A lattice model for liquid transport in cracked unsaturated heterogeneous porous materials

VIII International Conference on Fracture Mechanics of Concrete and Concrete StructuresFraMCoS-8

J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang (Eds)

A LATTICE MODEL FOR LIQUID TRANSPORT IN CRACKEDUNSATURATED HETEROGENEOUS POROUS MATERIALS

Peter Grassl∗, Caroline Fahy∗, Domenico Gallipoli† and John Bolander‡

∗University of GlasgowSchool of Engineering, Glasgow, UK,

Rankine Building, University of Glasgow, G12 8QQ,emails: [email protected], [email protected]

†Universite de Pau et des Pays de l’Adour, FranceLaboratoire SIAME, France

email: [email protected]

‡University of California, DavisOne Shields Avenue, Davis, CA 95616, USA

email: [email protected]

Key words: Concrete, Fracture mechanics, Unsaturated flow, Meso-scale, Lattice, Coupling

Abstract. Fracture increases the permeability of cementitious materials, which is known to acceleratetheir deterioration when exposed to aggressive environments. The coupling of fracture and flow istherefore important for modelling the durability of these materials. Lattice models are suitable forthis in that they describe well the discontinuities that arise from fracture processes and are capableof modelling flow accurately. This paper describes progress made in developing an approach tocouple irregular lattices of beam-like elements representing mechanical behaviour (i.e. elasticityand fracture) and conduit elements representing mass transport. The transport part of the model isused to describe absorption of water in concrete using principles developed originally for unsaturatedsoils. The effect of fracture on transport is considered by adjusting the transport properties of theconduit elements according to a cubic function of the crack opening. The modelling is applied toseveral benchmarks and extended to consider the meso-structure of concrete in the form of randomlyarranged circular inclusions. The influence of inclusion volume fraction on the change of watercontent is studied.

1 Introduction

Concrete is one of the most common build-ing materials to suffer from water ingress.Given that concrete is widely used in all as-pects of construction, transport models are of-ten needed for predicting the extent of moisturemovement and, hence, to gauge the condition ofstructures. Furthermore, it is known that frac-ture facilitates the water ingress. In this context,a mass transport model is proposed to describe

the movement of liquid through partly saturatedporous fractured concrete. The proposed modelis based on a lattice approach that uses a ran-domly generated dual Delaunay and Voronoitessellation of the material domain [1, 2]. Theconstitutive retention and permeability laws ofthe porous material are based on the hydraulicfunctions originally proposed by [3] and [4] forunsaturated soils and subsequently modified by[5]. The effect of fracture on transport is con-

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Peter Grassl, Caroline Fahy, Domenico Gallipoli and John Bolander

sidered by adjusting the transport properties ofthe conduit elements according to a cubic func-tion of the crack opening [6]. The lattice modelincorporating the above constitutive laws is ap-plied to two benchmark problems (in 1D and2D, respectively) to demonstrate its numericalrobustness and ability to describe liquid move-ment through porous media.

2 Lattice modelThe present two-dimensional numerical

model for the coupling of the movement of liq-uid under the gradient of hydraulic potential andfracture is based on lattices of one-dimensionalstructural and conduit elements. The spatialarrangement of the lattice elements and theircross-sectional properties are based on Delau-nay and Voronoi tessellations of the domainshown in Figure 1.

For the mechanical lattice, the elements areplaced on the edges of the Delaunay triangles.The geometry of the mid cross-section of thelattice elements is determined by the corre-sponding edge of the Voronoi polygon of lengthl. Each node has three degrees of freedom,that is two translations u and v and one rota-tion φ, which determine the displacement jumpat the centroid C of the element’s mid cross-section (Figure 2). The displacement jumps aretransformed into strains by the element lengthh. An isotropic damage model is used to re-late the strains to stresses. The evolution ofdamage is controlled by a stress-crack openingcurve, so that the mechanical response is in-dependent of the length of the lattice elementsused. A more detailed description of the me-chanical lattice and constitutive model can befound in [1] and [2], respectively. Transport ele-ments are placed along the facets of the Voronoipolygons and their cross-sectional areas are cal-culated from the length of the correspondingedges of the dual Delaunay triangles. The mate-rial response of the transport elements is calcu-lated, as for the structural elements, at point C(Figure 2). This simplifies the transfer of in-formation from the structural to transport latticeand vice versa.

