Hang Deng, Jeffrey P. Fitts, Catherine A. Peters *

Supplementary Material for “Quantifying Fracture Geometry with X-ray Tomography: Technique of Iterative Local Thresholding (TILT) for 3D Image Segmentation”

Computational Geosciences

Hang Deng, Jeffrey P. Fitts, Catherine A. Peters * Department of Civil & Environmental Engineering Princeton University, Princeton, NJ 08540, U.S.A.

* email: [email protected], phone 609-258-5645

1. Data acquisition and reconstruction

Here, we present details about the X-ray CT scanning and data acquisition used to generate the testbed dataset for this manuscript. This dataset is part of the data generated for an experimental study of reactive flows in fractured cores of Indiana Limestone [1]. As shown in figure 1, the object being scanned is the core and core holder. A carbon fiber core-holder was used to minimize imaging interferences, as carbon fiber is a material with very low attenuation capacity. The carbon fiber core- holder is better than a stainless steel core-holder, because it enables the penetration of the x-ray beam to the specimen. In this work, the microfocus x-ray tube was used with energy of 220 keV and current of 300 µA. Each scan took approximately 90 mins.

The setup for the North Star Imaging M-5000 X-ray CT scanner at the National Energy Technology Laboratory (NETL) in Morgantown, WV is as follows. The attenuated beam intensity is recorded by a high-resolution flat panel detector and saved as a 2D sinogram / radiograph. To capture the spatial variations of attenuation in 3D, the sample is rotated, and 2D sinograms at each angular position are collected. The 2D sonograms are then mapped to a 3D tomographic image by reconstruction, a mathematical process that inversely solves for attenuation values at each voxel of the 3D volume image in Cartesian coordinates.

For each scan in our experiments, the sample was rotated for 180 degrees in 0.25 degree steps to ensure the fine angular increment that is required for reliable reconstruction [2]. The series of radiographs were imported into the NSI (North Star Imaging) proprietary CT software efX for calibration, optimization and cone-beam reconstruction. The reconstructed 3D volume image was exported as 2D 32bit grayscale images with voxel resolution of 28 µm. The 2D image represents a slice of the fractured core on the plane perpendicular to the core axis with a thickness equal to the spatial resolution. The grayscale values of the 2D images are a linear map of the voxel attenuation, and provide the basis for subsequent image processing.

1 Figure 1. The setup of the flow-through experimental apparatus in the xCT scanner.

2. Pre-processing - Ring Artifacts Reduction and Noise Removal

Some of the xCT images showed evident ring artifacts (figure 2(a)), which affect subsequent segmentation of the xCT images (figure 2(b)). The combined wavelet-Fourier filter was applied to reduce the ring artifacts (figure 2(f)). This filter first converts the vertical stipe information into the vertical detail coefficients generated by the multi-scale wavelet decomposition, and then further converts the striping information into the horizontal direction of the Fourier domain by applying discrete Fourier transformation on the vertical detail coefficients. The condensed striping noise is dampened using a Gaussian function, and then the image is reconstructed.

In our application, we first identified and isolated the ring artifacts (figure 2(d)), and converted the region of ring artifacts into polar coordinates (figure 2(e)), because polar coordinates have been reported to give better results than Cartesian coordinates in reduction of ring artifacts [3]. Second, after the coordinate conversion, the ring artifacts were transformed into vertical stripes, and the combined wavelet-FFT filter was applied to reduce the vertical stripes (figure 2(f)). A range of parameters required for the filtering processes were tested to identify the best reduction. Lastly, the treated images were converted back to Cartesian coordinates (figure 2(g)). Figure 2 (a) A grayscale xCT image with presence of ring artifact, binary images of the grayscale xCT image affected by ring artifact (b) before and (c) after applying the destriping filter, blow-up of the ring artifact region in (d) (g) Cartesian coordinates, and (e) (f) polar coordinates before and after applying the destriping filter.

For noise removal, the median filter is a rank filter that uses the median value of the neighborhood specified by the kernel for the reference voxel. Unlike mean filters which cause blurring to the images, the median filter is edge-preserving [4-8]. It has been shown to deliver robust results, and is more computationally efficient than other appealing noise reduction and edge-preserving filters such as the anisotropic diffusion filter. In our study, a 3D median filter1, 3D anisotropic diffusion filter2, and 3D non-local means filter were tested on a subsection of a scan. A range of parameters were explored for each filter3, and the visually best filtered images were compared among three filters. As shown in figure 3, the 3D anisotropic diffusion filter and 3D non-local means filter do not offer significantly better improvements over the 3D median filter. Therefore, in our study, 3D median filter was used to denoise images.

1 Codes from Damien Garcia, website: http://www.biomecardio.com, downloaded at http://www.mathworks.com/matlabcentral/fileexchange/30853-field-mapping-toolbox/content/medfilt3.m

2 Codes from Daniel Simoes Lopes, ICIST, Instituto Superior Tecnico - Universidade Tecnica de Lisboa, website: http://www.civil.ist.utl.pt/~danlopes, downloaded at http://www.mathworks.com/matlabcentral/fileexchange/14995-anisotropic-diffusion--perona---malik-

3 Codes from Dirk-Jan Kroon, downloaded at http://www.mathworks.com/matlabcentral/fileexchange/27395- fast-non-local-means-1d--2d-color-and-3d

3 Figure 3. Grayscale images after denoising using (a) a 3D median filter, (b) a 3D anisotropic diffusion filter, and (c) a non-local means filter

3. Multi-scale Hessian Fracture (MHF) Filter

The Multi-scale Hessian Fracture (MHF) filter4 is designed to capture fractures with fine apertures. It takes advantage of the geometrical characteristics of the objects that are quantified in the sheetness parameters (figure 4(a)) to distinguish fracture from rock matrix. Segmenting the normalized sheetness parameters generates binary images (figure 4(b)) that capture the fractures and planar- shaped pores. The fracture can be isolated by applying a 3D connectivity operator (figure 4(c)).

There are, however, limitations to this method. The results are sensitive to the Gaussian scales used to generate the Hessian matrix. As shown in figure 5, the resulting aperture distributions, especially for fracture geometries after reaction when the presence of preferential flow paths creates large aperture variations, are highly subject to the range of Gaussian scales used, with the majority of the determined apertures falling between 0 to 2.

4 In this study, the realization of MHF filter builds on the codes written by Dirk-Jan Kroon. The original codes were downloaded at http://www.mathworks.com/matlabcentral/fileexchange/24409-hessian-based-frangi-vesselness- filter/content/FrangiFilter3D.m Figure 4. Major steps of multi-scale Hessian filter (MHF): (a) normalized sheetness parameter, (b) binary image generated by applying Otsu’s method on the sheetness parameter matrix, (c) isolated fracture using a 3D connectivity filter.

Figure 5. Aperture distributions generated from the multi-scale Hessian fracture filter with different ranges of Gaussian scales for (a) before and (b) after reaction.

5 4. Sub-voxel fine apertures

Different aperture values were assigned to the contact areas to explore the uncertainties caused by mischaracterized sub-voxel fine apertures. The simulation results of the modified aperture maps based on the aperture map generated by TILT with 3D thresholding are shown in figure 6. The hydrodynamic properties of different aperture maps are in great agreement with one another, with the symbols lying on top of each other.

Figure 6. Fracture permeabilities with different aperture values assigned to the contact area. The geometry used is determined from the 3D TILT routine.

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