Category: Porous Media

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
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
image analysis Porous Media

On the Analysis of Spatial Binary Images

C. Lang, J. Ohser, R. Hilfer

Journal of Microscopy 203, 303 (2001)
https://doi.org/10.1046/j.1365-2818.2001.00899.x

submitted on
Tuesday, March 28, 2000

This paper deals with the analysis of spatial images taken from microscopically heterogeneous but macroscopically homogeneous microstructures. A new method is presented, which is strictly based on integral‐geometric formulae such as Crofton’s intersection formulae and Hadwiger’s recursive definition of the Euler number. By means of this approach the quermassdensities can be expressed as the inner products of two vectors where the first vector carries the ‘integrated local knowledge’ about the microstructure and the second vector depends on the lateral resolution of the image as well as the quadrature rules used in the discretization of the integral‐geometric formulae. As an example of application we consider the analysis of spatial microtomographic images obtained from natural sandstones.



For more information see

url

url
pdf

pdf
html

more
Categories
image analysis Lattice Models Porous Media

Erosion-Dilation Analysis for Experimental and Synthetic Microstructures of Sedimentary Rock

A. Tscheschel, D. Stoyan, R. Hilfer

Physica A 284, 46 (2000)
https://doi.org/10.1016/S0378-4371(00)00116-3

submitted on
Thursday, February 17, 2000

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.



For more information see

url

url
pdf

pdf
html

more
Categories
image analysis Porous Media review article

Review on Scale Dependent Characterization of the Microstructure of Porous Media

R. Hilfer

Transport in Porous Media 46, 373 (2002)
https://doi.org/10.1023/A:1015014302642

submitted on
Tuesday, February 8, 2000

The paper discusses local porosity theory and its relation with other geometric characterization methods for porous media such as correlation functions and contact distributions. Special emphasis is placed on the charcterization of geometric observables through Hadwigers theorem in stochastic geometry. The four basic Minkowski functionals are introduced into local porosity theory, and for the first time a relationship is established between the Euler characteristic and the local percolation probabilities. Local porosity distributions and local percolation probabilities provide a scale dependent characterization of the microstructure of porous media that can be used in an effective medium approach to predict transport.



For more information see

url

url
pdf

pdf
html

html
html

more
Categories
Porous Media

Stochastic Reconstruction of Sandstones

C. Manwart, S. Torquato, R. Hilfer

Physical Review E 62, 893 (2000)
https://doi.org/10.1103/PhysRevE.62.893

submitted on
Friday, February 4, 2000

A simulated annealing algorithm is employed to generate a stochastic model for a Berea sandstone and a Fontainebleau sandstone, with each a 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 significant differences of the geometrical connectivity between the reconstructed and the experimental samples.



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
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.



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
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.



For more information see

url

url
pdf

pdf
html

more
Categories
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.



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
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.



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
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.



For more information see

url

url
pdf

pdf
html

more