Category: Uncategorized

R. Steinle, R. Hilfer

Physical Review E 95, 043112 (2017)
https://doi.org/10.1103/PhysRevE.95.043112

submitted on
Wednesday, December 21, 2016

Traditional Darcy theory for two-phase flow in porous media is shown to predict the propagation of nonmonotone saturation profiles, also known as saturation overshoot. The phenomenon depends sensitively on the constitutive parameters, on initial conditions, and on boundary conditions. Hysteresis in relative permeabilities is needed to observe the effect. Two hysteresis models are discussed and compared. The shape of overshoot solutions can change as a function of time or remain fixed and time independent. Traveling-wave-like overshoot profiles of fixed width exist in experimentally accessible regions of parameter space. They are compared quantitatively against experiment.

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R. Hilfer, P.E. Øren

Transport in Porous Media 22, 53 (1996)
https://doi.org/10.1007/BF00974311

submitted on
Wednesday, July 27, 1994

A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a ‘macroscopic capillary number’ which differs from the usual microscopic capillary number in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure. It can be related to the microscopic capillary number and the Leverett-J-function. Previous dimensional analyses contain a tacit assumption which amounts to setting the macroscopic capillary number equal to unity. This fact has impeded quantitative upscaling in the past. Our definition, however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at values around unity. The length scale related difference between the macroscopic capillary number for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment.

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R. Hilfer

Zeitschrift für Physik B: Condensed Matter 96, 63 (1994)
https://doi.org/10.1007/BF01313016

submitted on
Tuesday, February 1, 1994

Finite size scaling theory and hyperscaling are analyzed in the ensemble limit which differs from the finite size scaling limit. Different scaling limits are discussed. Hyperscaling relations are related to the identification of thermodynamics as the infinite volume limit of statistical mechanics. This identification combined with finite ensemble scaling leads to the conclusion that hyperscaling relations cannot be violated for phase transitions with strictly positive specific heat exponent. The ensemble limit allows to derive analytical expressions for the universal part of the finite size scaling functions at the critical point. The analytical expressions are given in terms of general H-functions, scaling dimensions and a new universal shape parameter. The universal shape parameter is found to characterize the type of boundary conditions, symmetry and other universal influences on critical behaviour. The critical finite size scaling functions for the order parameter distribution are evaluated numerically for the cases delta = 3, delta = 5 and delta = 15 where delta is the equation of state exponent. Using a tentative assignment of periodic boundary conditions to the universal shape parameter yields good agreement between the analytical prediction and Monte-Carlo simulations for the two dimensional Ising model. Analytical expressions for critical amplitude ratios are derived in terms of critical exponents and the universal shape parameters. The paper offers an explanation for the numerical discrepancies and the pathological behaviour of the renormalized coupling constant in mean field theory. Low order moment ratios of difference variables are proposed and calculated which are independent of boundary conditions, and allow to extract estimates for a critical exponent.

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R. Hilfer, P.E. Øren

Two Phase Flow and Relative Permeabilities
Statoil Publication Nr. F{\&}U-LoU-94001, Trondheim, 1993

submitted on
Saturday, November 11, 1911

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T. Johnsen, Th. Walmann, R. Hilfer, P. Meakin, T. Jøssang, J. Feder

Fracton A/S, Oslo, 1993

submitted on
Thursday, November 11, 1993

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R. Hilfer, B.Nøst, E.Haslund, Th.Kautzsch, B.Virgin, B.D.Hansen

Physica A 207, 19 (1994)
https://doi.org/10.1016/0378-4371(94)90350-6

submitted on
Monday, August 9, 1993

We report preliminary results for the application of local porosity theory to dielectric response measurements on two classes of inhomogeneous systems. One class of systems are mixtures of insulators and conductors realized experimentally as sintered glass bead porous media saturated with salt water. In this case the response arises from the Maxwell-Wagner effect. The second class are mixtures of insulators realized experimentally in polymer blends where the response arises from the relaxation of atomic or molecular dipole moments. For the case of water saturated sintered glass bead systems two-dimensional local porosity distributions have been determined from digital image analysis. These measurements allow for the first time semiquantitative comparisons to previous theoretical approaches and with experiment. The dielectric measurements are used to extract the total fraction of percolating cells in the mixture. For the polymer case we show that recent concentration fluctuation models for the dielectric α-relaxation arise as special cases of local porosity theory. Furthermore it is exemplified how information from static Monte-Carlo simulations of polymer blends may be useful in comparing theoretical calculations to experiment.

