Category: Disordered Systems

Effective transport coefficients of anisotropic disordered materials

Post author By R Hilfer
Post date January 4, 2022

R. Hilfer, J. Hauskrecht

European Physical Journal B 95, 117 (2022)
https://doi.org/10.1140/epjb/s10051-022-00338-5

submitted on
Tuesday, January 4, 2022

A novel effective medium theory for homogenized transport coefficients of anisotropic mixtures of possibly anisotropic materials is developed. Existing theories for isotropic systems cannot be easily extended, because that would require geometric characterizations of anisotropic connectivity. In this work anisotropic connectivity is characterized by introducing a tensor that is constructed from a histogram of local percolating directions. The construction is inspired by local porosity theory. A large number of known and unknown generalized effective medium approximations for anisotropic media are obtained as limiting special cases from the new theory. Among these limiting cases the limit of strong cylindrical anisotropy is of particular interest. The parameter space of the generalized theory is explored, and the advanced results are applied to experiment.

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fluid flow Porous Media review article Two-Phase Flow

A Brief Review of Capillary Number and its Use in Capillary Desaturation Curves

Post author By R Hilfer
Post date August 9, 2021

H. Guo, K. Song, R. Hilfer

Transport in Porous Media 144, 3-31 (2022)
https://doi.org/10.1007/s11242-021-01743-7

submitted on
Monday, August 9, 2021

Capillary number, understood as the ratio of viscous force to capillary force, is one of the most important parameters in enhanced oil recovery (EOR). It continues to attract the interest of scientists and engineers, because the nature and quantification of macroscopic capillary forces remains controversial. At least 41 different capillary numbers have been collected here from the literature. The ratio of viscous and capillary force enters crucially into capillary desaturation experiments. Although the ratio is length scale dependent, not all definitions of capillay number depend on length scale, indicating potential inconsistencies between various applications and publications. Recently, new numbers have appeared and the subject continues to be actively discussed. Therefore, a short review seems appropriate and pertinent.

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Mathematical Physics Porous Media Two-Phase Flow

Existence and Uniqueness of Nonmonotone Solutions in Porous Media Flow

Post author By R Hilfer
Post date May 5, 2021

R. Steinle, T. Kleiner, P. Kumar, R. Hilfer

Axioms 11, 327 (2022)
https://doi.org/10.3390/axioms11070327

submitted on
Thursday, May 5, 2022

Existence and uniqueness of solutions for a simplified model of immiscible two-phase flow in porous media are obtained in this paper. The mathematical model is a simplified physical model with hysteresis in the flux functions. The resulting semilinear hyperbolic-parabolic equation is expected from numerical work to admit non-monotone imbibition-drainage fronts. We prove the local existence of imbibition-drainage fronts. The uniqueness, global existence, maximal regularity and boundedness of the solutions are also discussed. Methodically, the results are established by means of semigroup theory and fractional interpolation spaces.

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Disordered Systems Heterogeneous Materials Percolation Porous Media

Percolativity of Porous Media

Post author By R Hilfer
Post date April 19, 2021

R. Hilfer, J. Hauskrecht

Transport in Porous Media 145, 1-12 (2022)
https://doi.org/10.1007/s11242-021-01735-7

submitted on
Monday, April 19, 2021

Connectivity and connectedness are non-additive geometric functionals on the set of pore scale structures. They determine transport of mass, volume or momentum in porous media, because without connectivity there cannot be transport. Percolativity of porous media is introduced here as a geometric descriptor of connectivity, that can be computed from the pore scale and persists to the macroscale through a suitable upscaling limit. It is a measure that combines local percolation probabilities with a probability density of ratios of eigenvalues of the tensor of local percolating directions. Percolativity enters directly into generalized effective medium approximations. Predictions from these generalized effective medium approximations are found to be compatible with apparently anisotropic Archie correlations observed in experiment.

