**1. Introduction**

20 Numerical Simulations

468 Computational Simulations and Applications

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Fluid flow phenomena in porous materials can be found in many important processes in nature and in society. In particular, fluid flow through a porous medium contribute to several technological problems, e.g. extraction of oil or gas from porous rocks, spreading of contaminants in fluid-saturated soils and certain separation processes, such as filtration (Torquato, 2001). In paper and wood industry single and multi phase fluid flow properties in porous media play important roles related to manufacturing process and product development.

The general laws describing creeping fluid flows are well known. However, a detailed study of fluid flow in porous heterogeneous media is complicated. This is a direct consequence of the often very complex, internal micro-scale structures of these materials. That is, the interplay between fluid flow and complex internal structure at the micro-scale gives rise to the effective fluid flow properties at the macro-scale. Traditionally, efforts for analysing fluid flow properties by means of modelling are based on using regular pore geometries that may possess the bulk properties of the actual medium and are simple enough to allow for analytic solution of the relevant transport equations. However, the development of imaging techniques based on computerised x-ray micro-tomography (CXµT) together with advanced numerical techniques have made it possible to analyse structural and transport properties of complex materials based on 3D digitalisation of their real microstructures (Coles et al., 1998; Samuelsen et al., 2001; Goel et al., 2002; Thibault & Bloch, 2002; Holmstad et al., 2003; Rolland et al., 2005; Goel et al., 2006; Stock, 2009). X-ray tomography is a non-invasive and non-destructive imaging method where individual x-ray images recorded from different viewing directions are used for reconstructing the internal 3D structure of the object of interest (Stock, 2009). Although a great opportunity to materials research, CXµT also poses new challenges. The imaging method produces noise, edge blurring, and various other artefacts that may distort the 3D reconstruction of the sample structure and thus result in unrealistic analysis results.

In various industrial and scientific applications an effective material property, permeability, is used for describing the ability of porous materials to transmit fluids. Permeability coefficient for single phase creeping fluid flow through a porous media is defined by the phenomenological law by Henry Darcy as the proportionality constant between the average fluid velocity and applied pressure gradient (Darcy, 1856). The analytical approaches to analyse permeability are often confined to simplified sample geometries. Some of the

The Effect of Tomography Imaging Artefacts on

mechanical inaccuracy of the system.

**2.2.1 Sample geometry preparation** 

Visualisations of the REVs are presented in Fig. 1.

450x450x420 voxels.

(c) the sandstone samples.

around 7 pixels.

**2.2.2 Edge blurring and imaging noise** 

removal by hardware optimisation or robust analysis software.

Structural Analysis and Numerical Permeability Simulations 471

imaging artefacts are noise and edge blurring caused by optics or the non-optimal light source of laboratory scale devices. In addition, the reconstruction procedure can add more artefacts to the tomographic reconstructions (Stock, 2009). The main reconstruction -based artefacts are rings, streaks and shadows caused by hardening of x-rays (Stock, 2009) or

Many image processing tools have been developed in order to overcome the problems related to the imaging artefacts, see e.g. Stock (2009) and references therein. However, most of the artefacts cannot be fully removed algorithmically and they thus require either

Representative elementary volumes (REVs) of the sample geometries were cropped out of the full CXµT reconstructions. In this study the REVs were determined in a deterministic way by evaluating the porosity of larger and larger sample volumes always centred on the same image voxel (Drugan & Willis, 1996; Rolland du Roscoat et al., 2007). A REV size thus obtained for the wool fibre web sample was (in XxYxZ -directions, see Fig. 1) 300x300x360 voxels, for the sand stone sample 500x500x500 voxels and for the packaging board sample

The sample REVs were filtered by variance-weighted mean filter (Gonzales & Woods, 2002) and later thresholded to yield binary images including the solid material and the pore space.

Fig. 1. Tomographic reconstructions of (a) the wool fibre web, (b) the packaging board and

Especially in laboratory scale CXµT systems, like SkyScan and Xradia, the image quality of the material edges in the reconstructed geometry is limited by optical properties of the system. The edge spreading is caused by the non-zero aperture diameter of the x-ray source, the optics in between the source and detector, and scintillator, i.e. the component that converts the x-rays into visible light. Thus, instead of sharp transition between different material phases, there is a smooth curve called edge spread function (ESF). In Fig. 2, the edge smoothness can be seen in the intensity profile plot. The width of the ESF in this case is

numerical permeability studies are based on analysis made for computationally generated models of porous media (Rasi et al., 1999; Aaltosalmi et. al., 2004; Belov et al., 2004; Holmstad et al. 2005; Lundstrom et al., 2004; Verleye et al., 2005; Verleye et al., 2007). Tomographic reconstructions are increasingly utilised in combination with numerical methods to analyse permeability of porous materials (Manwart et al., 2002; Martys & Hagedorn, 2002; Aaltosalmi et al., 2004; Kutay et al., 2006; Fourie et al., 2007). According to our knowledge, only a few studies are based on analysing the effect of tomographic image properties on numerical permeability results (Aaltosalmi et al., 2004; Holmstad, 2005)

The effects of imaging noise, imaging artefacts and the quality of image segmentation on flow permeability found by using direct numerical flow simulation by a specific implementation of finite-difference method (FDM) (Wiegmann, 2007) are studied. The specific surface area of the samples and the features of pore geometry are also analysed. The analyses are done for four different sample types. First, an artificial sample geometry, comprised of hexagonal array of cylinders with known analytical permeability result, is used to analyse the effect of added random noise and edge blurring on the analysis results. Second, CXµT reconstructions of wool fibre web, packaging board and sandstone samples are used to illustrate the effects of different artefact removal and image segmentation methods on permeability results. Finally, the numerically simulated values of flow permeability are compared with experimental results for the same material.