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R. Hilfer

International Journal of Modern Physics B 7, 4371 (1993)
https://doi.org/10.1142/S0217979293003711

submitted on
Tuesday, April 27, 1993

A refined classification theory for phase transitions in thermodynamics and statistical mechanics in terms of their orders is introduced and analyzed. The refined thermodynamic classification is based on two independent generalizations of Ehrenfests traditional classification scheme. The statistical mechanical classification theory is based on generalized limit theorems for sums of random variables from probability theory and the newly defined block ensemble limit. The block ensemble limit combines thermodynamic and scaling limits and is similar to the finite size scaling limit. The statistical classification scheme allows for the first time a derivation of finite size scaling without renormalization group methods. The classification distinguishes two fundamentally different types of phase transitions. Phase transitions of order λ larger than 1 correspond to well known equilibrium phase transitions, while phase transitions with order λ less than 1 represent a new class of transitions termed anequilibrium transitions. The generalized order λ varies inversely with the strength of fluctuations. First order and second order transitions play a special role in both classification schemes. First order transitions represent a limiting case separating equilibrium and anequilibrium transitions. The special role or second order transitions is shown to be related to the breakdown of hyperscaling. For anequilibrium transitions the nature of the heat bath in the canonical ensemble becomes important.

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R. Hilfer

in: Random Magnetism and High-Temperature Superconductivity
edited by: W.P. Beyermann and N.L. Huang-Liu and D.E. MacLaughlin
World Scientific Publ. Co., Singapore, 85 (1994)
https://doi.org/10.1142/2378
978-981-4550-80-2

submitted on
Friday, March 19, 1993

A recently introduced classification theory for phase transitions characterizes each phase transition by its generalized noninteger order and a slowly varying function. Thermodynamically this characterization arises from generalizing the classification scheme of Ehrenfest. The same characterization emerges in statistical mechanics from generalizing the finite size scaling limit. The classification theory predicts an unusual class of phase transitions characterized by fractional orders less than unity. Examples are found in unstable models of statistical mechanics. Finally it is shown how the statistical classification theory gives rise to a classification of macroscopic dynamical behaviour based on a generalization of the stationarity concept.

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R. Hilfer

Dynamic and geometric correlation effects in disordered systems
Habilitationsschrift, Universität Mainz, 1992

submitted on
Sunday, December 20, 1992

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

Materials Research Society Proceedings 290, 185 (1993)
https://doi.org/10.1557/PROC-290-185

submitted on
Monday, November 30, 1992

A recent study [1] of the dielectric frequency response of a two component composite performed on a single specimen shows that local porosity theory LPT [2] represents a substantial improvement compared with other theories predicting the complex dielectric dispersion [3,4,5]. The purpose of the present work is to extend this investigation to a systematic study on several specimens with different compositions.

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R. Hilfer

Thermal Field Theory
UR.Nr. 451/92 Weirich, Ingelheim, 1992

submitted on
Saturday, November 11, 1911

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R. Hilfer

Physica A 194, 406 (1993)
https://doi.org/10.1016/0378-4371(93)90372-B

submitted on
Sunday, August 2, 1992

The current status of local porosity theory for transport in porous media is briefly reviewed. Local porosity theory provides a simple and general method for the geometric characterization of stochastic geometries with correlated disorder. Combining this geometric characterization with effective medium theory allows for the first time to understand a large variety of electrical and hydrodynamical flow experiments on porous rocks from a single unified theoretical framework. Rather than reproducing or rephrasing the original results the present review attempts instead to place local porosity theory within the context of other current developments in theory and experiment.

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R. Hilfer

Mod. Phys. Lett. B 6, 773 (1992)
https://doi.org/10.1142/S0217984992000855

submitted on
Monday, May 11, 1992

The recent classification theory for phase transitions (R. Hilfer, Physica Scripta 44, 321 (1991)) and its relation with the foundations of statistical physics is reviewed. First it is outlined how Ehrenfests classification scheme can be generalized into a general thermodynamic classification theory for phase transitions. The classification theory implies scaling and multiscaling thereby eliminating the need to postulate the scaling hypothesis as a fourth law of thermodynamics. The new classification has also led to the discovery and distinction of nonequilibrium transitions within equilibrium statistical physics. Nonequilibrium phase transitions are distinguished from equilibrium transitions by orders less than unity and by the fact that equilibrium thermodynamics and statistical mechanics become inapplicable at the critical point. The latter fact requires a change in the Gibbs assumption underlying the canonical and grandcanonical ensembles in order to recover the thermodynamic description in the critical limit.

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R. Hilfer

Physical Review E 48, 2466 (1993)
10.1103/PhysRevE.48.2466

submitted on
Monday, March 16, 1992

The paper introduces a classification of phase transitions in which each transition is characterized through its generalized order and a slowly varying function. This characterization is shown to be applicable in statistical mechanics as well as in thermodynamics albeit for different mathematical reasons. By introducing the block ensemble limit the statistical classification is based on the theory of stable laws from probability theory. The block ensemble limit combines scaling limit and thermodynamic limit. The thermodynamic classification on the other hand is based on generalizing Ehrenfest’s traditional classification scheme. Both schemes imply the validity of scaling at phase transitions without the need to invoke renormalizaton-group arguments. The statistical classification scheme allows derivation of a form of finite-size scaling for the distributions of statistical averages while the thermodynamic classification gives rise to multiscaling of thermodynamic potentials. The different nature of the two classification theories is also apparent from the fact that the generalized thermodynamic order is unbounded while the statistical order is restricted to values less than 2. This fact is found to be related to the breakdown of hyperscaling relations. Both classification theories predict the possible existence of phase transitions having orders less than unity. Such transitions are termed anequilibrium transitions. Systems near anequilibrium transitions cannot be described by conventional equilibrium thermodynamics or equilibrium statistical mechanics because of very strong fluctuations. Anequilibrium transitions are found to exist in statistical-mechanical model systems. The identification of the Lagrange parameter β in the canonical ensemble becomes invalid if a reservoir and a system of the same substance are in thermal contact and anequilibrium transitions are present. Based on the ergodic hypothesis and the theory of convolution semigroups it is shown that near anequilibrium transitions the equations of motion for macroscopic observables of infinite systems may involve modified time derivatives as generators of the macroscopic time evolution. The general solution to the modified equations of motion exhibits very slow dynamics as frequently observed in a nonequilibrium experiment.