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Fractional Calculus Functional analysis Glasses Mathematical Physics Mathematics Special Functions

Fractional glassy relaxation and convolution modules of distributions

Post author By R Hilfer
Post date September 30, 2020

T. Kleiner, R. Hilfer

Analysis and Mathematical Physics 11, 130 (2021)
https://doi.org/10.1007/s13324-021-00504-5

submitted on
Wednesday, September 30, 2020

Solving fractional relaxation equations requires precisely characterized domains of definition for applications of fractional differential and integral operators. Determining these domains has been a longstanding problem. Applications in physics and engineering typically require extension from domains of functions to domains of distributions. In this work convolution modules are constructed for given sets of distributions that generate distributional convolution algebras. Convolutional inversion of fractional equations leads to a broad class of multinomial Mittag-Leffler type distributions. A comprehensive asymptotic analysis of these is carried out. Combined with the module construction the asymptotic analysis yields domains of distributions, that guarantee existence and uniqueness of solutions to fractional differential equations. The mathematical results are applied to anomalous dielectric relaxation in glasses. An analytic expression for the frequency dependent dielectric susceptibility is applied to broadband spectra of glycerol. This application reveals a temperature independent and universal dynamical scaling exponent.

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fluid flow Porous Media Two-Phase Flow

A Critical Review of Capillary Number and its Application in Enhanced Oil Recovery

Post author By R Hilfer
Post date August 30, 2020

H. Guo, K. Song, R. Hilfer

SPE Conference Proceedings 2020, SPE-200419-MS (2020)
https://doi.org/10.2118/200419-MS

submitted on
Sunday, August 30, 2020

Capillary number (Ca), defined as dimensionless ratio of viscous force to capillary force, is one of the most important parameters in enhanced oil recovery (EOR). The ratio of viscous and capillary force is scale-dependent. At least 33 different Cas have been proposed, indicating inconsistencies between various applications and publications. The most concise definition containing velocity, interfacial tension and viscosity is most widely used in EOR. Many chemical EOR applications are thus based on the correlation between residual oil saturation (ROS) and Ca, which is also known as capillary desaturation curve (CDC). Various CDCs lead to a basic conclusion of using surfactant to reduce interfacial tension to ultra-low values to get a minimum ROS and maximum displacement efficiency. However, after a deep analysis of Ca and recent new experimental observations, the traditional definition of Ca was found to have many limitations and based on misunderstandings. First, the basic object in EOR is a capillary-trapped oil ganglia, thus Darcy’s law is only valid under certain conditions. Further, many recent tests reported results contradicting previous ones. It seems most Cas cannot account for mixed-wet CDC. The influence of wettability on two-phase flow is important but not reflected in the definition of the Ca. Then, it is certainly very peculiar that, when the viscous and capillary forces acting on a blob are equal, the current most widely used classic Ca is equal to 2.2* 10−3. Ideally, the condition Ca ∼ 1 marks the transition from capillary dominated to viscous-dominated flow, but most Cas cannot fulfill this expectation. These problems are caused by scale dependent flow characterization. It has been proved that the traditional Ca is of microscopic nature. Based on the dynamic characterization of the change of capillary force and viscous force on the macroscopic scale, a macroscopic Ca can well explain these complex results. The requirement of ultra-low IFT from microscopic Ca for surfactant flood is not supported by macroscopic Ca. The effect of increasing water viscosity to EOR is much higher than reducing IFT. Realizing the microscopic nature of the traditional Ca and using CDCs based on the more reasonable macroscopic Ca helps to update screening criteria for chemical flooding.

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Porous Media Two-Phase Flow

Capillary number correlations for two-phase flow in porous media

Post author By R Hilfer
Post date August 20, 2020

R. Hilfer

Physical Review E 102, 053103 (2020)
https://doi.org/10.1103/PhysRevE.102.053103

submitted on
Thursday, August 20, 2020

Relative permeabilities and capillary number correlations are widely used for quantitative estimates of enhanced water flood performance in porous media. They enter as essential parameters into reservoir simulations. Experimental capillary number correlations for seven different reservoir rocks and 21 pairs of wetting and nonwetting fluids are analyzed. The analysis introduces generalized local macroscopic capillary number correlations. It eliminates shortcomings of conventional capillary number correlations. Surprisingly, the use of capillary number correlations on reservoir scales may become inconsistent in the sense that the limits of applicability of the underlying generalized Darcy law are violated. The results show that local macroscopic capillary number correlations can distinguish between rock types. The experimental correlations are ordered systematically using a three-parameter fit function combined with a novel fluid pair based figure of merit.