For the application of the lattice model in thiswork, both homogeneous and heterogeneousmaterial properties are considered. The hetero-geneity of the material is represented by ran-domly arranged circular inclusions with varyingdiameter which are projected onto the lattice.Depending on the position of lattice elementsrelative to the inclusions, the properties of theelements represent either those of the matrix orthe inclusions. For the interface between ma-trix and inclusions, the lattice nodes are placedat special locations, such that the mid cross-sections of the mechanical lattice elements formthe boundaries between inclusions and matrix[7]. The arrangement of the mechanical latticeelements with respect to the inclusions is shownin Figure 3a. In the present work, inclusionsare assumed to be impermeable and are, there-fore, not discretised by transport elements (Fig-ure 3b).

Transport elements are idealised as one-dimensional conductive pipes [7]. The gra-dient of hydraulic head, which governs flowrate along each transport element, is determinedfrom the capillary pressures Pc at the two nodes.Figure 2 shows one transport element togetherwith the capillary pressures acting at the nodesand the associated cross-sectional area as ob-tained from the dual Delaunay triangle. In thiswork we restrict our attention to the case wherepore water is in tension and, in the following,liquid pressure is therefore referred to as cap-illary pressure. The mass balance equation de-scribes the change in moisture inside a porouselement as a consequence of liquid flow andsolid-liquid retention. A positive sign is as-sumed for liquid tension, unlike the conventionof soil mechanics which assumes compressionpositive. For the balance equation, the follow-ing differential expression is obtained:

c∂Pc

∂t− kdiv

(∇(Pc

g− ρz

))= 0 (1)

where c is the mass capacity, k is the hydraulicconductivity, g is the acceleration of gravity, zis the capillary height and t is the time. Bound-ary conditions are imposed either as prescribed

2


values of capillary pressure (on boundary Γ1)or as prescribed values of flux (on boundaryΓ2). The latter boundary condition can thenbe related to the gradient of capillary pressurethrough Darcy’s law. This results in the follow-ing two mathematical constraints on Pc:

Pc = g (x) on Γ1 and∂Pc

∂n= f (x) on Γ2

(2)where n denotes the direction normal to theboundary while g(x) and f(x) are functions ofthe spatial coordinate vector x.

The discrete form of the differential equationfor mass transport in (1) for a one-dimensionaltransport element is

αePc + Ce∂Pc

∂t= fe (3)

where Pc is a vector containing the nodal val-ues of the capillary pressure, αe is the conduc-tivity matrix, Ce is the capacity matrix and feis the nodal flow rate vector. The influence ofthe capillary height z in (1) is modelled by aflow vector analogous to the way of consider-ing body forces in structural problems.

The capacity matrix is

Ce =A`

12c

(2 11 2

)(4)

where c is the capacity of the material. Further-more, the conductivity matrix is defined as

αe =A

`k

(1 −1−1 1

)(5)

where k is the hydraulic conductivity and A =hb is the cross-sectional area. Here, b is theout-of-plane thickness of the element assuminga rectangular cross-section.

3 Constitutive lawsThe mass transport equation in (1) and its

discrete form in (3) are based on the constitu-tive laws for the capacity c and the hydraulicconductivity k. The capacity c is defined as

c = −ρ ∂θ∂Pc

(6)

where θ is the volumetric water content whichis calculated by a modified version of vanGenuchten’s retention model proposed in [5].The presence of a crack in an element does notinfluence the capacity in the present model. Thevolumetric water content is

θ = Se (θs − θr) + θr (7)

where θr and θs are the residual and saturatedwater contents corresponding to effective satu-ration values of Se = 0 and Se = 1, respec-tively. The effective degree of saturation Se isdefined as

Se =

Ξ

(1 +

(Pc

a

) 11−m

)−m

if Pc ≥ Pc(aev)

1 if Pc < Pc(aev)

(8)in which a, m are model parameters. In (8),

Ξ =θm − θrθs − θr

(9)

where θm is an additional model parameter andPc(aev) is the air-entry value of capillary pres-sure which separates saturated (Pc < Pc(aev))from unsaturated states (Pc > Pc(aev)). Fromthe physical point of view, air breaks into an ini-tially saturated material when capillary tensionexceeds Pc(aev) (hence inducing a drop of de-gree of saturation). Conversely, water floods allpores of an initially unsaturated material whencapillary tension falls below Pc(aev) (hence at-taining a degree of saturation of one). It is intu-itive that the smaller the pore size of the mate-rial, the larger the value of Pc(aev) will be. Therole of parameter Ξ is illustrated in Figure 4which provides a schematic representation ofthe saturation-pressure law in (8).

