HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS		QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

This Article Abstract Full Text Free Full Text (PDF) Free Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via ISI Web of Science (4) Citing Articles via Google Scholar Google Scholar Articles by Lehmann, P. Articles by Flühler, H. Search for Related Content PubMed Articles by Lehmann, P. Articles by Flühler, H. GeoRef GeoRef Citation Agricola Articles by Lehmann, P. Articles by Flühler, H. Related Collections Tomography Pore-Scale Modeling Multiphase Fluid Flow

Tomographical Imaging and Mathematical Description of Porous Media Used for the Prediction of Fluid Distribution

^a Institute of Terrestrial Ecology, Swiss Federal Institute of Technology, ETH Zurich, Switzerland
^b Centre for Non-destructive Testing, Swiss Federal Laboratories for Materials Testing and Research, EMPA, Switzerland
^c Spallation Neutron Source Division, Paul Scherrer Institute, PSI, Switzerland
^d Institut für Computeranwendungen im Bauingenieurwesen, TU Braunschweig, Germany
⁵ GKSS-Research Centre, Geesthacht, Germany

View larger version (78K): [in a new window]   Fig. 1. Reconstructed geometries of scanned samples. Sand samples were scanned using X-rays from (A) tubes or (B) synchrotrons with a voxel size of 60 and 3.5 µm and a sample diameter of 25 and 5 mm, respectively. The glass bead columns with diameters of (C) 52 mm and (D) 40 mm, respectively, were scanned using thermal neutrons with a voxel size of 167 µm. In the latter case, the pore space was water filled at the bottom of the column.

View larger version (144K): [in a new window]   Fig. 2. In the case of industrial X-ray tomography, the voxel size was 70 by 70 by 70 µm3. But the gray value of each voxel corresponded to the material density in a cuboid of size 70 by 70 by 210 µm3. We reconstructed the material density in cubic voxels of size 70 by 70 by 70 µm3 using the Fourier transformation technique. In the first row, vertical cross sections through the scanned sand sample are shown (A) before and (B) after noise filtering and inverse Fourier transformation. In the last image (C), the image is segmented into solid phase (black) and pore space (white). In the second row, the calculated density distribution in a vertical cross section of a simulated sample is shown. In the first image (D), the gray values were calculated for cubes of size 70 by 70 by 70 µm3. In the next image (E), the density in cuboids of size 70 by 70 by 210 µm3 is given. Due to the anisotropic mass contribution, the shape of the spheres is slightly distorted in the vertical direction. (F) From this asymmetric density distribution, the material density in cubes was calculated using Fourier transformation. The cross section shown in Part F is similar to the "true" structure shown in D. The same technique was applied for the measured data shown above. The numbers denote the volume that determined the gray values of the voxels.

View larger version (61K): [in a new window]   Fig. 3. In the first row, a gray-scale value image of the material density (left) and the corresponding black (particles) and white (pores) image is shown. The figures show a sector of the sand model with a side length of 50 voxels and voxel size of 60 µm. This voxel size equals the resolution of the measurement with micro-X-ray tomography. Not all voxels determined as pores are entirely within the pore space (gray value 0). The fraction of these gray voxels is given in the figure in the second row. This fraction decreases with decreasing voxel size. The large symbols denote the resolution of the tomography with X-rays from tubes (black) or thermal neutrons (gray), respectively. The positions of the arrows correspond to the resolution with 50% of voxels denoted as pores with a gray value of 0. For the tomography of the glass bead column, this requirement was fulfilled.

View larger version (123K): [in a new window]   Fig. 4. Cross sections through the density map (top row) and the segmented image (middle row) for the simulated sand media (left) and numerically generated glass bead packings (right). The numbers in the images denote the voxel sizes. In the first two rows, the images calculated with the voxel size of the tomography with X-rays from tubes or thermal neutrons, respectively, are shown. The last row gives the density map calculated with the highest resolution (10003 voxels).