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dielectric relaxation Disordered Systems Glasses Transport Processes

Excess wing physics and nearly constant loss in glasses

Post author By R Hilfer
Post date May 31, 2019

R. Hilfer

Journal of Statistical Mechanics: Theory and Experiment 2019, 104007 (2019)
https://doi.org/10.1088/1742-5468/ab38bc

submitted on
Friday, May 31, 2019

Excess wings and nearly constant loss are almost universal nonequilibrium phenomena in glass formers. Both lack an accepted theoretical foundation. A model-free and unified theoretical description for these phenomena is presented that encompasses also fast β-processes, emergent Debye peaks, and the relaxation strength of the boson peak. The theory is model-free in the same way as the classical Debye relaxation equation for orientational polarisation. It is based on generalizing time flow from translation semigroups to composite time translation-convolution semigroups. Composite translation-convolution fits have less parameters than traditional fits. They need only one dynamic scaling exponent, while four are needed in Havriliak-Negami fits. For glycerol the single dynamic exponent in the translation-convolution fit is found to be temperature-independent.

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Tags 2019, Excess wing

Porous Media Two-Phase Flow

Stable Propagation of Saturation Overshoots for Two-Phase Flow in Porous Media

Post author By R Hilfer
Post date March 21, 2017

M. Schneider, T. Köppl, R. Helmig, R. Steinle, R. Hilfer

Transport in Porous Media 121, 621-641 (2018)
https://doi.org/10.1007/s11242-017-0977-y

submitted on
Tuesday, March 21, 2017

Propagation of saturationovershoots for two-phaseflow of immiscible and incompressible fluids in porous media is analyzed using different computational methods. In particular, it is investigated under which conditions a given saturation overshoot remains stable while moving through a porous medium. Two standard formulations are employed in this investigation, a fractional flow formulation and a pressure–saturation formulation. Neumann boundary conditions for pressure are shown to emulate flux boundary conditions in homogeneous media. Gravity driven flows with Dirichlet boundary conditions for pressure that model infiltration into heterogeneous media with position-dependent permeability are found to exhibit pronounced saturation overshoots very similar to those seen in experiment.

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Tags 2018, Hysteresis, saturation overshoot

Heterogeneous Materials Mathematical Physics Percolation Porous Media

Multiscale local porosity theory, weak limits, and dielectric response in composite and porous media

Post author By R Hilfer
Post date December 22, 2016

R. Hilfer

Journal of Mathematical Physics 59, 103511 (2018)
https://doi.org/10.1063/1.5063466

submitted on
Thursday, December 22, 2016

A mathematical scaling approach to macroscopic heterogeneity of composite and porous media is introduced. It is based on weak limits of uniformly bounded measurable functions. The limiting local porosity distributions, that were introduced in Advances in Chemical Physics, vol XCII, p. 299-424 (1996), are found to be related to Young measures of a weakly convergent sequence of local volume fractions. The Young measures determine frequency dependent complex dielectric functions of multiscale media within a generalized selfconsistent effective medium approximation. The approach separates scales by scale factor functions of regular variation. It renders upscaled results independent of the shape of averaging windows upon reaching the scaling limit.

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Tags 2018, local percolation probability, local porosity theory, Multiscaling, Scaling Limit

fluid flow Porous Media Simulations Two-Phase Flow

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

fluid flow Porous Media Two-Phase Flow

Pore-scale displacement mechanisms as a source of hysteresis for two-phase flow in porous media

Post author By R Hilfer
Post date October 16, 2015

S. Schlüter, S. Berg, M. Rücker, R. Armstrong, H.-J. Vogel, R. Hilfer, D. Wildenschild

Water Resources Research 52, 2194-2205 (2016)
https://doi.org/10.1002/2015WR018254

