[Show abstract][Hide abstract] ABSTRACT: 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 20483 voxel has been calculated.
[Show abstract][Hide abstract] ABSTRACT: The paper compares a theory for immiscible displacement based on distinguishing percolating and non-percolating fluid parts (Physica A, 371, 209 (2006)) with experimental observations from multistep outflow experiments (Water Resources Res, 27, 2113 (1991)). The paper focusses 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 non-percolating fluid parts provides a physical basis to predict quantitatively the hysterestic 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.
[Show abstract][Hide abstract] ABSTRACT: One-dimensional traveling wave solutions for imbibition processes into a homogeneous porous medium are found within a recent generalized theory of macro- scopic 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 rep- resentation. 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 rep- resentation 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.
Transport in Porous Media 06/2013; 99(3). DOI:10.1007/s11242-013-0196-0 · 1.43 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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 saturation 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 single-valued 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.
Vadose Zone Journal 08/2012; 11:-. DOI:10.2136/vzj2012.0021 · 1.78 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Physical Review E 07/2012; 86(1):016317. DOI:10.1103/PhysRevE.86.016317 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Analysis of excess wings in broadband dielectric spectroscopy data of glass
forming materials is found to provide 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. We show 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 $\alpha$-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 6 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. Finally we report new and computable expressions of
time dependent relaxation functions corresponding to the frequency dependent
dielectric susceptibilities.
[Show abstract][Hide abstract] ABSTRACT: 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.
New Journal of Physics 12/2011; 13(12):123030. DOI:10.1088/1367-2630/13/12/123030 · 3.56 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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 32768^{3}=35184372088832 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.
[Show abstract][Hide abstract] ABSTRACT: 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 diagenetically 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 (<1 mu m 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 mu m 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.
[Show abstract][Hide abstract] ABSTRACT: A recent study [1] of the dielectric frequency response of a two component composite performed on a single specimen shows that local porosity theory LPT [2] represents a substantial improvement compared with other theories predicting the complex dielectric dispersion [3,4,5]. The purpose of the present work is to extend this investigation to a systematic study on several specimens with different compositions.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we study a certain family of generalized Riemann–Liouville fractional derivative operators of order α and type β, which were introduced and investigated in several earlier works [R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000; R. Hilfer, Fractional time evolution, in Applications of Fractional Calculus in Physics, R. Hilfer, ed., World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000, pp. 87–130; R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, J. Chem. Phys. 284 (2002), pp. 399–408; R. Hilfer, Threefold introduction to fractional derivatives, in Anomalous Transport: Foundations and Applications, R. Klages, G. Radons, and I.M. Sokolov, eds., Wiley-VCH Verlag, Weinheim, 2008, pp. 17–73; R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E 51 (1995), pp. R848–R851; R. Hilfer, Y. Luchko, and Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12 (2009), pp. 299–318; F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal. 10 (2007), pp. 269–308; T. Sandev and Ž. Tomovski, General time fractional wave equation for a vibrating string, J. Phys. A Math. Theor. 43 (2010), 055204; H.M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009), pp. 198–210]. In particular, we derive various compositional properties, which are associated with Mittag–Leffler functions and Hardy-type inequalities for the generalized fractional derivative operator . Furthermore, by using the Laplace transformation methods, we provide solutions of many different classes of fractional differential equations with constant and variable coefficients and some general Volterra-type differintegral equations in the space of Lebesgue integrable functions. Particular cases of these general solutions and a brief discussion about some recently investigated fractional kinetic equations are also given.
