Categories
image analysis Porous Media

Quantitative Analysis of Experimental and Synthetic Microstructures for Sedimentary Rock

B. Biswal, C. Manwart, R. Hilfer, S. Bakke, P.E. Øren

Physica A 273, 452 (1999)
https://doi.org/10.1016/S0378-4371(99)00248-4

submitted on
Monday, May 17, 1999

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 microstructures by a method based on local porosity theory. Differences are found in the degree of anisotropy, and in fluctuations of porosity and connectivity. The stochastic models differ strongly from the real sandstone in their connectivity properties, and hence need further refinement when used to model transport.



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Porous Media review article Transport Processes

Local Porosity Theory and Stochastic Reconstruction for Porous Media

R. Hilfer

in: Räumliche Statistik und Statistische Physik
edited by: D. Stoyan and K. Mecke
Springer, Berlin, 203 (2000)
10.1007/3-540-45043-2
ISBN: 978-3-642-08725-7

submitted on
Tuesday, February 23, 1999

The paper reviews recent developments in local porosity theory, and discusses its application to the analysis of stochastic reconstruction models for sedimentary rocks. Special emphasis is placed on the geometric observables in local porosity theory and their relation with the Hadwiger theorem from stochastic geometry. In addition recent results for the exact calculation of effective physical transport properties are given for a Fontainebleau sandstone. The calculations pertain to potential type problems such as electrical conduction, dielectric relaxation, diffusion or Darcy flow. The exact results are compared to the approximate parameterfree predictions from local porosity, and are found to be in good agreement.



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dielectric relaxation electrical conductivity fluid flow Porous Media

Exact and Approximate Calculations for Conductivity of Sandstones

J. Widjajakusuma, C. Manwart, B. Biswal, R. Hilfer

Physica A 270, 325 (1999)
https://doi.org/10.1016/S0378-4371(99)00141-7

submitted on
Tuesday, January 5, 1999

We analyze a three-dimensional pore space reconstruction of Fontainebleau sandstone and calculate from it the effective conductivity using local porosity theory. We compare this result with an exact calculation of the effective conductivity that solves directly the disordered Laplace equation. The prediction of local porosity theory is in good quantitative agreement with the exact result.



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dielectric relaxation diffusion electrical conductivity fluid flow Heterogeneous Materials Porous Media

Quantitative Prediction of Effective Material Properties of Heterogeneous Media

J. Widjajakusuma, B. Biswal, R. Hilfer

Computational Materials Science 16, 70 (1999)
https://doi.org/10.1016/S0927-0256(99)00047-6

submitted on
Thursday, October 8, 1998

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 approximation. Contrary to the predictions from effective medium theory, the predictions of LPT are in better quantitative agreement with the exact results.



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Porous Media Simulations

Reconstruction of Random Media Using Monte Carlo Methods

C. Manwart, R. Hilfer

Physical Review E 59, 5596 (1999)
https://doi.org/10.1103/PhysRevE.59.5596

submitted on
Tuesday, September 8, 1998

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 anisotropy that is originally not present.



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image analysis Porous Media Simulations

Microstructure Analysis of Reconstructed Porous Media

B. Biswal, R. Hilfer

Physica A 266, 307 (1999)
https://doi.org/10.1016/S0378-4371(98)00607-4

submitted on
Wednesday, July 15, 1998

We compare the quantitative microstructural properties of Berea Sandstone with stochastic reconstructions of the same sandstone. The comparison is based on local porosity theory. The reconstructions employ Fourier space filtering of Gaussian random fields and match the average porosity and two-point correlation function of the experimental model. Connectivity properties of the stochastic models differ significantly from the experimental model. Reconstruction models with different levels of coarse graining also show different average local connectivity.



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diffusion Fractional Calculus Random Walks

On Fractional Diffusion and its Relation with Continuous Time Random Walks

R. Hilfer

in: Anomalous Diffusion: From Basis to Applications
edited by: R. Kutner, A. Pekalski and K. Sznajd-Weron
Lecture Notes in Physics, vol. 519,Springer, Berlin, 77 (1999)
10.1007/BFb0106828
978-3-662-14242-4

submitted on
Friday, May 22, 1998

Time evolutions whose infinitesimal generator is a fractional time derivative arise generally in the long time limit. Such fractional time evolutions are considered here for random walks. An exact relationship is established between the fractional master equation and a separable continuous time random walk of the Montroll-Weiss type. The waiting time density can be expressed using a generalized Mittag-Leffier function. The first moment of the waiting density does not exist.



