Transport properties of heterogeneous materials. Combining computerised X-ray micro-tomography and...
9
Transport properties of heterogeneous materials. Combining computerised X-ray micro-tomography and direct numerical simulations Viivi Koivu a *, Maxime Decain b,c , Christian Geindreau b , Keijo Mattila a , Jean-Francis Bloch b and Markku Kataja a a Department of Physics, University of Jyva ¨skyla ¨, Jyva ¨skyla ¨, Finland; b Papermaking Process Laboratory of Grenoble, St Martin D’Heres, Grenoble, France; c Laboratory 3S-R, University of Grenoble-CNRS, Grenoble, France (Received 24 February 2009; final version received 22 February 2010) Feasibility of a method for finding flow permeability of porous materials, based on combining computerised X-ray micro-tomography and numerical simulations, is assessed. The permeability is found by solving fluid flow through the complex 3D pore structures obtained by tomography for actual material samples. We estimate overall accuracy of the method and compare numerical and experimental results. Factors contributing to uncertainty of the method include numerical error arising from the finite resolution of tomographic images and the rather small sample size available with the present tomographic techniques. The total uncertainty of computed values of permeability is, however, not essentially larger than that of experimental results. We conclude that the method provides a feasible alternative for finding fluid flow properties of the kind of materials studied. It can be used to estimate all components of permeability tensor and is useful in cases where direct measurements are not achievable. Analogous methods can be applied to other modes of transport, such as diffusion and heat conduction. Keywords: permeability; X-ray micro-tomography; porous media; numerical methods; lattice-Boltzmann; finite difference Introduction An important problem in analysing and modelling many complex natural and industrial processes is to determine realistic effective material laws and values of material parameters that would properly represent, e.g. the relevant transport properties of materials involved. Various methods ranging from purely experimental correlations to modelling based on fundamental microscopic dynamics, and lately also to numerical simulation, have been used (see e.g. Saar and Manga (1999), Belov et al. (2003), Lundstro¨m et al. (2004), Arns et al. (2004), Kerwald (2004), Levitz (2005), Nordlund et al. (2005), Verleye et al. (2005), Eker and Akin (2006), Keehm et al. (2006), Hellstro¨m and Lundstro¨m (2006), Lundstro¨m et al. (2006), White et al. (2006), Verleye et al. (2007), Jaganathan et al. (2007), Nordlund and Lundstro¨m (2007), Vidal et al. (2008), Andersson et al. (2009) and Frishfelds et al. (2009)). An essential factor affecting properties of many heterogeneous and porous materials is their specific internal microscopic or mesoscopic structure. That structure is often very complex and difficult to characterise in a manner that would readily be used in correlating structural and transport properties of these materials. Computerised X-ray micro-tomography (CXmT) techniques based on synchrotron radiation sources have been used in probing the internal structure of heterogeneous materials for more than a decade. The techniques involve taking a large number of X-ray images of the material sample from different angles. Within the so called absorption mode imaging, numer- ical reconstruction algorithm is then used to obtain the 3D spatial distribution of X-ray absorption coefficient which closely correlates with the local density distribu- tion. Visualisation of this distribution gives the 3D ‘image’ of the material. During the recent years, the method has become more feasible as also table-top tomographic scanners utilising X-ray tubes and capable of resolution in the sub-micron range have become available. Indeed, the utilisation of non-destructive 3D examination using CXmT techniques has opened new perspectives in heterogeneous materials research (see e.g. Thibault and Bloch (2002), Manwart et al. (2002), Aaltosalmi et al. (2004), Ramaswamy et al. (2004), Rolland du Roscoat et al. (2005), Hyva¨luoma et al. (2006) and Fourie et al. (2007)). Analysis of structural properties such as total amount and spatial distribution of various material components, porosity and pore size distribution, specific surface area, etc. are typical and rather straightforward examples of application of *Corresponding author. Email: [email protected].fi International Journal of Computational Fluid Dynamics Vol. 23, No. 10, December 2009, 713–721 ISSN 1061-8562 print/ISSN 1029-0257 online Ó 2009 Taylor & Francis DOI: 10.1080/10618561003727512 http://www.informaworld.com Downloaded By: [UJF - INP Grenoble SICD 1] At: 14:18 16 April 2010
Transcript of Transport properties of heterogeneous materials. Combining computerised X-ray micro-tomography and...
Transport properties of heterogeneous materials Combining computerised X-ray
micro-tomography and direct numerical simulations
Viivi Koivua Maxime Decainbc Christian Geindreaub Keijo Mattilaa Jean-Francis Blochb and Markku Katajaa
aDepartment of Physics University of Jyvaskyla Jyvaskyla Finland bPapermaking Process Laboratory of Grenoble St MartinDrsquoHeres Grenoble France cLaboratory 3S-R University of Grenoble-CNRS Grenoble France
(Received 24 February 2009 final version received 22 February 2010)
Feasibility of a method for finding flow permeability of porous materials based on combining computerised X-raymicro-tomography and numerical simulations is assessed The permeability is found by solving fluid flow throughthe complex 3D pore structures obtained by tomography for actual material samples We estimate overall accuracyof the method and compare numerical and experimental results Factors contributing to uncertainty of the methodinclude numerical error arising from the finite resolution of tomographic images and the rather small sample sizeavailable with the present tomographic techniques The total uncertainty of computed values of permeability ishowever not essentially larger than that of experimental results We conclude that the method provides a feasiblealternative for finding fluid flow properties of the kind of materials studied It can be used to estimate all componentsof permeability tensor and is useful in cases where direct measurements are not achievable Analogous methods canbe applied to other modes of transport such as diffusion and heat conduction
Keywords permeability X-ray micro-tomography porous media numerical methods lattice-Boltzmann finitedifference
Introduction
An important problem in analysing and modellingmany complex natural and industrial processes is todetermine realistic effective material laws and values ofmaterial parameters that would properly represent egthe relevant transport properties of materials involvedVarious methods ranging from purely experimentalcorrelations to modelling based on fundamentalmicroscopic dynamics and lately also to numericalsimulation have been used (see eg Saar and Manga(1999) Belov et al (2003) Lundstrom et al (2004)Arns et al (2004) Kerwald (2004) Levitz (2005)Nordlund et al (2005) Verleye et al (2005) Eker andAkin (2006) Keehm et al (2006) Hellstrom andLundstrom (2006) Lundstrom et al (2006) Whiteet al (2006) Verleye et al (2007) Jaganathan et al(2007) Nordlund and Lundstrom (2007) Vidalet al (2008) Andersson et al (2009) and Frishfeldset al (2009)) An essential factor affecting properties ofmany heterogeneous and porous materials is theirspecific internal microscopic or mesoscopic structureThat structure is often very complex and difficult tocharacterise in a manner that would readily be used incorrelating structural and transport properties of thesematerials
Computerised X-ray micro-tomography (CXmT)techniques based on synchrotron radiation sourceshave been used in probing the internal structure ofheterogeneous materials for more than a decade Thetechniques involve taking a large number of X-rayimages of the material sample from different anglesWithin the so called absorption mode imaging numer-ical reconstruction algorithm is then used to obtain the3D spatial distribution of X-ray absorption coefficientwhich closely correlates with the local density distribu-tion Visualisation of this distribution gives the 3Dlsquoimagersquo of the material During the recent years themethod has become more feasible as also table-toptomographic scanners utilising X-ray tubes and capableof resolution in the sub-micron range have becomeavailable Indeed the utilisation of non-destructive 3Dexamination using CXmT techniques has opened newperspectives in heterogeneous materials research (seeeg Thibault and Bloch (2002) Manwart et al (2002)Aaltosalmi et al (2004) Ramaswamy et al (2004)Rolland du Roscoat et al (2005) Hyvaluoma et al(2006) and Fourie et al (2007)) Analysis of structuralproperties such as total amount and spatial distributionof various material components porosity and pore sizedistribution specific surface area etc are typical andrather straightforward examples of application of
Corresponding author Email viivikoivuphysjyufi
International Journal of Computational Fluid Dynamics
Vol 23 No 10 December 2009 713ndash721
ISSN 1061-8562 printISSN 1029-0257 online
2009 Taylor amp Francis
DOI 10108010618561003727512
httpwwwinformaworldcom
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
CXmT Combining X-ray tomography with numericalsimulation enables more advanced analysis of alsodynamic characteristics such as transport and elasticproperties of materials An important advantage ofthe method is that various analyses of quite differentnature can be made based on the same basic data ndash the3D digital image of the material structure The objectiveof this work is to develop and evaluate a methodfor characterising the properties of fluid flow throughporous materials by combining tomographic techniqueswith numerical simulations in the micro-scaleMore precisely the method is applied in finding flowpermeability of selected fibrous porous materials
The flow permeability coefficient k is generally atensor valued measure of the ability of a porousmaterial to transmit fluids It is defined for slowsteady-state isothermal Newtonian fluid flow througha porous medium by Darcysrsquos law (Darcy (1856) seeeg Bear (1972))
~q frac14 1mk rj eth1THORN
where ~q is the (superficial) volume flux vector m isthe dynamic viscosity of the fluid rj frac14 rp r~g p isthe pressure r is the density of the fluid and ~g is theacceleration due to a body force In what follows weconcentrate on diagonal components of k and applyEquation (1) in cases where 7rj is in the direction ofmean flow that takes place in three mutually perpen-dicular coordinate directions (to be defined later) Inthis case Equation (1) is reduced in a diagonalcomponent form as
qj frac14 kjjmjxj
j frac14 1 2 3 eth2THORN
In principle permeability coefficients for a givenmaterial sample can be found by numerically solvingincompressible creeping Newtonian flow through the3D pore space obtained by CXmT Obviously the keyissue in such an approach is the accuracy by which theactual pore structure of the medium can be reproducedby tomographic techniques This accuracy is affectedby properties of the CXmT hardware used its spatialresolution in particular choice of imaging parametersand by 3D image post-processing methods that may beused to enhance image quality Many natural andman-made porous materials are heterogeneous show-ing variation of properties in multiple scales Largeenough samples should thus be used in order to obtainstatistically meaningful results In practice this maynecessitate compromising between high resolution andlarge sample size
The finite resolution of the tomographic image asset by hardware limitations or by trade-off between
resolution and sample size also sets a practical upperlimit for the useful density of the grid for numericalsolution of flow In this work we use most straightfor-ward method for grid generation in which thenumerical grid is a regular cubic lattice based on thetomographic image itself The unit cell of the grid thusequals the cubic lsquovoxelrsquo of the digital tomographicimage In order to estimate the accuracy of the solutionobtained using such a relatively coarse numerical gridwe first compute the values of permeability coefficientsfor a model porous material that consists of regularlattice of straight cylinders for which also analyticalresults can be found In the next step we apply themethods in two heterogeneous fibrous materialsnamely plastic felt and hardwood paper A table-topCXmT device and a tomographic imaging facility basedon synchrotron radiation were used to obtain tomo-graphic images for felt and paper samples respectivelyThe numerical values of flow permeability obtainedusing the numerical grids based on these tomographicimages were compared with experimental results ofpermeability for the same materials
In this work creeping flow through the pore spaceof materials was solved using the lattice-Boltzmannmethod (LBM) It is based on solving the discreteBoltzmann equation instead of the standard conti-nuum flow equations The solution thus yields velocitydistribution function which is then used to obtainhydrodynamic variables as its moments (Succi 2001)The standard LBM involves an explicit time iterationscheme with a constant time step uniform grid andlocal data dependencies It is ideal for parallelcomputing As demonstrated earlier (Succi et al1989 Koponen et al 1998 Manwart et al 2002Martys and Hagedorn 2002 Belov et al 2003 Kerwald2004 Eker and Akin 2006 Kutay et al 2006 Pan et al2006 White et al 2006 Vidal et al 2008) the LBM isparticularly well suited for solving fluid flow incomplex geometries such as in the pore space ofirregular porous media This advantage of the methodfollows mainly from straightforward implementationof the no-slip boundary condition on fluidndashsolidinterface Here we use the standard halfway bounce-back boundary condition Furthermore we use theparticular D3Q19 implementation of LBM withthe linear two relaxation time approximation of theBoltzmann equation introduced by Ginzburg et al(2008) It appears that this approach avoids theproblem of dependence of permeability on viscositythat is encountered within more simple models such asthe single relaxation time Bhatnagar-Gross-Krook(BKG) model (Ginzburg and drsquoHumieres 2003Ginzburg 2008) Within LBM a simple regulargrid that consists of the centre points of the cubicunit cells (voxels of the tomographic image) was used
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The finite difference method (FDM) implementedin the Geodict software (httpwwwgeodictcom) wasused for reference in solving the flow through theregular geometries and for selected porous materialsamples Within this method a staggered grid was usedwhere the values of velocity and pressure were given atcentre points of the faces and volume of the cubic unitcells respectively The partial differential equations forincompressible Stokes flow are solved by using thelsquoFFF-Stokes solverrsquo based on Fast Fourier TransformThis solver appears to be fast and memory efficient forlarge computations characteristic to the presentapproach (Wiegmann 2007)
Estimate of discretisation error for regular geometries
As stated above a major potential source of un-certainty within the present approach is the numericalerror caused by the finite resolution of the tomographicimages and thus of the numerical grids used Here wewish to estimate the order of magnitude of this error bysolving the flow through regular model porousmaterials with varying values of porosity Solutionsare found for numerical grids of different sizes and theresulting values of permeability coefficient are com-pared also with analytical results To this end we findnumerical solution of flow through regular geometriesthat consist of square array of parallel straight rodsaligned in the z direction at four different porositylevels (06 07 08 and 09) For each such geometryincompressible creeping flow is induced by a small butotherwise arbitrary constant body force r~g in x and zcoordinate directions (see Figure 1) Periodic boundaryconditions were used at the boundary of the computa-tion region in all coordinate directions and no-slipboundary condition was enforced at all solidndashfluidinterfaces Numerical values of the diagonal elementsof the permeability tensor were then calculated fromEquation (2) using the mean flow rate given by thenumerical solution Notice that due to symmetry
reasons the permeability tensor is isotropic in the x-yplane and that z direction is one of its principaldirections The diagonal elements kxx kyy (frac14kxx) andkzz are thus also the principal values of the perme-ability tensor
The solution was found using a high-density gridand a low-density grid For the high-density grid thelength of the unit cell was 400 grid units This was highenough resolution to provide an accurate solution andan adequate independence of the actual grid size Forthe low-density grid the diameter of the rods was 7 gridunits which equals the mean diameter of the actualfibres of the plastic felt sample measured in units ofimage voxel size (see below) Example of solved flowfield using high and low density grids is shown inFigure 2 for mean flow in the x direction
A number of analytical results have been derivedfor permeability of arrays of cylinders with circularcross section and for flow parallel and perpendicular tothe cylinders (see eg Jackson and James (1986) andreferences therein) For flow parallel to the cylinders(z-direction) Drummond and Tahir (1984) propose anapproximative formula
kzzr2 1
4eth1fTHORN lneth1fTHORNKthorn2eth1fTHORN1
2eth1fTHORN2
eth4THORN
where f is porosity r is radius of the cylinders and K isa constant that depends on array geometry For asquare array it has the value 1476 Equation (4) isuseful at least in the porosity range 03 5 f 5 099Sangani and Acrivos (1982) used a unit cell approachto find an exact solution for flow perpendicular to thecylinders (x-y plane) Their solution was given in termsof series expansions where the effect of boundaryconditions must be found numerically This result cannot be expressed in closed form but Sangani andAcrivos provide tabulated numerical results that canbe readily applied in the present case
The computed results for permeability obtainedwith high and low density grids for square array ofcircular cylinders at different values of porosity areshown in Figure 3 together with correspondingtheoretical results The numerical results obtainedusing the high density grid closely agree with thetheoretical predictions The numerical results obtainedwith the low-density grid deviate from the theoreticaland more accurate numerical results Including all thecases analysed the maximum relative deviation be-tween the values of permeability obtained using low-density numerical grid as compared to either theore-tical or high resolution numerical results was less than10 In addition to results shown in Figure 3 similar
Figure 1 Unit cell of a square array of parallel rods withcircular cross-section together with the coordinateconvention used
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results were obtained for hexagonal arrays of circularrods This analysis was also repeated for square andhexagonal arrays of rods with square cross-section Inthis case theoretical results are not available butaccurate numerical solution can be found usingsufficiently dense grid Finally the numerical resultsobtained with LBM were verified independently usingFDM for a subset of regular geometry cases
Permeability of porous materials
We apply the analysis discussed in the previoussections for samples of fibrous porous materials ie
felt made of plastic fibres and hand sheet paper madeof chemical hardwood pulp The actual microscopicpore geometry of the samples was obtained by CXmTThe results will be compared with experimental datafor the same materials The felt material consists ofnearly cylindrical smooth fibres of average diameter 34mm The fibres are curved to some extent and have onlyweak tendency to be oriented in the plane of the feltWithin that plane the orientation is isotropic inaverage The structure of paper and the internal shapeof individual wood fibres are considerably morecomplicated than those of plastic felt Wood fibrestypically include a lumen and a cell wall with flattenedirregular cross-section The wall thickness and theaverage diameter of the fibres were found to beapproximately 4 mm and 20 mm respectively Thefibres are strongly oriented towards the plane of paperbut are also isotropically oriented within that plane(Rolland du Roscoat et al 2007)
The tomographic image of a plastic felt sampleused in this work was obtained using a table-toptomographic scanner (SkyScan 1172) with voxelresolution 484 mm The scanned felt sample was acircular section of diameter 6 mm In order toreproduce the internal structure of paper material incomparable detail considerably higher resolution mustbe used The paper sample used in this work wasscanned at ID19 laboratory at European SynchrotronRadiation Facility (ESRF Grenoble France) with avoxel size of 07 mm In this case the sample was arectangular section of size 14 6 14 mm2 The sam-ples were carefully cut from larger sheets of felt andpaper to avoid any damage to the structure due tocutting The locations from which the samples wereextracted were selected usi ng a grey-scale transillumi-nation image such that the samples would represent
Figure 2 Flow speed solved using high density grid (a) and low density grid (b) for an array of circular cylinders (porosities70 and 71 respectively) The mean flow is in the x direction Red and blue colours indicate high and low flow speedrespectively Available in colour online
Figure 3 Numerical and analytical values of dimensionlesspermeability in three principal directions for square array ofcircular cylinders at different porosity levels Analyticalvalues for flow parallel to the cylinders (z direction) aregiven by Equation (4) and the semianalytical results for flowperpendicular to the cylinders (in x-y plane) by tabulatedresults of Sangani and Acrivos see Tables 1 and 2 in Sanganiand Acrivos (1982)
