Category: Transport Processes

Categories
fluid flow Porous Media Simulations

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

A. Narvaez, Th. Zauner, F. Raischel, R. Hilfer, J. Harting

Journal of Statistical Mechanics 2010, P11026 (2010)
https://doi.org/10.1088/1742-5468/2010/11/P11026

submitted on
Sunday, May 30, 2010

During the last decade, lattice-Boltzmann 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 set-up. We present a detailed discussion of possible simulation set-ups and quantitative studies of the influence of simulation parameters. We illustrate our results by applying the algorithm to a Fontainebleau sandstone and by comparing our benchmark studies to other numerical permeability measurements in the literature.



For more information see

url

url
pdf

pdf
html

more
Categories
Porous Media Simulations Two-Phase Flow

Numerical solutions of a generalized theory for macroscopic capillarity

F. Doster, P. Zegeling, R. Hilfer

Physical Review E 81, 036307 (2010)
https://doi.org/10.1103/PhysRevE.81.036307

submitted on
Thursday, March 12, 2009

A recent macroscopic theory of biphasic flow in porous media [R. Hilfer, Phys. Rev. E 73, 016307 (2006)] has proposed to treat microscopically percolating fluid regions differently from microscopically nonpercolating regions. Even in one dimension the theory reduces to an analytically intractable set of ten coupled nonlinear partial differential equations. This paper reports numerical solutions for three different initial and boundary value problems that simulate realistic laboratory experiments. All three simulations concern a closed column containing a homogeneous porous medium filled with two immiscible fluids of different densities. In the first simulation the column is raised from a horizontal to a vertical orientation inducing a buoyancy-driven fluid flow that separates the two fluids. In the second simulation the column is first raised from a horizontal to a vertical orientation and subsequently rotated twice by 180° to compare the resulting stationary saturation profiles. In the third simulation the column is first raised from horizontal to vertical orientation and then returned to its original horizontal orientation. In all three simulations imbibition and drainage processes occur simultaneously inside the column. This distinguishes the results reported here from conventional simulations based on existing theories of biphasic flows. Existing theories are unable to predict flow processes where imbibition and drainage occur simultaneously. The approximate numerical results presented here show the same process dependence and hysteresis as one would expect from an experiment.



For more information see

url

url
pdf

pdf
html

more
Categories
Percolation Porous Media Two-Phase Flow

Percolation as a basic concept for macroscopic capillarity

R. Hilfer, F. Doster

Transport in Porous Media 82, 507 (2010)
https://doi.org/10.1007/s11242-009-9395-0

submitted on
Wednesday, November 12, 2008

The concepts of relative permeability and capillary pressure are crucial for the accepted traditional theory of two phase flow in porous media. Recently, a theoretical approach was introduced that does not require these concepts as input (Hilfer, Physica A, 359:119–128, 2006a; Phys. Rev. E, 73:016307, 2006b). Instead it was based on the concept of hydraulic percolation of fluid phases. This paper presents the first numerical solutions of the coupled nonlinear partial differential equations introduced in Hilfer (Phys. Rev. E, 73:016307, 2006b). Approximate numerical results for saturation profiles in one spatial dimension have been calculated. Long time limits of dynamic time-dependent profiles are compared to stationary solutions of the traditional theory. The initial and boundary conditions are chosen to model the displacement processes that occur when a closed porous column containing two immiscible fluids of different density is raised from a horizontal to a vertical position in a gravitational field. The nature of the displacement process may change locally in space and time between drainage and imbibition. The theory gives local saturations for nonpercolating trapped fluids near the endpoint of the displacement.



For more information see

url

url
pdf

pdf
html

html
html

more
Categories
Porous Media Simulations Two-Phase Flow

Modeling and simulation of macrocapillarity

R. Hilfer

in: CP1091, Modeling and Simulation of Materials
edited by: P. Garrido and P. Hurtado and J. Marro
American Institute of Physics, New York, 141 (2009)
https://doi.org/10.1063/1.3082273

submitted on
Monday, November 3, 2008

Macroscopic capillarity. or macrocapillarity for short, refers to capillary phenomena occurring during twophase and multiphase flow in porous media. Wetting phenomena and hysteresis in porous media are at present poorly understood in the sense that neither in physics nor in engineering a fully predictive theory seems to exist, that can describe or predict all observations. This paper extends the consitutive assumptions of a recent approach based on the concept of hydraulic percolation of fluid phases. The theory proposed here allows prediction of residual saturations. It can describe displacement processes in which imbibition and drainage occur simultaneously. This contrasts with the established traditional theory where capillary forces are lumped into capillary pressure function and relative permeabilities, and these functions need to be specified for each displacement process as input. Contrary to the traditional theory the approach advanced here allows to predict capillary pressure saturation relations as output.



