Continuum Monte Carlo simulation of phase transitions in rod-like molecules at surfaces

Stiff rod-like chain molecules with harmonic bond length potentials and trigonometric bond angle potentials are used to model Langmuir monolayers at high densities. One end of the rod-like molecules is strongly bound to a flat two-dimensional substrate which represents the air-water interface. A ground-state analysis is performed which suggests phase transitions between phases with and without collective uniform tilt. Large-scale off-lattice Monte Carlo simulations over a wide temperature range show in addition to the tilting transition the presence of a strongly constrained melting transition at high temperatures. The latter transition appears to be related to two-dimensional melting of th…

Lattice-Boltzmann and finite difference simulations for the permeability of three-dimensional porous media

Numerical micropermeametry is performed on three dimensional porous samples having a linear size of approximately 3 mm and a resolution of 7.5 $\mu$m. One of the samples is a microtomographic image of Fontainebleau sandstone. Two of the samples are stochastic reconstructions with the same porosity, specific surface area, and two-point correlation function as the Fontainebleau sample. The fourth sample is a physical model which mimics the processes of sedimentation, compaction and diagenesis of Fontainebleau sandstone. The permeabilities of these samples are determined by numerically solving at low Reynolds numbers the appropriate Stokes equations in the pore spaces of the samples. The physi…

Multicanonical Monte Carlo study and analysis of tails for the order-parameter distribution of the two-dimensional Ising model.

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.

ON FRACTIONAL RELAXATION

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.

Experimental evidence for fractional time evolution in glass forming materials

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…

On fractional diffusion and continuous time random walks

Abstract 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 diffusion equations with a fractional time derivative are in general not asymptotically equivalent.

Random walks with short memory in a disordered environment

Etude du modele du saut en arriere pour le cas d'un reseau desordonne. Le modele du saut en arriere est un modele de la marche aleatoire de proches voisins correlee dans lequel le marcheur possede une probabilite de transition differente pour les sauts vers le site qu'il a visite anterieurement, par rapport aux sauts vers tous les autres sites proches voisins. La formulation standard de ce modele doit etre modifiee si le desordre est introduit au niveau de l'equation directrice habituelle. On discute des difficultes rencontrees avec la formulation standard. L'equation maitresse du premier ordre pour le modele du saut en arriere desordonne est etablie, et a partir d'elle, on derive une equat…

A comparison between simulation and experiment for hysteretic phenomena during two-phase immiscible displacement

[1] The paper compares a theory for immiscible displacement based on distinguishing percolating and nonpercolating fluid parts with experimental observations from multistep outflow experiments. The theory was published in 2006 in Physica A, volume 371, pages 209–225; the experiments were published in 1991 in Water Resources Research, volume 27, pages 2113. The present paper focuses on hysteretic phenomena resulting from repeated cycling between drainage and imbibition processes in multistep pressure experiments. Taking into account, the hydraulic differences between percolating and nonpercolating fluid parts provides a physical basis to predict quantitatively the hysteretic phenomena observ…

Transport and Relaxation Phenomena in Porous Media

Stochastic reconstruction of sandstones

A simulated annealing algorithm is employed to generate a stochastic model for a Berea and a Fontainebleau sandstone with 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 s…

Local porosity theory for electrical and hydrodynamical transport through porous media

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.

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

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.

EXACT SOLUTIONS FOR A CLASS OF FRACTAL TIME RANDOM WALKS

Fractal time random walks with generalized Mittag-Leffler functions as waiting time densities are studied. This class of fractal time processes is characterized by a dynamical critical exponent 0<ω≤1, and is equivalently described by a fractional master equation with time derivative of noninteger order ω. Exact Greens functions corresponding to fractional diffusion are obtained using Mellin transform techniques. The Greens functions are expressible in terms of general H-functions. For ω<1 they are singular at the origin and exhibit a stretched Gaussian form at infinity. Changing the order ω interpolates smoothly between ordinary diffusion ω=1 and completely localized behavior in the …

Analytical representations for relaxation functions of glasses

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 represented 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.

Reconstruction of random media using Monte Carlo methods.