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the mean properties (especially density) of the materialas closely as possible Both samples were scannedslightly compressed in the z direction Compressionwas obtained by plastic tubular sample holdersincluding a piston and a screw The size of theoriginal tomographic grey scale images was1240 6 1240 6 277 voxels and 2000 6 2000 6 121voxels for the felt and paper samples respectively Theimage of the felt sample was processed with variance-weighted mean filter (Gonzales and Woods 2001)before thresholding to a binary image For this samplethresholding was rather straightforward due to highquality of the tomographic image and regular shape ofthe plastic fibres The image of the paper sample wasdenoised by nonlinear anisotropic diffusion filter andfurther thresholded using seeded region growingmethod (Rolland du Roscoat et al 2005) Thethickness and porosity of the samples were determinedfrom the filtered and segmented tomographic images(and the same values of thickness were used inexperimental measurement of permeability see below)Porosity was determined as the total proportion ofopen void space in the tomographic subsamples Thevalues of porosity thus found were 616 for the feltand 486 for the paper sample
For numerical flow solution five non-overlappingsubvolumes were extracted from the original tomo-graphic images The dimensions of these subvolumeswere 400 6 400 6 277 pixels (physical size1930 6 1930 6 1341 mm3) and 300 6 300 6 121pixels (physical size 210 6 210 6 85 mm3) for thefelt and paper samples respectively Visualisation ofone of the subvolumes used in numerical analysis forfelt and paper are shown in Figure 4 Numericalsolution of flow through all the subsamples with meanflow in the three coordinate directions was thenobtained With the sizes of subsamples given abovethe total central processing unit (CPU) time requiredfor solution varied between 2 and 6 h in HP Proliant
DL145G2 18 GHz with 8 GB random access memory(RAM) nodes Figure 5 illustrates the flow velocityfields on a single 2D layer inside the subsamples shownin Figure 4 The detailed 3D numerical solution wasthen used to find the values of the diagonal compo-nents of permeability tensor for the subsamples as inthe case of regular geometries
Experimental values of permeability were obtainedby using a permeability measurement device (PMD)constructed originally for measuring transverse perme-ability (kzz) of soft porous sheet-like samples undermechanical compression (Koivu et al 2009a) For thepresent purpose the device was modified such that alsoin-plane measurement (kxx kyy) is possible (Koivuet al 2009b) Both liquids and gases can be used aspermeating fluids In transverse flow measurement acircular sample sheet of diameter 90 mm is placedbetween smooth sintered stainless steel compressionplates having a central area of diameter 60 mm openfor flow For in-plane measurements a rectangularsample sheet of size 70 mm 6 5 mm was sealedbetween solid stainless steel compression plates thatallow lateral flow through the sample The sampleswere then compressed to the same thickness (porosity)as the samples scanned by X-ray tomography Experi-ments were conducted with air flow in order to preventstructure changes due to swelling All measurementswere repeated for five macroscopically identicalsamples
Numerical and experimental results of permeabilityin three orthogonal directions for plastic felt andhardwood paper samples are summarised in Table 1Notice that in this case the values given in Table 1 arenot necessarily the principal values of permeabilitytensor but merely the values of diagonal elements inthe selected coordinate system In this case thenumerical results obtained by the two methods LBMand FDM deviate from each other more than in thecase of regular arrays Most likely this is just an
Figure 4 3D rendering of tomographic images of plastic felt (a) and hardwood paper (b) subsamples
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indication of both methods being less accurate in thepresent complex geometry Notice that the boundaryof the flow domain (pore space) is in this case knownonly to the accuracy of the tomographic imageTherefore using a numerical grid essentially finerthan the voxel matrix of the 3D tomographic digitalimage does not in general improve the overall validityof the solution and of the computed value ofpermeability coefficient It is thus obvious that therelatively coarse cubic grid provided directly by thetomographic images does not allow for very accuratesolution of flow especially near the solid boundariesand in narrow passages Nevertheless the largest
deviation between numerical and experimental resultsis less than 30 This is of the same order ofmagnitude than the overall uncertainty inherent alsoin the used experimental method for thin compressedfibre mats (Koivu et al 2009a)
The most important single source of uncertainty inthe present numerical results most likely is the ratherpoor statistics obtained due to the low number andsmall size of (sub)samples Notice however that eventhough several subsamples were analysed the variationof permeability value thus obtained (and indicated inTable 1) does not necessarily correspond to the totallarge scale variation of the sample materials but merely
Table 1 Numerical and experimental values of permeability in three orthogonal directions (in physical units) for plastic fibresample and hard wood paper sample
Sample
Permeability kjj [m2 E-13]
kxx kyy kzz
LBM FDM Experimental LBM FDM Experimental LBM FDM Experimental
Plasticfibre mat
290 + 10 352 311 + 9 290 + 30 332 304 + 9 270 + 30 305 270 + 5
Hardwoodsample
26 + 03 260 31 + 01 27 + 03 284 31 + 01 076 + 005 085 063 + 001
The numerical results obtained with LBM are given as mean values of results based on five non-overlapping subvolumes and the experimentalresults as mean values of five independent measurements with standard deviation
Figure 5 Numerically solved flow field in plastic felt (a) and hardwood paper (b) Shown is the scaled flow speed on single 2Dcross sections (x-y plane) of the full 3D solution space within the actual pore geometry of material subsamples as obtained by X-ray micro-tomography (see Figure 4) Red and blue colours indicate high and low flow velocity respectively Solid material iscoloured black In (a) mean velocity is in the direction perpendicular to the plane shown (z direction) In (b) mean flow is in theplane shown and from up to down (y direction) Notice also the pure fluid layers visible at the upstream and downstream sides ofthe paper sample Inserts show details of the pore structure and flow velocity pattern with resolution of the numerical grid givenby the tomographic images (484 mm and 07 mm for (a) and (b) respectively) Available in colour online
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the variation within the yet rather small originalsamples scanned The samples used in experimentsare however much larger and the results are thereforeexpected to better represent mean values (apart frompossible systematic errors remaining)
Discussion
In this work we develop and critically evaluate amethod for finding flow permeability of porousmaterials by numerically solving creeping fluid flowthrough the complex pore space of actual materialsamples Two conceptually different numericalschemes namely LBM and finite difference methodwere used The regular cubic grid used in numericalsolution is based directly on the 3D images of materialsamples obtained by two different states of the art X-ray microtomographic techniques
In many practical cases the main source ofuncertainty within the present approach is the finiteresolution of the tomographic image that provides thenumerical grid This may be due to hardware limita-tions or due to the need of compromising resolutionfor sample size for heterogeneous materials The orderof magnitude of the discretisation error arising fromthe relatively sparse numerical grid used was firstevaluated solving flows through a set of digitallygenerated regular geometries of straight cylinders Thevalues of permeability coefficients were computedusing grids of density comparable to that used withactual materials (relative to cylinderfibre diameter)The results were compared with analytical resultsavailable and with more accurate numerical resultsobtained using grids of sufficiently high density Forthe regular geometries studied the maximum deviationof the results obtained with sparse grid from analyticaland more accurate numerical results was less than10 This perhaps even surprisingly good accuracyobtained for regular geometries indicates that as far asdiscretisation error alone is concerned the methodmay be useful also for actual materials We mayhowever expect the discretisation error to be some-what higher for actual material samples due to theirmore complex pore structure Notice also that addi-tional error may arise due to the finite resolution of thetomographic techniques whereby details of the struc-ture (eg surface roughness of fibres and very narrowpassages in the porous space) may not be reproducedby the images Furthermore noise and other artefactspresent in the original reconstructed images as well asimage conditioning methods used to remove orminimise them may affect the results for eachindividual sample These are the main factors thatcurrently limit the applicability of the method espe-cially for fine-structured materials
In order to assess the overall uncertainty inherentin the computed results we applied the method infinding the diagonal components of permeabilitytensor for two fibrous materials plastic felt andhand-sheet paper under steady mechanical compres-sion and compared the results with correspondingexperimental values for the same materials Themaximum deviation between the numerical resultsand the experimental results was found to be less than30 In addition to numerical uncertainties arisingfrom discretisation and image quality discussed abovethis deviation includes statistical variation arising fromrelatively small size and low number of samples used incomputation In principle the accuracy can beimproved by using higher resolution tomographicimages and better statistics The former conditionmay become feasible with the present rapid develop-ment of X-ray tomographic techniques The latter canbe obtained by larger size or larger number of samplesObviously these improvements come with the cost ofhigher computational effort which however does notappear critical from the point of view of applicabilityof the method
To conclude the method of finding values of flowpermeability of porous materials based on numericallysolving fluid flow through the actual 3D pore geometryof material samples found using X-ray micro-tomo-graphy appears useful especially in cases whereexperimental results are not available or are difficultto obtain An example of such an application is to findthe permeability of various layers in a layer-structuredmaterial for which the layers can not be separated forindependent measurement (Koivu et al 2009b)Another advantage of the method is that it can beused to find all components of the permeability tensorWith the X-ray tomographic techniques presentlyused the overall accuracy of the computed values ofpermeability is moderate for the kind of materials andflow conditions studied here However this uncer-tainty is not essentially larger than the typical totaluncertainty of corresponding experimental results
Here we have demonstrated the method inevaluating a particular macroscopic transport propertyof porous materials the flow permeability Similarprocedure can be applied in other modes of transportsuch as diffusion and heat conduction in heterogeneousmaterials
Acknowledgements
The authors thank Tuomas Turpeinen MSc for hishelp related to the segmentation of tomographicimages and Jarno Alaraudanjoki MSc for thework done for experimental permeability measure-ments The authors also express their gratitude to the
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ID19rsquos team of ESRF for help with the acquisition ofthe images and to Sabine Rolland du Roscoat for herenthusiastic help concerning both the acquisition andthe image analysis of the studied samples
References
Aaltosalmi U et al 2004 Numerical analysis of fluid flowthrough fibrous porous materials Journal of Pulp andPaper Science 30 (9) 251ndash255
Andersson AG et al 2009 Flow through a two-scaleporosity material Research Letters in Materials Science2009 (701512) 4 pp
Arns C et al 2004 Virtual permeametry on microtomo-graphic images Journal of Petroleum Science andEngineering 45 41ndash46
Bear J 1972 Dynamics of fluid in porous media New YorkDover Publications inc
Belov EB et al 2003 Modelling of permeability of textilereinforcements lattice Boltzmann method CompositesScience and Technology 64 1069ndash1080
Darcy H 1856 Les Fontaines Publiques de la Ville de DijonParis Victor Dalmont
Drummond JE and Tahir MI 1984 Laminar viscousflow through regular arrays of parallel solid cylindersInternational Journal of Multiphase Flow 10 (5) 515ndash525
Eker E and Akin S 2006 Lattice-Boltzmann simulation offluid flow in synthetic fractures Transport in porousmedia 65 363ndash384
Fourie W et al 2007 The simulation of pore scale fluidflow with real world geometries obtained from X-raycomputed tomography European COMSOL Multiphy-sics Conference 23ndash24 October 2007 Boston GrenobleCOMSOL inc
Frishfelds et al 2009 Flow-induced deformation of non-crimp fabrics during composites manufacturing Proceed-ings in Interdisciplinary Transport Phenomena IV(ITP2009) Volterra ITP-09-52
Ginzburg I 2008 Consistent lattice Boltzmann schemesfor the Brinkman model of porous flow and finiteChapman-Enskog expansion Physical Review E 77066704
Ginzburg I and drsquoHumieres D 2003 Multireflectionboundary conditions for lattice Boltzmann modelsPhysical Review E 68 (6) 066614
Ginzburg I Verhaeghe F and drsquoHumieres D 2008 Two-relaxation-time lattice-boltzmann scheme about para-metrization velocity pressure and mixed boundaryconditions Communications in Computational Physics 3(2) 427ndash478
Gonzales RC and Woods RE 2001 Digital imageprocessing 2nd edn New Jersey Prentice-Hall Inc237ndash241
Hellstrom JGI and Lundstrom TS 2006 Flow throughPorous Media at Moderate Reynolds Number Interna-tional Scientific Colloquim Modelling for MaterialProcessing Riga Latvia Faculty of Physics and Mathe-matics University of Latvia 129ndash134
Hyvaluoma J et al 2006 Simulation of liquid penetrationin paper Physical Review E 73 (3) 036705
Jackson GW and James DF 1986 The permeability offibrous porous media Canadian Journal of ChemicalEngineering 64 364ndash373
Jaganathan S Vahedi Tafreshi H and Pourdeyhimi B2007 A realistic approach for modeling permeabilityof fibrous media 3-D imaging coupled with CFDsimulation Chemical Engineering science 63 (1) 224ndash252
Keehm Y Sternlof K and Mukerji T 2006 Computa-tional estimation of compaction band permeability insandstone Geosciences Journal 10 (4) 499ndash505
Kerwald D 2004 Parallel lattice Boltzmann simulation ofcomplex flow NAFEMS Seminar simulation of complexflows (CFD) ndash applications and trends NiedernhausenWiesbaden
Koivu V Mattila K and Kataja M 2009a A method formeasuring darcian flow permeability of thin fibre matsNordic Pulp and Paper Research Journal 24 (4) 395ndash402
Koivu V et al 2009b Flow permeability of fibrous porousmaterials Micro-tomography and numerical simulationsIn SJ IrsquoAnson ed Advances in pulp and paper researchTransactions on 14th Research Fundamental SymposiumOxford September 14ndash18 2009 Manchester The Pulpand Paper Fundamental Research Society 437ndash454
Koponen A et al 1998 Permeability of three-dimensionalrandom fiber webs Physical Review Letters 80 (4) 716
Kutay ME Aydilek AH and Masad E 2006 Labora-tory validation of lattice Boltzmann method for model-ing pore-scale flow in granular materials ComputationalGeotechnics 33 381ndash395
Levitz P 2005 Toolbox for 3D imaging and modeling ofporous media relationship with transport propertiesCement and Concrete Research 37 351ndash359
Lundstrom TS Frishfelds V and Jakovics A 2004 Astatistical approach to permeability of clustered fibrereinforcements Journal of Composite Materials 38 (13)1137ndash1149
Lundstrom TS Sundlof H and Holmberg JA 2006Modeling of power-law fluid flow through fiber bedsJournal of Composite Materials 40 (3) 283ndash296
Manwart C et al 2002 Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media Physical Review E 66 (1)016702
Martys NS and Hagedorn JG 2002 Multiscale modelingof fluid transport in heterogeneous materials usingdiscrete Boltzmann methods Journal of Materials andStructures 35 (10) 650
Nordlund M et al 2005 Permeability network model fornon-crimp fabrics Composites Part A 37 826ndash835
Nordlund M and Lundstrom TS 2007 Effect ofmulti-scale porosity in local permeability modelling ofnon-crimp fabrics Transport in Porous Media 73 109ndash124
Pan C Luo L-S and Miller CT 2006 An evaluation oflattice Boltzmann schemes for porous medium flowsimulation Computers amp Fluids 35 (8ndash9) 898
Ramaswamy S et al 2004 The 3D structure of fabric andits relationship to liquid and vapor transport Colloidsand Surfaces A Physicochemical and Engineering As-pects 241 323ndash333
Rolland du Roscoat S Bloch J-F and Thibault X 2005Synchrotron Radiation microtomography applied topaper investigation Journal of Physics D AppliedPhysics 38 A78ndashA84
Rolland du Roscoat S et al 2007 Estimation ofmicrostructural properties from synchrotron X-ray microtomography and determination of the REV in papermaterials Acta Materiala 55 (8) 2841ndash2850
720 V Koivu et al
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Saar MO and Manga M 1999 Permeability-porosityrelationship in vesicular basalts Geophysical ResearchLetters 26 (1) 111ndash114
Sangani AS and Acrivos A 1982 Slow flow past periodicarrays of cylinders with application to heat transferInternational Journal of Multiphase Flow 8 193ndash206
Succi S 2001 The lattice Boltzmann equation for fluiddynamics and beyond Gloucestershire Clarendon Press
Succi S Foti E and Higuera F 1989 Three-dimensionalflows in complex geometries with the lattice Boltzmannmethod Europhysics Letters 10 (5) 432ndash438
Thibault X and Bloch J-F 2002 Structural analysis of X-ray microtomography of a strained nonwoven paper-maker felt Textile Research Journal 72 (6) 480ndash485
Verleye B et al 2005 Computation of permeability oftextile reinforcements Proceedings in Scientific Computa-tion IMACS 2005 Paris France
Verleye B et al 2007 Permeability of textile reinforce-ments simulation influence of shear and validationComposite Science and Technology 68 (1) 2804ndash2810
Vidal D et al 2008 Effect od particle size distribution andpacking compression on fluid permeability as predictedby lattice-Boltzmann simulations Computers andChemical Engineering 33 (1) 256ndash266
White JA Borja IR and Fredrich JT 2006 Calculat-ing the effective permeability of sandstone with multi-scale lattice-Boltzmannfinite element simulations ActaGeotechnica 1 195ndash209
Wiegman A et al 2007 Computation of the permeabilityof porous materials from their microstructure by FFT-stokes Report of the Fraunhofer ITWM 129
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CXmT Combining X-ray tomography with numericalsimulation enables more advanced analysis of alsodynamic characteristics such as transport and elasticproperties of materials An important advantage ofthe method is that various analyses of quite differentnature can be made based on the same basic data ndash the3D digital image of the material structure The objectiveof this work is to develop and evaluate a methodfor characterising the properties of fluid flow throughporous materials by combining tomographic techniqueswith numerical simulations in the micro-scaleMore precisely the method is applied in finding flowpermeability of selected fibrous porous materials
The flow permeability coefficient k is generally atensor valued measure of the ability of a porousmaterial to transmit fluids It is defined for slowsteady-state isothermal Newtonian fluid flow througha porous medium by Darcysrsquos law (Darcy (1856) seeeg Bear (1972))
~q frac14 1mk rj eth1THORN
where ~q is the (superficial) volume flux vector m isthe dynamic viscosity of the fluid rj frac14 rp r~g p isthe pressure r is the density of the fluid and ~g is theacceleration due to a body force In what follows weconcentrate on diagonal components of k and applyEquation (1) in cases where 7rj is in the direction ofmean flow that takes place in three mutually perpen-dicular coordinate directions (to be defined later) Inthis case Equation (1) is reduced in a diagonalcomponent form as
qj frac14 kjjmjxj
j frac14 1 2 3 eth2THORN
In principle permeability coefficients for a givenmaterial sample can be found by numerically solvingincompressible creeping Newtonian flow through the3D pore space obtained by CXmT Obviously the keyissue in such an approach is the accuracy by which theactual pore structure of the medium can be reproducedby tomographic techniques This accuracy is affectedby properties of the CXmT hardware used its spatialresolution in particular choice of imaging parametersand by 3D image post-processing methods that may beused to enhance image quality Many natural andman-made porous materials are heterogeneous show-ing variation of properties in multiple scales Largeenough samples should thus be used in order to obtainstatistically meaningful results In practice this maynecessitate compromising between high resolution andlarge sample size
The finite resolution of the tomographic image asset by hardware limitations or by trade-off between
resolution and sample size also sets a practical upperlimit for the useful density of the grid for numericalsolution of flow In this work we use most straightfor-ward method for grid generation in which thenumerical grid is a regular cubic lattice based on thetomographic image itself The unit cell of the grid thusequals the cubic lsquovoxelrsquo of the digital tomographicimage In order to estimate the accuracy of the solutionobtained using such a relatively coarse numerical gridwe first compute the values of permeability coefficientsfor a model porous material that consists of regularlattice of straight cylinders for which also analyticalresults can be found In the next step we apply themethods in two heterogeneous fibrous materialsnamely plastic felt and hardwood paper A table-topCXmT device and a tomographic imaging facility basedon synchrotron radiation were used to obtain tomo-graphic images for felt and paper samples respectivelyThe numerical values of flow permeability obtainedusing the numerical grids based on these tomographicimages were compared with experimental results ofpermeability for the same materials