For more information see

url

url
pdf

pdf
html

html
html

more
Categories
fluid flow Porous Media

Numerical Modeling of Fluid Flow in Porous Media and in Driven Colloidal Suspensions

J. Harting, T. Zauner, R. Weber, R. Hilfer

in: High Performance Computing in Science and Engineering ’08
edited by: W. Nagel and D. Kröner and M. Resch
Springer, Berlin, 349 (2009)
10.1007/978-3-540-88303-6
ISBN: 978-3-540-88301-2

submitted on
Tuesday, July 1, 2008

This article summarizes some of our main efforts performed on the computing facilities provided by the high performance computing centers in Stuttgart and Karlsruhe. At first, large scale lattice Boltzmann simulations are utilized to support resolution dependent analysis of geometrical and transport properties of a porous sandstone model. The second part of this report focuses on Brownian dynamics simulations of optical tweezer experiments where a large colloidal particle is dragged through a polymer solution and a colloidal crystal. The aim of these simulations is to improve our understanding of structuring effects, jamming behavior and defect formation in such colloidal systems.



For more information see

url

url
pdf

pdf
html

more
Categories
Porous Media Two-Phase Flow

Dimensional Analysis of Two-phase Flow Including a Rate-dependent Capillary Pressure-Saturation Relationship

S. Manthey, M. Hassanizadeh, R. Helmig, R. Hilfer

Advances in Water Resources 31, 1137 (2008)
https://doi.org/10.1016/j.advwatres.2008.01.021

submitted on
Thursday, March 15, 2007

The macroscopic modelling of two-phase flow processes in subsurface hydrosystems or industrial applications on the Darcy scale usu ally requires a constitutive relationship between capillary pressure and saturation, the Pc(Sw) relationship. Traditionally, it is assumed that a unique relation between Pc and Sw exists independently of the flow conditions as long as hysteretic effects can be neglected. Recently, this assumption has been questioned and alternative formulations have been suggested. For example, the extended Pc(Sw) relationship by Hassanizadeh and Gray [Hassanizadeh SM, Gray WG. Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resources 1990;13(4):169–86] proposes that the difference between the phase pressures to the equilibrium capillary pressure is a linear function of the rate of change of saturation, thereby introducing a constant of proportionality, the coefficient s. It is desirable to identify cases where the extended relationship needs to be considered. Consequently, a dimensional analysis is performed on the basis of the two-phase balance equations. In addition to the well-known capillary and gravitational number, the dimensional analysis yields a new dimensionless number. The dynamic number Dy quantifies the ratio of dynamic capillary to viscous forces. Relating the dynamic to the capillary as well as the gravitational number gives the new numbers DyC and DyG, respectively. For given sets of fluid and porous medium parameters, the dimensionless numbers Dy and DyC are interpreted as functions of the characteristic length and flow velocity. The simulation of an imbibition process provides insight into the interpretation of the characteristic length scale. The most promising choice for this length scale seems to be the front width. We conclude that consideration of the extended Pc(Sw) relationship may be important for porous media with high permeability, small entry pressure and high coefficient s when systems with a small characteristic length (e.g. steep front) and small characteristic time scale are under investigation.



