Three-dimensional liquid transport in concrete cracks

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2004; 28:671–687 (DOI: 10.1002/nag.373)

Three-dimensional liquid transport in concrete cracks

J. Carmeliet1,2,n,y, J.-F. Delerue1, K. Vandersteen1 and S. Roels1

1Department of Civil Engineering, Laboratory of Building Physics, Catholic University of Leuven, The Netherlands2Department of Building and Architecture, Physical Aspects of the built Environment,

Eindhoven University of Technology, The Netherlands

SUMMARY

Cracks in concrete are measured in 3D by microfocus X-ray computer tomography. The tomographicimages are thresholded matching a characteristic of the measured crack attenuation profile. Amethodology is proposed to convert the 3D measured voxel data into a network of parallel plates. Themethodology is based on the determination of the aperture map and skeleton of the void space, and thesegmentation of the void space in crack segments. The segmentation and network approach allows to studycrack aperture and connectivity distributions of the crack. Static invasion percolation and moving fronttechnique are used to analyse liquid flow in cracks. One-dimensional simulations of transport in a crackwith variable crack width exemplify the retardation effect of narrow passages. In 2D, the narrow passagescan be by-passed resulting in preferential flow patterns, where coarse crack zones remain unfilled. Meshsensitivity in the network approach is studied showing a limited influence of the mesh size on the fillingpatterns, caused by a change of connectivity when refining the mesh. Comparison of 3D and 2Dsimulations indicates that flow in 2D crack sections can strongly underestimate possible fluid penetrationdepths. Finally the network model is validated analysing water uptake in a fractured brick sample.Copyright # 2004 John Wiley & Sons, Ltd.

KEY WORDS: discrete fracture flow model; moving front; X-ray computer tomography; image analysis;network construction

1. INTRODUCTION

Liquid flow in fractured media is an important phenomenon in resource recovery (mining,petroleum production, geothermal and water supply reservoirs), environmental protection(chemical and nuclear contamination) and engineering structures (slopes, dams and surfacestorage reservoirs, underground structures and buildings). Fractures or cracks may showcomplex structures including crack bridging, crack branching and bifurcation, side-bandcracking and secondary cracking parallel to main cracks. Especially in composite materials asconcrete, composed of a random arrangement of aggregates, a surrounding matrix with built-inair voids and a weaker interfacial zone, crack networks with varying crack width andconnectivity may develop. Cracks are preferential pathways for liquid water leading to anaccelerated wetting of the porous material surrounding the crack. Continuity and variations incrack width play an important role in the penetration of these fluids in the crack structure.

Received 18 July 2003Revised 20 February 2004Copyright # 2004 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: J. Carmeliet, Department of Civil Engineering, Laboratory of Building Physics, Catholic Universityof Leuven, The Netherlands.

In durability mechanics, poromechanical models have been presented to address coupledhygrothermal, mechanical and chemical related damage processes in building structures [1–7].An essential key factor in these models is the prediction of cracking and the related increase ofthe permeability. Most models are continuum-based formulations, where localization is handledusing smeared or higher order approaches}rather a crack strain over a band governs thedamage process than a crack width. Direct information on crack widths is thus not available.Empirical laws for relating crack strains or crack widths in concrete to permeability are given inliterature [8–10]. Simplified analytical formulas are formulated for taking into account crackwidth variation. Zimmerman et al. [11] investigated a number of analytical solutions for theprediction of the saturated hydraulic conductivity of rough-walled fractures. They concludedthat the fracture’s roughness has to be included to calculate the saturated hydraulic conductivity.