Integral Transforms and Special Functions 11/2010; 21(11-11):797-814. DOI:10.1080/10652461003675737 · 0.72 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: During the last decade, lattice-Boltzmann (LB) 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 setup. We present a detailed discussion of possible simulation setups 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. Comment: 14 pages, 11 figures
Journal of Statistical Mechanics Theory and Experiment 05/2010; 2010(11). DOI:10.1088/1742-5468/2010/11/P11026 · 2.40 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Specific internal surface areas and other morphological descriptors of complex disordered systems can be estimated from threedimensional tomographics images using methods of stochastic geometry [1]. Often such data are unavailable for realistic media because these exhibit structural features on multiple length scales. A method to reconstruct stochastic morphologies for multiscale media was developed in [2]. The method is particularly suited for modeling sedimentary rocks that exhibit porosity and grain structure covering several decades in length scales. It combines crystallite information from two-dimensional high-resolution images with sedimentary correlations from a three-dimensional low-resolution micro-CT image [3]. The mathematical model reproduces correlations with primordial depositional textures, scale dependent intergranular porosity over many decades, vuggy porosity, a percolating pore space, a percolating matrix space, and resolution dependence of both physical and morphological descriptors. In [3] the method has been applied to Fontainebleau sandstone. Synthetic micro-CT images of the reconstructed model match well with experimental micro-CT images at different resolutions. Specific surface area and other morphological descriptors are found to be in good agreement with experiment. [1] C. Lang, J. Ohser, R. Hilfer, Journal of Microscopy, vol 203, 303 (2001) [2] B. Biswal, P. Øren, R. Held, S. Bakke, R.Hilfer, Phys.Rev.E, vol 75, 061303 (2007) [3] B. Biswal, R. Held, V. Khanna, J. Wang, R. Hilfer Physical Review E, vol 80, 041301 (2009)
[Show abstract][Hide abstract] ABSTRACT: The concepts of relative permeability and capillary pressure are crucial for the accepted traditional theory of two phase flow in porous media. Recently a theoretical approach was introduced that does not require these concepts as input [1][2][3]. Instead it was based on the concept of hydraulic percolation of fluid phases. The presentation will describe this novel approach. It allows to simulate processes with simultaneous occurence of drainage and imbibition. Furthermore, it predicts residual saturations and their spatiotemporal changes during two phase immiscible displacement [1][2][3][4][5]. [1] R. Hilfer. Capillary Pressure, Hysteresis and Residual Saturation in Porous Media, Physica A, vol. 359, pp. 119, 2006. [2] R. Hilfer. Macroscopic Capillarity and Hysteresis for Flow in Porous Media, Physical Review E, vol. 73, pp. 016307, 2006. [3] R. Hilfer. Macroscopic capillarity without a constitutive capillary pressure function, Physica A, vol. 371, pp. 209, 2006. [4] R. Hilfer. Modeling and Simulation of Macrocapillarity, in: P. Garrido et al. (eds.) Modeling and Simulation of Materials vol. CP1091, pp. 141, American Institute of Physcis, New York, 2009. [5] R. Hilfer and F. Doster. Percolation as a basic concept for macroscopic capillarity, Transport in Porous Media, DOI 10.1007/s11242-009-9395-0, in print, 2009.
[Show abstract][Hide abstract] ABSTRACT: The concepts of relative permeability and capillary pressure are crucial for the accepted traditional theory of two phase
flow in porous media. Recently, a theoretical approach was introduced that does not require these concepts as input (Hilfer,
Physica A, 359:119–128, 2006a; Phys. Rev. E, 73:016307, 2006b). Instead it was based on the concept of hydraulic percolation
of fluid phases. This paper presents the first numerical solutions of the coupled nonlinear partial differential equations
introduced in Hilfer (Phys. Rev. E, 73:016307, 2006b). Approximate numerical results for saturation profiles in one spatial
dimension have been calculated. Long time limits of dynamic time-dependent profiles are compared to stationary solutions of
the traditional theory. The initial and boundary conditions are chosen to model the displacement processes that occur when
a closed porous column containing two immiscible fluids of different density is raised from a horizontal to a vertical position
in a gravitational field. The nature of the displacement process may change locally in space and time between drainage and
imbibition. The theory gives local saturations for nonpercolating trapped fluids near the endpoint of the displacement.
KeywordsCapillarity-Hysteresis-Residual saturation-Multiphase flow-Porous media-Immiscible displacement
Transport in Porous Media 04/2010; 82(3):507-519. DOI:10.1007/s11242-009-9395-0 · 1.43 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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-CTimages at different resolutions from a laboratory scale model of Fontainebleau sandstone are made available to the scientific community for resolution dependent petrophysical analysis.
Physica A: Statistical Mechanics and its Applications 04/2010; 389(8):1607-1618. DOI:10.1016/j.physa.2009.12.006 · 1.73 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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 degrees 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.
[Show abstract][Hide abstract] ABSTRACT: Using recent investigated integral representations for the generalized alternating Math- ieu series ˜ S (α,β) μ r; {an} ∞ n=1 r,α,β,μ, {an} ∞ n=1 ∈ R + (9,14,18) with an = nγ, γ ∈ R+ and Mellin-Laplace type integral transforms for the generalized hypergeometric functions and the Bessel function offirstkind, somebounding inequalities for ˜ S (α,β) μ r; {nγ }∞ n=1 are presented. Namely, it is shown that the series ˜ S (α,β) μ r; {nγ }∞ n=1 under some conditions for parameters α, β, γ and μ are bounded with constants which do not depend on α ,β and γ but only depend on r and μ,i.e. ˜ S( α,β) μ r; nγ ∞ n=1 2 (1 + r2)μ .