Category: Simulations

Hysteresis in relative permeabilities suffices for propagation of saturation overshoot: A quantitative comparison with experiment

Post author By R Hilfer
Post date December 21, 2016

R. Steinle, R. Hilfer

Physical Review E 95, 043112 (2017)
https://doi.org/10.1103/PhysRevE.95.043112

submitted on
Wednesday, December 21, 2016

Traditional Darcy theory for two-phase flow in porous media is shown to predict the propagation of nonmonotone saturation profiles, also known as saturation overshoot. The phenomenon depends sensitively on the constitutive parameters, on initial conditions, and on boundary conditions. Hysteresis in relative permeabilities is needed to observe the effect. Two hysteresis models are discussed and compared. The shape of overshoot solutions can change as a function of time or remain fixed and time independent. Traveling-wave-like overshoot profiles of fixed width exist in experimentally accessible regions of parameter space. They are compared quantitatively against experiment.

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Tags 2017, Hysteresis, saturation overshoot

Disordered Systems Porous Media Precision Simulations Simulations

Differential porosimetry and permeametry for random porous media

Post author By R Hilfer
Post date January 5, 2015

R. Hilfer, A. Lemmer

Physical Review E 92, 013305 (2015)
https://doi.org/10.1103/PhysRevE.92.013305

submitted on
Monday, January 5, 2015

Accurate determination of geometrical and physical properties of natural porous materials is notoriously difficult. Continuum multiscale modeling has provided carefully calibrated realistic microstructure models of reservoir rocks with floating point accuracy. Previous measurements using synthetic microcomputed tomography (μ-CT) were based on extrapolation of resolution-dependent properties for discrete digitized approximations of the continuum microstructure. This paper reports continuum measurements of volume and specific surface with full floating point precision. It also corrects an incomplete description of rotations in earlier publications. More importantly, the methods of differential permeametry and differential porosimetry are introduced as precision tools. The continuum microstructure chosen to exemplify the methods is a homogeneous, carefully calibrated and characterized model for Fontainebleau sandstone. The sample has been publicly available since 2010 on the worldwide web as a benchmark for methodical studies of correlated random media. High-precision porosimetry gives the volume and internal surface area of the sample with floating point accuracy. Continuum results with floating point precision are compared to discrete approximations. Differential porosities and differential surface area densities allow geometrical fluctuations to be discriminated from discretization effects and numerical noise. Differential porosimetry and Fourier analysis reveal subtle periodic correlations. The findings uncover small oscillatory correlations with a period of roughly 850 μm, thus implying that the sample is not strictly stationary. The correlations are attributed to the deposition algorithm that was used to ensure the grain overlap constraint. Differential permeabilities are introduced and studied. Differential porosities and permeabilities provide scale-dependent information on geometry fluctuations, thereby allowing quantitative error estimates.

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Porous Media Simulations

Parallel domain decomposition method with non-blocking communication for flow through porous media

Post author By R Hilfer
Post date December 19, 2013

A. Lemmer, R. Hilfer

Journal of Computational Physics 281, 970-981 (2015)
https://doi.org/10.1016/j.jcp.2014.08.032

submitted on
Thursday, December 19, 2013

This paper introduces a domain decomposition method for numerically solving the Stokes equation for very large, complex geometries. Examples arise from realistic porous media. The computational method is based on the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm which uses a finite-differences approach for discretizing the underlying equations. It achieves comparable speed and efficiency as lattice Boltzmann methods. The domain decomposition method splits a large three-dimensional region into slices that can be processed in parallel on multi-processor computation environments with only minimal communication between the computation nodes. With this method, the flow through a porous medium with grid sizes up to 2048 x 2048 x 2048 voxel has been calculated.

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Tags domain decomposition, parallelization

image analysis Porous Media

High precision synthetic computed tomography of reconstructed porous media

Post author By R Hilfer
Post date July 26, 2011

R. Hilfer, T. Zauner

Physical Review E 84, 062301 (2011)
https://doi.org/10.1103/PhysRevE.84.062301

submitted on
Tuesday, July 26, 2011

Multiscale simulation of transport in disordered and porous media requires microstructures covering several decades in length scale. X-ray and synchrotron computed tomography are presently unable to resolve more than one decade of geometric detail. Recent advances in pore scale modeling [Biswal, Held, Khanna, Wang, and Hilfer, Phys. Rev. E 80, 041301 (2009)] provide strongly correlated microstructures with several decades in microstructural detail. A carefully calibrated microstructure model for Fontainebleau sandstone has been discretized into a suite of three-dimensional microstructures with resolutions from roughly 128 μm down to roughly 500 nm. At the highest resolution the three-dimensional image consists of 35 184 372 088 832 discrete cubic volume elements with gray values between 0 and 216. To the best of our knowledge, this synthetic image is the largest computed tomogram of a porous medium available at present.