Flow in fractured media can also be handled by replacing the fractured domain by anequivalent continuum [12–17]. The continuum models, however, are not appropriate when thescale of interest is small or the water-bearing fractures are few leading to preferential penetrationof fluids. In these cases, a discrete model that represents individual fractures is more advisable.In discrete fracture flow models, a fractured medium is characterized as a network of flowchannels in which saturated flow is described using the Poiseuille equation or Darcy’s equationin combination with a mass balance equation. Different ways of representing the flow channelsexist: parallel plates with constant aperture [18] or variable apertures [19–21], discs [22–24],circular tubes [18], line elements [25] or pipe elements [26]. Several studies make use of astochastically generated description of the variable fracture aperture to numerically investigatevarious aspects of two-phase flow in rough-walled fractures using various types of networkmodelling. Pruess and Tsang [27], Mendoza [28] developed static invasion percolation modelsfor the determination of relative permeability versus saturation relationships and capillarypressure versus saturation relationships. Murphy and Thomson [29] developed a discrete modelfor simulating two-phase flow in a single fracture with variable aperture in two dimensions.Esposito and Thomson [30] modified this model in order to incorporate diffusion of dissolvedcontaminants into the rock matrix.

Discrete models for flow in cracks require that the geometry and topology of the fracturenetwork is specified explicitly. Information on crack geometry and topology may be acquired bytwo-dimensional section analysis. Using stereological methods this 2D information can be used toinfer some of the 3D characteristics. Techniques such as serial sectioning and more recently X-raycomputed tomography allow however direct access to the three-dimensional structure of cracks.

The objective of this study consists in analysing the flow characteristics in three-dimensionalconcrete cracks. In a first part, the 3D fracture is measured using microfocus computer X-raytomography ðmCTÞ: Using a calibrated threshold and segmentation technique a 3D networkrepresentation of crack space is constructed. Finally, static invasion percolation and the movingfront technique are respectively used to study mercury intrusion and water uptake processes inconcrete cracks.

2. IMAGE ACQUISITION, SEGMENTATION AN NETWORK MODELLING

2.1. Image acquisition

Microfocus computer X-ray tomography ðmCTÞ is used to obtain a 3D image of a crack inconcrete. To create a fracture, a cylindrical specimen (diameter of 110 mm and a thickness of

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J. CARMELIET ET AL.672

20 mm) is loaded in a tensile splitting test until failure. The concrete specimen is imaged on a 3DAEA mCT apparatus developed at K.U.Leuven (Figure 1). A polychromatic X-ray cone beamproduced by an X-ray source with a focal spot size around 10 mm; irradiates the specimen,which is placed on a translation/rotation manipulator. The specimen is scanned using 25 keVX-rays. After conversion of the attenuated X-rays to light by a fluorescent screen, they arerecorded by a 12 bit CCD camera of 1024� 1024 pixels and 4096 intensity levels (grey scales).We used an optical magnification of 1.5 according to a source-to-object distance of 886 mm andobject-to-detector distance of 455 mm; resulting in a pixel size sp of 114 mm for the recordedimage. The images obtained by the cone beam configuration are automatically transferred into aparallel beam configuration during measurement. For each sample 180 2D images were takenover a field of view of 1798: The images are further corrected for spatial distortion and scatter[31]. Beam hardening artefacts were corrected using a linearization procedure [20]. A 3D filteredbackprojection algorithm is used to reconstruct a 3D image of the sample from the set of 2Dprojections.

The 3D reconstruction was limited to a region of interest around the crack (Figure 2(a)). Theimages are further processed to remove artefacts, such as outlining artefacts, ring artefacts and

Figure 1. Experimental set-up of the X-ray projection method.

Figure 2. (a) Location of the region of interest around the crack; and (b) three-dimensional reconstructedX-ray image of cracked concrete specimen loaded in a tensile splitting test.

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line artefacts (for a detailed description, see Van Geet [31]). The size of the concrete sampleshown in Figure 2(b) is 55� 27:5� 20 mm: The aggregates (high density) appear in light grey,the matrix is darker, while the crack appears black. This image shows that the crack is in essencea three-dimensional feature, since the crack attempts to follow the weaker aggregate-mortarinterface, which position strongly depends on the aggregate distribution within the sample.

