-order phase transitions that does not end in a critical point.

For more information see

url

pdf

Fractional Calculus Fractional Time Theory of Time

Strange Kinetics

Post author By R Hilfer
Post date July 8, 2002

R. Hilfer, R. Metzler, A. Blumen, J. Klafter(eds)

Chemical Physics 284, 1 (2002)
https://doi.org/10.1016/S0301-0104(02)00801-7

submitted on
Monday, July 8, 2002

The term strange kinetics originally referred to the dynamics of Hamiltonian systems which, in the limit of weak chaos, display superdiffusion and Levy-walk characteristics. Here we employ the term strange kinetics in a generalized sense to denote all forms of slow kinetics or anomalous dynamics, such as sub-diffusion, superdiffusion, non-Debye relaxation, Levy walks or fractional time evolutions.

For more information see

url

pdf

Fractional Calculus Fractional Time Theory of Time

Remarks on Fractional Time

Post author By R Hilfer
Post date July 1, 2002

R. Hilfer

in: Time, Quantum and Information
edited by: L. Castell and O. Ischebeck
Springer, Berlin, 235 (2003)
10.1007/978-3-662-10557-3
ISBN: 978-3-540-44033-8

submitted on
Monday, July 1, 2002

It is not possible to repeat an experiment in the past. The underlying philosophical truth in this observation is the difference between certainty of the past and potentiality of the future. This difference is discussed, for example, in C.F.v. Weizsäcker’s papers and it was often pointed out by him in our discussions in the years 1983-1986 in the Starnberg institute. The perennial philosophical problem related to this difference between past and future is the question whether time is real or not.

For more information see

url

pdf

Simulations

Physics on the Computer

Post author By R Hilfer
Post date June 28, 2002

R. Hilfer

ICP, Stuttgart, 2002

submitted on
Friday, June 28, 2002

For more information see

url

dielectric relaxation diffusion electrical conductivity Heterogeneous Materials Porous Media

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

Post author By R Hilfer
Post date June 4, 2002

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.

For more information see

url

pdf

dielectric relaxation Fractional Calculus Fractional Time Glasses

Experimental Evidence for Fractional Time Evolution in Glass Forming Materials

Post author By R Hilfer
Post date December 7, 2001

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.

For more information see