View larger version (15K): [in a new window]   Fig. 5. Scaled surface area of the particles as a function of voxel size analyzed with simulated media. (A) The calculated surface area, divided by the analytically determined surface of the spheres, converges to the true value (1.0) with decreasing voxel size. The large symbols indicate the voxel size of the measurements with X-rays from tubes (black) or thermal neutrons (gray), respectively. (B) The absolute value of the change of area with decreasing voxel size is constant or decreases for voxel sizes smaller than 10 to 15% of the mean particle radius (arrows). The resolutions of the tomography did not fulfill this condition.

View larger version (16K): [in a new window]   Fig. 6. Effect of image resolution on image properties analyzed with simulated media. (A) The mean breadth of the mapped particles, divided by the analytically determined breadth of the spheres, becomes positive for resolutions with voxel sizes smaller than 23 to 25% of the mean particle radius (arrows). The large symbols indicate the voxel size of the measurements with X-rays from tubes (black) or thermal neutrons (gray), respectively. For the tomography of the glass beads using neutron transmission technique, the voxel size was 16.7% of the mean particle radius and this requirement was fulfilled. (B) The Euler-Poincaré characteristic (EPC) of the solid phase divided by the number of particles. For high resolutions, the spheres in the image become separated and a positive EPC results. The resolutions of the tomography did not fulfill this condition.

View larger version (19K): [in a new window]   Fig. 7. Effect of resolution on pore-size distribution for simulated media. The cumulative pore-size distributions for (A) the sand model and (B) the generated glass bead packings were determined. The numbers in the legend are the resolution of the images in micrometers. The bold black lines with the large symbols show the size distribution computed with the resolution of the measurements with X-rays from tubes or thermal neutrons, respectively. The black curves without symbols were calculated with the highest resolution. The gray lines without symbols correspond to the largest voxel size fulfilling the criterion of a second derivative changing from positive to negative values with increasing pore size.

View larger version (106K): [in a new window]   Fig. 8. A sample of dry sand was scanned using micro-X-ray tomography with a resolution of 60 µm. (A) The particles in a cubic section of size 3 mm are shown. The water and air distributions in this cube for different pressure conditions at the lower boundary were calculated using the Lattice-Boltzmann approach. With this method, the Navier-Stokes equation for the liquid and the gas phase are solved numerically. For a drainage process, the resulting (B) air and (C) water distributions are shown. (D) For the same applied suction during a wetting process, the water distribution is different.

View larger version (110K): [in a new window]   Fig. 9. Dry sand samples were mapped with synchrotron radiation tomography (left: HASYLAB, right: SLS). Particles are shown in black. The sizes of the particles range (A and C) from 300 to 900 µm and (B and D) from 100 to 200 µm. (A and B) First, the pore sizes for all voxels of the pore space were determined. Bright values indicate large pore sizes. A small section indicated by the white border is enlarged to explain the determination of the pore size. First, the distance from the solid phase is calculated (A1). This distance corresponds to the radius of a sphere. A voxel of the pore space can be an element of different spheres and the radius of the largest sphere determines the pore size of the voxel (A2). The grid shows the size of the voxels. In the bottom row, the water (gray) and air (white) distributions computed with a morphological pore network model are shown. A head of (C) –15 cm and (D) –50 cm was applied at the lower boundary in the numerical experiment. The computed water saturation of the pore space was 56% for the coarse sand and 49% for the fine sand.

View larger version (15K): [in a new window]   Fig. 10. The effect of pore size and pore connectivity on the water retention curve. (A) The drainage was calculated with the morphological pore network model for the sand material with particles ranging from 300 to 900 µm ("network"). The dry sand sample was analyzed with X-rays from synchrotrons with a resolution of 11.5 µm. The size of the image was 900 by 900 by 300 voxels. The modeled retention curve corresponds well to the water retention curve measured in the laboratory by Ursino and Gimmi (2004). The computed and measured drainage curves are compared with the cumulated pore-size distribution ("size"). In the latter case, connectivity is neglected and all pores with low capillary forces drain out. (B) The cumulative pore-size distribution ("size") was compared with the computed drainage curve ("network") for the sand sample with fine particles of size 100 to 200 µm. The pore structure of the dry sand sample was measured with a resolution of 3.5 µm using X-rays from synchrotrons. The image size was 1000 by 1000 by 100 voxels. By neglecting the connectivity of the pore space, the water saturation was underestimated.