3d Liquid Transport

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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 the

segmentation 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 surface

storage reservoirs, underground structures and buildings). Fractures or cracks may show

complex structures including crack bridging, crack branching and bifurcation, side-band

cracking and secondary cracking parallel to main cracks. Especially in composite materials as

concrete, composed of a random arrangement of aggregates, a surrounding matrix with built-in

air voids and a weaker interfacial zone, crack networks with varying crack width and

connectivity may develop. Cracks are preferential pathways for liquid water leading to an

accelerated 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 coupled

hygrothermal, 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 of

the permeability. Most models are continuum-based formulations, where localization is handled

using 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 in

literature [8–10]. Simplified analytical formulas are formulated for taking into account crack

width variation. Zimmerman et al . [11] investigated a number of analytical solutions for the

prediction of the saturated hydraulic conductivity of rough-walled fractures. They concluded

that 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 an

equivalent continuum [12–17]. The continuum models, however, are not appropriate when the

scale of interest is small or the water-bearing fractures are few leading to preferential penetration

of 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 flow

channels 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 channels

exist: 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 a

stochastically generated description of the variable fracture aperture to numerically investigate

various aspects of two-phase flow in rough-walled fractures using various types of network

modelling. Pruess and Tsang [27], Mendoza [28] developed static invasion percolation models

for the determination of relative permeability versus saturation relationships and capillary

pressure versus saturation relationships. Murphy and Thomson [29] developed a discrete model

for 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 dissolved

contaminants 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 by

two-dimensional section analysis. Using stereological methods this 2D information can be used to

infer some of the 3D characteristics. Techniques such as serial sectioning and more recently X-ray

computed 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-dimensional

concrete cracks. In a first part, the 3D fracture is measured using microfocus computer X-ray

tomography ðmCTÞ: Using a calibrated threshold and segmentation technique a 3D network

representation of crack space is constructed. Finally, static invasion percolation and the moving

front technique are respectively used to study mercury intrusion and water uptake processes in

concrete 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 in

concrete. To create a fracture, a cylindrical specimen (diameter of 110 mm and a thickness of

Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2004; 28:671–687

J. CARMELIET ET AL.672



20 mm) is loaded in a tensile splitting test until failure. The concrete specimen is imaged on a 3D

AEA mCT apparatus developed at K.U.Leuven (Figure 1). A polychromatic X-ray cone beam

produced 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 keV

X-rays. After conversion of the attenuated X-rays to light by a fluorescent screen, they arerecorded by a 12 bit CCD camera of 1024 Â 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 and

object-to-detector distance of 455 mm; resulting in a pixel size s p of 114 mm for the recorded

image. The images obtained by the cone beam configuration are automatically transferred into a

parallel beam configuration during measurement. For each sample 180 2D images were taken

over 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 filtered

backprojection algorithm is used to reconstruct a 3D image of the sample from the set of 2D

projections.

The 3D reconstruction was limited to a region of interest around the crack (Figure 2(a)). The

images 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.


LIQUID TRANSPORT IN CONCRETE CRACKS 673



line artefacts (for a detailed description, see Van Geet [31]). The size of the concrete sample

shown in Figure 2(b) is 55 Â 27:5 Â 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 essence

a three-dimensional feature, since the crack attempts to follow the weaker aggregate-mortar

interface, 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 fracture

with an (ideal) parallel-plate configuration. Theoretically, the attenuation profile over such a

fracture is Gaussian resulting from the convolution of a theoretical rectangular fracture profile

with the line spread function (LSF) (Figure 3). The line spread function is the spatial derivative

of the edge response function (ERF) representing the smooth transition at the edge material/air

and which is a measure for the spatial resolution. Knowledge of the LSF or ERF enables the

determination of the crack aperture by deconvolution of the measured profile. In reality, thetheoretical profile is contaminated by noise and the application of deconvolution results in

highly uncertain fracture widths. Therefore, characteristic parameters such as peak height (PH),

the full width at half maximum (FWHM) and missing attenuation (MA)}surface under the

attenuation profile}of the fracture attenuation profile are used as calibration parameters for

the quantification of the fracture aperture. Vandersteen [20] showed these techniques to be

advantageous for the determination of planar crack structures when high quality images

showing a good signal-to-noise ratio are available. For non-planar fracture structures showing

high variations in crack direction, crack width and connectivity, the automatic application of

these techniques is less straightforward. A second technique consist in thresholding the grey

scale 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).





