Category: fluid flow

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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

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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fluid flow Porous Media Two-Phase Flow

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

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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fluid flow Porous Media Simulations Two-Phase Flow

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

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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fluid flow Porous Media Two-Phase Flow

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

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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fluid flow Porous Media Two-Phase Flow

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

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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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.



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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.



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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.



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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.



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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.



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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.



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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.



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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.



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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.



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Disordered Systems fluid flow Porous Media Transport Processes

Local Porosity Theory for Flow in Porous Media

R. Hilfer

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

submitted on
Thursday, March 28, 1991

A recently introduced geometric characterization of porous media based on local-porosity distributions and local-percolation probabilities is used to calculate dc permeabilities for porous media. The disorder in porous media is found to be intimately related to the percolation concept. The geometric characterization is shown to open a possibility for understanding experimentally observed scaling relations between permeability, formation factor, specific internal surface, and porosity. In particular, Kozeny’s equation relating effective permeability and bulk porosity and the power lawrelation between permeability and formation factor are analyzed. A simple and general consolidation model is introduced. It is based on the reduction of local porosities and emphasizes the general applicability and flexibility of the local-porosity concept. The theoretical predictions are compared with the experimentally observed range for the exponents, and are found to be in excellent agreement.



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diffusion fluid flow Nonequilibrium Pattern Formation

Pattern Formation at Liquid Interfaces

B. Heidel, C. Knobler, R. Hilfer, R. Bruinsma

Physical Review Letters 60, 2492 (1988)
10.1103/PhysRevLett.60.2492

submitted on
Monday, January 11, 1988

Although many examples of pattern formation resulting from chemical reactions at liquid interfaces are known, few have been studied in detail. We report a quantitative study of patterns formed by the photoproduction of Fe++ and its subsequent reaction to form Turnbull’s Blue. The experiment leads to the postulation of a mechanism in which autocatalysis is enhanced by double diffusion. The phase diagram contains a line of phase transitions whose critical behavior is discussed.



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