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Special Section Articles

# Flows Through Real Porous Media: X-Ray Computed Tomography, Experiments, and Numerical Simulations

[+] Author and Article Information
Wim-Paul Breugem

Laboratory for Aero and Hydrodynamics,
Delft University of Technology,
Leeghwaterstraat 21,
2628 CA Delft, The Netherlands
e-mail: w.p.breugem@tudelft.nl

Vincent van Dijk

IHC Merwede,
Smitweg 6,
2961 AW Kinderdijk, The Netherlands
e-mail: v.vandijk@mtiholland.com

René Delfos

Laboratory for Aero and Hydrodynamics,
Delft University of Technology,
Leeghwaterstraat 21,
2628 CA Delft, The Netherlands
e-mail: r.delfos@tudelft.nl

The global should be distinguished from the local porosity, which could be calculated from $ɛ≡∫VmγdV$ with an appropriate choice of the weighting function m. The local porosity is expected to vary with the radial distance to the wall of the permeability cell. This causes radial variations in the flow near the wall and is known as the wall-channeling effect [19].

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 6, 2013; final manuscript received August 27, 2013; published online February 28, 2014. Assoc. Editor: Elias Balaras.

J. Fluids Eng 136(4), 040902 (Feb 28, 2014) (8 pages) Paper No: FE-13-1138; doi: 10.1115/1.4025311 History: Received March 06, 2013; Revised August 27, 2013

## Abstract

Two different direct-forcing immersed boundary methods (IBMs) were applied for the purpose of simulating slow flow through a real porous medium: the volume penalization IBM and the stress IBM. The porous medium was a random close packing of about 9000 glass beads in a round tube. The packing geometry was determined from an X-ray computed tomography (CT) scan in terms of the distribution of the truncated solid volume fraction (either 0 or 1) on a three-dimensional Cartesian grid. The scan resolution corresponded to 19.3 grid cells over the mean bead diameter. A facility was built to experimentally determine the permeability of the packing. Numerical simulations were performed for the same packing based on the CT scan data. For both IBMs the numerically determined permeability based on the Richardson extrapolation was just 10% lower than the experimentally found value. As expected, at finite grid resolution the stress IBM appeared to be the most accurate IBM.