surface of the specimen. The threshold limit was determined using the first calibration

technique. Errors may be included by the fact that the location at the surface is not exactly

matched (the mCT measurement is always just beneath the surface) and due to inaccuracies in

the visual microscopic technique, as an exact delineation of the edges of the fracture is difficult

to evaluate. Using the thresholding technique the spatial resolution for identifying crack widths

equals the pixel size of 114 mm: A more detailed discussion on the spatial resolution which can

be 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 using

the threshold technique. At the right hand side (i.e. the middle of the concrete sample), a main

crack is observed. Detailed analysis shows that the crack width is almost uniform. The crack

follows the aggregate boundaries in an ellipsoidal shape. At the left side (i.e. the surface of the

specimen), the crack closes and shows a poor connectivity. The transition zone is characterised

by a complex structure. We observe one main crack and a lot of secondary side-band cracking

and macroporosity following the aggregates and air voids present in the specimen. The

secondary cracking shows to be less connected with high variations in crack aperture. Figure

5(a) shows as an example 2D cross-sections at two different positions (2Da and 2Db). We notice

in section 2Db that the main crack shows higher variations in crack width and more secondary

cracking compared to section 2Da. The influence of these features (3D versus 2D sections, crack

width 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 converted

into 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.





space V or to the solid matrix. The aim of the segmentation procedure is to divide the void space

into 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 to

the same class of a local crack aperture size. The local aperture size bð pÞ for a point p of the void

space V is defined as the diameter d of the maximum ball B with centre c; which can be built in

the void space including the point p (Figure 6(a)), or

bð pÞ ¼ maxfd j Bðc; d Þ & V ; p 2 Bðc; d Þg ð1Þ

The segmentation of the void space into crack segments is based on two steps. First we search

for 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 centred

on the skeleton of the void space. The void skeleton is determined using a Voronoi tessellation

procedure [32]. Using the determined skeleton, we find the arrangement of non-overlapping

maximum 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 are

simultaneously expanded to crack segments using a growing region algorithm (Figure 6(c)). In

this expansion process the shape of the balls is not conserved, but as if balloons were inflated in

the void space the crack segments adopt a more irregular shape (or with each ball one crack

segment 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.





an expanded ball or crack segment: a crack segment map of the void space is obtained. Efficient

algorithms for extracting a crack segment map from a voxel representation of the void space are

proposed 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 at

the centre of the interface between two crack segments i and j (Figure 7). The interface has a

surface S ij : The distance between the centre of crack segment i and the centre of interface ij is

denoted 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 E ij connects

the nodes i and j and is composed of two parallel plate elements: PPEij between node i and the

centre of the interface ij and PPE ji ; 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 the

crack segment i : A network element thus consists of two parallel plate elements PPE ij and PPE ji ;;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 lines

between the centres of the crack segments represented as balls. The size of the ball resembles the

crack aperture. Figure 5(c) gives the network for the 2D sections.

Based on the network representation the crack width and connectivity can be evaluated. The

crack width distribution gives the relative volume of crack segments with a crack width bi

between ð2i À 1Þsp and ð2i þ 1Þsp; with sp the pixel size. We observe in Figure 9(a) that in 3D a

higher volume fraction of small cracks is present compared to the 2D distributions. The

connectivity cij of the element E ij is defined as the number of elements connected to the element

E ij : 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. Connectivity

cij ¼ 2 indicates a one way flow through the crack segment, while higher values show multiple

flow connections. We observe in Figure 9(b) that the connectivity values in 3D range from 2 to

15 showing a high degree of connectivity. In 2D a higher fraction of isolated and dead end crack

segments 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.





We conclude that in 3D the crack shows a higher connectivity and a higher fraction of small

cracks. In 2D, information on connectivity is lost resulting in a higher fraction of isolated and

dead 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.





3. STATIC AND DYNAMIC FLOW IN CRACKS

The transport characteristics of the crack can be calculated using a static or dynamic network

approach. We apply the static approach for simulating mercury intrusion into a crack and the

dynamic 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 the

wetting and non-wetting fluid at different imposed capillary pressures pc (the capillary pressure

is defined as the pressure difference between the two fluids). The basic rule of filling of a PPE by

a 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. For

simplicity we assume further that in the case of liquid water the liquid perfectly wets the porous

material (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 this

case 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 can

only be occupied by the infiltrating fluid (mercury) if a continuous mercury path exists from the

source (the outside) to the crack segment under consideration (accessibility criterion). Figure 10

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

d i s t r i b u t i o n

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

connectivity (-)