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Porous Media Two-Phase Flow

Trapping and Mobilization of Residual Fluid During Capillary Desaturation in Porous Media

L. Anton, R. Hilfer

Physical Review E 59, 6819 (1999)
https://doi.org/10.1103/PhysRevE.59.6819

submitted on
Tuesday, April 21, 1998

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.



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image analysis Porous Media Simulations

Threedimensional Local Porosity Analysis of Porous Media

B. Biswal, C. Manwart, R. Hilfer

Physica A 255, 221 (1998)
https://doi.org/10.1016/S0378-4371(98)00111-3

submitted on
Thursday, February 5, 1998

A quantitative comparison of the pore space geometry for three natural sandstones is presented. The comparison is based on local porosity theory which provides a geometric characterization of stochastic microstructures. The characterization focusses on porosity and connectivity fluctuations. Porosity fluctuations are measured using local porosity distributions while connectivity fluctuations are measured using local percolation probabilities. We report the first measurement of local percolation probability functions for experimentally obtained three-dimensional pore space reconstructions. Our results suggest the use of local porosity distributions and percolation probabilities as a quantitative method to compare microstructures of models and experiment.



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Porous Media Two-Phase Flow

Macroscopic Equations of Motion for Two Phase Flow in Porous Media

R. Hilfer

Physical Review E 58, 2090 (1998)
https://doi.org/10.1103/PhysRevE.58.2090

submitted on
Tuesday, January 20, 1998

The usual 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 that 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 that exhibit a complex dependence on the state variables. A capillary pressure function can be identified in equilibrium that shows the qualitative saturation dependence known from experiment. In addition, the proposed equations include a description of the spatiotemporal changes of residual saturations during immiscible displacement.



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fluid flow Porous Media Two-Phase Flow

Old Problems and New Solutions for Multiphase Flow in Porous Media

R. Hilfer, H. Besserer

in: Porous Media: Physics, Mo\-dels, Simulation
edited by: A. Dmitrievsky and M. Panfilov
World Scientific Publ. Co., Singapore, 133-144 (2000)
https://doi.org/10.1142/9789812817617_0008
ISBN: 978-981-02-4126-1

submitted on
Thursday, November 20, 1997

The existing macroscopic equations of motion for multiphase flow in porous media are unsatisfactory in two general respects. On the one hand characteristic experimental features, such as relationships between capillary pressure and saturations, cannot be predicted. On the other hand the theoretical derivation of the equations from the well-known laws of hydrodynamics has not yet been accomplished. In this paper we discuss these deficiencies and present an alternative description which is based on energy balances. Our description includes surface tensions as parameters and interface areas as a new macroscopic state variable. The equations are obtained from general multiphase mixture theory by explicitly accounting for the pairwise character of interfacial energies. For the special case of two immiscible fluids in a porous medium the most important ingredient is the distinction between a connected and a disconnected subphase of each fluid phase. In this way it becomes possible to handle also the spatiotemporal variation of residual saturations. The connection between the new approach and the established formulation is given by identifying a generalized Darcy Law with generalized relative permeabilities. The new equations reproduce qualitatively the saturation dependent behaviour of capillary pressure in gravitational equilibrium.



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Fractional Calculus review article

Fractional Derivatives in Static and Dynamic Scaling

R. Hilfer

in: Scale Invariance and Beyond
edited by: B. Dubrulle and F. Graner and D. Sornette
Springer, Berlin, 53 (1997)
10.1007/978-3-662-09799-1
978-3-540-64000-4

submitted on
Tuesday, March 11, 1997

The paper is a brief review of recent applications of fractional calculus in physics with emphasis on static and dynamic scaling.



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Categories
Lattice Models Nonequilibrium Simulations Stochastic Processes

Statistical Prediction of Corrosion Front Penetration

T. Johnsen, R. Hilfer

Phys.Rev. E 55, 5433 (1997)
https://doi.org/10.1103/PhysRevE.55.5433

submitted on
Wednesday, September 18, 1996

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 based in the theory of stochastic processes. It allows the estimation of conditional probability densities for penetration depth, pitting factor, residual lifetimes, and corrosion rates which are of technological interest.



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Heterogeneous Materials Percolation Porous Media

Local Entropy Characterization of Correlated Random Microstructures

C. Andraud, A. Beghdadi, E. Haslund, R. Hilfer, J. Lafait, B. Virgin

Physica A 235, 307 (1997)
https://doi.org/10.1016/S0378-4371(96)00354-8

submitted on
Tuesday, August 13, 1996

A rigorous connection is established between the local porosity entropy introduced by Boger et al. (Physica A 187 (1992) 55) and the configurational entropy of Andraud et al. (Physica A 207 (1994) 208). These entropies were introduced as morphological descriptors derived from local volume fluctuations in arbitrary correlated microstructures occurring in porous media, composites or other heterogeneous systems. It is found that the entropy lengths at which the entropies assume an extremum become identical for high enough resolution of the underlying configurations. Several examples of porous and heterogeneous media are given which demonstrate the usefulness and importance of this morphological local entropy concept.