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image analysis Porous Media

Particle-based Rendering for Porous Media

Post author By R Hilfer
Post date October 28, 2010

S. Grottel, G. Reina, T. Zauner, R. Hilfer, T. Ertl

in: Proceedings of SIGRAD 2010: Content aggregation and visualization
edited by: K. Jää-Aro and T. Larsson
Link{\”o}ping Electronic Conference Proceedings, vol. 52,Link{\”o}ping University Electronic Press, Linköping, 45 (2010)

submitted on
Thursday, October 28, 2010

Particle-based modeling and simulation of granular or porous media is a widely-used tool in physics and material science to study behavior like fracture and failure under external force. Classical models use spherical particles. However, up to 108 polyhedral-shaped particles are required to achieve realistic results comparable to laboratory experiments. As contact points and exposed surfaces play important roles for the analysis, a meaningful visualization aiding the numeric analysis has to represent the exact particle shapes. For particle-based data sets with spherical particles, ray tracing has been established as the state-of-the-art approach yielding high rendering performance, optimal visual quality and good scalability. However, when rendering polyhedral-shaped particles, there is no issue with visual quality comparing polygon-based rendering approaches and ray casting, whereas the polygon-based approaches cause significantly lower fragment load. The paper at hand investigates the advantages and drawbacks of both approaches by analyzing the performance of state-of-the-art rendering methods employing vertex-buffer objects, hardware-supported instancing, geometry shader, and GPU-based ray casting.

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fluid flow Porous Media Simulations

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

Post author By R Hilfer
Post date May 30, 2010

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.

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Porous Media Precision Simulations Simulations

Continuum-based rock model of a reservoir dolostone with four orders of magnitude in pore sizes

Post author By R Hilfer
Post date May 28, 2010

S. Roth, B. Biswal, G. Afshar, R. Held, P. Øren, L. Berge, R. Hilfer

AAPG Bulletin 95, 925 (2011)
DOI:10.1306/12031010092

submitted on
Friday, May 28, 2010

A continuum-based pore-scale representation of a dolomite reservoir rock is presented, containing several orders of magnitude in pore sizes within a single rock model. The macroscale rock fabric from a low-resolution x-ray microtomogram was combined with microscale information gathered from high- resolution two-dimensional electron microscope images. The low-resolution x-ray microtomogram was segmented into six separate rock phases in terms of mineralogy, matrix appearances, and open- versus crystal-filled molds. These large-scale rock phases were decorated (modeled) with geometric objects, such as different dolomite crystal types and anhydrite, according to the high-resolution information gathered from the electron microscope images. This procedure resulted in an approximate three-dimensional representation of the diage- netically transformed rock sample with respect to dolomite crystal sizes, porosity, appearance, and volume of different matrix phases and pore/matrix/cement ratio. The resulting rock model contains a pore-size distribution ranging from moldic macropores (several hundred micrometers in diameter) down to mudstone micropores ( less than 1 mm in diameter). This allows us to study the effect and contribution of different pore classes to the petrophysical properties of the rock. Higher resolution x-ray tomographs of the same rock were used as control volumes for the pore-size distribution of the model. The pore-size analysis and percolation tests performed in three dimensions at various discretization resolutions indicate pore-throat radii of 1.5 to 6 mm for the largest interconnected pore network. This also highlights the challenge to determine appropriate resolutions for x-ray imaging when the exact rock microstructure is not known.

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image analysis Porous Media Precision Simulations Simulations

Towards precise prediction of transport properties from synthetic computer tomography of reconstructed porous media

Post author By R Hilfer
Post date April 4, 2009

B. Biswal, R.J. Held, V. Khanna, J. Wang, R. Hilfer

Physical Review E 80, 041301 (2009)
https://doi.org/10.1103/PhysRevE.80.041301

submitted on
Saturday, April 4, 2009

Transport properties of a multiscale carbonate rock are predicted from pore scale models, reconstructed using a continuum geometrical modeling technique. The method combines crystallite information from two-dimensional high-resolution images with sedimentary correlations from a three-dimensional low-resolution microcomputed tomography micro-CT-image to produce a rock sample with calibrated porosity, structural correlation, and transport properties at arbitrary resolutions. Synthetic micro-CT images of the reconstructed model match well with experimental micro-CT images at different resolutions, making it possible to predict physical transport parameters at higher resolutions.

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Porous Media Simulations Two-Phase Flow

Numerical solutions of a generalized theory for macroscopic capillarity

Post author By R Hilfer
Post date March 12, 2009

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.