In this work creeping flow through the pore spaceof materials was solved using the lattice-Boltzmannmethod (LBM) It is based on solving the discreteBoltzmann equation instead of the standard conti-nuum flow equations The solution thus yields velocitydistribution function which is then used to obtainhydrodynamic variables as its moments (Succi 2001)The standard LBM involves an explicit time iterationscheme with a constant time step uniform grid andlocal data dependencies It is ideal for parallelcomputing As demonstrated earlier (Succi et al1989 Koponen et al 1998 Manwart et al 2002Martys and Hagedorn 2002 Belov et al 2003 Kerwald2004 Eker and Akin 2006 Kutay et al 2006 Pan et al2006 White et al 2006 Vidal et al 2008) the LBM isparticularly well suited for solving fluid flow incomplex geometries such as in the pore space ofirregular porous media This advantage of the methodfollows mainly from straightforward implementationof the no-slip boundary condition on fluidndashsolidinterface Here we use the standard halfway bounce-back boundary condition Furthermore we use theparticular D3Q19 implementation of LBM withthe linear two relaxation time approximation of theBoltzmann equation introduced by Ginzburg et al(2008) It appears that this approach avoids theproblem of dependence of permeability on viscositythat is encountered within more simple models such asthe single relaxation time Bhatnagar-Gross-Krook(BKG) model (Ginzburg and drsquoHumieres 2003Ginzburg 2008) Within LBM a simple regulargrid that consists of the centre points of the cubicunit cells (voxels of the tomographic image) was used
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The finite difference method (FDM) implementedin the Geodict software (httpwwwgeodictcom) wasused for reference in solving the flow through theregular geometries and for selected porous materialsamples Within this method a staggered grid was usedwhere the values of velocity and pressure were given atcentre points of the faces and volume of the cubic unitcells respectively The partial differential equations forincompressible Stokes flow are solved by using thelsquoFFF-Stokes solverrsquo based on Fast Fourier TransformThis solver appears to be fast and memory efficient forlarge computations characteristic to the presentapproach (Wiegmann 2007)
Estimate of discretisation error for regular geometries
As stated above a major potential source of un-certainty within the present approach is the numericalerror caused by the finite resolution of the tomographicimages and thus of the numerical grids used Here wewish to estimate the order of magnitude of this error bysolving the flow through regular model porousmaterials with varying values of porosity Solutionsare found for numerical grids of different sizes and theresulting values of permeability coefficient are com-pared also with analytical results To this end we findnumerical solution of flow through regular geometriesthat consist of square array of parallel straight rodsaligned in the z direction at four different porositylevels (06 07 08 and 09) For each such geometryincompressible creeping flow is induced by a small butotherwise arbitrary constant body force r~g in x and zcoordinate directions (see Figure 1) Periodic boundaryconditions were used at the boundary of the computa-tion region in all coordinate directions and no-slipboundary condition was enforced at all solidndashfluidinterfaces Numerical values of the diagonal elementsof the permeability tensor were then calculated fromEquation (2) using the mean flow rate given by thenumerical solution Notice that due to symmetry
reasons the permeability tensor is isotropic in the x-yplane and that z direction is one of its principaldirections The diagonal elements kxx kyy (frac14kxx) andkzz are thus also the principal values of the perme-ability tensor
The solution was found using a high-density gridand a low-density grid For the high-density grid thelength of the unit cell was 400 grid units This was highenough resolution to provide an accurate solution andan adequate independence of the actual grid size Forthe low-density grid the diameter of the rods was 7 gridunits which equals the mean diameter of the actualfibres of the plastic felt sample measured in units ofimage voxel size (see below) Example of solved flowfield using high and low density grids is shown inFigure 2 for mean flow in the x direction
A number of analytical results have been derivedfor permeability of arrays of cylinders with circularcross section and for flow parallel and perpendicular tothe cylinders (see eg Jackson and James (1986) andreferences therein) For flow parallel to the cylinders(z-direction) Drummond and Tahir (1984) propose anapproximative formula
kzzr2 1
4eth1fTHORN lneth1fTHORNKthorn2eth1fTHORN1
2eth1fTHORN2
eth4THORN
where f is porosity r is radius of the cylinders and K isa constant that depends on array geometry For asquare array it has the value 1476 Equation (4) isuseful at least in the porosity range 03 5 f 5 099Sangani and Acrivos (1982) used a unit cell approachto find an exact solution for flow perpendicular to thecylinders (x-y plane) Their solution was given in termsof series expansions where the effect of boundaryconditions must be found numerically This result cannot be expressed in closed form but Sangani andAcrivos provide tabulated numerical results that canbe readily applied in the present case
The computed results for permeability obtainedwith high and low density grids for square array ofcircular cylinders at different values of porosity areshown in Figure 3 together with correspondingtheoretical results The numerical results obtainedusing the high density grid closely agree with thetheoretical predictions The numerical results obtainedwith the low-density grid deviate from the theoreticaland more accurate numerical results Including all thecases analysed the maximum relative deviation be-tween the values of permeability obtained using low-density numerical grid as compared to either theore-tical or high resolution numerical results was less than10 In addition to results shown in Figure 3 similar
Figure 1 Unit cell of a square array of parallel rods withcircular cross-section together with the coordinateconvention used
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results were obtained for hexagonal arrays of circularrods This analysis was also repeated for square andhexagonal arrays of rods with square cross-section Inthis case theoretical results are not available butaccurate numerical solution can be found usingsufficiently dense grid Finally the numerical resultsobtained with LBM were verified independently usingFDM for a subset of regular geometry cases
Permeability of porous materials
We apply the analysis discussed in the previoussections for samples of fibrous porous materials ie
felt made of plastic fibres and hand sheet paper madeof chemical hardwood pulp The actual microscopicpore geometry of the samples was obtained by CXmTThe results will be compared with experimental datafor the same materials The felt material consists ofnearly cylindrical smooth fibres of average diameter 34mm The fibres are curved to some extent and have onlyweak tendency to be oriented in the plane of the feltWithin that plane the orientation is isotropic inaverage The structure of paper and the internal shapeof individual wood fibres are considerably morecomplicated than those of plastic felt Wood fibrestypically include a lumen and a cell wall with flattenedirregular cross-section The wall thickness and theaverage diameter of the fibres were found to beapproximately 4 mm and 20 mm respectively Thefibres are strongly oriented towards the plane of paperbut are also isotropically oriented within that plane(Rolland du Roscoat et al 2007)
The tomographic image of a plastic felt sampleused in this work was obtained using a table-toptomographic scanner (SkyScan 1172) with voxelresolution 484 mm The scanned felt sample was acircular section of diameter 6 mm In order toreproduce the internal structure of paper material incomparable detail considerably higher resolution mustbe used The paper sample used in this work wasscanned at ID19 laboratory at European SynchrotronRadiation Facility (ESRF Grenoble France) with avoxel size of 07 mm In this case the sample was arectangular section of size 14 6 14 mm2 The sam-ples were carefully cut from larger sheets of felt andpaper to avoid any damage to the structure due tocutting The locations from which the samples wereextracted were selected usi ng a grey-scale transillumi-nation image such that the samples would represent
Figure 2 Flow speed solved using high density grid (a) and low density grid (b) for an array of circular cylinders (porosities70 and 71 respectively) The mean flow is in the x direction Red and blue colours indicate high and low flow speedrespectively Available in colour online
Figure 3 Numerical and analytical values of dimensionlesspermeability in three principal directions for square array ofcircular cylinders at different porosity levels Analyticalvalues for flow parallel to the cylinders (z direction) aregiven by Equation (4) and the semianalytical results for flowperpendicular to the cylinders (in x-y plane) by tabulatedresults of Sangani and Acrivos see Tables 1 and 2 in Sanganiand Acrivos (1982)
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the mean properties (especially density) of the materialas closely as possible Both samples were scannedslightly compressed in the z direction Compressionwas obtained by plastic tubular sample holdersincluding a piston and a screw The size of theoriginal tomographic grey scale images was1240 6 1240 6 277 voxels and 2000 6 2000 6 121voxels for the felt and paper samples respectively Theimage of the felt sample was processed with variance-weighted mean filter (Gonzales and Woods 2001)before thresholding to a binary image For this samplethresholding was rather straightforward due to highquality of the tomographic image and regular shape ofthe plastic fibres The image of the paper sample wasdenoised by nonlinear anisotropic diffusion filter andfurther thresholded using seeded region growingmethod (Rolland du Roscoat et al 2005) Thethickness and porosity of the samples were determinedfrom the filtered and segmented tomographic images(and the same values of thickness were used inexperimental measurement of permeability see below)Porosity was determined as the total proportion ofopen void space in the tomographic subsamples Thevalues of porosity thus found were 616 for the feltand 486 for the paper sample
For numerical flow solution five non-overlappingsubvolumes were extracted from the original tomo-graphic images The dimensions of these subvolumeswere 400 6 400 6 277 pixels (physical size1930 6 1930 6 1341 mm3) and 300 6 300 6 121pixels (physical size 210 6 210 6 85 mm3) for thefelt and paper samples respectively Visualisation ofone of the subvolumes used in numerical analysis forfelt and paper are shown in Figure 4 Numericalsolution of flow through all the subsamples with meanflow in the three coordinate directions was thenobtained With the sizes of subsamples given abovethe total central processing unit (CPU) time requiredfor solution varied between 2 and 6 h in HP Proliant
DL145G2 18 GHz with 8 GB random access memory(RAM) nodes Figure 5 illustrates the flow velocityfields on a single 2D layer inside the subsamples shownin Figure 4 The detailed 3D numerical solution wasthen used to find the values of the diagonal compo-nents of permeability tensor for the subsamples as inthe case of regular geometries
Experimental values of permeability were obtainedby using a permeability measurement device (PMD)constructed originally for measuring transverse perme-ability (kzz) of soft porous sheet-like samples undermechanical compression (Koivu et al 2009a) For thepresent purpose the device was modified such that alsoin-plane measurement (kxx kyy) is possible (Koivuet al 2009b) Both liquids and gases can be used aspermeating fluids In transverse flow measurement acircular sample sheet of diameter 90 mm is placedbetween smooth sintered stainless steel compressionplates having a central area of diameter 60 mm openfor flow For in-plane measurements a rectangularsample sheet of size 70 mm 6 5 mm was sealedbetween solid stainless steel compression plates thatallow lateral flow through the sample The sampleswere then compressed to the same thickness (porosity)as the samples scanned by X-ray tomography Experi-ments were conducted with air flow in order to preventstructure changes due to swelling All measurementswere repeated for five macroscopically identicalsamples
Numerical and experimental results of permeabilityin three orthogonal directions for plastic felt andhardwood paper samples are summarised in Table 1Notice that in this case the values given in Table 1 arenot necessarily the principal values of permeabilitytensor but merely the values of diagonal elements inthe selected coordinate system In this case thenumerical results obtained by the two methods LBMand FDM deviate from each other more than in thecase of regular arrays Most likely this is just an
Figure 4 3D rendering of tomographic images of plastic felt (a) and hardwood paper (b) subsamples
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indication of both methods being less accurate in thepresent complex geometry Notice that the boundaryof the flow domain (pore space) is in this case knownonly to the accuracy of the tomographic imageTherefore using a numerical grid essentially finerthan the voxel matrix of the 3D tomographic digitalimage does not in general improve the overall validityof the solution and of the computed value ofpermeability coefficient It is thus obvious that therelatively coarse cubic grid provided directly by thetomographic images does not allow for very accuratesolution of flow especially near the solid boundariesand in narrow passages Nevertheless the largest
deviation between numerical and experimental resultsis less than 30 This is of the same order ofmagnitude than the overall uncertainty inherent alsoin the used experimental method for thin compressedfibre mats (Koivu et al 2009a)
The most important single source of uncertainty inthe present numerical results most likely is the ratherpoor statistics obtained due to the low number andsmall size of (sub)samples Notice however that eventhough several subsamples were analysed the variationof permeability value thus obtained (and indicated inTable 1) does not necessarily correspond to the totallarge scale variation of the sample materials but merely
Table 1 Numerical and experimental values of permeability in three orthogonal directions (in physical units) for plastic fibresample and hard wood paper sample
Sample
Permeability kjj [m2 E-13]
kxx kyy kzz
LBM FDM Experimental LBM FDM Experimental LBM FDM Experimental
Plasticfibre mat
290 + 10 352 311 + 9 290 + 30 332 304 + 9 270 + 30 305 270 + 5
Hardwoodsample
26 + 03 260 31 + 01 27 + 03 284 31 + 01 076 + 005 085 063 + 001
The numerical results obtained with LBM are given as mean values of results based on five non-overlapping subvolumes and the experimentalresults as mean values of five independent measurements with standard deviation
Figure 5 Numerically solved flow field in plastic felt (a) and hardwood paper (b) Shown is the scaled flow speed on single 2Dcross sections (x-y plane) of the full 3D solution space within the actual pore geometry of material subsamples as obtained by X-ray micro-tomography (see Figure 4) Red and blue colours indicate high and low flow velocity respectively Solid material iscoloured black In (a) mean velocity is in the direction perpendicular to the plane shown (z direction) In (b) mean flow is in theplane shown and from up to down (y direction) Notice also the pure fluid layers visible at the upstream and downstream sides ofthe paper sample Inserts show details of the pore structure and flow velocity pattern with resolution of the numerical grid givenby the tomographic images (484 mm and 07 mm for (a) and (b) respectively) Available in colour online
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the variation within the yet rather small originalsamples scanned The samples used in experimentsare however much larger and the results are thereforeexpected to better represent mean values (apart frompossible systematic errors remaining)
Discussion
In this work we develop and critically evaluate amethod for finding flow permeability of porousmaterials by numerically solving creeping fluid flowthrough the complex pore space of actual materialsamples Two conceptually different numericalschemes namely LBM and finite difference methodwere used The regular cubic grid used in numericalsolution is based directly on the 3D images of materialsamples obtained by two different states of the art X-ray microtomographic techniques
In many practical cases the main source ofuncertainty within the present approach is the finiteresolution of the tomographic image that provides thenumerical grid This may be due to hardware limita-tions or due to the need of compromising resolutionfor sample size for heterogeneous materials The orderof magnitude of the discretisation error arising fromthe relatively sparse numerical grid used was firstevaluated solving flows through a set of digitallygenerated regular geometries of straight cylinders Thevalues of permeability coefficients were computedusing grids of density comparable to that used withactual materials (relative to cylinderfibre diameter)The results were compared with analytical resultsavailable and with more accurate numerical resultsobtained using grids of sufficiently high density Forthe regular geometries studied the maximum deviationof the results obtained with sparse grid from analyticaland more accurate numerical results was less than10 This perhaps even surprisingly good accuracyobtained for regular geometries indicates that as far asdiscretisation error alone is concerned the methodmay be useful also for actual materials We mayhowever expect the discretisation error to be some-what higher for actual material samples due to theirmore complex pore structure Notice also that addi-tional error may arise due to the finite resolution of thetomographic techniques whereby details of the struc-ture (eg surface roughness of fibres and very narrowpassages in the porous space) may not be reproducedby the images Furthermore noise and other artefactspresent in the original reconstructed images as well asimage conditioning methods used to remove orminimise them may affect the results for eachindividual sample These are the main factors thatcurrently limit the applicability of the method espe-cially for fine-structured materials
In order to assess the overall uncertainty inherentin the computed results we applied the method infinding the diagonal components of permeabilitytensor for two fibrous materials plastic felt andhand-sheet paper under steady mechanical compres-sion and compared the results with correspondingexperimental values for the same materials Themaximum deviation between the numerical resultsand the experimental results was found to be less than30 In addition to numerical uncertainties arisingfrom discretisation and image quality discussed abovethis deviation includes statistical variation arising fromrelatively small size and low number of samples used incomputation In principle the accuracy can beimproved by using higher resolution tomographicimages and better statistics The former conditionmay become feasible with the present rapid develop-ment of X-ray tomographic techniques The latter canbe obtained by larger size or larger number of samplesObviously these improvements come with the cost ofhigher computational effort which however does notappear critical from the point of view of applicabilityof the method
To conclude the method of finding values of flowpermeability of porous materials based on numericallysolving fluid flow through the actual 3D pore geometryof material samples found using X-ray micro-tomo-graphy appears useful especially in cases whereexperimental results are not available or are difficultto obtain An example of such an application is to findthe permeability of various layers in a layer-structuredmaterial for which the layers can not be separated forindependent measurement (Koivu et al 2009b)Another advantage of the method is that it can beused to find all components of the permeability tensorWith the X-ray tomographic techniques presentlyused the overall accuracy of the computed values ofpermeability is moderate for the kind of materials andflow conditions studied here However this uncer-tainty is not essentially larger than the typical totaluncertainty of corresponding experimental results
Here we have demonstrated the method inevaluating a particular macroscopic transport propertyof porous materials the flow permeability Similarprocedure can be applied in other modes of transportsuch as diffusion and heat conduction in heterogeneousmaterials
Acknowledgements
The authors thank Tuomas Turpeinen MSc for hishelp related to the segmentation of tomographicimages and Jarno Alaraudanjoki MSc for thework done for experimental permeability measure-ments The authors also express their gratitude to the
International Journal of Computational Fluid Dynamics 719
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ID19rsquos team of ESRF for help with the acquisition ofthe images and to Sabine Rolland du Roscoat for herenthusiastic help concerning both the acquisition andthe image analysis of the studied samples
References
Aaltosalmi U et al 2004 Numerical analysis of fluid flowthrough fibrous porous materials Journal of Pulp andPaper Science 30 (9) 251ndash255
Andersson AG et al 2009 Flow through a two-scaleporosity material Research Letters in Materials Science2009 (701512) 4 pp
Arns C et al 2004 Virtual permeametry on microtomo-graphic images Journal of Petroleum Science andEngineering 45 41ndash46
Bear J 1972 Dynamics of fluid in porous media New YorkDover Publications inc
Belov EB et al 2003 Modelling of permeability of textilereinforcements lattice Boltzmann method CompositesScience and Technology 64 1069ndash1080
Darcy H 1856 Les Fontaines Publiques de la Ville de DijonParis Victor Dalmont