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Figure 1: Dual lattice model for fracture and transport:Delaunay triangulation (solid lines) and Voronoi tessella-tion (dashed lines).

1 2h

Figure 2: Dual lattice model for fracture and transport:Mechanical and transport elements.

(a)

(b)

Figure 3: Lattice element arrangement around inclusions:(a) Mechanical elements and (b) transport elements.

00 Pc

Modified1

Se

Figure 4: Effective degree of saturation Se versus capil-lary pressure Pc for the modified van Genuchten model.

By setting Se = 1 in (8), the air-entry valueof capillary pressure Pc = Pc(aev) is calculatedas

Pc(aev) = a(Ξ1/m − 1

)1−m(10)

The hydraulic conductivity k in (5) consistsof

k = k0 + kc (11)

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where k0 is the hydraulic conductivity of the in-tact material and kc is the additional conductiv-ity due to cracking. The part k0 is expressed interms of intrinsic conductivity κ, relative con-ductivity κr, density of water ρ and absolute(dynamic) viscosity of water µ as

k0 =ρg

µκκr (12)

The relative conductivity κr is a function of theeffective degree of saturation and is defined as

κr =√Se

1−

[1−

(Se

Ξ

)1/m]m

1−

[1−

(1

Ξ

)1/m]m

2

(13)

If θm = θs, equation (9) reduces to Ξ = 1and (13) reduces to the expression of the rela-tive conductivity of the original van Genuchtenmodel. The cracking part is

kc = ξρg

µ

w3c

12h(14)

where ξ is a tortuosity factor taking into accountthe roughness of the crack surface, wc is theequivalent crack opening of the mechanical lat-tice and h is the length of the mechanical ele-ment.

4 Analyses4.1 1D analysis

To verify that the transport constitutive equa-tions of Section 3 are correctly implemented,the 1D capillary infiltration problem investi-gated by [5] was modelled again in the presentstudy. Table 1 gives the parameter values of themodified van Genuchten model [5] as used inthe simulations while Figure 5 shows the corre-sponding variations of volumetric water contentθ, water capacity c and relative conductivity κrwith capillary pressure Pc.

Prior to the start of the analysis, the 1 m soilcolumn in Figure 6a is assumed to be in hy-draulic equilibrium with an imposed capillary

pressure of 0.0981 MPa at the bottom (see Fig-ure 6b). This value of capillary pressure corre-sponds to an imposed pressure head of −10 mas specified in [5]. At time zero, atmosphericpressure was imposed at the bottom of the col-umn combined with a zero flux at the top (Fig-ure 6c), leading to an upward infiltration (i.e.an infiltration against gravity). As can be seenin Figure 7, the wetting fronts predicted by thelattice approach agree well with those presentedin [5].

0.25

0.3

0.35

0.4

0.0001 0.001 0.01 0.1 1 10

Vo

lum

etr

ic w

ate

r co

nte

nt,

θ

Pressure, Pc [MPa]

(a)

0

5

10

15

0.0001 0.001 0.01 0.1 1 10Wa

ter

ca

pa

city,

× 1

0-6

c (

1/m

m)

Pressure, Pc [MPa]

(b)

0.0001

0.001

0.01

0.1

1

10

0.0001 0.001 0.01 0.1 1 10

Re

lative

co

nd

uctivity,

κr

Pressure, Pc [MPa]

(c)

Figure 5: Constitutive responses used in the 1D analy-sis: (a) Volumetric water content versus capillary pres-sure, (b) water capacity versus capillary pressure and (c)relative conductivity versus capillary pressure.

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Table 1: Parameters for 1D soil column analysis.

θs 0.380θr 0.068θm 0.3803µ [t/(mms)] 1 × 10−9

κ [mm2] 5.6634 × 10−8

m 0.0826a [MPa] 0.0123

Figure 6: (a) Geometry, (b) Initial pressure distribution,i.e. prior to start (t < 0) and (c) Boundary conditions atstart of the analysis (t = 0).