2.2. Determination of the void space

For the determination of the void space occupied by the fracture in the 3D tomographic image,we used two different techniques.

The first technique is based on the particular shape of the attenuation profile across a fracturewith an (ideal) parallel-plate configuration. Theoretically, the attenuation profile over such afracture is Gaussian resulting from the convolution of a theoretical rectangular fracture profilewith the line spread function (LSF) (Figure 3). The line spread function is the spatial derivativeof the edge response function (ERF) representing the smooth transition at the edge material/airand which is a measure for the spatial resolution. Knowledge of the LSF or ERF enables thedetermination of the crack aperture by deconvolution of the measured profile. In reality, thetheoretical profile is contaminated by noise and the application of deconvolution results inhighly uncertain fracture widths. Therefore, characteristic parameters such as peak height (PH),the full width at half maximum (FWHM) and missing attenuation (MA)}surface under theattenuation profile}of the fracture attenuation profile are used as calibration parameters forthe quantification of the fracture aperture. Vandersteen [20] showed these techniques to beadvantageous for the determination of planar crack structures when high quality imagesshowing a good signal-to-noise ratio are available. For non-planar fracture structures showinghigh variations in crack direction, crack width and connectivity, the automatic application ofthese techniques is less straightforward. A second technique consist in thresholding the greyscale tomographic image to a black and white image by matching a known crack aperture at the

source

object

with crack

detector

ERF

LSF

attenuation profile PHFWHM

Figure 3. Due to the finite size of the X-ray source the specimen edge is on the detector notseen as a jump but as a smooth attenuation function (edge-response function, ERF).

Differentiation of the ERF gives the line-spread function (LSF).

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surface of the specimen. The threshold limit was determined using the first calibrationtechnique. Errors may be included by the fact that the location at the surface is not exactlymatched (the mCT measurement is always just beneath the surface) and due to inaccuracies inthe visual microscopic technique, as an exact delineation of the edges of the fracture is difficultto evaluate. Using the thresholding technique the spatial resolution for identifying crack widthsequals the pixel size of 114 mm: A more detailed discussion on the spatial resolution which canbe obtained using the ERF function can be found in Van Geet [31].

Figure 4 shows the 3D crack structure determined from the tomographic concrete image usingthe threshold technique. At the right hand side (i.e. the middle of the concrete sample), a maincrack is observed. Detailed analysis shows that the crack width is almost uniform. The crackfollows the aggregate boundaries in an ellipsoidal shape. At the left side (i.e. the surface of thespecimen), the crack closes and shows a poor connectivity. The transition zone is characterisedby a complex structure. We observe one main crack and a lot of secondary side-band crackingand macroporosity following the aggregates and air voids present in the specimen. Thesecondary cracking shows to be less connected with high variations in crack aperture. Figure5(a) shows as an example 2D cross-sections at two different positions (2Da and 2Db). We noticein section 2Db that the main crack shows higher variations in crack width and more secondarycracking compared to section 2Da. The influence of these features (3D versus 2D sections, crackwidth variation, connectivity) on flow characteristics is analysed in subsequent sections.

2.3. Segmentation of a pore space into pore objects

Using the threshold algorithm the grey-scale mCT-image of the fractured specimen is convertedinto a binary map where the value of each voxel (0 or 1) indicates if the voxel belongs to the void

Figure 4. Image of the crack obtained from the X-ray image using a threshold technique. At the right sidea main crack is visible. In the middle zone side-band secondary cracking appears, while at the left side the

crack closes and becomes less connected.