A simulated annealing algorithm is applied to the reconstruction of two-dimensional porous media with prescribed correlation functions. The experimental correlation function of an isotropic sample of Fontainebleau sandstone and a synthetic correlation function with damped oscillations are used in the reconstructions. To reduce the numerical effort we follow a proposal suggesting the evaluation of the correlation functions only along certain directions. The results show that this simplification yields significantly different microstructures as compared to a full evaluation of the correlation function. In particular, we find that the simplified reconstruction method introduces an artificial a…

Capillary pressure, hysteresis and residual saturation in porous media

Abstract 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.

ON FRACTIONAL RELAXATION

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-Leer functions. The associated frequency dependent susceptibilities are related to symmetrically broadened ColeCole susceptibilities occurring as Johari Goldstein -relaxation in many glass formers. The generalized susceptibilities exhibit a high frequency wing and strong minimum enhancement.

Threefold Introduction to Fractional Derivatives

Dielectriic Dispersion Measurements of Salt-Water Saturated Porous Glass Compared with Local Porosity Theory.

AbstractA 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.

CLASSIFICATION THEORY FOR PHASE TRANSITIONS

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 reno…

Generalized Buckley–Leverett theory for two-phase flow in porous media

Hysteresis and fluid entrapment pose unresolved problems for the theory of flow in porous media. A generalized macroscopic mixture theory for immiscible two-phase displacement in porous media (Hilfer 2006b Phys. Rev. E 73 016307) has introduced percolating and nonpercolating phases. It is studied here in an analytically tractable hyperbolic limit. In this limit a fractional flow formulation exists, that resembles the traditional theory. The Riemann problem is solved analytically in one dimension by the method of characteristics. Initial and boundary value problems exhibit shocks and rarefaction waves similar to the traditional Buckley-Leverett theory. However, contrary to the traditional th…

Quantitative prediction of effective material properties of heterogeneous media

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 app…

Multiscaling and the classification of continuous phase transitions

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.

Quantitative analysis of numerical estimates for the permeability of porous media from lattice-Boltzmann simulations

During the last decade, lattice-Boltzmann (LB) simulations have been improved to become an efficient tool for determining the permeability of porous media samples. However, well known improvements of the original algorithm are often not implemented. These include for example multirelaxation time schemes or improved boundary conditions, as well as different possibilities to impose a pressure gradient. This paper shows that a significant difference of the calculated permeabilities can be found unless one uses a carefully selected setup. We present a detailed discussion of possible simulation setups and quantitative studies of the influence of simulation parameters. We illustrate our results b…

Thermodynamic potentials for the infinite range Ising model with strong coupling

Abstract 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.

Macroscopic two-phase flow in porous media

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.

FOUNDATIONS OF FRACTIONAL DYNAMICS

Time flow in dynamical systems is reconsidered in the ultralong time limit. The ultralong time limit is a limit in which a discretized time flow is iterated infinitely often and the discretization time step is infinite. The new limit is used to study induced flows in ergodic theory, in particular for subsets of measure zero. Induced flows on subsets of measure zero require an infinite renormalization of time in the ultralong time limit. It is found that induced flows are given generically by stable convolution semigroups and not by the conventional translation groups. This could give new insight into the origin of macroscopic irreversibility. Moreover, the induced semigroups are generated …

Stochastic multiscale model for carbonate rocks.

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 $2\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ contains roughly $42\ifmmode\times\else\texttimes\fi{}{10}^{6}$ crystallites and pore diameters varying over many decades. Petrophysical parameters are com…

Modelling of Orientational Ordering in Lipid Monolayers

Lipid monolayers at high densities are modelled as rigid rods grafted to an interface at the sites of a regular lattice. The transition between the state where the rods are uniformly tilted to a disordered state with no (average) tilt is studied by computer simulation methods. For the one-dimensional model, the molecular dynamics approach is found much less suitable to equilibrate the system rather than Monte Carlo methods. Both in d=2 discretized versions of Monte Carlo codes are much more efficient than continuum Monte Carlo methods, in spite of huge storage requirements. While in d=l the transition occurs at temperature T=0 via the spontaneous creation of solitons, at d=2 a finite temper…

Dimensional analysis of pore scale and field scale immiscible displacement

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’\(\overline {Ca}\) which differs from the usual microscopic capillary number Ca in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number\(\overline {Ca}\) is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure.\(\overline {Ca}\) can be related to the microscopic capillary number Ca and the LeverettJ-function. Previous dimensional analyses contain a tacit assumption which amounts to setting…