0

0.02

0.04

0.06

0.08

0.1

0.12

0 200 400 600 800 1000

Pc[MPa]

y-coord [mm]

latticemodelVogel et al. (2000)

Figure 7: Comparison of capillary pressure versus spec-imen height at different times obtained by the latticemodel and against results reported by Vogel et al. [5] us-ing the modified van Genuchten model.

4.2 2D analysisThe dual lattice modelling approach pre-

sented in the previous section is applied to

model water absorption in cracked and un-cracked concrete and compared to experimen-tal results reported by Wittmann and co-workersin [8]. This experimental study consists of twoparts in which the bottom surfaces of uncrackedand cracked concrete beams are subjected tocontact with water. The absorption of water intothe unsaturated concrete was measured in theexperiments by means of neutron radiography.The idealised geometry for the present 2D mod-elling of the experiments is shown in Figure 8.For the cracked concrete beam, an equivalentcrack opening of wc = 0.35 mm at the bot-tom of the beam was used in accordance withthe experimental study. The model parametersused for the transport analyses are shown in Ta-ble 2. Here, the parameters a and m were takenfrom [9]. Since θs = θm in Table 2, the ex-pressions of capacity and relative conductivityin (6) and (13), respectively, reduce to those ofthe original van Genuchten model.

Table 2: Parameters for 2D concrete beam analysis.

θs 0.0924θr 0θm 0.0924µ [t/(mms)] 1×10−9

κ [mm2] 1×10−13

m 0.4396a [MPa] 18.6237ρ [t/(mm3)] 1×10−9

In the numerical analyses, the beams wereassumed to be partially saturated with an initialeffective saturation of Se = 0.5 and the resultsare presented in the form of a change of wa-ter content. The parameter θs = 0.0924 in Ta-ble 2 was chosen assuming a maximum changeof water content of ∆θ = 0.0462 (taken from[8]) and an initial saturation of S = 0.5. Thevalue for water density and dynamic viscosityare obtained from standard tables. The tortuos-ity parameter ξ was taken from [10]. Only theintrinsic permeability was adjusted to match thewater content distribution curves reported in theexperiments at a time of 15 min. The chosen in-

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trinsic permeability lies within the range of val-ues observed in experiments.

For the uncracked case the results are pre-sented in Figure 9 in the form of a contour plotof the capillary pressure Pc. Only the centreof the beam is shown. The water content dis-tribution obtained in the analysis for a verticalsection in the middle of the uncracked beam(dashed line in Figure 8b) is compared to ex-perimental results in Figure 10 for water ab-sorption up to 8 h. For the first 4 time steps,the water content curves are represented wellby the model, which is expected as the intrin-sic conductivity was chosen to fit the water con-tent profile for a time of 15 min. However, forgreater times the lattice model overestimates thewater absorption.

The second part of the analyses involved themodelling of water absorption for a crackedconcrete beam. Firstly, the mechanical latticewas used to simulate the three point bendingtest shown in Figure 8a. This loading setup re-sults in one main crack propagating from thebottom in the middle of the beam to the top. Ata maximum equivalent crack opening of wc =0.35 mm, the mass transport analysis was per-formed. The resulting contour plots of the cap-illary pressure Pc for the water transport analy-sis of the cracked beam are shown in Figure 11.Only the centre of the beam is shown. It can beseen that water is transported through the crackinto the concrete beam. In addition, transportfrom the crack into the surrounding concrete isvisible. The model and experimental results arecompared in the form of change of water con-tent versus the x-coord along the length of thebeam at mid-height (Figure 12). The change ofwater content within the crack (x = 0) is over-estimated by the model. However, the penetra-tion depth into the surrounding concrete is cap-tured well by the numerical approach.

(a)

(b)

Figure 8: Geometry and loading setup for the three-pointbending beam: (a) mechanical loading to induce crack-ing, (b) transport.

(a)

(b)

(c)

Figure 9: Contour plots of the distribution of capillarypressure for three time steps for the uncracked beam.Black corresponds to Pc = 0 MPa (S = 1) and thelightest gray shade corresponds to Pc = 39.58 MPa(S = 0.5).