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image analysis Porous Media Simulations

Continuum reconstruction of the pore scale microstructure for Fontainebleau sandstone

Post author By R Hilfer
Post date March 11, 2009

F. Latief, B. Biswal, U. Fauzi, R. Hilfer

Physica A 389, 1607 (2010)
https://doi.org/10.1016/j.physa.2009.12.006

submitted on
Wednesday, March 11, 2009

A stochastic geometrical modeling technique is used to reconstruct a laboratory scale Fontainebleau sandstone with a sidelength of 1.5 cm. The model reconstruction is based on crystallite properties and diagenetic parameters determined from two-dimensional images. The three-dimensional pore scale microstructure of the sandstone is represented by a list of quartz crystallites defined geometrically and placed in the continuum. This allows generation of synthetic μ-CT images of the rock model at arbitrary resolutions. Quantitative microstructure comparison based on Minkowski functionals, two-point correlation function and local porosity theory indicates that this modeling technique can provide more realistic and accurate models of sandstones than many existing techniques used currently. Synthetic μ-CT images at different resolutions from a laboratory scale model of Fon- tainebleau sandstone are made available to the scientific community for resolution dependent petrophysical analysis.

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Porous Media Precision Simulations

Modeling of Multiscale Porous Media

Post author By R Hilfer
Post date January 30, 2009

B. Biswal, P.E. Øren, R.J. Held, S. Bakke, R. Hilfer

Image Analysis and Stereology 28, 23-34 (2009)
DOI: 10.5566/ias.v28.p23-34

submitted on
Friday, January 30, 2009

A stochastic geometrical modeling method for reconstructing three dimensional pore scale microstructures of multiscale porous media is presented. In this method the porous medium is represented by a random but spatially correlated structure of objects placed in the continuum. The model exhibits correlations with the sedimentary textures, scale dependent intergranular porosity over many decades, vuggy or dissolution porosity, a percolating pore space, a fully connected matrix space, strong resolution dependence and wide variability in the permeabilities and other properties. The continuum representation allows discretization at arbitrary resolutions providing synthetic micro-computertomographic images for resolution dependent fluid flow simulation. Model implementations for two different carbonate rocks are presented. The method can be used to generate pore scale models of a wide class of multiscale porous media.

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Porous Media Simulations Two-Phase Flow

Modeling and simulation of macrocapillarity

Post author By R Hilfer
Post date November 3, 2008

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.

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image analysis Porous Media Simulations

Stochastic Multiscale Model for Carbonate Rocks

Post author By R Hilfer
Post date January 9, 2007

B. Biswal, P.E. Øren, R. Held, S. Bakke, R. Hilfer

Physical Review E 75, 061303 (2007)
https://doi.org/10.1103/PhysRevE.75.061303

submitted on
Tuesday, January 9, 2007

A multiscale model for the diagenesis of carbonate rocks is proposed. It captures important pore scale characteristics of carbonate rocks: wide range of length scales in the pore diameters; large variability in the permeability; and strong dependence of the geometrical and transport parameters on the resolution. A pore scale microstructure of an oolithic dolostone with generic diagenetic features is successfully generated. The continuum representation of a reconstructed cubic sample of sidelength 2 mm contains roughly 42⫻ 106 crystallites and pore diameters varying over many decades. Petrophysical parameters are computed on discretized samples of sizes up to 10003. The model can be easily adapted to represent the multiscale microstructure of a wide variety of carbonate rocks.

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image analysis Porous Media

Characterization of Porous Media by Local Porosities, Minkowski- and Non-Minkowski Functionals

Post author By R Hilfer
Post date April 16, 2004

R. Hilfer

Microscopy and Microanalysis 10, 72 (2004)
DOI: 10.1017/S1431927604884472

submitted on
Friday, April 16, 2004

The presentation reviews local porosity theory. Recent results and new developments are discussed.

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Simulations

Physics on the Computer

Post author By R Hilfer
Post date June 28, 2002

R. Hilfer

ICP, Stuttgart, 2002

submitted on
Friday, June 28, 2002

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fluid flow Simulations

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

Post author By R Hilfer
Post date November 28, 2001

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.

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Simulations

Physics on fast computers: What are computer experiments good for?

Post author By R Hilfer
Post date September 1, 2000

R. Hilfer, S. Schwarzer, F. Gähler, J. Roth

Stuttgarter Uni Kurier 86 2/2000, 53 (2000)

submitted on
Friday, September 1, 2000

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image analysis Porous Media

On the Analysis of Spatial Binary Images

Post author By R Hilfer
Post date March 28, 2000

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.

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image analysis Lattice Models Porous Media

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

Post author By R Hilfer
Post date February 17, 2000

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.

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image analysis Porous Media review article

Review on Scale Dependent Characterization of the Microstructure of Porous Media

Post author By R Hilfer
Post date February 8, 2000

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.

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