Category: Transport Processes

Categories
electrical conductivity fluid flow Porous Media

Permeability and Conductivity for Reconstruction Models of Porous Media

R. Hilfer, C. Manwart

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

submitted on
Friday, October 27, 2000

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



For more information see

url

url
pdf

pdf
html

more
Categories
diffusion Fractional Calculus

Fractional Diffusion based on Riemann-Liouville Fractional Derivatives

R. Hilfer

The Journal of Physical Chemistry B 104, 3914-3917 (2000)
DOI: 10.1021/jp9936289

submitted on
Tuesday, October 12, 1999

A fractional diffusion equation based on Riemann−Liouville fractional derivatives is solved exactly. The initial values are given as fractional integrals. The solution is obtained in terms of H-functions. It differs from the known solution of fractional diffusion equations based on fractional integrals. The solution of fractional diffusion based on a Riemann−Liouville fractional time derivative does not admit a probabilistic interpretation in contrast with fractional diffusion based on fractional integrals. While the fractional initial value problem is well defined and the solution finite at all times, its values for t → 0 are divergent.



For more information see

url

url
pdf

pdf
html

more
Categories
dielectric relaxation diffusion electrical conductivity fluid flow Porous Media

Effective Physical Properties of Sandstones

J. Widjajakusuma, R. Hilfer

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

submitted on
Wednesday, October 6, 1999

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



For more information see

url

url
pdf

pdf
html

more
Categories
Porous Media Two-Phase Flow

Macroscopic Two Phase Flow in Porous Media

R. Hilfer, H. Besserer

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

submitted on
Tuesday, July 6, 1999

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



For more information see

url

url
pdf

pdf
html

more
Categories
dielectric relaxation electrical conductivity Heterogeneous Materials

Macroscopic Dielectric Constant for Microstructures of Sedimentary Rocks

R. Hilfer, J. Widjajakusuma, B. Biswal

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

submitted on
Friday, May 21, 1999

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



For more information see

url

url
pdf

pdf
html

more
Categories
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.



For more information see

url

url
pdf

pdf
html

html
html

more
Categories
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.



For more information see

url

url
pdf

pdf
html

more
Categories
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.



For more information see

url

url
pdf

pdf
html

more
Categories
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.



For more information see

url

url
pdf

pdf
html

more

Categories
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.



For more information see

url

url
pdf

pdf
html

more
Categories
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.



For more information see

url

url
pdf

pdf
html

more
Categories
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.



For more information see

url

url
pdf

pdf
html

more
Categories
Porous Media Transport Processes

Transport and Relaxation Phenomena in Porous Media

R. Hilfer

Advances in Chemical Physics XCII, 299 (1996)
ISBN: 978-0-470-14204-2

submitted on
Tuesday, May 9, 1995

Almost all studies of transport and relaxation in porous media are motivated by one central question. How are the effektive macroscopic transport parameters influenced by the microscopic geometric structure of the medium?



For more information see

url

url
pdf

pdf
html

html
html

more
Categories
dielectric relaxation Disordered Systems Porous Media

Measurement of Local Porosities and Dielectric Dispersion for a Water Saturated Porous Medium

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

Journal of Applied Physics 76, 5473 (1994)
https://doi.org/10.1063/1.357205

submitted on
Monday, October 4, 1993

The frequency‐dependent conductivity and dielectric constant of a salt‐water‐saturated porous glass specimen have been measured. The measurements cover the full frequency range of the Maxwell–Wagner dispersion. The experimental results have been compared with the recently introduced local porosity theory and with previous theories. For the purpose of comparing with the local porosity theory experimental measurements of local porosity distributions from digitized pore space images are presented. These experimental porosity distributions are then used for a first experimental test of local porosity theory. The comparison with previous theoretical expressions for the frequency‐dependent effective dielectric function shows that local porosity theory constitutes a significant improvement in the quantitative agreement.



For more information see

url

url
pdf

pdf
html

more
Categories
Disordered Systems fluid flow Porous Media Transport Processes

Local Porosity Theory for Flow in Porous Media

R. Hilfer

Physical Review B 45, 7115 (1992)
https://doi.org/10.1103/PhysRevB.45.7115

submitted on
Thursday, March 28, 1991

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 relating effective permeability and bulk porosity and the power lawrelation between permeability and formation factor are analyzed. A simple and general consolidation model is introduced. It is based on the reduction of local porosities and emphasizes the general applicability and flexibility of the local-porosity concept. The theoretical predictions are compared with the experimentally observed range for the exponents, and are found to be in excellent agreement.



For more information see

url

url
pdf

pdf
html

more
Categories
dielectric relaxation Porous Media

Geometric and Dielectric Characterization of Porous Media

R. Hilfer

Physical Review B 44, 60 (1991)
https://doi.org/10.1103/PhysRevB.44.60

submitted on
Friday, October 12, 1990

This paper introduces local porosity distributions and local percolation probabilities as well-defined and experimentally observable geometric characteristics of general porous media. Based on these concepts the dielectric response is analyzed using the effective-medium approximation and percolation scaling theory. The theoretical origin of static and dynamic scaling laws for the dielectric response including Archie’s law in the low-porosity limit are elucidated. The zero-frequency real dielectric constant is found to diverge as as a power law in the high-porosity limit with an exponent analogous to the cementation exponent. Model calculations are presented for the interplay between geometric characteristics and the frequency-dependent dielectric response. Three purely geometric mechanisms are identified, each of which can give rise to a large dielectric enhancement.