Drummond JE and Tahir MI 1984 Laminar viscousflow through regular arrays of parallel solid cylindersInternational Journal of Multiphase Flow 10 (5) 515ndash525
Eker E and Akin S 2006 Lattice-Boltzmann simulation offluid flow in synthetic fractures Transport in porousmedia 65 363ndash384
Fourie W et al 2007 The simulation of pore scale fluidflow with real world geometries obtained from X-raycomputed tomography European COMSOL Multiphy-sics Conference 23ndash24 October 2007 Boston GrenobleCOMSOL inc
Frishfelds et al 2009 Flow-induced deformation of non-crimp fabrics during composites manufacturing Proceed-ings in Interdisciplinary Transport Phenomena IV(ITP2009) Volterra ITP-09-52
Ginzburg I 2008 Consistent lattice Boltzmann schemesfor the Brinkman model of porous flow and finiteChapman-Enskog expansion Physical Review E 77066704
Ginzburg I and drsquoHumieres D 2003 Multireflectionboundary conditions for lattice Boltzmann modelsPhysical Review E 68 (6) 066614
Ginzburg I Verhaeghe F and drsquoHumieres D 2008 Two-relaxation-time lattice-boltzmann scheme about para-metrization velocity pressure and mixed boundaryconditions Communications in Computational Physics 3(2) 427ndash478
Gonzales RC and Woods RE 2001 Digital imageprocessing 2nd edn New Jersey Prentice-Hall Inc237ndash241
Hellstrom JGI and Lundstrom TS 2006 Flow throughPorous Media at Moderate Reynolds Number Interna-tional Scientific Colloquim Modelling for MaterialProcessing Riga Latvia Faculty of Physics and Mathe-matics University of Latvia 129ndash134
Hyvaluoma J et al 2006 Simulation of liquid penetrationin paper Physical Review E 73 (3) 036705
Jackson GW and James DF 1986 The permeability offibrous porous media Canadian Journal of ChemicalEngineering 64 364ndash373
Jaganathan S Vahedi Tafreshi H and Pourdeyhimi B2007 A realistic approach for modeling permeabilityof fibrous media 3-D imaging coupled with CFDsimulation Chemical Engineering science 63 (1) 224ndash252
Keehm Y Sternlof K and Mukerji T 2006 Computa-tional estimation of compaction band permeability insandstone Geosciences Journal 10 (4) 499ndash505
Kerwald D 2004 Parallel lattice Boltzmann simulation ofcomplex flow NAFEMS Seminar simulation of complexflows (CFD) ndash applications and trends NiedernhausenWiesbaden
Koivu V Mattila K and Kataja M 2009a A method formeasuring darcian flow permeability of thin fibre matsNordic Pulp and Paper Research Journal 24 (4) 395ndash402
Koivu V et al 2009b Flow permeability of fibrous porousmaterials Micro-tomography and numerical simulationsIn SJ IrsquoAnson ed Advances in pulp and paper researchTransactions on 14th Research Fundamental SymposiumOxford September 14ndash18 2009 Manchester The Pulpand Paper Fundamental Research Society 437ndash454
Koponen A et al 1998 Permeability of three-dimensionalrandom fiber webs Physical Review Letters 80 (4) 716
Kutay ME Aydilek AH and Masad E 2006 Labora-tory validation of lattice Boltzmann method for model-ing pore-scale flow in granular materials ComputationalGeotechnics 33 381ndash395
Levitz P 2005 Toolbox for 3D imaging and modeling ofporous media relationship with transport propertiesCement and Concrete Research 37 351ndash359
Lundstrom TS Frishfelds V and Jakovics A 2004 Astatistical approach to permeability of clustered fibrereinforcements Journal of Composite Materials 38 (13)1137ndash1149
Lundstrom TS Sundlof H and Holmberg JA 2006Modeling of power-law fluid flow through fiber bedsJournal of Composite Materials 40 (3) 283ndash296
Manwart C et al 2002 Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media Physical Review E 66 (1)016702
Martys NS and Hagedorn JG 2002 Multiscale modelingof fluid transport in heterogeneous materials usingdiscrete Boltzmann methods Journal of Materials andStructures 35 (10) 650
Nordlund M et al 2005 Permeability network model fornon-crimp fabrics Composites Part A 37 826ndash835
Nordlund M and Lundstrom TS 2007 Effect ofmulti-scale porosity in local permeability modelling ofnon-crimp fabrics Transport in Porous Media 73 109ndash124
Pan C Luo L-S and Miller CT 2006 An evaluation oflattice Boltzmann schemes for porous medium flowsimulation Computers amp Fluids 35 (8ndash9) 898
Ramaswamy S et al 2004 The 3D structure of fabric andits relationship to liquid and vapor transport Colloidsand Surfaces A Physicochemical and Engineering As-pects 241 323ndash333
Rolland du Roscoat S Bloch J-F and Thibault X 2005Synchrotron Radiation microtomography applied topaper investigation Journal of Physics D AppliedPhysics 38 A78ndashA84
Rolland du Roscoat S et al 2007 Estimation ofmicrostructural properties from synchrotron X-ray microtomography and determination of the REV in papermaterials Acta Materiala 55 (8) 2841ndash2850
720 V Koivu et al
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
Saar MO and Manga M 1999 Permeability-porosityrelationship in vesicular basalts Geophysical ResearchLetters 26 (1) 111ndash114
Sangani AS and Acrivos A 1982 Slow flow past periodicarrays of cylinders with application to heat transferInternational Journal of Multiphase Flow 8 193ndash206
Succi S 2001 The lattice Boltzmann equation for fluiddynamics and beyond Gloucestershire Clarendon Press
Succi S Foti E and Higuera F 1989 Three-dimensionalflows in complex geometries with the lattice Boltzmannmethod Europhysics Letters 10 (5) 432ndash438
Thibault X and Bloch J-F 2002 Structural analysis of X-ray microtomography of a strained nonwoven paper-maker felt Textile Research Journal 72 (6) 480ndash485
Verleye B et al 2005 Computation of permeability oftextile reinforcements Proceedings in Scientific Computa-tion IMACS 2005 Paris France
Verleye B et al 2007 Permeability of textile reinforce-ments simulation influence of shear and validationComposite Science and Technology 68 (1) 2804ndash2810
Vidal D et al 2008 Effect od particle size distribution andpacking compression on fluid permeability as predictedby lattice-Boltzmann simulations Computers andChemical Engineering 33 (1) 256ndash266
White JA Borja IR and Fredrich JT 2006 Calculat-ing the effective permeability of sandstone with multi-scale lattice-Boltzmannfinite element simulations ActaGeotechnica 1 195ndash209
Wiegman A et al 2007 Computation of the permeabilityof porous materials from their microstructure by FFT-stokes Report of the Fraunhofer ITWM 129
International Journal of Computational Fluid Dynamics 721
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The finite difference method (FDM) implementedin the Geodict software (httpwwwgeodictcom) wasused for reference in solving the flow through theregular geometries and for selected porous materialsamples Within this method a staggered grid was usedwhere the values of velocity and pressure were given atcentre points of the faces and volume of the cubic unitcells respectively The partial differential equations forincompressible Stokes flow are solved by using thelsquoFFF-Stokes solverrsquo based on Fast Fourier TransformThis solver appears to be fast and memory efficient forlarge computations characteristic to the presentapproach (Wiegmann 2007)
Estimate of discretisation error for regular geometries
As stated above a major potential source of un-certainty within the present approach is the numericalerror caused by the finite resolution of the tomographicimages and thus of the numerical grids used Here wewish to estimate the order of magnitude of this error bysolving the flow through regular model porousmaterials with varying values of porosity Solutionsare found for numerical grids of different sizes and theresulting values of permeability coefficient are com-pared also with analytical results To this end we findnumerical solution of flow through regular geometriesthat consist of square array of parallel straight rodsaligned in the z direction at four different porositylevels (06 07 08 and 09) For each such geometryincompressible creeping flow is induced by a small butotherwise arbitrary constant body force r~g in x and zcoordinate directions (see Figure 1) Periodic boundaryconditions were used at the boundary of the computa-tion region in all coordinate directions and no-slipboundary condition was enforced at all solidndashfluidinterfaces Numerical values of the diagonal elementsof the permeability tensor were then calculated fromEquation (2) using the mean flow rate given by thenumerical solution Notice that due to symmetry
reasons the permeability tensor is isotropic in the x-yplane and that z direction is one of its principaldirections The diagonal elements kxx kyy (frac14kxx) andkzz are thus also the principal values of the perme-ability tensor
The solution was found using a high-density gridand a low-density grid For the high-density grid thelength of the unit cell was 400 grid units This was highenough resolution to provide an accurate solution andan adequate independence of the actual grid size Forthe low-density grid the diameter of the rods was 7 gridunits which equals the mean diameter of the actualfibres of the plastic felt sample measured in units ofimage voxel size (see below) Example of solved flowfield using high and low density grids is shown inFigure 2 for mean flow in the x direction
A number of analytical results have been derivedfor permeability of arrays of cylinders with circularcross section and for flow parallel and perpendicular tothe cylinders (see eg Jackson and James (1986) andreferences therein) For flow parallel to the cylinders(z-direction) Drummond and Tahir (1984) propose anapproximative formula
kzzr2 1
4eth1fTHORN lneth1fTHORNKthorn2eth1fTHORN1
2eth1fTHORN2
eth4THORN
where f is porosity r is radius of the cylinders and K isa constant that depends on array geometry For asquare array it has the value 1476 Equation (4) isuseful at least in the porosity range 03 5 f 5 099Sangani and Acrivos (1982) used a unit cell approachto find an exact solution for flow perpendicular to thecylinders (x-y plane) Their solution was given in termsof series expansions where the effect of boundaryconditions must be found numerically This result cannot be expressed in closed form but Sangani andAcrivos provide tabulated numerical results that canbe readily applied in the present case
The computed results for permeability obtainedwith high and low density grids for square array ofcircular cylinders at different values of porosity areshown in Figure 3 together with correspondingtheoretical results The numerical results obtainedusing the high density grid closely agree with thetheoretical predictions The numerical results obtainedwith the low-density grid deviate from the theoreticaland more accurate numerical results Including all thecases analysed the maximum relative deviation be-tween the values of permeability obtained using low-density numerical grid as compared to either theore-tical or high resolution numerical results was less than10 In addition to results shown in Figure 3 similar
Figure 1 Unit cell of a square array of parallel rods withcircular cross-section together with the coordinateconvention used
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results were obtained for hexagonal arrays of circularrods This analysis was also repeated for square andhexagonal arrays of rods with square cross-section Inthis case theoretical results are not available butaccurate numerical solution can be found usingsufficiently dense grid Finally the numerical resultsobtained with LBM were verified independently usingFDM for a subset of regular geometry cases
Permeability of porous materials
We apply the analysis discussed in the previoussections for samples of fibrous porous materials ie
felt made of plastic fibres and hand sheet paper madeof chemical hardwood pulp The actual microscopicpore geometry of the samples was obtained by CXmTThe results will be compared with experimental datafor the same materials The felt material consists ofnearly cylindrical smooth fibres of average diameter 34mm The fibres are curved to some extent and have onlyweak tendency to be oriented in the plane of the feltWithin that plane the orientation is isotropic inaverage The structure of paper and the internal shapeof individual wood fibres are considerably morecomplicated than those of plastic felt Wood fibrestypically include a lumen and a cell wall with flattenedirregular cross-section The wall thickness and theaverage diameter of the fibres were found to beapproximately 4 mm and 20 mm respectively Thefibres are strongly oriented towards the plane of paperbut are also isotropically oriented within that plane(Rolland du Roscoat et al 2007)
The tomographic image of a plastic felt sampleused in this work was obtained using a table-toptomographic scanner (SkyScan 1172) with voxelresolution 484 mm The scanned felt sample was acircular section of diameter 6 mm In order toreproduce the internal structure of paper material incomparable detail considerably higher resolution mustbe used The paper sample used in this work wasscanned at ID19 laboratory at European SynchrotronRadiation Facility (ESRF Grenoble France) with avoxel size of 07 mm In this case the sample was arectangular section of size 14 6 14 mm2 The sam-ples were carefully cut from larger sheets of felt andpaper to avoid any damage to the structure due tocutting The locations from which the samples wereextracted were selected usi ng a grey-scale transillumi-nation image such that the samples would represent
Figure 2 Flow speed solved using high density grid (a) and low density grid (b) for an array of circular cylinders (porosities70 and 71 respectively) The mean flow is in the x direction Red and blue colours indicate high and low flow speedrespectively Available in colour online
Figure 3 Numerical and analytical values of dimensionlesspermeability in three principal directions for square array ofcircular cylinders at different porosity levels Analyticalvalues for flow parallel to the cylinders (z direction) aregiven by Equation (4) and the semianalytical results for flowperpendicular to the cylinders (in x-y plane) by tabulatedresults of Sangani and Acrivos see Tables 1 and 2 in Sanganiand Acrivos (1982)
716 V Koivu et al
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the mean properties (especially density) of the materialas closely as possible Both samples were scannedslightly compressed in the z direction Compressionwas obtained by plastic tubular sample holdersincluding a piston and a screw The size of theoriginal tomographic grey scale images was1240 6 1240 6 277 voxels and 2000 6 2000 6 121voxels for the felt and paper samples respectively Theimage of the felt sample was processed with variance-weighted mean filter (Gonzales and Woods 2001)before thresholding to a binary image For this samplethresholding was rather straightforward due to highquality of the tomographic image and regular shape ofthe plastic fibres The image of the paper sample wasdenoised by nonlinear anisotropic diffusion filter andfurther thresholded using seeded region growingmethod (Rolland du Roscoat et al 2005) Thethickness and porosity of the samples were determinedfrom the filtered and segmented tomographic images(and the same values of thickness were used inexperimental measurement of permeability see below)Porosity was determined as the total proportion ofopen void space in the tomographic subsamples Thevalues of porosity thus found were 616 for the feltand 486 for the paper sample
For numerical flow solution five non-overlappingsubvolumes were extracted from the original tomo-graphic images The dimensions of these subvolumeswere 400 6 400 6 277 pixels (physical size1930 6 1930 6 1341 mm3) and 300 6 300 6 121pixels (physical size 210 6 210 6 85 mm3) for thefelt and paper samples respectively Visualisation ofone of the subvolumes used in numerical analysis forfelt and paper are shown in Figure 4 Numericalsolution of flow through all the subsamples with meanflow in the three coordinate directions was thenobtained With the sizes of subsamples given abovethe total central processing unit (CPU) time requiredfor solution varied between 2 and 6 h in HP Proliant
DL145G2 18 GHz with 8 GB random access memory(RAM) nodes Figure 5 illustrates the flow velocityfields on a single 2D layer inside the subsamples shownin Figure 4 The detailed 3D numerical solution wasthen used to find the values of the diagonal compo-nents of permeability tensor for the subsamples as inthe case of regular geometries
Experimental values of permeability were obtainedby using a permeability measurement device (PMD)constructed originally for measuring transverse perme-ability (kzz) of soft porous sheet-like samples undermechanical compression (Koivu et al 2009a) For thepresent purpose the device was modified such that alsoin-plane measurement (kxx kyy) is possible (Koivuet al 2009b) Both liquids and gases can be used aspermeating fluids In transverse flow measurement acircular sample sheet of diameter 90 mm is placedbetween smooth sintered stainless steel compressionplates having a central area of diameter 60 mm openfor flow For in-plane measurements a rectangularsample sheet of size 70 mm 6 5 mm was sealedbetween solid stainless steel compression plates thatallow lateral flow through the sample The sampleswere then compressed to the same thickness (porosity)as the samples scanned by X-ray tomography Experi-ments were conducted with air flow in order to preventstructure changes due to swelling All measurementswere repeated for five macroscopically identicalsamples
Numerical and experimental results of permeabilityin three orthogonal directions for plastic felt andhardwood paper samples are summarised in Table 1Notice that in this case the values given in Table 1 arenot necessarily the principal values of permeabilitytensor but merely the values of diagonal elements inthe selected coordinate system In this case thenumerical results obtained by the two methods LBMand FDM deviate from each other more than in thecase of regular arrays Most likely this is just an
Figure 4 3D rendering of tomographic images of plastic felt (a) and hardwood paper (b) subsamples
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indication of both methods being less accurate in thepresent complex geometry Notice that the boundaryof the flow domain (pore space) is in this case knownonly to the accuracy of the tomographic imageTherefore using a numerical grid essentially finerthan the voxel matrix of the 3D tomographic digitalimage does not in general improve the overall validityof the solution and of the computed value ofpermeability coefficient It is thus obvious that therelatively coarse cubic grid provided directly by thetomographic images does not allow for very accuratesolution of flow especially near the solid boundariesand in narrow passages Nevertheless the largest
deviation between numerical and experimental resultsis less than 30 This is of the same order ofmagnitude than the overall uncertainty inherent alsoin the used experimental method for thin compressedfibre mats (Koivu et al 2009a)
The most important single source of uncertainty inthe present numerical results most likely is the ratherpoor statistics obtained due to the low number andsmall size of (sub)samples Notice however that eventhough several subsamples were analysed the variationof permeability value thus obtained (and indicated inTable 1) does not necessarily correspond to the totallarge scale variation of the sample materials but merely
Table 1 Numerical and experimental values of permeability in three orthogonal directions (in physical units) for plastic fibresample and hard wood paper sample
Sample
Permeability kjj [m2 E-13]
kxx kyy kzz
LBM FDM Experimental LBM FDM Experimental LBM FDM Experimental
Plasticfibre mat
290 + 10 352 311 + 9 290 + 30 332 304 + 9 270 + 30 305 270 + 5
Hardwoodsample
26 + 03 260 31 + 01 27 + 03 284 31 + 01 076 + 005 085 063 + 001
The numerical results obtained with LBM are given as mean values of results based on five non-overlapping subvolumes and the experimentalresults as mean values of five independent measurements with standard deviation
Figure 5 Numerically solved flow field in plastic felt (a) and hardwood paper (b) Shown is the scaled flow speed on single 2Dcross sections (x-y plane) of the full 3D solution space within the actual pore geometry of material subsamples as obtained by X-ray micro-tomography (see Figure 4) Red and blue colours indicate high and low flow velocity respectively Solid material iscoloured black In (a) mean velocity is in the direction perpendicular to the plane shown (z direction) In (b) mean flow is in theplane shown and from up to down (y direction) Notice also the pure fluid layers visible at the upstream and downstream sides ofthe paper sample Inserts show details of the pore structure and flow velocity pattern with resolution of the numerical grid givenby the tomographic images (484 mm and 07 mm for (a) and (b) respectively) Available in colour online
718 V Koivu et al
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the variation within the yet rather small originalsamples scanned The samples used in experimentsare however much larger and the results are thereforeexpected to better represent mean values (apart frompossible systematic errors remaining)
Discussion
In this work we develop and critically evaluate amethod for finding flow permeability of porousmaterials by numerically solving creeping fluid flowthrough the complex pore space of actual materialsamples Two conceptually different numericalschemes namely LBM and finite difference methodwere used The regular cubic grid used in numericalsolution is based directly on the 3D images of materialsamples obtained by two different states of the art X-ray microtomographic techniques