Phase Transitions in Dense Lipid Monolayers Grafted to a Surface:  Monte Carlo Investigation of a Coarse-Grained Off-Lattice Model

Semiflexible amphiphilic molecules end-grafted at a flat surface are modeled by a bead-spring chain with stiff bond angle potentials. Constant density Monte Carlo simulations are performed varying temperature, density, and chain length of the molecules, whose effective monomers interact with Lennard-Jones potentials. For not too large densities and low temperatures the monolayer is in a quasi-two-dimensional crystalline state, characterized by uniform tilt of the (stretched) chains. Raising the temperature causes a second-order transition into a (still solid) phase with no tilt. For the first time, finite size scaling concepts are applied to a model of a surfactant monolayer, and it is foun…

Analysis of multilayer adsorption models without screening

A class of recently introduced irreversible multilayer adsorption models without screening is analysed. The basic kinetic process of these models leads to power law behaviour for the decay of the jamming coverage as a function of height. The authors find the exact value for the power law exponent. An approximate analytical treatment of these models and previous Monte Carlo simulations are found to be in good agreement.

Trapping and mobilization of residual fluid during capillary desaturation in porous media

We discuss the problem of trapping and mobilization of nonwetting fluids during immiscible two-phase displacement processes in porous media. Capillary desaturation curves give residual saturations as a function of capillary number. Interpreting capillary numbers as the ratio of viscous to capillary forces the breakpoint in experimental curves contradicts the theoretically predicted force balance. We show that replotting the data against a novel macroscopic capillary number resolves the problem for discontinuous mode displacement.

Classification theory for anequilibrium phase transitions

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 transition…

Local-porosity theory for flow in porous media

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 k\ensuremath{\propto}\ensuremath{\varphi}\ifmmode\bar\else\textasciimacron\fi{} $^{\mathit{b}}$ between effective permeability and bulk porosity and the relation k\ensuremath{\…

Numerical simulation of creeping fluid flow in reconstruction models of porous media

Abstract 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, specific 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 find that in spite of the similarity in the geometrical pro…

Fractional master equations and fractal time random walks

Fractional master equations containing fractional time derivatives of order 0\ensuremath{\le}1 are introduced on the basis of a recent classification of time generators in ergodic theory. It is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks. The corresponding waiting time density \ensuremath{\psi}(t) is obtained exactly as \ensuremath{\psi}(t)=(${\mathit{t}}^{\mathrm{\ensuremath{\omega}}\mathrm{\ensuremath{-}}1}$/C)${\mathit{E}}_{\mathrm{\ensuremath{\omega}},\mathrm{\ensuremath{\omega}}}$(-${\mathit{t}}^{\mathrm{\ensuremath{\omega}}}$/C), where ${\mathit{E}}_{\mathrm{\ensuremath{\omega}},\mathrm{\ensuremat…

Exact and approximate calculations for the conductivity of sandstones

We analyze a three-dimensional pore space reconstruction of Fontainebleau sandstone and calculate from it the eective conductivity using local porosity theory. We compare this result with an exact calculation of the eective conductivity that solves directly the disordered Laplace equation. The prediction of local porosity theory is in good quantitative agreement with the exact result. c 1999 Elsevier Science B.V. All rights reserved.

Multicanonical Simulations of the Tails of the Order-Parameter Distribution of the Two-Dimensional Ising Model

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.

Macroscopic equations of motion for two-phase flow in porous media

The established macroscopic equations of motion for two phase immiscible displacement in porous media are known to be physically incomplete because they do not contain the surface tension and surface areas governing capillary phenomena. Therefore a more general system of macroscopic equations is derived here which incorporates the spatiotemporal variation of interfacial energies. These equations are based on the theory of mixtures in macroscopic continuum mechanics. They include wetting phenomena through surface tensions instead of the traditional use of capillary pressure functions. Relative permeabilities can be identified in this approach which exhibit a complex dependence on the state v…

SCALING THEORY AND THE CLASSIFICATION OF PHASE TRANSITIONS

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 transiti…

Quantitative Analysis of Experimental and Synthetic Microstructures for Sedimentary Rock