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Percolation Porous Media

Local Percolation Probabilities for a Natural Sandstone

R. Hilfer, T. Rage, B. Virgin

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

submitted on
Thursday, July 25, 1996

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



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Disordered Systems Fractals Porous Media

Probabilistic Methods, Upscaling and Fractal Statistics in Porous Media

R. Hilfer

Zentralblatt für Geologie und Paläontologie, Teil I 11/12, 1035 (1997)

submitted on
Friday, May 24, 1996

This contribution gives a brief introduction to local porosity theory. It is shown that fractal porosity fluctuations may arise for macroscopic media in a suitable upscaling limit.

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Critical phenomena Equilibrium Simulations

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

F. M. Haas, R. Hilfer, K. Binder

The Journal of Physical Chemistry 100 (37), 15290-15300 (1996)
DOI: 10.1021/jp9610980

submitted on
Friday, April 12, 1996

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 found that the technique in this case again is useful to locate the transition more precisely. For comparison, also a one-dimensional version of the model is studied, and directions for future extensions of this modeling are discussed.



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Percolation Porous Media

Rescaling Relations between Two- and Three Dimensional Local Porosity Distributions for Natural and Artificial Porous Media

B. Virgin, E. Haslund, R. Hilfer

Physica A 232, 1-10 (1996)
https://doi.org/10.1016/0378-4371(96)00131-8

submitted on
Friday, March 29, 1996

Local porosity distributions for a three-dimensional porous medium and local porosity distributions for a two-dimensional plane-section through the medium are generally different. However, for homogeneous and isotropic media having finite correlation lengths, a good degree of correspondence between the two sets of local porosity distributions can be obtained by rescaling lengths, and the mapping associating corresponding distributions can be found from two-dimensional observations alone. The agreement between associated distributions is good as long as the linear extent of the measurement cells involved is somewhat larger than the correlation length, and it improves as the linear extent increases. A simple application of the central limit theorem shows that there must be a correspondence in the limit of very large measurement cells, because the distributions from both sets approach normal distributions. A normal distribution has two independent parameters: the mean and the variance. If the sample is large enough, LPDs from both sets will have the same mean. Therefore corresponding distributions are found by matching variances of two and three dimensional local porosity distributions. The variance can be independently determined from correlation functions. Equating variances leads to a scaling relation for lengths in this limit. Three particular systems are examined in order to show that this scaling behavior persists at smaller length-scales.



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Percolation Porous Media

Local Porosity Theory for the Transition from Microscales to Macroscales in Porous Media

R. Hilfer, B. Virgin, T. Rage

ERCOFTAC Bull. 28, 6 (1996)

submitted on
Monday, February 12, 1996

A quantitative understanding of fluid flow and other transport processes in porous media remains a prerequisite for progress in many disciplines such as hydrology, petroleum technology, chemical engineering, environmental protection, nuclear waste storage, drug transport in biological tissues, catalysis, paleontology, filtration and separation technology to name but a few. While the microscopic equations governing flow and transport in porous media are often well known, the macroscopic laws are usually different and much less understood. Most approaches in computational fluid dynamics for porous media avoid to discuss or control the problems arising in the transition from a microscale (pores) to the macroscale (field or laboratory). As a consequence the upscaling of transport processes, particularly for immiscible fluid-fluid displacement, has remained difficult.



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Critical phenomena Simulations

Continuum Monte Carlo Simulation at Constant Pressure of Stiff Chain Molecules at Surfaces

F. M. Haas, R. Hilfer

Journal of Chemical Physics 105, 3859 (1996)
https://doi.org/10.1063/1.472206

submitted on
Thursday, August 31, 1995

Continuum Monte-Carlo simulations at constant pressure are performed on short chain molecules at surfaces. The rodlike chains, consisting of seven effective monomers, are attached at one end to a flat twodimensional substrate. It is found that the model exhibits phases similar to the liquid condensed and liquid expanded phases of Langmuir monolayers. The model is investigated here for a wide range of pressures and temperatures using a special form of constant pressure simulation compatible with the symmetry breaking during tilting transitions in the liquid condensed phases. At low pressures the chains undergo a tilting transition exhibiting tilt directions towards nearest and also next nearest neighbours depending on temperature. At elevated temperatures and low pressure the film enters a fluidlike phase similar to the liquid expanded phase observed in experiment.



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