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space V or to the solid matrix. The aim of the segmentation procedure is to divide the void spaceinto topological objects called crack segments. A crack segment is defined as a connected clusterof voxels, which has at least two boundaries with the solid matrix and where all voxels belong tothe same class of a local crack aperture size. The local aperture size bðpÞ for a point p of the voidspace V is defined as the diameter d of the maximum ball B with centre c; which can be built inthe void space including the point p (Figure 6(a)), or

bðpÞ ¼ maxfd j Bðc; dÞ � V ; p 2 Bðc; dÞg ð1Þ

The segmentation of the void space into crack segments is based on two steps. First we searchfor a unique arrangement of non-overlapping maximum balls whereby we favour bigger balls.In searching these balls, we make use of the interesting property that maximum balls are centredon the skeleton of the void space. The void skeleton is determined using a Voronoi tessellationprocedure [32]. Using the determined skeleton, we find the arrangement of non-overlappingmaximum balls by first positioning the biggest ball and then successively placing smaller andsmaller balls in the remaining void space (Figure 6(b)). In a second step, these balls aresimultaneously expanded to crack segments using a growing region algorithm (Figure 6(c)). Inthis expansion process the shape of the balls is not conserved, but as if balloons were inflated inthe void space the crack segments adopt a more irregular shape (or with each ball one cracksegment is associated). At the end of the process, every point of the void space is associated with

Figure 5. Water uptake in 2D cross-sections 2Da and 2Db of a crack in concrete: (a)tomographic image after thresholding; (b) crack segment map; (c) network where linesrepresent the connections between crack segments. The centres of the segments are representedas spheres. The size of the sphere resembles the crack aperture; and (d) Time evolution of the

saturation profiles in a crack during water uptake from the right side.

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an expanded ball or crack segment: a crack segment map of the void space is obtained. Efficientalgorithms for extracting a crack segment map from a voxel representation of the void space areproposed by Delerue et al. [32–34]. Figure 5(b) gives as example the crack segment maps for the2D cross-section 2Da and 2Db.

2.4. Network representation of the crack void space

The crack segment map is used to construct a network representation of the fracture void space.We locate nodes at the centre of the crack segments (i.e. the centre of the expanded balls) and atthe centre of the interface between two crack segments i and j (Figure 7). The interface has asurface Sij : The distance between the centre of crack segment i and the centre of interface ij isdenoted by Lij : The basic element of the hydraulic network is a parallel plate element (PPE),which most closely resembles the geometry of a crack segment. A network element Eij connectsthe nodes i and j and is composed of two parallel plate elements: PPEij between node i and thecentre of the interface ij and PPEji; which links the centre of the interface ij and node j: Thegeometry of the parallel plate element PPEij is defined by the length Lij and the aperture bi of thecrack segment i: A network element thus consists of two parallel plate elements PPEij and PPEji;;where one parallel plate element is made of one side of a crack segment (or expanded ball).

Figure 8(a) gives a 3D representation of the network. The links are represented as linesbetween the centres of the crack segments represented as balls. The size of the ball resembles thecrack aperture. Figure 5(c) gives the network for the 2D sections.

Based on the network representation the crack width and connectivity can be evaluated. Thecrack width distribution gives the relative volume of crack segments with a crack width bibetween ð2i � 1Þsp and ð2i þ 1Þsp; with sp the pixel size. We observe in Figure 9(a) that in 3D ahigher volume fraction of small cracks is present compared to the 2D distributions. Theconnectivity cij of the element Eij is defined as the number of elements connected to the elementEij : When cij ¼ 0; the crack segment is isolated and belongs to the closed porosity. Aconnectivity cij ¼ 1 means the crack segment belongs to the dead end fraction. Connectivitycij ¼ 2 indicates a one way flow through the crack segment, while higher values show multipleflow connections. We observe in Figure 9(b) that the connectivity values in 3D range from 2 to15 showing a high degree of connectivity. In 2D a higher fraction of isolated and dead end cracksegments is present and the connectivity is limited to a value of 7 compared to 15 in 3D.

Figure 6. Segmentation of a pore space into pore objects: (a) determination of aperture map;(b) arrangement of non-overlapping maximum balls; and (c) expansion of the balls to crack segments.

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We conclude that in 3D the crack shows a higher connectivity and a higher fraction of smallcracks. In 2D, information on connectivity is lost resulting in a higher fraction of isolated anddead end pores and lower connectivity values.