A quantitative comparison between the experimental microstructure of a sedimentary rock and three theoretical models for the same rock is presented. The microstructure of the rock sample (Fontainebleau sandstone) was obtained by microtomography. Two of the models are stochastic models based on correlation function reconstruction, and one model is based on sedimentation, compaction and diagenesis combined with input from petrographic analysis. The porosity of all models closely match that of the experimental sample and two models have also the same two point correlation function as the experimental sample. We compute quantitative differences and similarities between the various microstructur…

Continuum reconstruction of the pore scale microstructure for Fontainebleau sandstone

Abstract A stochastic geometrical modeling technique is used to reconstruct a laboratory scale Fontainebleau sandstone with a sidelength of 1.5 cm. The model reconstruction is based on crystallite properties and diagenetic parameters determined from two-dimensional images. The three-dimensional pore scale microstructure of the sandstone is represented by a list of quartz crystallites defined geometrically and placed in the continuum. This allows generation of synthetic μ -CT images of the rock model at arbitrary resolutions. Quantitative microstructure comparison based on Minkowski functionals, two-point correlation function and local porosity theory indicates that this modeling technique c…

Microstructural sensitivity of local porosity distributions

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.

Erosion–dilation analysis for experimental and synthetic microstructures of sedimentary rock

Abstract 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.

Applications and Implications of Fractional Dynamics for Dielectric Relaxation

This article summarizes briefly the presentation given by the author at the NATO Advanced Research Workshop on “Broadband Dielectric Spectroscopy and its Advanced Technological Applications”, held in Perpignan, France, in September 2011. The purpose of the invited presentation at the workshop was to review and summarize the basic theory of fractional dynamics (Hilfer, Phys Rev E 48:2466, 1993; Hilfer and Anton, Phys Rev E Rapid Commun 51:R848, 1995; Hilfer, Fractals 3(1):211, 1995; Hilfer, Chaos Solitons Fractals 5:1475, 1995; Hilfer, Fractals 3:549, 1995; Hilfer, Physica A 221:89, 1995; Hilfer, On fractional diffusion and its relation with continuous time random walks. In: Pekalski et al. …

Quantitative comparison of mean field mixing laws for conductivity and dielectric constants of porous media

Abstract Exact numerical solution of the electrostatic disordered potential problem is carried out for four fully discretized three-dimensional experimental reconstructions of sedimentary rocks. The measured effective macroscopic dielectric constants and electrical conductivities are compared with parameter-free 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.

Frequency-dependent response and dynamic disorder

Abstract This paper discusses selected aspects of the application of dynamic percolation models to ionic transport in mixed-ion superionic conductors. The discussion is based on an AB lattice gas model with hard-core repulsions and a ratio of τ, 0 ⩽ τ ⩽ ∞, between the transition rates of particles A and B. The frequency-dependent conductivity for a tracer particle is calculated within an effective-medium theory. The motion of the background B-particles is regarded as providing a fluctuating disordered environment for the tracer particles A. A crossover frequency separating high-frequency and low-frequency response is found which scales with τ as ω c ∼ τ 1 2 . The results for the dc limit ar…

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

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.,…

Corrigendum: Generalized Buckley–Leverett theory for two-phase flow in porous media

Macroscopic capillarity without a constitutive capillary pressure function

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…

Permeability and conductivity for reconstruction models of porous media

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 o…

Absence of hyperscaling violations for phase transitions with positive specific heat exponent

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 generalH-fu…

Local Entropy Characterization of Correlated Random Microstructures

A rigorous connection is established between the local porosity entropy introduced by Boger et al. (Physica A 187, 55 (1992)) and the configurational entropy of Andraud et al. (Physica A 207, 208 (1994)). These entropies were introduced as morphological descriptors derived from local volume fluctuations in arbitrary correlated microstructures occuring 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 morphologic…

Statistical prediction of corrosion front penetration

A statistical method to predict the stochastic evolution of corrosion fronts has been developed. The method is based on recording material loss and maximum front depth. In this paper we introduce the method and test its applicability. In the absence of experimental data we use simulation data from a three-dimensional corrosion model for this test. The corrosion model simulates localized breakdown of a protective oxide layer, hydrolysis of corrosion product and repassivation of the exposed surface. In the long time limit of the model, pits tend to coalesce. For different model parameters the model reproduces corrosion patterns observed in experiment. The statistical prediction method is base…