For more information see

url

url
pdf

pdf
html

html
html

more
Categories
Disordered Systems Lattice Models Transport Processes

Correlation Effects on Hopping Transport in a Disordered Medium

R. Hilfer

in: Dynamical Processes in Condensed Molecular Systems
edited by: A. Blumen and J. Klafter and D. Haarer
World Scientific Publ.Co., Singapore, 302 (1990)
https://doi.org/10.1142/9789814540261
ISBN: 978-981-4540-26-1

submitted on
Monday, April 23, 1990

Correlated hopping transport through a disordered system is discussed in terms of a random walk model with memory correlations on a bond disordered lattice. Correlations will in general result in a difference between the transition rate to the previously occupied site and the rate for transitions to any other nearest neighbour site. Such a correlated process corresponds exactly to Fürth’s model for a random walk with a finite memory. This paper establishes a first order master equation for Fürth’s random walk on a bond disordered lattice. The equation is found to be equivalent to a symmetrized second order equation which was used previously as the starting point for an effective medium treatment.



For more information see

url

url
pdf

pdf
html

more
Categories
Disordered Systems electrical conductivity Percolation Transport Processes

Correlated Hopping in a Disordered Medium

R. Hilfer

Physical Review B 44, 628 (1991)
10.1103/PhysRevB.44.628

submitted on
Monday, March 6, 1989

This paper discusses random walks with memory on a percolating network as a model of correlated hopping transport through a disordered system. Correlations can arise from such sources as hard-core and Coulomb repulsions, correlated hops of groups of particles, or lattice-relaxation effects. In general these correlations will result in a difference between the hopping probability for return to the previously visited site and the probability to jump to another nearest neighbor of the currently occupied site. Thus the hopping process possesses a memory of its previous hop. Such a random walk is investigated in this paper for the case of bond percolation on a regular lattice. The frequency-dependent conductivity σ(ω) is calculated using a generalized effective-medium approximation. Results are presented for the linear chain and the hexagonal lattice. New features appear in both the real and the imaginary part of σ. These depend on the strength of the correlations and on the concentration of bonds. As an example, the possibility of a pronounced maximum in the real part of σ(ω) at finite frequencies is found, which is sometimes accompanied by a change of sign in the imaginary part. The results are found to agree qualitatively with experimental data on ionic transport in Na+ β-alumina, where both disorder and correlations are known to be important.



For more information see

url

url
pdf

pdf
html

more
Categories
Disordered Systems electrical conductivity Lattice Models Nonequilibrium Percolation Statistical Physics Transport Processes

Correlated Random Walks in Dynamically Disordered Systems

R. Hilfer, R. Orbach

in: Dynamical Processes in Condensed Molecular Systems
edited by: J. Klafter and J. Jortner and A. Blumen
World Scientific Publ.Co., Singapore, 175 (1989)
https://doi.org/10.1142/9789814434379_0009
ISBN: 978-981-4434-37-9

submitted on
Tuesday, November 22, 1988

We discuss correlated hopping motion in a dynamically disordered environment. Particles of type A with one hopping rate diffuse in a background of B-particles with a different hopping rate. Double occupancy of sites is forbidden. Without correlations the limit in which the ratio of hopping rates diverges corresponds to diffusion on a percolating network, while the case of equal hopping rates is that of self-diffusion in a lattice gas. We consider also the effect of correlations. In general these will change the transition rate of the A-particle to the previously occupied site as compared to the rate for transitions to all other neighbouring sites. We calculate the frequency dependent conductivity for this model with arbitrary ratio of hopping rates and correlation strength. Results are reported for the two dimensional hexagonal lattice and the three dimensional face centered cubic lattice. We obtain our results from a generalization of the effective medium approximation for frozen percolating networks. We predict the appearance of new features in real and imaginary part of the conductivity as a result of correlations. Crossover behaviour resulting from the combined effect of disorder and correlations leads to apparent power laws over roughly one to two decades in frequency. In addition we find a crossover between a low frequency regime where the response is governed by the rearrangements in the geometry and a high frequency regime where the geometry appears frozen. We calculate the correlation factor for the d.c. limit and check our results against Monte Carlo simulations on the hexagonal and face centered cubic lattices. In all cases we find good agreement.



For more information see

url

url
pdf

pdf
html

html
html

more

Categories
Disordered Systems electrical conductivity Percolation Random Walks Stochastic Processes Transport Processes

Continuous Time Random Walk Approach to Dynamic Percolation

R. Hilfer, R. Orbach

Chemical Physics 128, 275 (1988)
https://doi.org/10.1016/0301-0104(88)85076-6

submitted on
Friday, September 16, 1988

We present an approximate solution for time (frequency) dependent response under conditions of dynamic percolation which may be related to excitation transfer in some random structures. In particular, we investigate the dynamics of structures where one random component blocks a second (carrier) component. Finite concentrations of the former create a percolation network for the latter. When the blockers are allowed to move in time, the network seen by the carriers changes with time, allowing for long-range transport even if the instantaneous carrier site availability is less than pc, the critical percolation concentration. A specific example of this situation is electrical transport in sodium β”-alumina. The carriersare Na+ ions which can hop on a two-dimensional honeycomb lattice. The blockers are ions of much higher activation energy, such as Ba++. We study the frequency dependence of the conductivity for such a system. Given a fixed Ba++ hopping rate the Na+ ions experience a frozen site percolation environment for frequencies larger than the inverse hopping rate. At frequencies smaller than the inverse hopping rate, the Na+ ions experience a dynamic environment which allows long-rangetransport, even below the percoltion threshold. A continuous time random walk mode1 combined with an effective medium approximation allows us to arrive at a numerical solution for the frequency-dependent Na+ conductivity which clearly exhibits the crossover from frozen to dynamic environment.



For more information see

url

url
pdf

pdf
html

more