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F. Boger, J. Feder, R. Hilfer, T. Jøssang

Physica A 187, 55 (1992)
https://doi.org/10.1016/0378-4371(92)90408-I

submitted on
Wednesday, December 4, 1991

The recently introduced concept of local porosity distributions for the geometric characterization of arbitrary porous media is scrutinized using computer generated pore space images. The paper presents the first direct determination of local porosity distributions from digital images. Pore space images with identical two point correlation functions are employed to analyse the geometrical sensitivity of the local porosity concept. The main finding is that local distributions can be used to discriminate between images which are indistinguishable using standard correlation functions. We also discuss the question of length scales associated with the local porosity concept.

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R. Hilfer, P. Meakin

Zeitschrift für Physik B 88, 223 (1992)
https://doi.org/10.1007/BF01323576

submitted on
Monday, November 25, 1991

The paper introduces and discusses an idealized competitive growth model with nucleation for the microstructure formation during dense branching phase separation in thin Al/Ge films. Grain size and grain length distributions for the new model are obtained analytically and by simulation. These distributions exhibit a characteristic scaling form similar to cluster size distributions in many other growth models. The cutoff functions in these scaling forms and their influence on the determination of effective exponents are studied in detail. It is found that nucleation introduces a new length scale into the other-wise selfsimilar competitive growth model. This length scale appears only inside the cutoff function and diverges algebraically as the nucleation rate vanishes. We find both analytically and by simulation that the cutoff functions can exhibit stretched exponential behaviour ∼exp(−x α) for large arguments. Our analytical and simulation results for grain size and grain length distributions are in excellent quantitative agreement.

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M. Scheringer, R. Hilfer, K. Binder

Journal of Chemical Physics 96, 2269 (1992)
https://doi.org/10.1063/1.462077

submitted on
Wednesday, October 23, 1991

A simple model for lipid monolayers on water surfaces at high spreading pressure is investigated in this work. In this model, the hydrophilic head group of the lipid molecules form a rigid regular triangular lattice, and the hydrophobic alkane chains (assumed to be in an all‐trans state) are represented by rigid rods with two angular degrees of freedom (θ, φ). The rods consist of ‘‘effective monomers,’’ and between the effective monomers on neighboring rods a Lennard‐Jones interaction is assumed. The model is studied by exact ground‐state calculations, mean‐field theory, and Monte Carlo simulations. Basic parameters are rod length a and lattice constant b. The ground‐state phase diagram shows the following phases: for small b, the rods are oriented perpendicularly to the monolayer plane (no‐tilt phase, 〈θ〉=0); for somewhat larger b, a sixfold degenerate uniform‐tilt state occurs with all rods tilted towards one of their next‐nearest neighbors. For still larger b, the rods are tilted nonuniformly and form a ‘‘striped’’ structure. These unexpected phases do not occur if we allow a rectangular distortion of the lattice. For T>0, the simplest mean‐field theory predicts a gradual disordering of the uniform‐tilt state via a second‐order phase transition. For the transition region, the Monte Carlo results disagree with this picture. Instead they show a strong asymmetric first‐order phase transition with pronounced hysteresis. The transition temperature increases with increasing rod length a, qualitatively similar to experiment.

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R. Hilfer

Physica Scripta T44, 51 (1992)
10.1088/0031-8949/1992/t44/007

submitted on
Sunday, September 15, 1991

Local porosity distributions and local percolation probabilities have been proposed as well defined and experimentally observable geometric characteristics of general porous media. Based on these concepts the dielectric response is analysed using the effective medium approximation and percolation scaling theory. The theoretical origin of static and dynamic scaling laws for the dielectric dispersion and enhancement including Archie’s law in the low porosity limit are elucidated. These well known experimental facts are unified within a theoretical framework based on a quantitative characterization of the pore space geometry. In the high porosity limit the zero frequency real dielectric constant is predicted to diverge as ε'(0) ∝ (1 − phgr)-m’ where phgr denotes porosity and m’ is analogous to the cementation exponent.

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R. Hilfer

Physical Review Letters 68, 190 (1992)
10.1103/PhysRevLett.68.190

submitted on
Monday, August 5, 1991

Multiscaling of the free energy is obtained by generalizing the classification of phase transitions proposed by Ehrenfest. The free energy is found to obey a new generalized scaling form which contains as special cases standard and multiscaling forms. The results are obtained by analytic continuation from the classification scheme of Ehrenfest.

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