For more information see

url

url
pdf

pdf
html

more
Categories
diffusion Fractional Calculus Porous Media Two-Phase Flow

Modeling Infiltration by Means of a Nonlinear Fractional Diffusion Model

E. Gerolymatou, I. Vardoulakis, R. Hilfer

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

submitted on
Thursday, May 18, 2006

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



For more information see

url

url
pdf

pdf
html

more
Categories
Porous Media Two-Phase Flow

Macroscopic capillarity without a constitutive capillary pressure function

R. Hilfer

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

submitted on
Friday, January 20, 2006

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



For more information see

url

url
pdf

pdf
html

more
Categories
diffusion Fractional Calculus Porous Media Two-Phase Flow

Simulating the Saturation Front Using a Fractional Diffusion Model

E. Gerolymatou, I. Vardoulakis, R. Hilfer

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

submitted on
Thursday, June 30, 2005

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



For more information see

url

url
pdf

pdf
html

more
Categories
Porous Media Two-Phase Flow

Macroscopic Capillarity and Hysteresis for Flow in Porous Media

R. Hilfer

Physical Review E 73, 016307 (2006)
https://doi.org/10.1103/PhysRevE.73.016307

submitted on
Friday, May 27, 2005

A macroscopic theory for capillarity in porous media is presented, challenging the established view that capillary pressure and relative permeability are constitutive parameter functions. The capillary pressure function in the present theory is not an input parameter but an outcome. The theoretical approach is based on introducing the residual saturations explicitly as state variables [as in Phys. Rev. E 58, 2090 (1998)]. Capillary pressure and relative permeability functions are predicted to exist for special cases. They exhibit hysteresis and process dependence as known from experiment.



For more information see

url

url
pdf

pdf
html

html
html

more
Categories
Porous Media Two-Phase Flow

Capillary Pressure, Hysteresis and Residual Saturation in Porous Media

R. Hilfer

Physica A 359, 119 (2006)
https://doi.org/10.1016/j.physa.2005.05.086

submitted on
Friday, May 20, 2005

A macroscopic theory for capillarity in porous media is presented. The capillary pressure function in this theory is not an input parameter but an outcome. The theory is based on introducing the trapped or residual saturations as state variables. It allows to predict spatiotemporal changes in residual saturation. The theory yields process dependence and hysteresis in capillary pressure as its main result.



For more information see

url

url
pdf

pdf
html

more
Categories
Heterogeneous Materials Porous Media Two-Phase Flow

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

R. Hilfer, R. Helmig

Advances in Water Resources 27, 1033 (2004)
https://doi.org/10.1016/j.advwatres.2004.07.003

submitted on
Monday, March 15, 2004

Dimensional analysis of the traditional equations of motion for two-phase flow in porous media allows to quantify the influence of heterogeneities. The heterogeneities are represented by position dependent capillary entry pressures and position dependent permeabilities. Dimensionless groups quantifying the influence of random heterogeneities are identified. For the case of heterogeneities with piecewise constant constitutive parameters (e.g. permeabilities, capillary pressures) we find that the upscaling ratio defined as the ratio of system size and the scale at which the constitutive parameters are known has to be smaller than the fluctuation strength of the heterogeneities defined e.g. as the ratio of the standard deviation to the mean value of a fluctuating quantity.



For more information see

url

url
pdf

pdf
html

more
Categories
diffusion Fractional Calculus

On fractional diffusion and continuous time random walks

R. Hilfer

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

submitted on
Thursday, May 22, 2003

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



For more information see

url

url
pdf

pdf
html

html
html

more
Categories
fluid flow Porous Media

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

C. Manwart, R. Hilfer

Physica A 314, 706 (2002)
https://doi.org/10.1016/S0378-4371(02)01193-7

submitted on
Sunday, March 30, 2003

In this paper we examine representative examples of realistic three-dimensional models for porous media by comparing their geometry and permeability with those of the original experimental specimen. The comparison is based on numerically exact evaluations of permeability, porosity, speci/c internal surface, mean curvature, Euler number and local percolation probabilities. The experimental specimen is a three-dimensional computer tomographic image of Fontainebleau sandstone. The three models are stochastic reconstructions for which many of the geometrical characteristics coincide with those of the experimental specimen. We /nd that in spite of the similarity in the geometrical properties the permeability and formation factor can differ greatly between models and the experiment.