Figure 7. Network construction: (a) crack segments with interface; and (b) parallel plate representation.

Figure 8. Water uptake in a 3D crack in concrete: (a) network where lines represent theconnections between crack segments. The centres of the segments are represented as spheres.The size of the sphere resembles the crack aperture; (b) tomographic image after thresholding(view from side); and (c) Time evolution of the saturation profiles in a crack during water

uptake from the right side. Final profile is represented in bold.

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3. STATIC AND DYNAMIC FLOW IN CRACKS

The transport characteristics of the crack can be calculated using a static or dynamic networkapproach. We apply the static approach for simulating mercury intrusion into a crack and thedynamic model for simulating transient water uptake into a crack.

3.1. Mercury intrusion into a crack

The static network or percolation approach assumes a series of stationary distributions of thewetting and non-wetting fluid at different imposed capillary pressures pc (the capillary pressureis defined as the pressure difference between the two fluids). The basic rule of filling of a PPE bya wetting or non-wetting fluid (also called the occupancy criterion) is based on Laplace formula

pcðbÞ ¼2g cos y

bð2Þ

with y the contact angle and g the interfacial surface tension between the two fluids. Forsimplicity we assume further that in the case of liquid water the liquid perfectly wets the porousmaterial (or cos y ¼ 08), which is a valid assumption for most porous building materials. If theimposed capillary pressure exceeds pcðbÞ the PPE is occupied by the non-wetting fluid (in thiscase mercury, y ¼ 1308). In the reverse case, the wetting fluid (air) occupies the segment.Invasion percolation also accounts for the physics of two-phase flow: i.e. a crack segment canonly be occupied by the infiltrating fluid (mercury) if a continuous mercury path exists from thesource (the outside) to the crack segment under consideration (accessibility criterion). Figure 10

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(a) (b)

Figure 9. (a) Volume distribution of crack width for 3D crack and 2D cross sections; and (b) connectivitydistribution for 3D crack and 2D cross sections.

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shows the intrusion of mercury into the crack from the right side at increasing pressures. Weassume no flow at the lateral boundaries and no trapping of air, since it is supposed that air canalways escape from a fracture segment. We observe that mercury preferentially fills theconnected coarse crack segments, while only at high pressures the finer crack fraction is filled bymercury. The secondary cracks are only filled at higher pressures when fine crack segmentsbecome filled and create passages to them (this phenomenon is known as the ink-bottle effect).The example clearly shows that the filling mechanism has a 3D character: not the average crackwidth, but the particular composition of small and coarse crack segments, by-passingconnective crack pathways highly influences the resulting mercury intrusion process.

3.2. Infiltration of water into a crack

In the dynamic network approach we consider the transient mass transfer of water in a crack.To solve the liquid water uptake in a fracture, we use the moving front technique, which tracksthe movement of liquid waterfronts in the active links of the network. We assume that air caneasily escape and remains at constant atmospheric pressure. Liquid water transport in the crackcan then be modelled as a one-phase problem, where a waterfront moves through the void spaceof the crack. The suction of the cement matrix surrounding the crack is neglected. In Reference[35] it was shown that only for cracks with a crack width smaller than 0:02 mm; the suction ofthe matrix results in a retardation of the movement of the water front into the crack. The crack

Figure 10. Mercury intrusion from right side into the crack. Filling patterns atincreasing intrusion pressure.

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analysed in this paper has a minimal crack width of around 0:1 mm (the spatial resolution is114 mm), which means matrix suction may be neglected.