3D

2Da

2Db

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5 3 3.5 4


0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5 3 3.5 4


0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5 3 3.5 4

crack width (mm)


3D

2Da

2Db

(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.





shows the intrusion of mercury into the crack from the right side at increasing pressures. We

assume no flow at the lateral boundaries and no trapping of air, since it is supposed that air can

always escape from a fracture segment. We observe that mercury preferentially fills the

connected 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 segments

become 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 crack

width, but the particular composition of small and coarse crack segments, by-passing

connective 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 tracks

the 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 crack

can then be modelled as a one-phase problem, where a waterfront moves through the void space

of 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 of

the 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.





analysed in this paper has a minimal crack width of around 0 :1 mm (the spatial resolution is

114 mm), which means matrix suction may be neglected.

During the water uptake process in the crack network, an element can be in three different

states: 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 during

a time step Dt: The x-axis is defined as the local axis in the length direction of the link. The

progress of the water front Dx during the time step Dt is calculated according to Poiseuille’s

equation:

ux ¼@x

@tffi

Dx

Dt¼ À

k

rl

D pl

Dxþ r g cos f

; k ¼

rl b2

12ml

ð3Þ

with ux the velocity, k the permeability, g the gravity constant, f the slope with the vertical

direction, ml the dynamic viscosity, rl the density of the liquid and D pl the liquid pressure

difference over the water filled fraction of the active crack segment (xactive). It is noted that we

implicitly 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 the

assumption that the fracture consists of two smooth, parallel plates. In Vandersteen [20] a

detailed literature analysis on the validity of cubic law is given, concluding that cubic law can be

used under the assumption of small variations both in tortuosity and aperture. Several

conditions are formulated for evaluation this assumptions. Based on this analysis we may

conclude 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 is

imposed. The capillary pressure pcðbÞ is given by Laplace’s equation (2). We assume every gas

pressure build-up to vanish rapidly, so that the gas pressure in the non-filled part remains

constant. The nodal pressures in Equation (3) are obtained by solving the mass balance equation

for the nodes belonging to the water filled fraction of the network:

XQij ¼ 0 with Qij ¼ S ij

rl b2i

12ml

D pij

Lij

þ rl g cos fij

ð4Þ

At the water uptake surface of the crack a prescribed liquid pressure is assumed ð pl ¼ 0Þ: We

assume no flow at the lateral boundaries.

t n ∆ xn

t n+∆t n

active element

filled element

unfilled element

i

i-1

i+1

pc

pl,i

pl,i-1

x

xactive

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





An adaptive time-stepping procedure is used. The time step Dtnþ1 ¼ tnþ1 À tn is adapted as

follows [30, 35]:

Dtnþ1 ¼xactiveDtn

p1 maxðDxn; xactive= p2Þð5Þ

with xactive the length of the active fracture segment, Dxn the progress of the waterfront in the

fracture during the previous time step Dtn; and p1 and p2 parameters controlling the time stepping

process. The parameter p2 controls the number of time steps per fracture segment, while

parameter p1 scales the new time step based on the previous time step. Since cumulative inflow

during a water uptake process is proportional to the square root of time, the time-step can

increase during the simulation. When the waterfront reaches a following crack segment, the water

height is set to a very small value in the next fracture segment before starting the next pressure

calculation (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 water

uptake into a crack. The crack in this analysis has a length of 74 mm: The variable crack width

is 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 varied

the standard deviation. Figure 12(b) gives the evolution in time of the height of the waterfront

for the two cracks and compares the results with the simulation of water uptake in a crack with

constant crack width (0:86 mm). The crack with constant crack aperture overestimates the

height of the waterfront. A succession of crack segments with varying crack width results in a

retardation of the waterfront movement in the crack. With increasing standard deviation the

retardation effect increases. Retardation occurs when the waterfront passes from crack segments

with small crack width to coarse crack segments. This retardation effect is further exemplified by

incorporating a narrowing (b ¼ 0:25 mm instead of 0:42 mm) in the middle of the crack with st

dev ¼ 0:3 mm (Figure 12(a)). We observe that when the waterfront passes through the

narrowing (time t1 in Figure 12(b)), the water uptake speeds up due to higher capillary pressure

in 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 (see

Equation (3)). The narrowing acts as a high resistance against liquid flow. In addition the

capillary pressures in the coarse crack segments after the narrowing are lower enlarging the

retardation effect. The retardation effect is most pronounced in 1D simulations. It is however

readily understood that in 2D and 3D retardation zones can be by-passed by crack pathways

with more uniform crack width. The influence of dimensionality of the crack is analysed in the

subsequent sections.