In many practical cases the main source ofuncertainty within the present approach is the finiteresolution of the tomographic image that provides thenumerical grid This may be due to hardware limita-tions or due to the need of compromising resolutionfor sample size for heterogeneous materials The orderof magnitude of the discretisation error arising fromthe relatively sparse numerical grid used was firstevaluated solving flows through a set of digitallygenerated regular geometries of straight cylinders Thevalues of permeability coefficients were computedusing grids of density comparable to that used withactual materials (relative to cylinderfibre diameter)The results were compared with analytical resultsavailable and with more accurate numerical resultsobtained using grids of sufficiently high density Forthe regular geometries studied the maximum deviationof the results obtained with sparse grid from analyticaland more accurate numerical results was less than10 This perhaps even surprisingly good accuracyobtained for regular geometries indicates that as far asdiscretisation error alone is concerned the methodmay be useful also for actual materials We mayhowever expect the discretisation error to be some-what higher for actual material samples due to theirmore complex pore structure Notice also that addi-tional error may arise due to the finite resolution of thetomographic techniques whereby details of the struc-ture (eg surface roughness of fibres and very narrowpassages in the porous space) may not be reproducedby the images Furthermore noise and other artefactspresent in the original reconstructed images as well asimage conditioning methods used to remove orminimise them may affect the results for eachindividual sample These are the main factors thatcurrently limit the applicability of the method espe-cially for fine-structured materials
In order to assess the overall uncertainty inherentin the computed results we applied the method infinding the diagonal components of permeabilitytensor for two fibrous materials plastic felt andhand-sheet paper under steady mechanical compres-sion and compared the results with correspondingexperimental values for the same materials Themaximum deviation between the numerical resultsand the experimental results was found to be less than30 In addition to numerical uncertainties arisingfrom discretisation and image quality discussed abovethis deviation includes statistical variation arising fromrelatively small size and low number of samples used incomputation In principle the accuracy can beimproved by using higher resolution tomographicimages and better statistics The former conditionmay become feasible with the present rapid develop-ment of X-ray tomographic techniques The latter canbe obtained by larger size or larger number of samplesObviously these improvements come with the cost ofhigher computational effort which however does notappear critical from the point of view of applicabilityof the method
To conclude the method of finding values of flowpermeability of porous materials based on numericallysolving fluid flow through the actual 3D pore geometryof material samples found using X-ray micro-tomo-graphy appears useful especially in cases whereexperimental results are not available or are difficultto obtain An example of such an application is to findthe permeability of various layers in a layer-structuredmaterial for which the layers can not be separated forindependent measurement (Koivu et al 2009b)Another advantage of the method is that it can beused to find all components of the permeability tensorWith the X-ray tomographic techniques presentlyused the overall accuracy of the computed values ofpermeability is moderate for the kind of materials andflow conditions studied here However this uncer-tainty is not essentially larger than the typical totaluncertainty of corresponding experimental results
Here we have demonstrated the method inevaluating a particular macroscopic transport propertyof porous materials the flow permeability Similarprocedure can be applied in other modes of transportsuch as diffusion and heat conduction in heterogeneousmaterials
Acknowledgements
The authors thank Tuomas Turpeinen MSc for hishelp related to the segmentation of tomographicimages and Jarno Alaraudanjoki MSc for thework done for experimental permeability measure-ments The authors also express their gratitude to the
International Journal of Computational Fluid Dynamics 719
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ID19rsquos team of ESRF for help with the acquisition ofthe images and to Sabine Rolland du Roscoat for herenthusiastic help concerning both the acquisition andthe image analysis of the studied samples
References
Aaltosalmi U et al 2004 Numerical analysis of fluid flowthrough fibrous porous materials Journal of Pulp andPaper Science 30 (9) 251ndash255
Andersson AG et al 2009 Flow through a two-scaleporosity material Research Letters in Materials Science2009 (701512) 4 pp
Arns C et al 2004 Virtual permeametry on microtomo-graphic images Journal of Petroleum Science andEngineering 45 41ndash46
Bear J 1972 Dynamics of fluid in porous media New YorkDover Publications inc
Belov EB et al 2003 Modelling of permeability of textilereinforcements lattice Boltzmann method CompositesScience and Technology 64 1069ndash1080
Darcy H 1856 Les Fontaines Publiques de la Ville de DijonParis Victor Dalmont
Drummond JE and Tahir MI 1984 Laminar viscousflow through regular arrays of parallel solid cylindersInternational Journal of Multiphase Flow 10 (5) 515ndash525
Eker E and Akin S 2006 Lattice-Boltzmann simulation offluid flow in synthetic fractures Transport in porousmedia 65 363ndash384
Fourie W et al 2007 The simulation of pore scale fluidflow with real world geometries obtained from X-raycomputed tomography European COMSOL Multiphy-sics Conference 23ndash24 October 2007 Boston GrenobleCOMSOL inc
Frishfelds et al 2009 Flow-induced deformation of non-crimp fabrics during composites manufacturing Proceed-ings in Interdisciplinary Transport Phenomena IV(ITP2009) Volterra ITP-09-52
Ginzburg I 2008 Consistent lattice Boltzmann schemesfor the Brinkman model of porous flow and finiteChapman-Enskog expansion Physical Review E 77066704
Ginzburg I and drsquoHumieres D 2003 Multireflectionboundary conditions for lattice Boltzmann modelsPhysical Review E 68 (6) 066614
Ginzburg I Verhaeghe F and drsquoHumieres D 2008 Two-relaxation-time lattice-boltzmann scheme about para-metrization velocity pressure and mixed boundaryconditions Communications in Computational Physics 3(2) 427ndash478
Gonzales RC and Woods RE 2001 Digital imageprocessing 2nd edn New Jersey Prentice-Hall Inc237ndash241
Hellstrom JGI and Lundstrom TS 2006 Flow throughPorous Media at Moderate Reynolds Number Interna-tional Scientific Colloquim Modelling for MaterialProcessing Riga Latvia Faculty of Physics and Mathe-matics University of Latvia 129ndash134
Hyvaluoma J et al 2006 Simulation of liquid penetrationin paper Physical Review E 73 (3) 036705
Jackson GW and James DF 1986 The permeability offibrous porous media Canadian Journal of ChemicalEngineering 64 364ndash373
Jaganathan S Vahedi Tafreshi H and Pourdeyhimi B2007 A realistic approach for modeling permeabilityof fibrous media 3-D imaging coupled with CFDsimulation Chemical Engineering science 63 (1) 224ndash252
Keehm Y Sternlof K and Mukerji T 2006 Computa-tional estimation of compaction band permeability insandstone Geosciences Journal 10 (4) 499ndash505
Kerwald D 2004 Parallel lattice Boltzmann simulation ofcomplex flow NAFEMS Seminar simulation of complexflows (CFD) ndash applications and trends NiedernhausenWiesbaden
Koivu V Mattila K and Kataja M 2009a A method formeasuring darcian flow permeability of thin fibre matsNordic Pulp and Paper Research Journal 24 (4) 395ndash402
Koivu V et al 2009b Flow permeability of fibrous porousmaterials Micro-tomography and numerical simulationsIn SJ IrsquoAnson ed Advances in pulp and paper researchTransactions on 14th Research Fundamental SymposiumOxford September 14ndash18 2009 Manchester The Pulpand Paper Fundamental Research Society 437ndash454
Koponen A et al 1998 Permeability of three-dimensionalrandom fiber webs Physical Review Letters 80 (4) 716
Kutay ME Aydilek AH and Masad E 2006 Labora-tory validation of lattice Boltzmann method for model-ing pore-scale flow in granular materials ComputationalGeotechnics 33 381ndash395
Levitz P 2005 Toolbox for 3D imaging and modeling ofporous media relationship with transport propertiesCement and Concrete Research 37 351ndash359
Lundstrom TS Frishfelds V and Jakovics A 2004 Astatistical approach to permeability of clustered fibrereinforcements Journal of Composite Materials 38 (13)1137ndash1149
Lundstrom TS Sundlof H and Holmberg JA 2006Modeling of power-law fluid flow through fiber bedsJournal of Composite Materials 40 (3) 283ndash296
Manwart C et al 2002 Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media Physical Review E 66 (1)016702
Martys NS and Hagedorn JG 2002 Multiscale modelingof fluid transport in heterogeneous materials usingdiscrete Boltzmann methods Journal of Materials andStructures 35 (10) 650
Nordlund M et al 2005 Permeability network model fornon-crimp fabrics Composites Part A 37 826ndash835
Nordlund M and Lundstrom TS 2007 Effect ofmulti-scale porosity in local permeability modelling ofnon-crimp fabrics Transport in Porous Media 73 109ndash124
Pan C Luo L-S and Miller CT 2006 An evaluation oflattice Boltzmann schemes for porous medium flowsimulation Computers amp Fluids 35 (8ndash9) 898
Ramaswamy S et al 2004 The 3D structure of fabric andits relationship to liquid and vapor transport Colloidsand Surfaces A Physicochemical and Engineering As-pects 241 323ndash333
Rolland du Roscoat S Bloch J-F and Thibault X 2005Synchrotron Radiation microtomography applied topaper investigation Journal of Physics D AppliedPhysics 38 A78ndashA84
Rolland du Roscoat S et al 2007 Estimation ofmicrostructural properties from synchrotron X-ray microtomography and determination of the REV in papermaterials Acta Materiala 55 (8) 2841ndash2850
720 V Koivu et al
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
Saar MO and Manga M 1999 Permeability-porosityrelationship in vesicular basalts Geophysical ResearchLetters 26 (1) 111ndash114
Sangani AS and Acrivos A 1982 Slow flow past periodicarrays of cylinders with application to heat transferInternational Journal of Multiphase Flow 8 193ndash206
Succi S 2001 The lattice Boltzmann equation for fluiddynamics and beyond Gloucestershire Clarendon Press
Succi S Foti E and Higuera F 1989 Three-dimensionalflows in complex geometries with the lattice Boltzmannmethod Europhysics Letters 10 (5) 432ndash438
Thibault X and Bloch J-F 2002 Structural analysis of X-ray microtomography of a strained nonwoven paper-maker felt Textile Research Journal 72 (6) 480ndash485
Verleye B et al 2005 Computation of permeability oftextile reinforcements Proceedings in Scientific Computa-tion IMACS 2005 Paris France
Verleye B et al 2007 Permeability of textile reinforce-ments simulation influence of shear and validationComposite Science and Technology 68 (1) 2804ndash2810
Vidal D et al 2008 Effect od particle size distribution andpacking compression on fluid permeability as predictedby lattice-Boltzmann simulations Computers andChemical Engineering 33 (1) 256ndash266
White JA Borja IR and Fredrich JT 2006 Calculat-ing the effective permeability of sandstone with multi-scale lattice-Boltzmannfinite element simulations ActaGeotechnica 1 195ndash209
Wiegman A et al 2007 Computation of the permeabilityof porous materials from their microstructure by FFT-stokes Report of the Fraunhofer ITWM 129
International Journal of Computational Fluid Dynamics 721
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results were obtained for hexagonal arrays of circularrods This analysis was also repeated for square andhexagonal arrays of rods with square cross-section Inthis case theoretical results are not available butaccurate numerical solution can be found usingsufficiently dense grid Finally the numerical resultsobtained with LBM were verified independently usingFDM for a subset of regular geometry cases
Permeability of porous materials
We apply the analysis discussed in the previoussections for samples of fibrous porous materials ie
felt made of plastic fibres and hand sheet paper madeof chemical hardwood pulp The actual microscopicpore geometry of the samples was obtained by CXmTThe results will be compared with experimental datafor the same materials The felt material consists ofnearly cylindrical smooth fibres of average diameter 34mm The fibres are curved to some extent and have onlyweak tendency to be oriented in the plane of the feltWithin that plane the orientation is isotropic inaverage The structure of paper and the internal shapeof individual wood fibres are considerably morecomplicated than those of plastic felt Wood fibrestypically include a lumen and a cell wall with flattenedirregular cross-section The wall thickness and theaverage diameter of the fibres were found to beapproximately 4 mm and 20 mm respectively Thefibres are strongly oriented towards the plane of paperbut are also isotropically oriented within that plane(Rolland du Roscoat et al 2007)
The tomographic image of a plastic felt sampleused in this work was obtained using a table-toptomographic scanner (SkyScan 1172) with voxelresolution 484 mm The scanned felt sample was acircular section of diameter 6 mm In order toreproduce the internal structure of paper material incomparable detail considerably higher resolution mustbe used The paper sample used in this work wasscanned at ID19 laboratory at European SynchrotronRadiation Facility (ESRF Grenoble France) with avoxel size of 07 mm In this case the sample was arectangular section of size 14 6 14 mm2 The sam-ples were carefully cut from larger sheets of felt andpaper to avoid any damage to the structure due tocutting The locations from which the samples wereextracted were selected usi ng a grey-scale transillumi-nation image such that the samples would represent
Figure 2 Flow speed solved using high density grid (a) and low density grid (b) for an array of circular cylinders (porosities70 and 71 respectively) The mean flow is in the x direction Red and blue colours indicate high and low flow speedrespectively Available in colour online
Figure 3 Numerical and analytical values of dimensionlesspermeability in three principal directions for square array ofcircular cylinders at different porosity levels Analyticalvalues for flow parallel to the cylinders (z direction) aregiven by Equation (4) and the semianalytical results for flowperpendicular to the cylinders (in x-y plane) by tabulatedresults of Sangani and Acrivos see Tables 1 and 2 in Sanganiand Acrivos (1982)
716 V Koivu et al
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the mean properties (especially density) of the materialas closely as possible Both samples were scannedslightly compressed in the z direction Compressionwas obtained by plastic tubular sample holdersincluding a piston and a screw The size of theoriginal tomographic grey scale images was1240 6 1240 6 277 voxels and 2000 6 2000 6 121voxels for the felt and paper samples respectively Theimage of the felt sample was processed with variance-weighted mean filter (Gonzales and Woods 2001)before thresholding to a binary image For this samplethresholding was rather straightforward due to highquality of the tomographic image and regular shape ofthe plastic fibres The image of the paper sample wasdenoised by nonlinear anisotropic diffusion filter andfurther thresholded using seeded region growingmethod (Rolland du Roscoat et al 2005) Thethickness and porosity of the samples were determinedfrom the filtered and segmented tomographic images(and the same values of thickness were used inexperimental measurement of permeability see below)Porosity was determined as the total proportion ofopen void space in the tomographic subsamples Thevalues of porosity thus found were 616 for the feltand 486 for the paper sample
For numerical flow solution five non-overlappingsubvolumes were extracted from the original tomo-graphic images The dimensions of these subvolumeswere 400 6 400 6 277 pixels (physical size1930 6 1930 6 1341 mm3) and 300 6 300 6 121pixels (physical size 210 6 210 6 85 mm3) for thefelt and paper samples respectively Visualisation ofone of the subvolumes used in numerical analysis forfelt and paper are shown in Figure 4 Numericalsolution of flow through all the subsamples with meanflow in the three coordinate directions was thenobtained With the sizes of subsamples given abovethe total central processing unit (CPU) time requiredfor solution varied between 2 and 6 h in HP Proliant
DL145G2 18 GHz with 8 GB random access memory(RAM) nodes Figure 5 illustrates the flow velocityfields on a single 2D layer inside the subsamples shownin Figure 4 The detailed 3D numerical solution wasthen used to find the values of the diagonal compo-nents of permeability tensor for the subsamples as inthe case of regular geometries
Experimental values of permeability were obtainedby using a permeability measurement device (PMD)constructed originally for measuring transverse perme-ability (kzz) of soft porous sheet-like samples undermechanical compression (Koivu et al 2009a) For thepresent purpose the device was modified such that alsoin-plane measurement (kxx kyy) is possible (Koivuet al 2009b) Both liquids and gases can be used aspermeating fluids In transverse flow measurement acircular sample sheet of diameter 90 mm is placedbetween smooth sintered stainless steel compressionplates having a central area of diameter 60 mm openfor flow For in-plane measurements a rectangularsample sheet of size 70 mm 6 5 mm was sealedbetween solid stainless steel compression plates thatallow lateral flow through the sample The sampleswere then compressed to the same thickness (porosity)as the samples scanned by X-ray tomography Experi-ments were conducted with air flow in order to preventstructure changes due to swelling All measurementswere repeated for five macroscopically identicalsamples
Numerical and experimental results of permeabilityin three orthogonal directions for plastic felt andhardwood paper samples are summarised in Table 1Notice that in this case the values given in Table 1 arenot necessarily the principal values of permeabilitytensor but merely the values of diagonal elements inthe selected coordinate system In this case thenumerical results obtained by the two methods LBMand FDM deviate from each other more than in thecase of regular arrays Most likely this is just an
Figure 4 3D rendering of tomographic images of plastic felt (a) and hardwood paper (b) subsamples
International Journal of Computational Fluid Dynamics 717
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indication of both methods being less accurate in thepresent complex geometry Notice that the boundaryof the flow domain (pore space) is in this case knownonly to the accuracy of the tomographic imageTherefore using a numerical grid essentially finerthan the voxel matrix of the 3D tomographic digitalimage does not in general improve the overall validityof the solution and of the computed value ofpermeability coefficient It is thus obvious that therelatively coarse cubic grid provided directly by thetomographic images does not allow for very accuratesolution of flow especially near the solid boundariesand in narrow passages Nevertheless the largest
deviation between numerical and experimental resultsis less than 30 This is of the same order ofmagnitude than the overall uncertainty inherent alsoin the used experimental method for thin compressedfibre mats (Koivu et al 2009a)
The most important single source of uncertainty inthe present numerical results most likely is the ratherpoor statistics obtained due to the low number andsmall size of (sub)samples Notice however that eventhough several subsamples were analysed the variationof permeability value thus obtained (and indicated inTable 1) does not necessarily correspond to the totallarge scale variation of the sample materials but merely
Table 1 Numerical and experimental values of permeability in three orthogonal directions (in physical units) for plastic fibresample and hard wood paper sample
Sample
Permeability kjj [m2 E-13]
kxx kyy kzz
LBM FDM Experimental LBM FDM Experimental LBM FDM Experimental
Plasticfibre mat
290 + 10 352 311 + 9 290 + 30 332 304 + 9 270 + 30 305 270 + 5
Hardwoodsample
26 + 03 260 31 + 01 27 + 03 284 31 + 01 076 + 005 085 063 + 001
The numerical results obtained with LBM are given as mean values of results based on five non-overlapping subvolumes and the experimentalresults as mean values of five independent measurements with standard deviation
Figure 5 Numerically solved flow field in plastic felt (a) and hardwood paper (b) Shown is the scaled flow speed on single 2Dcross sections (x-y plane) of the full 3D solution space within the actual pore geometry of material subsamples as obtained by X-ray micro-tomography (see Figure 4) Red and blue colours indicate high and low flow velocity respectively Solid material iscoloured black In (a) mean velocity is in the direction perpendicular to the plane shown (z direction) In (b) mean flow is in theplane shown and from up to down (y direction) Notice also the pure fluid layers visible at the upstream and downstream sides ofthe paper sample Inserts show details of the pore structure and flow velocity pattern with resolution of the numerical grid givenby the tomographic images (484 mm and 07 mm for (a) and (b) respectively) Available in colour online
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the variation within the yet rather small originalsamples scanned The samples used in experimentsare however much larger and the results are thereforeexpected to better represent mean values (apart frompossible systematic errors remaining)
Discussion
In this work we develop and critically evaluate amethod for finding flow permeability of porousmaterials by numerically solving creeping fluid flowthrough the complex pore space of actual materialsamples Two conceptually different numericalschemes namely LBM and finite difference methodwere used The regular cubic grid used in numericalsolution is based directly on the 3D images of materialsamples obtained by two different states of the art X-ray microtomographic techniques