During the water uptake process in the crack network, an element can be in three differentstates: completely filled with water, empty (filled with air) and partly filled with water(Figure 11). In a partly filled or active element, the water front moves over a distance Dx duringa time step Dt: The x-axis is defined as the local axis in the length direction of the link. Theprogress of the water front Dx during the time step Dt is calculated according to Poiseuille’sequation:

ux ¼@x

@tffi

DxDt

¼ �k

rl

DplDx

þ rg cos f� �

; k ¼rlb

2

12mlð3Þ

with ux the velocity, k the permeability, g the gravity constant, f the slope with the verticaldirection, ml the dynamic viscosity, rl the density of the liquid and Dpl the liquid pressuredifference over the water filled fraction of the active crack segment (xactive). It is noted that weimplicitly assume that the equation for the permeability k (known as cubic law) holds for eachpoint in the variable aperture fracture with rough walls. Cubic law is however derived under theassumption that the fracture consists of two smooth, parallel plates. In Vandersteen [20] adetailed literature analysis on the validity of cubic law is given, concluding that cubic law can beused under the assumption of small variations both in tortuosity and aperture. Severalconditions are formulated for evaluation this assumptions. Based on this analysis we mayconclude that cubic law in average sense holds for the cracks determined in this study.

At the water front, the liquid pressure corresponding to the capillary pressure pcðbÞ isimposed. The capillary pressure pcðbÞ is given by Laplace’s equation (2). We assume every gaspressure build-up to vanish rapidly, so that the gas pressure in the non-filled part remainsconstant. The nodal pressures in Equation (3) are obtained by solving the mass balance equationfor the nodes belonging to the water filled fraction of the network:

XQij ¼ 0 with Qij ¼ Sij

rlb2i

12ml

DpijLij

þ rlg cos fij

� �ð4Þ

At the water uptake surface of the crack a prescribed liquid pressure is assumed ðpl ¼ 0Þ: Weassume no flow at the lateral boundaries.

tn ∆xntn+∆tn

active element

filled element

unfilled element

i

i-1

i+1pc

pl,i

pl,i-1

x

xactive

Figure 11. Moving front technique for water transport in a crack.

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An adaptive time-stepping procedure is used. The time step Dtnþ1 ¼ tnþ1 � tn is adapted asfollows [30, 35]:

Dtnþ1 ¼xactiveDtn

p1 maxðDxn;xactive=p2Þð5Þ

with xactive the length of the active fracture segment, Dxn the progress of the waterfront in thefracture during the previous time step Dtn; and p1 and p2 parameters controlling the time steppingprocess. The parameter p2 controls the number of time steps per fracture segment, whileparameter p1 scales the new time step based on the previous time step. Since cumulative inflowduring a water uptake process is proportional to the square root of time, the time-step canincrease during the simulation. When the waterfront reaches a following crack segment, the waterheight is set to a very small value in the next fracture segment before starting the next pressurecalculation (Equation (4)). This is done in order to avoid singularities in the pressure equations.

4. RESULTS

4.1. One-dimensional simulations

In this section we demonstrate the important influence of the crack width variation on wateruptake into a crack. The crack in this analysis has a length of 74 mm: The variable crack widthis generated randomly assuming a lognormal crack width distribution. We simulated two cracks(Figure 12(a)) with the same mean fracture aperture (0:86 mm) and same trend, but only variedthe standard deviation. Figure 12(b) gives the evolution in time of the height of the waterfrontfor the two cracks and compares the results with the simulation of water uptake in a crack withconstant crack width (0:86 mm). The crack with constant crack aperture overestimates theheight of the waterfront. A succession of crack segments with varying crack width results in aretardation of the waterfront movement in the crack. With increasing standard deviation theretardation effect increases. Retardation occurs when the waterfront passes from crack segmentswith small crack width to coarse crack segments. This retardation effect is further exemplified byincorporating a narrowing (b ¼ 0:25 mm instead of 0:42 mm) in the middle of the crack with stdev ¼ 0:3 mm (Figure 12(a)). We observe that when the waterfront passes through thenarrowing (time t1 in Figure 12(b)), the water uptake speeds up due to higher capillary pressurein the active narrowing (see Laplace law Equation (2)). Once the narrowing is passed (time t2),the uptake process considerably slows down due to the low permeability of the narrowing (seeEquation (3)). The narrowing acts as a high resistance against liquid flow. In addition thecapillary pressures in the coarse crack segments after the narrowing are lower enlarging theretardation effect. The retardation effect is most pronounced in 1D simulations. It is howeverreadily understood that in 2D and 3D retardation zones can be by-passed by crack pathwayswith more uniform crack width. The influence of dimensionality of the crack is analysed in thesubsequent sections.