submitted on
Friday, October 16, 2015

The macroscopic description of the hysteretic behavior of two-phase flow in porous media remains a challenge. It is not obvious how to represent the underlying pore-scale processes at the Darcy-scale in a consistent way. Darcy-scale thermodynamic models do not completely eliminate hysteresis and our findings indicate that the shape of displacement fronts is an additional source of hysteresis that has not been considered before. This is a shortcoming because effective process behavior such as trapping efficiency of CO 2 or oil production during water flooding are directly linked to pore-scale displacement mechanisms with very different front shape such as capillary fingering, flat frontal displacement, or cluster growth. Here we introduce fluid topology, expressed by the Euler characteristic of the nonwetting phase, as a shape measure of displacement fronts. Using two high-quality data sets obtained by fast X-ray tomography, we show that the Euler characteristic is hysteretic between drainage and imbibition and characteristic for the underlying displacement pattern. In a more physical sense, the Euler characteristic can be interpreted as a parameter describing local fluid connectedness. It may provide the closing link between a topological characterization and macroscopic formulations of two-phase immiscible displacement in porous rock. Since fast X-ray tomography is currently becoming a mature technique, we expect a significant growth in high-quality data sets of real time fluid displacement processes in the future. The novel measures of fluid topology presented here have the potential to become standard metrics needed to fully explore them.

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Tags 2016, Hysteresis

Porous Media

Capillary saturation and desaturation

Post author By R Hilfer
Post date June 3, 2015

R. Hilfer, R. Armstrong, S. Berg, A. Georgiadis, H. Ott

Physical Review E 92, 063023 (2015)
https://doi.org/10.1103/PhysRevE.92.063023

submitted on
Wednesday, June 3, 2015

Capillary desaturation experiments produce disconnected (trapped) ganglia of mesoscopic sizes intermediate between pore size and system size. Experimental evidence for interactions between these mesoscale clusters during desaturation is analyzed and discussed within the established microscopic and macroscopic laws of Newton, Young-Laplace, and Darcy. A theoretical expression for capillary number correlations is introduced that seems to have remained unnoticed. It expresses capillary desaturation curves in terms of stationary capillary pressures and relative permeabilities. The theoretical expression shows that the plateau saturation in capillary desaturation curves may in general differ from the residual nonwetting saturation defined through the saturation limit of the main hysteresis loop. Hysteresis effects as well as the difference between wetting and nonwetting fluids are introduced into the analysis of capillary desaturation experiments. The article examines experiments with different desaturation protocols and discusses the existence of a mesoscopic length scale intermediate between pore scale and sample scale. The theoretical expression is derived entirely within the existing traditional theory of two-phase flow in porous media and compared to a recent experiment.

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Tags 2015, capillarity

Porous Media Two-Phase Flow

Non-monotonic Travelling Wave Fronts in a System of Fractional Flow Equations from Porous Media

Post author By R Hilfer
Post date May 31, 2015

O. Hönig, P. Zegeling, F. Doster, R. Hilfer

Transport in Porous Media 114, 309-340 (2016)
https://doi.org/10.1007/s11242-015-0618-2

submitted on
Sunday, May 31, 2015

Motivated by observations of saturation overshoot, this article investigates generic classes of smooth travelling wave solutions of a system of two coupled nonlinear parabolic partial differential equations resulting from a flux function of high symmetry. All boundary resp. limit value problems of the travelling wave ansatz, which lead to smooth travelling wave solutions, are systematically explored. A complete, visually and computationally useful representation of the five-dimensional manifold connecting wave velocities and boundary resp. limit data is found by using methods from dynamical systems theory. The travelling waves exhibit monotonic, non-monotonic or plateau-shaped behaviour. Special attention is given to the non-monotonic profiles. The stability of the travelling waves is studied by numerically solving the full system of the partial differential equations with an efficient and accurate adaptive moving grid solver.

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Tags 2016, 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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fluid flow Porous Media Two-Phase Flow

Inﬂuence of Initial Conditions on Propagation, Growth and Decay of Saturation Overshoot

Post author By R Hilfer
Post date November 24, 2014

R. Steinle, R. Hilfer

Transport in Porous Media 111, 369-380 (2016)
https://doi.org/10.1007/s11242-015-0598-2

submitted on
Monday, November 24, 2014

A sequence of drainage and imbibition shocks within the traditional theory of two-phase immiscible displacement can give rise to shallow non-monotone saturation profiles as shown in Hilfer and Steinle (Eur Phys J Spec Top 223:2323, 2014). This phenomenon depends sensitively on model parameters and initial conditions. The dependence of saturation overshoot on initial conditions is investigated more systematically in this article. The results allow to determine regions in the parameter space for the observation of saturation overshoot and to explore limitations of the underlying idealized hysteresis model. Numerical solutions of the nonlinear partial differential equations of motion reveal a strong dependence of the overshoot phenomenon on the boundary and initial conditions. Overshoot solutions with experimentally detectable height are shown to exist numerically. Extensive parameter studies reveal different classes of initial conditions for which the width of the overshoot region can decrease, increase or remain constant.