In many practical cases the main source ofuncertainty within the present approach is the finiteresolution of the tomographic image that provides thenumerical grid This may be due to hardware limita-tions or due to the need of compromising resolutionfor sample size for heterogeneous materials The orderof magnitude of the discretisation error arising fromthe relatively sparse numerical grid used was firstevaluated solving flows through a set of digitallygenerated regular geometries of straight cylinders Thevalues of permeability coefficients were computedusing grids of density comparable to that used withactual materials (relative to cylinderfibre diameter)The results were compared with analytical resultsavailable and with more accurate numerical resultsobtained using grids of sufficiently high density Forthe regular geometries studied the maximum deviationof the results obtained with sparse grid from analyticaland more accurate numerical results was less than10 This perhaps even surprisingly good accuracyobtained for regular geometries indicates that as far asdiscretisation error alone is concerned the methodmay be useful also for actual materials We mayhowever expect the discretisation error to be some-what higher for actual material samples due to theirmore complex pore structure Notice also that addi-tional error may arise due to the finite resolution of thetomographic techniques whereby details of the struc-ture (eg surface roughness of fibres and very narrowpassages in the porous space) may not be reproducedby the images Furthermore noise and other artefactspresent in the original reconstructed images as well asimage conditioning methods used to remove orminimise them may affect the results for eachindividual sample These are the main factors thatcurrently limit the applicability of the method espe-cially for fine-structured materials
In order to assess the overall uncertainty inherentin the computed results we applied the method infinding the diagonal components of permeabilitytensor for two fibrous materials plastic felt andhand-sheet paper under steady mechanical compres-sion and compared the results with correspondingexperimental values for the same materials Themaximum deviation between the numerical resultsand the experimental results was found to be less than30 In addition to numerical uncertainties arisingfrom discretisation and image quality discussed abovethis deviation includes statistical variation arising fromrelatively small size and low number of samples used incomputation In principle the accuracy can beimproved by using higher resolution tomographicimages and better statistics The former conditionmay become feasible with the present rapid develop-ment of X-ray tomographic techniques The latter canbe obtained by larger size or larger number of samplesObviously these improvements come with the cost ofhigher computational effort which however does notappear critical from the point of view of applicabilityof the method
To conclude the method of finding values of flowpermeability of porous materials based on numericallysolving fluid flow through the actual 3D pore geometryof material samples found using X-ray micro-tomo-graphy appears useful especially in cases whereexperimental results are not available or are difficultto obtain An example of such an application is to findthe permeability of various layers in a layer-structuredmaterial for which the layers can not be separated forindependent measurement (Koivu et al 2009b)Another advantage of the method is that it can beused to find all components of the permeability tensorWith the X-ray tomographic techniques presentlyused the overall accuracy of the computed values ofpermeability is moderate for the kind of materials andflow conditions studied here However this uncer-tainty is not essentially larger than the typical totaluncertainty of corresponding experimental results
Here we have demonstrated the method inevaluating a particular macroscopic transport propertyof porous materials the flow permeability Similarprocedure can be applied in other modes of transportsuch as diffusion and heat conduction in heterogeneousmaterials
Acknowledgements
The authors thank Tuomas Turpeinen MSc for hishelp related to the segmentation of tomographicimages and Jarno Alaraudanjoki MSc for thework done for experimental permeability measure-ments The authors also express their gratitude to the
International Journal of Computational Fluid Dynamics 719
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
ID19rsquos team of ESRF for help with the acquisition ofthe images and to Sabine Rolland du Roscoat for herenthusiastic help concerning both the acquisition andthe image analysis of the studied samples
References
Aaltosalmi U et al 2004 Numerical analysis of fluid flowthrough fibrous porous materials Journal of Pulp andPaper Science 30 (9) 251ndash255
Andersson AG et al 2009 Flow through a two-scaleporosity material Research Letters in Materials Science2009 (701512) 4 pp
Arns C et al 2004 Virtual permeametry on microtomo-graphic images Journal of Petroleum Science andEngineering 45 41ndash46
Bear J 1972 Dynamics of fluid in porous media New YorkDover Publications inc
Belov EB et al 2003 Modelling of permeability of textilereinforcements lattice Boltzmann method CompositesScience and Technology 64 1069ndash1080
Darcy H 1856 Les Fontaines Publiques de la Ville de DijonParis Victor Dalmont
Drummond JE and Tahir MI 1984 Laminar viscousflow through regular arrays of parallel solid cylindersInternational Journal of Multiphase Flow 10 (5) 515ndash525
Eker E and Akin S 2006 Lattice-Boltzmann simulation offluid flow in synthetic fractures Transport in porousmedia 65 363ndash384
Fourie W et al 2007 The simulation of pore scale fluidflow with real world geometries obtained from X-raycomputed tomography European COMSOL Multiphy-sics Conference 23ndash24 October 2007 Boston GrenobleCOMSOL inc
Frishfelds et al 2009 Flow-induced deformation of non-crimp fabrics during composites manufacturing Proceed-ings in Interdisciplinary Transport Phenomena IV(ITP2009) Volterra ITP-09-52
Ginzburg I 2008 Consistent lattice Boltzmann schemesfor the Brinkman model of porous flow and finiteChapman-Enskog expansion Physical Review E 77066704
Ginzburg I and drsquoHumieres D 2003 Multireflectionboundary conditions for lattice Boltzmann modelsPhysical Review E 68 (6) 066614
Ginzburg I Verhaeghe F and drsquoHumieres D 2008 Two-relaxation-time lattice-boltzmann scheme about para-metrization velocity pressure and mixed boundaryconditions Communications in Computational Physics 3(2) 427ndash478
Gonzales RC and Woods RE 2001 Digital imageprocessing 2nd edn New Jersey Prentice-Hall Inc237ndash241
Hellstrom JGI and Lundstrom TS 2006 Flow throughPorous Media at Moderate Reynolds Number Interna-tional Scientific Colloquim Modelling for MaterialProcessing Riga Latvia Faculty of Physics and Mathe-matics University of Latvia 129ndash134
Hyvaluoma J et al 2006 Simulation of liquid penetrationin paper Physical Review E 73 (3) 036705
Jackson GW and James DF 1986 The permeability offibrous porous media Canadian Journal of ChemicalEngineering 64 364ndash373
Jaganathan S Vahedi Tafreshi H and Pourdeyhimi B2007 A realistic approach for modeling permeabilityof fibrous media 3-D imaging coupled with CFDsimulation Chemical Engineering science 63 (1) 224ndash252
Keehm Y Sternlof K and Mukerji T 2006 Computa-tional estimation of compaction band permeability insandstone Geosciences Journal 10 (4) 499ndash505
Kerwald D 2004 Parallel lattice Boltzmann simulation ofcomplex flow NAFEMS Seminar simulation of complexflows (CFD) ndash applications and trends NiedernhausenWiesbaden
Koivu V Mattila K and Kataja M 2009a A method formeasuring darcian flow permeability of thin fibre matsNordic Pulp and Paper Research Journal 24 (4) 395ndash402
Koivu V et al 2009b Flow permeability of fibrous porousmaterials Micro-tomography and numerical simulationsIn SJ IrsquoAnson ed Advances in pulp and paper researchTransactions on 14th Research Fundamental SymposiumOxford September 14ndash18 2009 Manchester The Pulpand Paper Fundamental Research Society 437ndash454
Koponen A et al 1998 Permeability of three-dimensionalrandom fiber webs Physical Review Letters 80 (4) 716
Kutay ME Aydilek AH and Masad E 2006 Labora-tory validation of lattice Boltzmann method for model-ing pore-scale flow in granular materials ComputationalGeotechnics 33 381ndash395
Levitz P 2005 Toolbox for 3D imaging and modeling ofporous media relationship with transport propertiesCement and Concrete Research 37 351ndash359
Lundstrom TS Frishfelds V and Jakovics A 2004 Astatistical approach to permeability of clustered fibrereinforcements Journal of Composite Materials 38 (13)1137ndash1149
Lundstrom TS Sundlof H and Holmberg JA 2006Modeling of power-law fluid flow through fiber bedsJournal of Composite Materials 40 (3) 283ndash296
Manwart C et al 2002 Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media Physical Review E 66 (1)016702
Martys NS and Hagedorn JG 2002 Multiscale modelingof fluid transport in heterogeneous materials usingdiscrete Boltzmann methods Journal of Materials andStructures 35 (10) 650
Nordlund M et al 2005 Permeability network model fornon-crimp fabrics Composites Part A 37 826ndash835
Nordlund M and Lundstrom TS 2007 Effect ofmulti-scale porosity in local permeability modelling ofnon-crimp fabrics Transport in Porous Media 73 109ndash124
Pan C Luo L-S and Miller CT 2006 An evaluation oflattice Boltzmann schemes for porous medium flowsimulation Computers amp Fluids 35 (8ndash9) 898
Ramaswamy S et al 2004 The 3D structure of fabric andits relationship to liquid and vapor transport Colloidsand Surfaces A Physicochemical and Engineering As-pects 241 323ndash333
Rolland du Roscoat S Bloch J-F and Thibault X 2005Synchrotron Radiation microtomography applied topaper investigation Journal of Physics D AppliedPhysics 38 A78ndashA84
Rolland du Roscoat S et al 2007 Estimation ofmicrostructural properties from synchrotron X-ray microtomography and determination of the REV in papermaterials Acta Materiala 55 (8) 2841ndash2850
720 V Koivu et al
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
Saar MO and Manga M 1999 Permeability-porosityrelationship in vesicular basalts Geophysical ResearchLetters 26 (1) 111ndash114
Sangani AS and Acrivos A 1982 Slow flow past periodicarrays of cylinders with application to heat transferInternational Journal of Multiphase Flow 8 193ndash206
Succi S 2001 The lattice Boltzmann equation for fluiddynamics and beyond Gloucestershire Clarendon Press
Succi S Foti E and Higuera F 1989 Three-dimensionalflows in complex geometries with the lattice Boltzmannmethod Europhysics Letters 10 (5) 432ndash438
Thibault X and Bloch J-F 2002 Structural analysis of X-ray microtomography of a strained nonwoven paper-maker felt Textile Research Journal 72 (6) 480ndash485
Verleye B et al 2005 Computation of permeability oftextile reinforcements Proceedings in Scientific Computa-tion IMACS 2005 Paris France
Verleye B et al 2007 Permeability of textile reinforce-ments simulation influence of shear and validationComposite Science and Technology 68 (1) 2804ndash2810
Vidal D et al 2008 Effect od particle size distribution andpacking compression on fluid permeability as predictedby lattice-Boltzmann simulations Computers andChemical Engineering 33 (1) 256ndash266
White JA Borja IR and Fredrich JT 2006 Calculat-ing the effective permeability of sandstone with multi-scale lattice-Boltzmannfinite element simulations ActaGeotechnica 1 195ndash209
Wiegman A et al 2007 Computation of the permeabilityof porous materials from their microstructure by FFT-stokes Report of the Fraunhofer ITWM 129
International Journal of Computational Fluid Dynamics 721
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
the mean properties (especially density) of the materialas closely as possible Both samples were scannedslightly compressed in the z direction Compressionwas obtained by plastic tubular sample holdersincluding a piston and a screw The size of theoriginal tomographic grey scale images was1240 6 1240 6 277 voxels and 2000 6 2000 6 121voxels for the felt and paper samples respectively Theimage of the felt sample was processed with variance-weighted mean filter (Gonzales and Woods 2001)before thresholding to a binary image For this samplethresholding was rather straightforward due to highquality of the tomographic image and regular shape ofthe plastic fibres The image of the paper sample wasdenoised by nonlinear anisotropic diffusion filter andfurther thresholded using seeded region growingmethod (Rolland du Roscoat et al 2005) Thethickness and porosity of the samples were determinedfrom the filtered and segmented tomographic images(and the same values of thickness were used inexperimental measurement of permeability see below)Porosity was determined as the total proportion ofopen void space in the tomographic subsamples Thevalues of porosity thus found were 616 for the feltand 486 for the paper sample
For numerical flow solution five non-overlappingsubvolumes were extracted from the original tomo-graphic images The dimensions of these subvolumeswere 400 6 400 6 277 pixels (physical size1930 6 1930 6 1341 mm3) and 300 6 300 6 121pixels (physical size 210 6 210 6 85 mm3) for thefelt and paper samples respectively Visualisation ofone of the subvolumes used in numerical analysis forfelt and paper are shown in Figure 4 Numericalsolution of flow through all the subsamples with meanflow in the three coordinate directions was thenobtained With the sizes of subsamples given abovethe total central processing unit (CPU) time requiredfor solution varied between 2 and 6 h in HP Proliant
DL145G2 18 GHz with 8 GB random access memory(RAM) nodes Figure 5 illustrates the flow velocityfields on a single 2D layer inside the subsamples shownin Figure 4 The detailed 3D numerical solution wasthen used to find the values of the diagonal compo-nents of permeability tensor for the subsamples as inthe case of regular geometries
Experimental values of permeability were obtainedby using a permeability measurement device (PMD)constructed originally for measuring transverse perme-ability (kzz) of soft porous sheet-like samples undermechanical compression (Koivu et al 2009a) For thepresent purpose the device was modified such that alsoin-plane measurement (kxx kyy) is possible (Koivuet al 2009b) Both liquids and gases can be used aspermeating fluids In transverse flow measurement acircular sample sheet of diameter 90 mm is placedbetween smooth sintered stainless steel compressionplates having a central area of diameter 60 mm openfor flow For in-plane measurements a rectangularsample sheet of size 70 mm 6 5 mm was sealedbetween solid stainless steel compression plates thatallow lateral flow through the sample The sampleswere then compressed to the same thickness (porosity)as the samples scanned by X-ray tomography Experi-ments were conducted with air flow in order to preventstructure changes due to swelling All measurementswere repeated for five macroscopically identicalsamples
Numerical and experimental results of permeabilityin three orthogonal directions for plastic felt andhardwood paper samples are summarised in Table 1Notice that in this case the values given in Table 1 arenot necessarily the principal values of permeabilitytensor but merely the values of diagonal elements inthe selected coordinate system In this case thenumerical results obtained by the two methods LBMand FDM deviate from each other more than in thecase of regular arrays Most likely this is just an
Figure 4 3D rendering of tomographic images of plastic felt (a) and hardwood paper (b) subsamples
International Journal of Computational Fluid Dynamics 717
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
indication of both methods being less accurate in thepresent complex geometry Notice that the boundaryof the flow domain (pore space) is in this case knownonly to the accuracy of the tomographic imageTherefore using a numerical grid essentially finerthan the voxel matrix of the 3D tomographic digitalimage does not in general improve the overall validityof the solution and of the computed value ofpermeability coefficient It is thus obvious that therelatively coarse cubic grid provided directly by thetomographic images does not allow for very accuratesolution of flow especially near the solid boundariesand in narrow passages Nevertheless the largest
deviation between numerical and experimental resultsis less than 30 This is of the same order ofmagnitude than the overall uncertainty inherent alsoin the used experimental method for thin compressedfibre mats (Koivu et al 2009a)
The most important single source of uncertainty inthe present numerical results most likely is the ratherpoor statistics obtained due to the low number andsmall size of (sub)samples Notice however that eventhough several subsamples were analysed the variationof permeability value thus obtained (and indicated inTable 1) does not necessarily correspond to the totallarge scale variation of the sample materials but merely
Table 1 Numerical and experimental values of permeability in three orthogonal directions (in physical units) for plastic fibresample and hard wood paper sample
Sample
Permeability kjj [m2 E-13]
kxx kyy kzz
LBM FDM Experimental LBM FDM Experimental LBM FDM Experimental
Plasticfibre mat
290 + 10 352 311 + 9 290 + 30 332 304 + 9 270 + 30 305 270 + 5
Hardwoodsample
26 + 03 260 31 + 01 27 + 03 284 31 + 01 076 + 005 085 063 + 001
The numerical results obtained with LBM are given as mean values of results based on five non-overlapping subvolumes and the experimentalresults as mean values of five independent measurements with standard deviation
Figure 5 Numerically solved flow field in plastic felt (a) and hardwood paper (b) Shown is the scaled flow speed on single 2Dcross sections (x-y plane) of the full 3D solution space within the actual pore geometry of material subsamples as obtained by X-ray micro-tomography (see Figure 4) Red and blue colours indicate high and low flow velocity respectively Solid material iscoloured black In (a) mean velocity is in the direction perpendicular to the plane shown (z direction) In (b) mean flow is in theplane shown and from up to down (y direction) Notice also the pure fluid layers visible at the upstream and downstream sides ofthe paper sample Inserts show details of the pore structure and flow velocity pattern with resolution of the numerical grid givenby the tomographic images (484 mm and 07 mm for (a) and (b) respectively) Available in colour online
718 V Koivu et al
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
the variation within the yet rather small originalsamples scanned The samples used in experimentsare however much larger and the results are thereforeexpected to better represent mean values (apart frompossible systematic errors remaining)
Discussion
In this work we develop and critically evaluate amethod for finding flow permeability of porousmaterials by numerically solving creeping fluid flowthrough the complex pore space of actual materialsamples Two conceptually different numericalschemes namely LBM and finite difference methodwere used The regular cubic grid used in numericalsolution is based directly on the 3D images of materialsamples obtained by two different states of the art X-ray microtomographic techniques
In many practical cases the main source ofuncertainty within the present approach is the finiteresolution of the tomographic image that provides thenumerical grid This may be due to hardware limita-tions or due to the need of compromising resolutionfor sample size for heterogeneous materials The orderof magnitude of the discretisation error arising fromthe relatively sparse numerical grid used was firstevaluated solving flows through a set of digitallygenerated regular geometries of straight cylinders Thevalues of permeability coefficients were computedusing grids of density comparable to that used withactual materials (relative to cylinderfibre diameter)The results were compared with analytical resultsavailable and with more accurate numerical resultsobtained using grids of sufficiently high density Forthe regular geometries studied the maximum deviationof the results obtained with sparse grid from analyticaland more accurate numerical results was less than10 This perhaps even surprisingly good accuracyobtained for regular geometries indicates that as far asdiscretisation error alone is concerned the methodmay be useful also for actual materials We mayhowever expect the discretisation error to be some-what higher for actual material samples due to theirmore complex pore structure Notice also that addi-tional error may arise due to the finite resolution of thetomographic techniques whereby details of the struc-ture (eg surface roughness of fibres and very narrowpassages in the porous space) may not be reproducedby the images Furthermore noise and other artefactspresent in the original reconstructed images as well asimage conditioning methods used to remove orminimise them may affect the results for eachindividual sample These are the main factors thatcurrently limit the applicability of the method espe-cially for fine-structured materials
In order to assess the overall uncertainty inherentin the computed results we applied the method infinding the diagonal components of permeabilitytensor for two fibrous materials plastic felt andhand-sheet paper under steady mechanical compres-sion and compared the results with correspondingexperimental values for the same materials Themaximum deviation between the numerical resultsand the experimental results was found to be less than30 In addition to numerical uncertainties arisingfrom discretisation and image quality discussed abovethis deviation includes statistical variation arising fromrelatively small size and low number of samples used incomputation In principle the accuracy can beimproved by using higher resolution tomographicimages and better statistics The former conditionmay become feasible with the present rapid develop-ment of X-ray tomographic techniques The latter canbe obtained by larger size or larger number of samplesObviously these improvements come with the cost ofhigher computational effort which however does notappear critical from the point of view of applicabilityof the method
To conclude the method of finding values of flowpermeability of porous materials based on numericallysolving fluid flow through the actual 3D pore geometryof material samples found using X-ray micro-tomo-graphy appears useful especially in cases whereexperimental results are not available or are difficultto obtain An example of such an application is to findthe permeability of various layers in a layer-structuredmaterial for which the layers can not be separated forindependent measurement (Koivu et al 2009b)Another advantage of the method is that it can beused to find all components of the permeability tensorWith the X-ray tomographic techniques presentlyused the overall accuracy of the computed values ofpermeability is moderate for the kind of materials andflow conditions studied here However this uncer-tainty is not essentially larger than the typical totaluncertainty of corresponding experimental results