4.2. Two-dimensional analysis and mesh sensitivity

In this section we analyse characteristics of liquid water uptake in a 2D artificial crack withvarying crack width [20]. The crack is randomly generated from a correlated lognormaldistribution with a mean crack width of 0:1 mm; a variance of 0:05 mm and a correlation length

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constant σb = 0.2 mm σb = 0.3 mm

0.86 mm

narrowing

t1

t2

(a)

(b)

Figure 12. Water uptake in a 1D crack: (a) crack width variation in two cracks with sametrend: st dev ¼ 0:2 mm and st dev ¼ 0:3 mm: In a variant a narrowing (b ¼ 0:25 mm) isincorporated in the crack with st dev ¼ 0:3 mm; and (b) Front position as function of time fora crack with constant crack width, with a varying crack width and with a built-in narrowing.

Figure 13. Liquid water distributions in a 2D crack with varying crack width.

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of 0:4 mm: The crack is composed of 145� 290 planar square crack segments of equal sizeð0:055 mmÞ: Figure 13 shows the water filling at different time steps. We observe that parts ofthe crack become not filled by water. The unfilled parts are coarse crack zones where thecapillary pressure}the driving force for filling}is so low that no water will be sucked out offiner crack parts. The coarse crack segments are however by-passed by finer crack pathsresulting in preferential flow and fingering.

We also analyse the mesh sensitivity of the dynamic network approach. We generate a 2Dlognormal crack (mean crack width ¼ 0:076 mm; variance ¼ 0:002 mm) of 20� 20 crack segmentsof equal size (0:1 mm). The size of the reference network equals the crack segment map: a 20� 20network. A second finer network is constructed from the same crack segment map by dividing acrack segment in four (the aperture remains unchanged) resulting in a 40� 40 network. Figure 14compares the filling patterns for the two meshes. We observe a mesh sensitivity, but the deviationbetween the two results remains between acceptable limits. The mesh dependence is caused by theincrease of possible connective pathways between two nodes when refining the mesh.

4.3. Three-dimensional versus two-dimensional simulations

We simulate water uptake in the 3D crack network as determined in Section 2.4 (Figure 8(a)).Water uptake takes place from the right side, while at the lateral sides no flow conditions areimposed. We compare the 3D result with the 2D simulation of the sections 2Da and 2Db(Figure 5(c)). To represent the results we divide the crack in 10 equidistant zones and calculatethe average degree of saturation per zone (fraction of the void space filled by liquid water). Thesaturation profiles for different time steps are given in Figure 8(c) for the 3D case and in Figure5(d) for the 2D case. Also the final profile during uptake is given. We observe that the crack isonly partly filled (degree of saturation lower than 100%) and that after some time thepenetration completely halts. Detailed comparison with the crack network configuration

Figure 14. Mesh sensitivity analysis of the liquid water distribution in a 2D variable aperturecrack after 1 and 4 s: The solid lines give the water front position in the finest mesh.

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indicates that penetration stops, when coarse cracks segments are reached by the waterfront.In these coarse segments the capillary pressure}the driving force for water filling}is so lowthat no water will be sucked out of water-filled finer crack parts. In this case water-filling of acrack can only proceed, when the coarse crack segments are by-passed by smaller crackpathways. Figures 8(c) and 5(d) show that in 3D the water further penetrates into the crackcompared to the 2D cases. The higher connectivity values and higher fraction of fine cracks in3D (see Figure 9) explain the higher change of further water penetration by-passing coarse cracksegments. In 2D, we further observe that the water penetration highly depends on theconsidered section. In 2Da water penetrates until the fourth zone (compared to the eighth zonein 3D), while in 2Db the penetration is limited to the second zone and already stops after 1:7 s:

These simulations clearly show that water uptake in complex cracks, such as in concretewhere important secondary side-band cracking and crack bridging may occur, is essentially 3D.Simulations based on 2D sections limit too strongly possible pathways for by-passing flowleading to an important underestimation of penetration depths. The underestimation ofpenetration depth may strongly affect predictions of durability of cracked structures.