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Tags Hysteresis, saturation overshoot

dielectric relaxation Glasses

Excess wings in broadband dielectric spectroscopy

Post author By R Hilfer
Post date July 15, 2014

S. Candelaresi, R. Hilfer

AIP Conference Proceedings 1637, 1283 (2014)
https://doi.org/10.1063/1.4907293

submitted on
Tuesday, July 15, 2014

Analysis of excess wings in broadband dielectric spectroscopy data of glass forming materials provides evidence for anomalous time evolutions and fractional semigroups. Solutions of fractional evolution equations in frequency space are used to fit dielectric spectroscopy data of glass forming materials with a range between 4 and 10 decades in frequency. It is shown that with only three parameters (two relaxation times plus one exponent) excellent fits can be obtained for 5-methyl-2-hexanol and for methyl-m-toluate over up to 7 decades. The traditional Havriliak-Negami fit with three parameters (two exponents and one relaxation time) fits only 4-5 decades. Using a second exponent, as in Havriliak-Negami fits, the α-peak and the excess wing can be modeled perfectly with our theory for up to 10 decades for all materials at all temperatures considered here. Traditionally this can only be accomplished by combining two Havriliak-Negami functions with 7 parameters. The temperature dependent relaxation times are fitted with the Vogel-Tammann-Fulcher relation which provides the corresponding Vogel-Fulcher temperatures. The relaxation times turn out to obey almost perfectly the Vogel-Tammann-Fulcher law. Computable expressions of time dependent relaxation functions are also reported.

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Porous Media Two-Phase Flow

Saturation overshoot and hysteresis for twophase flow in porous media

Post author By R Hilfer
Post date April 3, 2014

R. Hilfer, R. Steinle

The European Physical Journal ST 223, 2323-2338 (2014)
https://doi.org/10.1140/epjst/e2014-02267-x

submitted on
Thursday, April 3, 2014

Saturation overshoot and hysteresis for two phase flow in porous media are briefly reviewed. Old and new challenges are discussed. It is widely accepted that the traditional Richards model for twophase flow in porous media does not support non-monotone travelling wave solutions for the saturation profile. As a concequence various extensions and generalizations have been recently discussed. The review highlights different limits within the traditional theory. It emphasizes the relevance of hysteresis in the Buckley–Leverett limit with jump-type hysteresis in the relative permeabilities. Reviewing the situation it emerges that the traditional theory may have been abandoned prematurely because of its inability to predict saturation overshoot in the Richards limit.

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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

Porous Media Two-Phase Flow

A comparison between simulation and experiment for hysteretic phenomena during two phase immiscible displacement

Post author By R Hilfer
Post date August 21, 2013

F. Doster, R. Hilfer

Water Resources Research 50, 681-686 (2014)
https://doi.org/10.1002/2013WR014619

submitted on
Wednesday, August 21, 2013

The paper compares a theory for immiscible displacement based on distinguishing percolating and nonpercolating fluid parts with experimental observations from multistep outflow experiments. The theory was published in 2006 in Physica A, volume 371, pages 209–225; the experiments were published in 1991 in Water Resources Research, volume 27, pages 2113. The present paper focuses on hysteretic phenomena resulting from repeated cycling between drainage and imbibition processes in multistep pressure experiments. Taking into account, the hydraulic differences between percolating and nonpercolating fluid parts provides a physical basis to predict quantitatively the hysteretic phenomena observed in the experiment. While standard hysteretic extensions of the traditional theory are nonlocal in time the theory used in this paper is local in time. Instead of storing the pressure and saturation history, it requires only the current state of the system to reach the same quantitative agreement.

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