Here we have demonstrated the method inevaluating a particular macroscopic transport propertyof porous materials the flow permeability Similarprocedure can be applied in other modes of transportsuch as diffusion and heat conduction in heterogeneousmaterials
Acknowledgements
The authors thank Tuomas Turpeinen MSc for hishelp related to the segmentation of tomographicimages and Jarno Alaraudanjoki MSc for thework done for experimental permeability measure-ments The authors also express their gratitude to the
International Journal of Computational Fluid Dynamics 719
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
ID19rsquos team of ESRF for help with the acquisition ofthe images and to Sabine Rolland du Roscoat for herenthusiastic help concerning both the acquisition andthe image analysis of the studied samples
References
Aaltosalmi U et al 2004 Numerical analysis of fluid flowthrough fibrous porous materials Journal of Pulp andPaper Science 30 (9) 251ndash255
Andersson AG et al 2009 Flow through a two-scaleporosity material Research Letters in Materials Science2009 (701512) 4 pp
Arns C et al 2004 Virtual permeametry on microtomo-graphic images Journal of Petroleum Science andEngineering 45 41ndash46
Bear J 1972 Dynamics of fluid in porous media New YorkDover Publications inc
Belov EB et al 2003 Modelling of permeability of textilereinforcements lattice Boltzmann method CompositesScience and Technology 64 1069ndash1080
Darcy H 1856 Les Fontaines Publiques de la Ville de DijonParis Victor Dalmont
Drummond JE and Tahir MI 1984 Laminar viscousflow through regular arrays of parallel solid cylindersInternational Journal of Multiphase Flow 10 (5) 515ndash525
Eker E and Akin S 2006 Lattice-Boltzmann simulation offluid flow in synthetic fractures Transport in porousmedia 65 363ndash384
Fourie W et al 2007 The simulation of pore scale fluidflow with real world geometries obtained from X-raycomputed tomography European COMSOL Multiphy-sics Conference 23ndash24 October 2007 Boston GrenobleCOMSOL inc
Frishfelds et al 2009 Flow-induced deformation of non-crimp fabrics during composites manufacturing Proceed-ings in Interdisciplinary Transport Phenomena IV(ITP2009) Volterra ITP-09-52
Ginzburg I 2008 Consistent lattice Boltzmann schemesfor the Brinkman model of porous flow and finiteChapman-Enskog expansion Physical Review E 77066704
Ginzburg I and drsquoHumieres D 2003 Multireflectionboundary conditions for lattice Boltzmann modelsPhysical Review E 68 (6) 066614
Ginzburg I Verhaeghe F and drsquoHumieres D 2008 Two-relaxation-time lattice-boltzmann scheme about para-metrization velocity pressure and mixed boundaryconditions Communications in Computational Physics 3(2) 427ndash478
Gonzales RC and Woods RE 2001 Digital imageprocessing 2nd edn New Jersey Prentice-Hall Inc237ndash241
Hellstrom JGI and Lundstrom TS 2006 Flow throughPorous Media at Moderate Reynolds Number Interna-tional Scientific Colloquim Modelling for MaterialProcessing Riga Latvia Faculty of Physics and Mathe-matics University of Latvia 129ndash134
Hyvaluoma J et al 2006 Simulation of liquid penetrationin paper Physical Review E 73 (3) 036705
Jackson GW and James DF 1986 The permeability offibrous porous media Canadian Journal of ChemicalEngineering 64 364ndash373
Jaganathan S Vahedi Tafreshi H and Pourdeyhimi B2007 A realistic approach for modeling permeabilityof fibrous media 3-D imaging coupled with CFDsimulation Chemical Engineering science 63 (1) 224ndash252
Keehm Y Sternlof K and Mukerji T 2006 Computa-tional estimation of compaction band permeability insandstone Geosciences Journal 10 (4) 499ndash505
Kerwald D 2004 Parallel lattice Boltzmann simulation ofcomplex flow NAFEMS Seminar simulation of complexflows (CFD) ndash applications and trends NiedernhausenWiesbaden
Koivu V Mattila K and Kataja M 2009a A method formeasuring darcian flow permeability of thin fibre matsNordic Pulp and Paper Research Journal 24 (4) 395ndash402
Koivu V et al 2009b Flow permeability of fibrous porousmaterials Micro-tomography and numerical simulationsIn SJ IrsquoAnson ed Advances in pulp and paper researchTransactions on 14th Research Fundamental SymposiumOxford September 14ndash18 2009 Manchester The Pulpand Paper Fundamental Research Society 437ndash454
Koponen A et al 1998 Permeability of three-dimensionalrandom fiber webs Physical Review Letters 80 (4) 716
Kutay ME Aydilek AH and Masad E 2006 Labora-tory validation of lattice Boltzmann method for model-ing pore-scale flow in granular materials ComputationalGeotechnics 33 381ndash395
Levitz P 2005 Toolbox for 3D imaging and modeling ofporous media relationship with transport propertiesCement and Concrete Research 37 351ndash359
Lundstrom TS Frishfelds V and Jakovics A 2004 Astatistical approach to permeability of clustered fibrereinforcements Journal of Composite Materials 38 (13)1137ndash1149
Lundstrom TS Sundlof H and Holmberg JA 2006Modeling of power-law fluid flow through fiber bedsJournal of Composite Materials 40 (3) 283ndash296
Manwart C et al 2002 Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media Physical Review E 66 (1)016702
Martys NS and Hagedorn JG 2002 Multiscale modelingof fluid transport in heterogeneous materials usingdiscrete Boltzmann methods Journal of Materials andStructures 35 (10) 650
Nordlund M et al 2005 Permeability network model fornon-crimp fabrics Composites Part A 37 826ndash835
Nordlund M and Lundstrom TS 2007 Effect ofmulti-scale porosity in local permeability modelling ofnon-crimp fabrics Transport in Porous Media 73 109ndash124
Pan C Luo L-S and Miller CT 2006 An evaluation oflattice Boltzmann schemes for porous medium flowsimulation Computers amp Fluids 35 (8ndash9) 898
Ramaswamy S et al 2004 The 3D structure of fabric andits relationship to liquid and vapor transport Colloidsand Surfaces A Physicochemical and Engineering As-pects 241 323ndash333
Rolland du Roscoat S Bloch J-F and Thibault X 2005Synchrotron Radiation microtomography applied topaper investigation Journal of Physics D AppliedPhysics 38 A78ndashA84
Rolland du Roscoat S et al 2007 Estimation ofmicrostructural properties from synchrotron X-ray microtomography and determination of the REV in papermaterials Acta Materiala 55 (8) 2841ndash2850
720 V Koivu et al
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
Saar MO and Manga M 1999 Permeability-porosityrelationship in vesicular basalts Geophysical ResearchLetters 26 (1) 111ndash114
Sangani AS and Acrivos A 1982 Slow flow past periodicarrays of cylinders with application to heat transferInternational Journal of Multiphase Flow 8 193ndash206
Succi S 2001 The lattice Boltzmann equation for fluiddynamics and beyond Gloucestershire Clarendon Press
Succi S Foti E and Higuera F 1989 Three-dimensionalflows in complex geometries with the lattice Boltzmannmethod Europhysics Letters 10 (5) 432ndash438
Thibault X and Bloch J-F 2002 Structural analysis of X-ray microtomography of a strained nonwoven paper-maker felt Textile Research Journal 72 (6) 480ndash485
Verleye B et al 2005 Computation of permeability oftextile reinforcements Proceedings in Scientific Computa-tion IMACS 2005 Paris France
Verleye B et al 2007 Permeability of textile reinforce-ments simulation influence of shear and validationComposite Science and Technology 68 (1) 2804ndash2810
Vidal D et al 2008 Effect od particle size distribution andpacking compression on fluid permeability as predictedby lattice-Boltzmann simulations Computers andChemical Engineering 33 (1) 256ndash266
White JA Borja IR and Fredrich JT 2006 Calculat-ing the effective permeability of sandstone with multi-scale lattice-Boltzmannfinite element simulations ActaGeotechnica 1 195ndash209
Wiegman A et al 2007 Computation of the permeabilityof porous materials from their microstructure by FFT-stokes Report of the Fraunhofer ITWM 129
International Journal of Computational Fluid Dynamics 721
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
indication of both methods being less accurate in thepresent complex geometry Notice that the boundaryof the flow domain (pore space) is in this case knownonly to the accuracy of the tomographic imageTherefore using a numerical grid essentially finerthan the voxel matrix of the 3D tomographic digitalimage does not in general improve the overall validityof the solution and of the computed value ofpermeability coefficient It is thus obvious that therelatively coarse cubic grid provided directly by thetomographic images does not allow for very accuratesolution of flow especially near the solid boundariesand in narrow passages Nevertheless the largest
deviation between numerical and experimental resultsis less than 30 This is of the same order ofmagnitude than the overall uncertainty inherent alsoin the used experimental method for thin compressedfibre mats (Koivu et al 2009a)
The most important single source of uncertainty inthe present numerical results most likely is the ratherpoor statistics obtained due to the low number andsmall size of (sub)samples Notice however that eventhough several subsamples were analysed the variationof permeability value thus obtained (and indicated inTable 1) does not necessarily correspond to the totallarge scale variation of the sample materials but merely
Table 1 Numerical and experimental values of permeability in three orthogonal directions (in physical units) for plastic fibresample and hard wood paper sample
Sample
Permeability kjj [m2 E-13]
kxx kyy kzz
LBM FDM Experimental LBM FDM Experimental LBM FDM Experimental
Plasticfibre mat
290 + 10 352 311 + 9 290 + 30 332 304 + 9 270 + 30 305 270 + 5
Hardwoodsample
26 + 03 260 31 + 01 27 + 03 284 31 + 01 076 + 005 085 063 + 001
The numerical results obtained with LBM are given as mean values of results based on five non-overlapping subvolumes and the experimentalresults as mean values of five independent measurements with standard deviation
Figure 5 Numerically solved flow field in plastic felt (a) and hardwood paper (b) Shown is the scaled flow speed on single 2Dcross sections (x-y plane) of the full 3D solution space within the actual pore geometry of material subsamples as obtained by X-ray micro-tomography (see Figure 4) Red and blue colours indicate high and low flow velocity respectively Solid material iscoloured black In (a) mean velocity is in the direction perpendicular to the plane shown (z direction) In (b) mean flow is in theplane shown and from up to down (y direction) Notice also the pure fluid layers visible at the upstream and downstream sides ofthe paper sample Inserts show details of the pore structure and flow velocity pattern with resolution of the numerical grid givenby the tomographic images (484 mm and 07 mm for (a) and (b) respectively) Available in colour online
718 V Koivu et al
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
the variation within the yet rather small originalsamples scanned The samples used in experimentsare however much larger and the results are thereforeexpected to better represent mean values (apart frompossible systematic errors remaining)
Discussion
In this work we develop and critically evaluate amethod for finding flow permeability of porousmaterials by numerically solving creeping fluid flowthrough the complex pore space of actual materialsamples Two conceptually different numericalschemes namely LBM and finite difference methodwere used The regular cubic grid used in numericalsolution is based directly on the 3D images of materialsamples obtained by two different states of the art X-ray microtomographic techniques
In many practical cases the main source ofuncertainty within the present approach is the finiteresolution of the tomographic image that provides thenumerical grid This may be due to hardware limita-tions or due to the need of compromising resolutionfor sample size for heterogeneous materials The orderof magnitude of the discretisation error arising fromthe relatively sparse numerical grid used was firstevaluated solving flows through a set of digitallygenerated regular geometries of straight cylinders Thevalues of permeability coefficients were computedusing grids of density comparable to that used withactual materials (relative to cylinderfibre diameter)The results were compared with analytical resultsavailable and with more accurate numerical resultsobtained using grids of sufficiently high density Forthe regular geometries studied the maximum deviationof the results obtained with sparse grid from analyticaland more accurate numerical results was less than10 This perhaps even surprisingly good accuracyobtained for regular geometries indicates that as far asdiscretisation error alone is concerned the methodmay be useful also for actual materials We mayhowever expect the discretisation error to be some-what higher for actual material samples due to theirmore complex pore structure Notice also that addi-tional error may arise due to the finite resolution of thetomographic techniques whereby details of the struc-ture (eg surface roughness of fibres and very narrowpassages in the porous space) may not be reproducedby the images Furthermore noise and other artefactspresent in the original reconstructed images as well asimage conditioning methods used to remove orminimise them may affect the results for eachindividual sample These are the main factors thatcurrently limit the applicability of the method espe-cially for fine-structured materials
In order to assess the overall uncertainty inherentin the computed results we applied the method infinding the diagonal components of permeabilitytensor for two fibrous materials plastic felt andhand-sheet paper under steady mechanical compres-sion and compared the results with correspondingexperimental values for the same materials Themaximum deviation between the numerical resultsand the experimental results was found to be less than30 In addition to numerical uncertainties arisingfrom discretisation and image quality discussed abovethis deviation includes statistical variation arising fromrelatively small size and low number of samples used incomputation In principle the accuracy can beimproved by using higher resolution tomographicimages and better statistics The former conditionmay become feasible with the present rapid develop-ment of X-ray tomographic techniques The latter canbe obtained by larger size or larger number of samplesObviously these improvements come with the cost ofhigher computational effort which however does notappear critical from the point of view of applicabilityof the method
To conclude the method of finding values of flowpermeability of porous materials based on numericallysolving fluid flow through the actual 3D pore geometryof material samples found using X-ray micro-tomo-graphy appears useful especially in cases whereexperimental results are not available or are difficultto obtain An example of such an application is to findthe permeability of various layers in a layer-structuredmaterial for which the layers can not be separated forindependent measurement (Koivu et al 2009b)Another advantage of the method is that it can beused to find all components of the permeability tensorWith the X-ray tomographic techniques presentlyused the overall accuracy of the computed values ofpermeability is moderate for the kind of materials andflow conditions studied here However this uncer-tainty is not essentially larger than the typical totaluncertainty of corresponding experimental results
Here we have demonstrated the method inevaluating a particular macroscopic transport propertyof porous materials the flow permeability Similarprocedure can be applied in other modes of transportsuch as diffusion and heat conduction in heterogeneousmaterials
Acknowledgements
The authors thank Tuomas Turpeinen MSc for hishelp related to the segmentation of tomographicimages and Jarno Alaraudanjoki MSc for thework done for experimental permeability measure-ments The authors also express their gratitude to the
International Journal of Computational Fluid Dynamics 719
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
ID19rsquos team of ESRF for help with the acquisition ofthe images and to Sabine Rolland du Roscoat for herenthusiastic help concerning both the acquisition andthe image analysis of the studied samples
References
Aaltosalmi U et al 2004 Numerical analysis of fluid flowthrough fibrous porous materials Journal of Pulp andPaper Science 30 (9) 251ndash255
Andersson AG et al 2009 Flow through a two-scaleporosity material Research Letters in Materials Science2009 (701512) 4 pp
Arns C et al 2004 Virtual permeametry on microtomo-graphic images Journal of Petroleum Science andEngineering 45 41ndash46
Bear J 1972 Dynamics of fluid in porous media New YorkDover Publications inc
Belov EB et al 2003 Modelling of permeability of textilereinforcements lattice Boltzmann method CompositesScience and Technology 64 1069ndash1080
Darcy H 1856 Les Fontaines Publiques de la Ville de DijonParis Victor Dalmont
Drummond JE and Tahir MI 1984 Laminar viscousflow through regular arrays of parallel solid cylindersInternational Journal of Multiphase Flow 10 (5) 515ndash525
Eker E and Akin S 2006 Lattice-Boltzmann simulation offluid flow in synthetic fractures Transport in porousmedia 65 363ndash384
Fourie W et al 2007 The simulation of pore scale fluidflow with real world geometries obtained from X-raycomputed tomography European COMSOL Multiphy-sics Conference 23ndash24 October 2007 Boston GrenobleCOMSOL inc
Frishfelds et al 2009 Flow-induced deformation of non-crimp fabrics during composites manufacturing Proceed-ings in Interdisciplinary Transport Phenomena IV(ITP2009) Volterra ITP-09-52
Ginzburg I 2008 Consistent lattice Boltzmann schemesfor the Brinkman model of porous flow and finiteChapman-Enskog expansion Physical Review E 77066704
Ginzburg I and drsquoHumieres D 2003 Multireflectionboundary conditions for lattice Boltzmann modelsPhysical Review E 68 (6) 066614
Ginzburg I Verhaeghe F and drsquoHumieres D 2008 Two-relaxation-time lattice-boltzmann scheme about para-metrization velocity pressure and mixed boundaryconditions Communications in Computational Physics 3(2) 427ndash478
Gonzales RC and Woods RE 2001 Digital imageprocessing 2nd edn New Jersey Prentice-Hall Inc237ndash241
Hellstrom JGI and Lundstrom TS 2006 Flow throughPorous Media at Moderate Reynolds Number Interna-tional Scientific Colloquim Modelling for MaterialProcessing Riga Latvia Faculty of Physics and Mathe-matics University of Latvia 129ndash134
Hyvaluoma J et al 2006 Simulation of liquid penetrationin paper Physical Review E 73 (3) 036705
Jackson GW and James DF 1986 The permeability offibrous porous media Canadian Journal of ChemicalEngineering 64 364ndash373
Jaganathan S Vahedi Tafreshi H and Pourdeyhimi B2007 A realistic approach for modeling permeabilityof fibrous media 3-D imaging coupled with CFDsimulation Chemical Engineering science 63 (1) 224ndash252
Keehm Y Sternlof K and Mukerji T 2006 Computa-tional estimation of compaction band permeability insandstone Geosciences Journal 10 (4) 499ndash505
Kerwald D 2004 Parallel lattice Boltzmann simulation ofcomplex flow NAFEMS Seminar simulation of complexflows (CFD) ndash applications and trends NiedernhausenWiesbaden
Koivu V Mattila K and Kataja M 2009a A method formeasuring darcian flow permeability of thin fibre matsNordic Pulp and Paper Research Journal 24 (4) 395ndash402
Koivu V et al 2009b Flow permeability of fibrous porousmaterials Micro-tomography and numerical simulationsIn SJ IrsquoAnson ed Advances in pulp and paper researchTransactions on 14th Research Fundamental SymposiumOxford September 14ndash18 2009 Manchester The Pulpand Paper Fundamental Research Society 437ndash454
Koponen A et al 1998 Permeability of three-dimensionalrandom fiber webs Physical Review Letters 80 (4) 716
Kutay ME Aydilek AH and Masad E 2006 Labora-tory validation of lattice Boltzmann method for model-ing pore-scale flow in granular materials ComputationalGeotechnics 33 381ndash395
Levitz P 2005 Toolbox for 3D imaging and modeling ofporous media relationship with transport propertiesCement and Concrete Research 37 351ndash359
Lundstrom TS Frishfelds V and Jakovics A 2004 Astatistical approach to permeability of clustered fibrereinforcements Journal of Composite Materials 38 (13)1137ndash1149
Lundstrom TS Sundlof H and Holmberg JA 2006Modeling of power-law fluid flow through fiber bedsJournal of Composite Materials 40 (3) 283ndash296
Manwart C et al 2002 Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media Physical Review E 66 (1)016702
Martys NS and Hagedorn JG 2002 Multiscale modelingof fluid transport in heterogeneous materials usingdiscrete Boltzmann methods Journal of Materials andStructures 35 (10) 650
Nordlund M et al 2005 Permeability network model fornon-crimp fabrics Composites Part A 37 826ndash835
Nordlund M and Lundstrom TS 2007 Effect ofmulti-scale porosity in local permeability modelling ofnon-crimp fabrics Transport in Porous Media 73 109ndash124
Pan C Luo L-S and Miller CT 2006 An evaluation oflattice Boltzmann schemes for porous medium flowsimulation Computers amp Fluids 35 (8ndash9) 898
Ramaswamy S et al 2004 The 3D structure of fabric andits relationship to liquid and vapor transport Colloidsand Surfaces A Physicochemical and Engineering As-pects 241 323ndash333
Rolland du Roscoat S Bloch J-F and Thibault X 2005Synchrotron Radiation microtomography applied topaper investigation Journal of Physics D AppliedPhysics 38 A78ndashA84
Rolland du Roscoat S et al 2007 Estimation ofmicrostructural properties from synchrotron X-ray microtomography and determination of the REV in papermaterials Acta Materiala 55 (8) 2841ndash2850
720 V Koivu et al
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
Saar MO and Manga M 1999 Permeability-porosityrelationship in vesicular basalts Geophysical ResearchLetters 26 (1) 111ndash114
Sangani AS and Acrivos A 1982 Slow flow past periodicarrays of cylinders with application to heat transferInternational Journal of Multiphase Flow 8 193ndash206
Succi S 2001 The lattice Boltzmann equation for fluiddynamics and beyond Gloucestershire Clarendon Press
Succi S Foti E and Higuera F 1989 Three-dimensionalflows in complex geometries with the lattice Boltzmannmethod Europhysics Letters 10 (5) 432ndash438
Thibault X and Bloch J-F 2002 Structural analysis of X-ray microtomography of a strained nonwoven paper-maker felt Textile Research Journal 72 (6) 480ndash485