4.4. Model validation

Water uptake in cracked concrete is a rapid process, which is difficult to experimentally monitorby non-destructive techniques like NMR, gamma- or X-ray attenuation. Water suction from thecrack to the matrix however can strongly retard the waterfront movement in the crack itself.Therefore, for model validation we monitor with X-ray radiography the water uptake process ina cracked material showing a high permeability of the matrix: e.g. ceramic brick has a capillaryabsorption coefficient (measure for the rate of water uptake) of one order of magnitude higherthan concrete. Figure 15 gives the evolution in time of the water distribution for a naturalfracture with mean aperture of 0:1 mm [35]. Besides the water present in the matrix, the filledpart of the fracture is clearly visible. The waterfront first quickly penetrates into the specimenand afterwards the crack acts as an extra water source for the surrounding matrix. Roels et al.[35] used the dynamic crack network model with varying crack width as presented above andcombined this discrete model with a continuum finite element model for simulating the suctionin the matrix (for details see Roels et al. [35]). In this paper, we present only the resultsindicating an experimental validation of the presented crack network model. For simplification,a perfectly vertical crack was assumed in the simulation limiting the space to half of thespecimen. Figure 15 gives the results and indicates a good agreement between measurement andsimulation for the height of rise as well as the water profiles in the matrix.

Figure 15. Measured (left) and simulated (right) waterfront in a naturally fractured bricksample with a mean aperture of approximately 0:1 mm: At the left side of the simulated liquid

water distribution the predicted height of the water front in the crack is presented.

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5. CONCLUSIONS

X-ray tomography can be used to monitor 3D cracks in concrete. A threshold procedure,calibrated with a more advanced crack aperture determination procedure, is used to construct avoxel map of the crack void space. The crack void space is divided in crack segments with anunique crack width using a segmentation technique: maximum non-overlapping balls arepositioned in the void space and subsequently expanded to crack segments. The crack segmentmap is used for constructing a hydraulic network of the crack using parallel plates as basicnetwork elements. Crack aperture and connectivity distributions can be determined from thesegmentation map. In a 2D cross-section, the amount of fine pores and the connectivity of thecrack are underestimated.

The network representation of cracks enables the analysis of flow characteristics of cracksusing static (invasion) percolation approaches or dynamic moving front techniques. Bothapproaches show that flow into concrete cracks is essentially a three-dimensional phenomenon.We observe partial filling of the crack space (air entrapment), preferential flow in crackssegments with uniform crack width, and retardation or halt of the water uptake process. One-dimensional simulation revealed that essentially small crack segments followed by coarser cracksegments results in an important retardation of the uptake process. Simulations in 2D and 3Dshow that the retardation (or halt) process can be by-passed when fine side-band cracks andsecondary cracks with more uniform crack width are present. Two-dimensional simulation ofwater uptake in complex crack systems can strongly underestimate the penetration depths offluids.

The findings of this paper indicate that models for reliable durability predictions should dealwith the stochastic nature of cracks: i.e. variations in crack width and connectivity. Thisstochastic nature of cracks originates from damage processes in the composite structure ofconcrete. Accurate prediction of crack development including details on crack topology andcrack geometry requires modelling on the mesoscale, where the different constituents of concreteare geometrically represented with their specific material properties. New potentials are situatedin the extension of macroscopic models to a multiscale approach, where simulations on themacroscale are enriched by information coming from detailed simulations on the mesoscale.

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