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0

0.01

0.02

0.03

0.04

0.05

0 10 20 30 40change

ofvolumetric

watercontent

y-coord [mm]

Figure 10: Change of water content versus y-coord alongthe ligament of the uncracked beam: Lattice results(lines) against experimental results (lines with symbols)reported in [8].

(a)

(b)

(c)

Figure 11: Contour plots of the distribution of capillarypressure for three time steps for the cracked beam. Blackcorresponds to Pc = 0 MPa (S = 1) and the lightest grayshade corresponds to Pc = 39.58 MPa (S = 0.5).

0

0.01

0.02

0.03

0.04

0.05

-30 -20 -10 0 10 20 30change

ofvolumetric

watercontent

x-coord [mm]

Figure 12: Change of water content versus x-coord alongthe mid-height of the cracked beam: Lattice results(lines) against experimental results (lines with symbols)reported in [8].

4.3 2D meso-scale analysisIn section 4.2, concrete was modelled as a

homogeneous material. Using this assumption,the lattice results agree well with the experi-mental data for the first hour of the absorptionprocess. However, for later stages, the modelpredictions overestimate the water absorptionobserved in experiments. In the recent workby Wang and Ueda in [11], the heterogeneity ofthe material in the form of impermeable inclu-sions has been considered for the modelling ofthe absorption of water by a meso-scale anal-ysis. In their study, the total amount of ab-sorbed water has been shown to be in goodagreement with experimental results even forlater stages for which in the present homoge-neous analyses the water absorption was over-estimated (Figure 10). However, no informa-tion on the influence of inclusions on the watercontent change profiles was presented in Wangand Ueda’s work [11].

In the present study, the influence of het-erogeneity on the water change profiles is in-vestigated in more detail. Three inclusion vol-ume fractions ρa = 0.2,0.3 and 0.4 are mod-elled to investigate the influence of inclusionson the absorption process. The lattices in Fig-ure 13 are finer than for the homogeneous anal-ysis in the previous section so that the closely

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spaced circular inclusions for the volume frac-tion ρ = 0.4 can be discretised. The inclu-sions are considered to be impermeable and are,therefore, not discretised by transport elements(see Figure 13). The profiles of change of wa-ter content is evaluated by subdividing the midregion of the beam (over a width of 0.1 m) bya regular grid of cells. The water content in thex-direction is averaged. The resulting distribu-tions of change of water content versus y-coordalong the ligament of the beam are shown inFigure 14.

The inclusions have a strong influence on theshape of the water content curves. The largerthe volume fraction, the lower is the water con-tent inside the beam. For the present 2D ideali-sation, the consideration of inclusions results ina strong fluctuations in the water content curve,which has not been reported in the experiments.It is expected that a 3D model would predict asmoother curve if the average of water contentsin the out of plane direction would be consid-ered. Alternatively, the response of multiple 2Dmeso-scale analyses could be averaged.

(a) (b)

(c) (d)

Figure 13: Meshes for the meso-scale transport analysisfor aggregate volume fractions of (a) ρ = 0, (b) 0.2, (c)0.3 and (d) 0.4.

0

0.01

0.02

0.03

0.04

0.05

0 10 20 30 40change

ofvolumetric

watercontent

y-coord [mm]

modelexp

Figure 14: Change of water content versus y-coord alongthe ligament of the beam for aggregate volume fractionsof ρ = 0, 0.2, 0.3 and 0.4 at a time of 8 h compared toexperimental results reported in [8].

5 ConclusionsIn the present work, a dual lattice model for

the coupling of fracture and water transport hasbeen presented. The main conclusions can besummarised as follows:

The transport part of the coupled modelbased on homogeneous material properties rep-resents well the transport of water through un-saturated porous concrete for the first few hoursafter the start of the absorption process. How-ever, at later stages the water absorption is over-estimated.

The model can simulate transport bothwithin cracks and from cracks into the sur-rounding material by means of a single lattice.

By consideration of the meso-structure ofconcrete in the form of impermeable inclusions,the water absorption at later stages is in bet-ter agreement with experimental results than forthe case of homogeneous properties.

AcknowledgementsThe simulations were performed with the

object-oriented finite element package OOFEM[12, 13] extended by the present author.

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A lattice model for liquid transport in cracked unsaturated heterogeneous porous materials

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Transcript of A lattice model for liquid transport in cracked unsaturated heterogeneous porous materials