Verleye B et al 2005 Computation of permeability oftextile reinforcements Proceedings in Scientific Computa-tion IMACS 2005 Paris France
Verleye B et al 2007 Permeability of textile reinforce-ments simulation influence of shear and validationComposite Science and Technology 68 (1) 2804ndash2810
Vidal D et al 2008 Effect od particle size distribution andpacking compression on fluid permeability as predictedby lattice-Boltzmann simulations Computers andChemical Engineering 33 (1) 256ndash266
White JA Borja IR and Fredrich JT 2006 Calculat-ing the effective permeability of sandstone with multi-scale lattice-Boltzmannfinite element simulations ActaGeotechnica 1 195ndash209
Wiegman A et al 2007 Computation of the permeabilityof porous materials from their microstructure by FFT-stokes Report of the Fraunhofer ITWM 129
International Journal of Computational Fluid Dynamics 721
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
the variation within the yet rather small originalsamples scanned The samples used in experimentsare however much larger and the results are thereforeexpected to better represent mean values (apart frompossible systematic errors remaining)
Discussion
In this work we develop and critically evaluate amethod for finding flow permeability of porousmaterials by numerically solving creeping fluid flowthrough the complex pore space of actual materialsamples Two conceptually different numericalschemes namely LBM and finite difference methodwere used The regular cubic grid used in numericalsolution is based directly on the 3D images of materialsamples obtained by two different states of the art X-ray microtomographic techniques
In many practical cases the main source ofuncertainty within the present approach is the finiteresolution of the tomographic image that provides thenumerical grid This may be due to hardware limita-tions or due to the need of compromising resolutionfor sample size for heterogeneous materials The orderof magnitude of the discretisation error arising fromthe relatively sparse numerical grid used was firstevaluated solving flows through a set of digitallygenerated regular geometries of straight cylinders Thevalues of permeability coefficients were computedusing grids of density comparable to that used withactual materials (relative to cylinderfibre diameter)The results were compared with analytical resultsavailable and with more accurate numerical resultsobtained using grids of sufficiently high density Forthe regular geometries studied the maximum deviationof the results obtained with sparse grid from analyticaland more accurate numerical results was less than10 This perhaps even surprisingly good accuracyobtained for regular geometries indicates that as far asdiscretisation error alone is concerned the methodmay be useful also for actual materials We mayhowever expect the discretisation error to be some-what higher for actual material samples due to theirmore complex pore structure Notice also that addi-tional error may arise due to the finite resolution of thetomographic techniques whereby details of the struc-ture (eg surface roughness of fibres and very narrowpassages in the porous space) may not be reproducedby the images Furthermore noise and other artefactspresent in the original reconstructed images as well asimage conditioning methods used to remove orminimise them may affect the results for eachindividual sample These are the main factors thatcurrently limit the applicability of the method espe-cially for fine-structured materials
In order to assess the overall uncertainty inherentin the computed results we applied the method infinding the diagonal components of permeabilitytensor for two fibrous materials plastic felt andhand-sheet paper under steady mechanical compres-sion and compared the results with correspondingexperimental values for the same materials Themaximum deviation between the numerical resultsand the experimental results was found to be less than30 In addition to numerical uncertainties arisingfrom discretisation and image quality discussed abovethis deviation includes statistical variation arising fromrelatively small size and low number of samples used incomputation In principle the accuracy can beimproved by using higher resolution tomographicimages and better statistics The former conditionmay become feasible with the present rapid develop-ment of X-ray tomographic techniques The latter canbe obtained by larger size or larger number of samplesObviously these improvements come with the cost ofhigher computational effort which however does notappear critical from the point of view of applicabilityof the method
To conclude the method of finding values of flowpermeability of porous materials based on numericallysolving fluid flow through the actual 3D pore geometryof material samples found using X-ray micro-tomo-graphy appears useful especially in cases whereexperimental results are not available or are difficultto obtain An example of such an application is to findthe permeability of various layers in a layer-structuredmaterial for which the layers can not be separated forindependent measurement (Koivu et al 2009b)Another advantage of the method is that it can beused to find all components of the permeability tensorWith the X-ray tomographic techniques presentlyused the overall accuracy of the computed values ofpermeability is moderate for the kind of materials andflow conditions studied here However this uncer-tainty is not essentially larger than the typical totaluncertainty of corresponding experimental results
Here we have demonstrated the method inevaluating a particular macroscopic transport propertyof porous materials the flow permeability Similarprocedure can be applied in other modes of transportsuch as diffusion and heat conduction in heterogeneousmaterials
Acknowledgements
The authors thank Tuomas Turpeinen MSc for hishelp related to the segmentation of tomographicimages and Jarno Alaraudanjoki MSc for thework done for experimental permeability measure-ments The authors also express their gratitude to the
International Journal of Computational Fluid Dynamics 719
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
ID19rsquos team of ESRF for help with the acquisition ofthe images and to Sabine Rolland du Roscoat for herenthusiastic help concerning both the acquisition andthe image analysis of the studied samples
References
Aaltosalmi U et al 2004 Numerical analysis of fluid flowthrough fibrous porous materials Journal of Pulp andPaper Science 30 (9) 251ndash255
Andersson AG et al 2009 Flow through a two-scaleporosity material Research Letters in Materials Science2009 (701512) 4 pp
Arns C et al 2004 Virtual permeametry on microtomo-graphic images Journal of Petroleum Science andEngineering 45 41ndash46
Bear J 1972 Dynamics of fluid in porous media New YorkDover Publications inc
Belov EB et al 2003 Modelling of permeability of textilereinforcements lattice Boltzmann method CompositesScience and Technology 64 1069ndash1080
Darcy H 1856 Les Fontaines Publiques de la Ville de DijonParis Victor Dalmont
Drummond JE and Tahir MI 1984 Laminar viscousflow through regular arrays of parallel solid cylindersInternational Journal of Multiphase Flow 10 (5) 515ndash525
Eker E and Akin S 2006 Lattice-Boltzmann simulation offluid flow in synthetic fractures Transport in porousmedia 65 363ndash384
Fourie W et al 2007 The simulation of pore scale fluidflow with real world geometries obtained from X-raycomputed tomography European COMSOL Multiphy-sics Conference 23ndash24 October 2007 Boston GrenobleCOMSOL inc
Frishfelds et al 2009 Flow-induced deformation of non-crimp fabrics during composites manufacturing Proceed-ings in Interdisciplinary Transport Phenomena IV(ITP2009) Volterra ITP-09-52
Ginzburg I 2008 Consistent lattice Boltzmann schemesfor the Brinkman model of porous flow and finiteChapman-Enskog expansion Physical Review E 77066704
Ginzburg I and drsquoHumieres D 2003 Multireflectionboundary conditions for lattice Boltzmann modelsPhysical Review E 68 (6) 066614
Ginzburg I Verhaeghe F and drsquoHumieres D 2008 Two-relaxation-time lattice-boltzmann scheme about para-metrization velocity pressure and mixed boundaryconditions Communications in Computational Physics 3(2) 427ndash478
Gonzales RC and Woods RE 2001 Digital imageprocessing 2nd edn New Jersey Prentice-Hall Inc237ndash241
Hellstrom JGI and Lundstrom TS 2006 Flow throughPorous Media at Moderate Reynolds Number Interna-tional Scientific Colloquim Modelling for MaterialProcessing Riga Latvia Faculty of Physics and Mathe-matics University of Latvia 129ndash134
Hyvaluoma J et al 2006 Simulation of liquid penetrationin paper Physical Review E 73 (3) 036705
Jackson GW and James DF 1986 The permeability offibrous porous media Canadian Journal of ChemicalEngineering 64 364ndash373
Jaganathan S Vahedi Tafreshi H and Pourdeyhimi B2007 A realistic approach for modeling permeabilityof fibrous media 3-D imaging coupled with CFDsimulation Chemical Engineering science 63 (1) 224ndash252
Keehm Y Sternlof K and Mukerji T 2006 Computa-tional estimation of compaction band permeability insandstone Geosciences Journal 10 (4) 499ndash505
Kerwald D 2004 Parallel lattice Boltzmann simulation ofcomplex flow NAFEMS Seminar simulation of complexflows (CFD) ndash applications and trends NiedernhausenWiesbaden
Koivu V Mattila K and Kataja M 2009a A method formeasuring darcian flow permeability of thin fibre matsNordic Pulp and Paper Research Journal 24 (4) 395ndash402
Koivu V et al 2009b Flow permeability of fibrous porousmaterials Micro-tomography and numerical simulationsIn SJ IrsquoAnson ed Advances in pulp and paper researchTransactions on 14th Research Fundamental SymposiumOxford September 14ndash18 2009 Manchester The Pulpand Paper Fundamental Research Society 437ndash454
Koponen A et al 1998 Permeability of three-dimensionalrandom fiber webs Physical Review Letters 80 (4) 716
Kutay ME Aydilek AH and Masad E 2006 Labora-tory validation of lattice Boltzmann method for model-ing pore-scale flow in granular materials ComputationalGeotechnics 33 381ndash395
Levitz P 2005 Toolbox for 3D imaging and modeling ofporous media relationship with transport propertiesCement and Concrete Research 37 351ndash359
Lundstrom TS Frishfelds V and Jakovics A 2004 Astatistical approach to permeability of clustered fibrereinforcements Journal of Composite Materials 38 (13)1137ndash1149
Lundstrom TS Sundlof H and Holmberg JA 2006Modeling of power-law fluid flow through fiber bedsJournal of Composite Materials 40 (3) 283ndash296
Manwart C et al 2002 Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media Physical Review E 66 (1)016702
Martys NS and Hagedorn JG 2002 Multiscale modelingof fluid transport in heterogeneous materials usingdiscrete Boltzmann methods Journal of Materials andStructures 35 (10) 650
Nordlund M et al 2005 Permeability network model fornon-crimp fabrics Composites Part A 37 826ndash835
Nordlund M and Lundstrom TS 2007 Effect ofmulti-scale porosity in local permeability modelling ofnon-crimp fabrics Transport in Porous Media 73 109ndash124
Pan C Luo L-S and Miller CT 2006 An evaluation oflattice Boltzmann schemes for porous medium flowsimulation Computers amp Fluids 35 (8ndash9) 898
Ramaswamy S et al 2004 The 3D structure of fabric andits relationship to liquid and vapor transport Colloidsand Surfaces A Physicochemical and Engineering As-pects 241 323ndash333
Rolland du Roscoat S Bloch J-F and Thibault X 2005Synchrotron Radiation microtomography applied topaper investigation Journal of Physics D AppliedPhysics 38 A78ndashA84
Rolland du Roscoat S et al 2007 Estimation ofmicrostructural properties from synchrotron X-ray microtomography and determination of the REV in papermaterials Acta Materiala 55 (8) 2841ndash2850
720 V Koivu et al
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
Saar MO and Manga M 1999 Permeability-porosityrelationship in vesicular basalts Geophysical ResearchLetters 26 (1) 111ndash114
Sangani AS and Acrivos A 1982 Slow flow past periodicarrays of cylinders with application to heat transferInternational Journal of Multiphase Flow 8 193ndash206
Succi S 2001 The lattice Boltzmann equation for fluiddynamics and beyond Gloucestershire Clarendon Press
Succi S Foti E and Higuera F 1989 Three-dimensionalflows in complex geometries with the lattice Boltzmannmethod Europhysics Letters 10 (5) 432ndash438
Thibault X and Bloch J-F 2002 Structural analysis of X-ray microtomography of a strained nonwoven paper-maker felt Textile Research Journal 72 (6) 480ndash485
Verleye B et al 2005 Computation of permeability oftextile reinforcements Proceedings in Scientific Computa-tion IMACS 2005 Paris France
Verleye B et al 2007 Permeability of textile reinforce-ments simulation influence of shear and validationComposite Science and Technology 68 (1) 2804ndash2810
Vidal D et al 2008 Effect od particle size distribution andpacking compression on fluid permeability as predictedby lattice-Boltzmann simulations Computers andChemical Engineering 33 (1) 256ndash266
White JA Borja IR and Fredrich JT 2006 Calculat-ing the effective permeability of sandstone with multi-scale lattice-Boltzmannfinite element simulations ActaGeotechnica 1 195ndash209
Wiegman A et al 2007 Computation of the permeabilityof porous materials from their microstructure by FFT-stokes Report of the Fraunhofer ITWM 129
International Journal of Computational Fluid Dynamics 721
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
ID19rsquos team of ESRF for help with the acquisition ofthe images and to Sabine Rolland du Roscoat for herenthusiastic help concerning both the acquisition andthe image analysis of the studied samples
References
Aaltosalmi U et al 2004 Numerical analysis of fluid flowthrough fibrous porous materials Journal of Pulp andPaper Science 30 (9) 251ndash255
Andersson AG et al 2009 Flow through a two-scaleporosity material Research Letters in Materials Science2009 (701512) 4 pp
Arns C et al 2004 Virtual permeametry on microtomo-graphic images Journal of Petroleum Science andEngineering 45 41ndash46
Bear J 1972 Dynamics of fluid in porous media New YorkDover Publications inc
Belov EB et al 2003 Modelling of permeability of textilereinforcements lattice Boltzmann method CompositesScience and Technology 64 1069ndash1080
Darcy H 1856 Les Fontaines Publiques de la Ville de DijonParis Victor Dalmont
Drummond JE and Tahir MI 1984 Laminar viscousflow through regular arrays of parallel solid cylindersInternational Journal of Multiphase Flow 10 (5) 515ndash525
Eker E and Akin S 2006 Lattice-Boltzmann simulation offluid flow in synthetic fractures Transport in porousmedia 65 363ndash384
Fourie W et al 2007 The simulation of pore scale fluidflow with real world geometries obtained from X-raycomputed tomography European COMSOL Multiphy-sics Conference 23ndash24 October 2007 Boston GrenobleCOMSOL inc
Frishfelds et al 2009 Flow-induced deformation of non-crimp fabrics during composites manufacturing Proceed-ings in Interdisciplinary Transport Phenomena IV(ITP2009) Volterra ITP-09-52
Ginzburg I 2008 Consistent lattice Boltzmann schemesfor the Brinkman model of porous flow and finiteChapman-Enskog expansion Physical Review E 77066704
Ginzburg I and drsquoHumieres D 2003 Multireflectionboundary conditions for lattice Boltzmann modelsPhysical Review E 68 (6) 066614
Ginzburg I Verhaeghe F and drsquoHumieres D 2008 Two-relaxation-time lattice-boltzmann scheme about para-metrization velocity pressure and mixed boundaryconditions Communications in Computational Physics 3(2) 427ndash478
Gonzales RC and Woods RE 2001 Digital imageprocessing 2nd edn New Jersey Prentice-Hall Inc237ndash241
Hellstrom JGI and Lundstrom TS 2006 Flow throughPorous Media at Moderate Reynolds Number Interna-tional Scientific Colloquim Modelling for MaterialProcessing Riga Latvia Faculty of Physics and Mathe-matics University of Latvia 129ndash134
Hyvaluoma J et al 2006 Simulation of liquid penetrationin paper Physical Review E 73 (3) 036705
Jackson GW and James DF 1986 The permeability offibrous porous media Canadian Journal of ChemicalEngineering 64 364ndash373
Jaganathan S Vahedi Tafreshi H and Pourdeyhimi B2007 A realistic approach for modeling permeabilityof fibrous media 3-D imaging coupled with CFDsimulation Chemical Engineering science 63 (1) 224ndash252
Keehm Y Sternlof K and Mukerji T 2006 Computa-tional estimation of compaction band permeability insandstone Geosciences Journal 10 (4) 499ndash505
Kerwald D 2004 Parallel lattice Boltzmann simulation ofcomplex flow NAFEMS Seminar simulation of complexflows (CFD) ndash applications and trends NiedernhausenWiesbaden
Koivu V Mattila K and Kataja M 2009a A method formeasuring darcian flow permeability of thin fibre matsNordic Pulp and Paper Research Journal 24 (4) 395ndash402
Koivu V et al 2009b Flow permeability of fibrous porousmaterials Micro-tomography and numerical simulationsIn SJ IrsquoAnson ed Advances in pulp and paper researchTransactions on 14th Research Fundamental SymposiumOxford September 14ndash18 2009 Manchester The Pulpand Paper Fundamental Research Society 437ndash454
Koponen A et al 1998 Permeability of three-dimensionalrandom fiber webs Physical Review Letters 80 (4) 716
Kutay ME Aydilek AH and Masad E 2006 Labora-tory validation of lattice Boltzmann method for model-ing pore-scale flow in granular materials ComputationalGeotechnics 33 381ndash395
Levitz P 2005 Toolbox for 3D imaging and modeling ofporous media relationship with transport propertiesCement and Concrete Research 37 351ndash359
Lundstrom TS Frishfelds V and Jakovics A 2004 Astatistical approach to permeability of clustered fibrereinforcements Journal of Composite Materials 38 (13)1137ndash1149
Lundstrom TS Sundlof H and Holmberg JA 2006Modeling of power-law fluid flow through fiber bedsJournal of Composite Materials 40 (3) 283ndash296
Manwart C et al 2002 Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media Physical Review E 66 (1)016702
Martys NS and Hagedorn JG 2002 Multiscale modelingof fluid transport in heterogeneous materials usingdiscrete Boltzmann methods Journal of Materials andStructures 35 (10) 650
Nordlund M et al 2005 Permeability network model fornon-crimp fabrics Composites Part A 37 826ndash835
Nordlund M and Lundstrom TS 2007 Effect ofmulti-scale porosity in local permeability modelling ofnon-crimp fabrics Transport in Porous Media 73 109ndash124
Pan C Luo L-S and Miller CT 2006 An evaluation oflattice Boltzmann schemes for porous medium flowsimulation Computers amp Fluids 35 (8ndash9) 898
Ramaswamy S et al 2004 The 3D structure of fabric andits relationship to liquid and vapor transport Colloidsand Surfaces A Physicochemical and Engineering As-pects 241 323ndash333
Rolland du Roscoat S Bloch J-F and Thibault X 2005Synchrotron Radiation microtomography applied topaper investigation Journal of Physics D AppliedPhysics 38 A78ndashA84
Rolland du Roscoat S et al 2007 Estimation ofmicrostructural properties from synchrotron X-ray microtomography and determination of the REV in papermaterials Acta Materiala 55 (8) 2841ndash2850
720 V Koivu et al
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
Saar MO and Manga M 1999 Permeability-porosityrelationship in vesicular basalts Geophysical ResearchLetters 26 (1) 111ndash114
Sangani AS and Acrivos A 1982 Slow flow past periodicarrays of cylinders with application to heat transferInternational Journal of Multiphase Flow 8 193ndash206
Succi S 2001 The lattice Boltzmann equation for fluiddynamics and beyond Gloucestershire Clarendon Press
Succi S Foti E and Higuera F 1989 Three-dimensionalflows in complex geometries with the lattice Boltzmannmethod Europhysics Letters 10 (5) 432ndash438
Thibault X and Bloch J-F 2002 Structural analysis of X-ray microtomography of a strained nonwoven paper-maker felt Textile Research Journal 72 (6) 480ndash485
Verleye B et al 2005 Computation of permeability oftextile reinforcements Proceedings in Scientific Computa-tion IMACS 2005 Paris France
Verleye B et al 2007 Permeability of textile reinforce-ments simulation influence of shear and validationComposite Science and Technology 68 (1) 2804ndash2810
Vidal D et al 2008 Effect od particle size distribution andpacking compression on fluid permeability as predictedby lattice-Boltzmann simulations Computers andChemical Engineering 33 (1) 256ndash266
White JA Borja IR and Fredrich JT 2006 Calculat-ing the effective permeability of sandstone with multi-scale lattice-Boltzmannfinite element simulations ActaGeotechnica 1 195ndash209
Wiegman A et al 2007 Computation of the permeabilityof porous materials from their microstructure by FFT-stokes Report of the Fraunhofer ITWM 129
International Journal of Computational Fluid Dynamics 721
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
Saar MO and Manga M 1999 Permeability-porosityrelationship in vesicular basalts Geophysical ResearchLetters 26 (1) 111ndash114
Sangani AS and Acrivos A 1982 Slow flow past periodicarrays of cylinders with application to heat transferInternational Journal of Multiphase Flow 8 193ndash206
Succi S 2001 The lattice Boltzmann equation for fluiddynamics and beyond Gloucestershire Clarendon Press
Succi S Foti E and Higuera F 1989 Three-dimensionalflows in complex geometries with the lattice Boltzmannmethod Europhysics Letters 10 (5) 432ndash438
Thibault X and Bloch J-F 2002 Structural analysis of X-ray microtomography of a strained nonwoven paper-maker felt Textile Research Journal 72 (6) 480ndash485
Verleye B et al 2005 Computation of permeability oftextile reinforcements Proceedings in Scientific Computa-tion IMACS 2005 Paris France
Verleye B et al 2007 Permeability of textile reinforce-ments simulation influence of shear and validationComposite Science and Technology 68 (1) 2804ndash2810
Vidal D et al 2008 Effect od particle size distribution andpacking compression on fluid permeability as predictedby lattice-Boltzmann simulations Computers andChemical Engineering 33 (1) 256ndash266
White JA Borja IR and Fredrich JT 2006 Calculat-ing the effective permeability of sandstone with multi-scale lattice-Boltzmannfinite element simulations ActaGeotechnica 1 195ndash209
Wiegman A et al 2007 Computation of the permeabilityof porous materials from their microstructure by FFT-stokes Report of the Fraunhofer ITWM 129
International Journal of Computational Fluid Dynamics 721
Downloaded By [UJF - INP Grenoble SICD 1] At 1418 16 April 2010