For more information see

url

url
pdf

pdf
html

more
Categories
dielectric relaxation diffusion electrical conductivity Heterogeneous Materials Porous Media

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

R. Hilfer, J. Widjajakusuma, B. Biswal

Physica A 318, 319 (2003)
https://doi.org/10.1016/S0378-4371(02)01197-4

submitted on
Tuesday, June 4, 2002

Exact numerical solution of the electrostatic disordered potential problem is carried out for four fully discretised threedimensional experimental reconstructions of sedimentary rocks. The measured effective macroscopic dielectric constants and electrical conductivities are compared with parameterfree predictions from several mean field type theories. All these theories give agreeable results for low contrast between the media. Predictions from Local porosity theory, however, match for the entire range of contrast.



For more information see

url

url
pdf

pdf
html

more
Categories
dielectric relaxation Fractional Calculus Fractional Time Glasses

Experimental Evidence for Fractional Time Evolution in Glass Forming Materials

R. Hilfer

Chem.Phys. 284, 399 (2002)
https://doi.org/10.1016/S0301-0104(02)00670-5

submitted on
Friday, December 7, 2001

The infinitesimal generator of time evolution in the standard equation for exponential (Debye) relaxation is replaced with the infinitesimal generator of composite fractional translations. Composite fractional translations are defined as a combination of translation and the fractional time evolution introduced in [Physica A, 221 (1995) 89]. The fractional differential equation for composite fractional relaxation is solved. The resulting dynamical susceptibility is used to fit broad band dielectric spectroscopy data of glycerol. The composite fractional susceptibility function can exhibit an asymmetric relaxation peak and an excess wing at high frequencies in the imaginary part. Nevertheless it contains only a single stretching exponent. Qualitative and quantitative agreement with dielectric data for glycerol is found that extends into the excess wing. The fits require fewer parameters than traditional fit functions and can extend over up to 13 decades in frequency.



For more information see

url

url
pdf

pdf
html

html
html

more
Categories
fluid flow Simulations

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

C. Manwart, U. Aaltosalmi, A. Koponen, R. Hilfer, J. Timonen

Physical Review E 66, 016702 (2002)
https://doi.org/10.1103/PhysRevE.66.016702

submitted on
Wednesday, November 28, 2001

Numerical micropermeametry is performed on three-dimensional porous samples having a linear size of approximately 3 mm and a resolution of 7.5 micrometer. 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 that 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 physical diagenesis model appears to reproduce the permeability of the real sandstone sample quite accurately, while the permeabilities of the stochastic reconstructions deviate from the latter by at least an order of magnitude. This finding confirms earlier qualitative predictions based on local porosity theory. Two numerical algorithms were used in these simulations. One is based on the lattice-Boltzmann method, and the other on conventional finite-difference techniques. The accuracy of these two methods is discussed and compared, also with experiment.



For more information see

url

url
pdf

pdf
html

more
Categories
dielectric relaxation Fractional Calculus Fractional Time Glasses

Fitting the excess wing in the dielectric α-relaxation of propylene carbonate

R. Hilfer

Journal of Physics: Condensed Matter 14, 2297 (2002)
https://doi.org/10.1088/0953-8984/14/9/318

submitted on
Wednesday, November 28, 2001

A novel fitting function for the complex frequency-dependent dielectric susceptibility is introduced and compared against other fitting functions for experimental broadband dielectric loss spectra of propylene carbonate taken from Schneider et al (Schneider U, Lunkenheimer P, Brand R and Loidl A 1999 Phys. Rev. E 59 6924). The fitting function contains a single stretching exponent similar to the familiar Cole–Davidson or Kohlrausch stretched exponential fits. It is compared to these traditional fits as well as to the Havriliak–Negami susceptibility and a susceptibility for a two-step Debye relaxation. The results for the novel fit are found to give superior agreement.



For more information see

url

url
pdf

pdf
html

html
html

more
Categories
dielectric relaxation Glasses Special Functions

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

R. Hilfer

Physical Review E 65, 061510 (2002)
https://doi.org/10.1103/PhysRevE.65.061510

submitted on
Thursday, June 28, 2001

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.



For more information see

url

url
pdf

pdf
html

html
html

more
Categories
dielectric relaxation Glasses Nonequilibrium Special Functions

Analytical representations for relaxation functions of glasses

R. Hilfer

Journal of Non-Crystalline Solids 305, 122 (2002)
https://doi.org/10.1016/S0022-3093(02)01088-8

submitted on
Friday, April 13, 2001

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 rep- resented 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.



For more information see

url

url
pdf

pdf
html

more