4.2. Two-dimensional analysis and mesh sensitivity

In this section we analyse characteristics of liquid water uptake in a 2D artificial crack with

varying crack width [20]. The crack is randomly generated from a correlated lognormal

distribution with a mean crack width of 0:1 mm; a variance of 0:05 mm and a correlation length





0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

time (s)

f r o n t p o s i t i o n ( m m )

-0.8

-0.4

0

0.4

0.8

0 10 20 30 40 50 60 70 80

x-axis (mm)

-0.8

-0.4

0

0.4

0.8σ

b = 0.2 mm

σ b

= 0.3 mmnarrowing (b=0.25 mm)

constant σ b

= 0.2 mm σ b

= 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.





of 0:4 mm: The crack is composed of 145 Â 290 planar square crack segments of equal size

ð0:055 mmÞ: Figure 13 shows the water filling at different time steps. We observe that parts of

the crack become not filled by water. The unfilled parts are coarse crack zones where the

capillary pressure}the driving force for filling}is so low that no water will be sucked out of

finer crack parts. The coarse crack segments are however by-passed by finer crack paths

resulting in preferential flow and fingering.

We also analyse the mesh sensitivity of the dynamic network approach. We generate a 2D

lognormal crack (mean crack width ¼ 0:076 mm; variance ¼ 0:002 mm) of 20 Â 20 crack segments

of equal size (0:1 mm). The size of the reference network equals the crack segment map: a 20 Â 20network. A second finer network is constructed from the same crack segment map by dividing a

crack segment in four (the aperture remains unchanged) resulting in a 40 Â 40 network. Figure 14

compares the filling patterns for the two meshes. We observe a mesh sensitivity, but the deviation

between the two results remains between acceptable limits. The mesh dependence is caused by the

increase 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 are

imposed. 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). The

saturation profiles for different time steps are given in Figure 8(c) for the 3D case and in Figure

5(d) for the 2D case. Also the final profile during uptake is given. We observe that the crack is

only partly filled (degree of saturation lower than 100%) and that after some time the

penetration 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.





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 low

that no water will be sucked out of water-filled finer crack parts. In this case water-filling of a

crack can only proceed, when the coarse crack segments are by-passed by smaller crack

pathways. Figures 8(c) and 5(d) show that in 3D the water further penetrates into the crack

compared 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 crack

segments. In 2D, we further observe that the water penetration highly depends on the

considered section. In 2Da water penetrates until the fourth zone (compared to the eighth zone

in 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 concrete

where 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 flow

leading to an important underestimation of penetration depths. The underestimation of

penetration 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 monitor

by non-destructive techniques like NMR, gamma- or X-ray attenuation. Water suction from the

crack 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 in

a cracked material showing a high permeability of the matrix: e.g. ceramic brick has a capillary

absorption coefficient (measure for the rate of water uptake) of one order of magnitude higher

than concrete. Figure 15 gives the evolution in time of the water distribution for a natural

fracture with mean aperture of 0:1 mm [35]. Besides the water present in the matrix, the filled

part of the fracture is clearly visible. The waterfront first quickly penetrates into the specimen

and 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 and

combined 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 results

indicating 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 the

specimen. Figure 15 gives the results and indicates a good agreement between measurement and

simulation 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.





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 a

voxel 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 are

positioned in the void space and subsequently expanded to crack segments. The crack segment

map is used for constructing a hydraulic network of the crack using parallel plates as basic

network elements. Crack aperture and connectivity distributions can be determined from the

segmentation map. In a 2D cross-section, the amount of fine pores and the connectivity of the

crack are underestimated.

The network representation of cracks enables the analysis of flow characteristics of cracks

using static (invasion) percolation approaches or dynamic moving front techniques. Both

approaches 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 cracks

segments 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 3D

show that the retardation (or halt) process can be by-passed when fine side-band cracks and

secondary cracks with more uniform crack width are present. Two-dimensional simulation of

water uptake in complex crack systems can strongly underestimate the penetration depths of

fluids.

The findings of this paper indicate that models for reliable durability predictions should deal

with the stochastic nature of cracks: i.e. variations in crack width and connectivity. This

stochastic nature of cracks originates from damage processes in the composite structure of

concrete. Accurate prediction of crack development including details on crack topology and

crack geometry requires modelling on the mesoscale, where the different constituents of concrete

are geometrically represented with their specific material properties. New potentials are situated

in 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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3d Liquid Transport