Category: Porous Media

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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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Mathematical Physics Porous Media Two-Phase Flow

Existence and Uniqueness of Nonmonotone Solutions in Porous Media Flow

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

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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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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Porous Media Two-Phase Flow

Capillary number correlations for two-phase flow in porous media

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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Porous Media Two-Phase Flow

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

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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Heterogeneous Materials Mathematical Physics Percolation Porous Media

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

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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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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Porous Media

Capillary saturation and desaturation

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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Porous Media Two-Phase Flow

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

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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Disordered Systems Porous Media Precision Simulations Simulations

Differential porosimetry and permeametry for random porous media

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

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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Porous Media Two-Phase Flow

Saturation overshoot and hysteresis for twophase flow in porous media

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

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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Porous Media Two-Phase Flow

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

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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Porous Media Two-Phase Flow

Travelling Wave Solutions in a Generalized Theory for Macroscopic Capillarity

O. Hönig, F. Doster, R. Hilfer

Transport in Porous Media 99, 467 (2013)
https://doi.org/10.1007/s11242-013-0196-0

submitted on
Thursday, December 20, 2012

One-dimensional traveling wave solutions for imbibition processes into a homogeneous porous medium are found within a recent generalized theory of macroscopic capillarity. The generalized theory is based on the hydrodynamic differences between percolating and nonpercolating fluid parts. The traveling wave solutions are obtained using a dynamical systems approach. An exhaustive study of all smooth traveling wave solutions for primary and secondary imbibition processes is reported here. It is made possible by introducing two novel methods of reduced graphical representation. In the first method the integration constant of the dynamical system is related graphically to the boundary data and the wave velocity. In the second representation the wave velocity is plotted as a function of the boundary data. Each of these two graphical representations provides an exhaustive overview over all one-dimensional and smooth solutions of traveling wave type, that can arise in primary and secondary imbibition. Analogous representations are possible for other systems, solution classes, and processes.



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Porous Media Two-Phase Flow

Generalized Buckley-Leverett theory for two phase flow in porous media

F. Doster, R. Hilfer

New Journal of Physics 13, 123030 (2011)
https://doi.org/10.1088/1367-2630/13/12/123030

submitted on
Wednesday, November 23, 2011

Hysteresis and fluid entrapment pose unresolved problems for the theory of flow in porous media. A generalized macroscopic mixture theory for immiscible two-phase displacement in porous media (Hilfer 2006b Phys. Rev. E 73 016307) has introduced percolating and nonpercolating phases. It is studied here in an analytically tractable hyperbolic limit. In this limit a fractional flow formulation exists, that resembles the traditional theory. The Riemann problem is solved analytically in one dimension by the method of characteristics. Initial and boundary value problems exhibit shocks and rarefaction waves similar to the traditional Buckley–Leverett theory. However, contrary to the traditional theory, the generalized theory permits simultaneous drainage and imbibition processes. Displacement processes involving flow reversal are equally allowed. Shock fronts and rarefaction waves in both directions in the percolating and the nonpercolating fluids are found, which can be compared directly to experiment.



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Porous Media Two-Phase Flow

Horizontal flows and capillarity driven redistribution

F. Doster, O. Hönig, R. Hilfer

Physical Review E 86, 016317 (2012)
https://doi.org/10.1103/PhysRevE.86.016317

submitted on
Tuesday, October 11, 2011

A recent macroscopic mixture theory for two-phase immiscible displacement in porous media has introduced percolating and nonpercolating phases. Quasi-analytic solutions are computed and compared to the traditional theory. The solutions illustrate physical insights and effects due to spatiotemporal changes of nonpercolating phases, and they highlight the differences from traditional theory. Two initial and boundary value problems are solved in one spatial dimension. In the first problem a fluid is displaced by another fluid in a horizontal homogeneous porous medium. The displacing fluid is injected with a flow rate that keeps the saturation constant at the injection point. In the second problem a horizontal homogeneous porous medium is considered which is divided into two subdomains with different but constant initial saturations. Capillary forces lead to a redistribution of the fluids. Errors in the literature are reported and corrected.



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Porous Media Two-Phase Flow

Nonmonotone Saturation Profiles for Hydrostatic Equilibrium in Homogeneous Media

R. Hilfer, F. Doster, P. Zegeling

Vadose Zone Journal 11, vzj2012.0021 (2012)
doi:10.2136/vzj2012.0021

submitted on
Friday, September 30, 2011

Recently, the observation of nonmonotonicity of traveling wave solutions for saturation profiles during constant flux infiltration experiments has highlighted the shortcomings of the traditional seventy year old mathematical model for immiscible displacement in porous media. Several recent modifications have been proposed to explain these observations. The present paper suggests, that nonmonotone saturations profiles might occur even at zero flux. Specifically, nonmonotonicity of saturation profiles is predicted for hydrostatic equilibrium, when both fluids are at rest. It is argued, that in traditional theories with the widely used singlevalued monotone constitutive functions nonmonotone profiles should not exist in hydrostatic equilibrium. The same applies to some modifications of the traditional theory. Nonmonotone saturation profiles in hydrostatic equilibrium arise within a generalized theory, that contains the traditional theory as a special case. The physical origin of the phenomenon is simultaneous occurrence of imbibition and drainage. It is argued, that indications for nonmonotone saturation profiles in hydrostatic equilibrium might have been observed in past experiments, and could become clearly observable in a closed column experiment.



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