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## References

Sahraoui, M. and Kaviany, M., 1992, “Slip and No-Slip Velocity Boundary Conditions at Interface of Porous, Plain Media,” Int. J. Heat Mass Transfer, 35(4), pp. 927–943.
Zick, A. A. and Homsy, G. M., 1982, “Stokes Flow Through Periodic Arrays of Spheres,” J. Fluid Mech., 115, pp. 13–26.
Breugem, W.-P., and Boersma, B., 2005, “Direct Numerical Simulations of Turbulent Flow Over a Permeable Wall Using a Direct and a Continuum Approach,” Phys. Fluids, 17(2), p. 025103.
Narsilio, G. A., Buzzi, O., Fityus, S., Yun, T. S., and Smith, D. W., 2009, “Upscaling of Navier–Stokes Equations in Porous Media: Theoretical, Numerical and Experimental Approach,” Comput. Geotech., 36, pp. 1200–1206.
Kaczmarczyk, J., Dohnalik, M., Zalewska, J., and Cnudde, V., 2010, “The Interpretation of X-ray Computed Microtomography Images of Rocks as an Application of Volume Image Processing and Analysis,” Proceedings of the WSCG 2010—Communication Papers, pp. 23–30.
Kaczmarczyk, J., Dohnalik, M., and Zalewska, J., 2010, “Three-Dimensional Pore Scale Fluid Flow Simulation Based on Computed Microtomography Carbonate Rocks' Images,” Fifth European Conference on Computational Fluid Dynamics (ECCOMAS CFD 2010).
Zaretskiy, Y., Geiger, S., Sorbie, K., and Förster, M., 2010, “Efficient Flow and Transport Simulations in Reconstructed 3D Pore Geometries,” Adv. Water Res., 33, pp. 1508–1516.
Ovaysi, S. and Piri, M., 2010, “Direct Pore-Level Modeling of Incompressible Fluid Flow in Porous Media,” J. Comput. Phys., 229, pp. 7456–7476.
Gerbaux, O., Buyens, F., Mourzenko, V. V., Memponteil, A., Vabre, A., Thovert, J.-F., and Adler, P., 2010, “Transport Properties of Real Metallic Foams,” J. Colloid Interface Sci., 342, pp. 155–165. [PubMed]
Mittal, R. and Iaccarino, G., 2005, “Immersed Boundary Methods,” Annu. Rev. Fluid Mech., 37, pp. 239–261.
Smolarkiewicz, P. K. and Winter, C. L., 2010, “Pores Resolving Simulation of Darcy Flows,” J. Comput. Phys., 229, pp. 3121–3133.
Lopez Penha, D. J., Geurts, B. J., Stolz, S., and Nordlund, M., 2011, “Computing the Apparent Permeability of an Array of Staggered Square Rods Using Volume-Penalization,” Comput. Fluids, 51, pp. 157–173.
Kajishima, T., Takiguchi, S., Hamasaki, H., and Miyake, Y., 2001, “Turbulence Structure of Particle-Laden Flow in a Vertical Plane Channel Due to Vortex Shedding,” JSME Int. J., Ser. B, 44(4), pp. 526–535.
Pourquie, M., Breugem, W.-P., and Boersma, B. J., 2009, “Some Issues Related to the Use of Immersed Boundary Methods to Represent Square Obstacles,” Int. J. Multiscale Comp. Eng., 7(6), pp. 509–522.
Bear, J., 1988, Dynamics of Fluids in Porous Media, Dover, New York.
Whitaker, S., 1999, The Method of Volume Averaging, Kluwer, Dordrecht, The Netherlands.
Bird, R. B., Stewart, W. E., and Lightfoot, E. N., 2002, Transport Phenomena, John Wiley and Sons, New York.
MacDonald, I. F., El-Sayed, M. S., Mow, K., and Dullien, F. A. L., 1979, “Flow Through Porous Media: The Ergun Equation Revisited,” Ind. Eng. Chem. Fundam., 18(3), pp. 199–208.
Gupte, A. R., 1970, “Experimentelle Untersuchung der Einflüsse von Porosität und Korngrößenverteilung im Widerstandsgesetz der Porenströmung,” Ph.D. thesis, Karlsruhe Institute of Technology, Karlsruhe, Germany.
Vafai, K., 1984, “Convective Flow and Heat Transfer in Variable-Porosity Porous Media,” J. Fluid Mech., 147, pp. 233–259.
Song, C., Wang, P., and Makse, H., 2008, “A Phase Diagram for Jammed Matter,” Nature (London), 453, p. 629632.
Cheng, N. S., 2008, “Formula for the Viscosity of a Glycerol-Water Mixture,” Ind. Eng. Chem. Res., 47(9), pp. 3285–3288.
Harlow, F. H. and Welch, J. E., 1965, “Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid With Free Surface,” Phys. Fluids, 8(12), pp. 2182–2189.
Wesseling, P., 2001, Principles of Computational Fluid Dynamics (Springer Series in Computational Mathematics), Vol. 29, Springer-Verlag, Berlin.
Scotti, A., 2006, “Direct Numerical Simulation of Turbulent Channel Flows With Boundary Roughened With Virtual Sandpaper,” Phys. Fluids, 18, p. 031701.
Belliard, M. and Fournier, C., 2010, “Penalized Direct Forcing and Projection Schemes for Navier–Stokes,” C.R. Acad. Sci., Ser. I, 348, pp. 1133–1136.
Ferziger, J. H. and Perić, M., 2002, Computational Methods for Fluid Dynamics, Springer-Verlag, Berlin.
Mustakis, I. and Kim, S., 1998, “Microhydrodynamics of Sharp Corners and Edges: Traction Singularities,” AIChE J., 44, pp. 1469–1483.

## Figures

Fig. 2

Measured permeability as a function of the packed bed Reynolds number. The horizontal and vertical error bars denote the estimated standard deviation in both quantities. The dashed line is the average value for the permeability. The dotted lines show the standard deviation in the average value based on the spreading in the data points.

Fig. 1

Close-up of the experimental setup showing the permeability cell, the cell holder, and the locations of the pressure taps and temperature sensor. Dimensions are given in mm.

Fig. 3

Histogram of the cell gray values as computed from the X-ray CT scan with a bin size of 24 in the gray value. The dashed line marks the threshold gray value used for determining the value of the cell solid volume fraction (αijk = 0 for air and 1 for glass).

Fig. 4

Illustration of the stress IBM of Breugem and Boersma [3] and Pourquie et al. [14] for rectangular-shaped obstacles. The crosses, dots, and open circles indicate the locations where additional forcing is imposed on the flow.

Fig. 5

(a) An xz cross section of the flow and pressure field inside the permeability cell, as obtained from a simulation with the stress IBM at 0.1 mm resolution (the same resolution as was used for the X-ray CT scan). The color denotes the pressure in Pa (assuming a fluid mass density of 1170 kg/m3) with a contour interval of 50 Pa. (b) Enlargement of the white box in (a). The reference vector represents a velocity of 50 mm/s.

Fig. 6

The same as in Fig. 5, but for an xy cross section of the permeability cell. The color denotes the streamwise velocity in mm/s.

Fig. 8

Percentage of error in the permeability (K) as a function of the grid resolution (dpx). The error is relative to the estimated value of the permeability at infinite grid resolution (Kr), as obtained from the Richardson extrapolation based on the three data points. Blue line/dots: volume penalization IBM. Red line/squares: stress IBM.

Fig. 7

The area-averaged intrinsic pressure 〈p〉 as a function of the streamwise distance z. Blue lines/dots: volume penalization IBM. Red line/squares: stress IBM. The lines represent different grid resolutions: —, dpx = 19.3; - - -, dpx = 38.6; …, dpx = 77.2. The arrows point in the direction of increasing grid resolution.

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