An immersed boundary-lattice Boltzmann model for biofilm growth in porous media

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An immersed boundary-lattice Boltzmann model for

biofilm growth in porous media

M. Benioug, F. Golfier, C. Oltéan, M.A. Buès, T. Bahar, J. Cuny

To cite this version:

M. Benioug, F. Golfier, C. Oltéan, M.A. Buès, T. Bahar, et al.. An immersed boundary-lattice

Boltzmann model for biofilm growth in porous media. Advances in Water Resources, Elsevier, 2017,

107, pp.65-82. �10.1016/j.advwatres.2017.06.009�. �hal-02968533�

ContentslistsavailableatScienceDirect

Advances

in

Water

Resources

journalhomepage:www.elsevier.com/locate/advwatres

Review

An

immersed

boundary-lattice

Boltzmann

model

for

biofilm

growth

in

porous

media

M.

Benioug

∗

,

F.

Golfier

,

C.

Oltéan

,

M.A.

Buès

,

T.

Bahar

,

J.

Cuny

University of Lorraine / CNRS / CREGU, GeoRessources Laboratory, BP 40, F54501 Vandoeuvre-les-Nancy, France

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 30 July 2016 Revised 18 May 2017 Accepted 8 June 2017 Available online 15 June 2017

Keywords:

Lattice Botzmann method Immersed boundary method Biofilm growth

Porous media Cellular automata

a

b

s

t

r

a

c

t

Inthispaper,wepresentatwo-dimensionalpore-scalenumericalmodeltoinvestigatethemain mecha-nismsgoverningbiofilmgrowthinporousmedia.Thefluidflowandsolutetransportequationsare cou-pledwithabiofilmevolutionmodel.Fluidflowissimulatedwithanimmersedboundary–lattice Boltz-mannmodelwhilesolutetransportisdescribedwithavolume-of-fluid-typeapproach.Acellular automa-tonalgorithmcombinedwithimmersedboundarymethodswasdevelopedtodescribethespreadingand distributionofbiomass.Bacterialattachmentanddetachmentmechanismsarealsotakenintoaccount. Thecapabilityofthismodeltodescribecorrectlythecouplingsinvolvedbetweenfluidcirculation, nu-trienttransportandbacterialgrowthistestedunderdifferenthydrostaticandhydrodynamicconditions (i)onaflat mediumand(ii) foracomplex porousmedium.Forthe secondcase,differentregimes of biofilmgrowthareidentifiedandarefoundtoberelatedtothedimensionlessparametersofthemodel, DamköhlerandPécletnumbersanddimensionlessshearstress.Finally,theimpactofbiofilmgrowthon themacroscopicpropertiesoftheporousmediumisinvestigatedandwediscusstheunicityofthe rela-tionshipsbetweenhydraulicconductivityandbiofilmvolumefraction.

© 2017ElsevierLtd.Allrightsreserved.

1. Introduction

Biofilm isa communityof one ormore speciesof bacteria or other microorganisms (fungi,algae, yeasts) associatedirreversibly witha liquidorsolid surface(water, biologicaltissues, solid sub-strates located in marine environments or freshwater) (Potera, 1999)andenclosedinamatrixofpolysaccharide(Kalmokoff etal., 2001; Prakash et al., 2003; Smith, 2005). Biofilm is composed primarily ofmicrobial cellsandextracellular polymeric substance (EPS). The EPSplays various rolesin theformation andstructure ofthebiofilmandalso,protectsthecellsbypreventingtheaccess ofantimicrobialandxenobioticcompoundsandconfersprotection against environmentalstressessuch asUV radiation,pHshift, os-moticshockanddesiccation(ChmielewskiandFrank,2003;Decho, 2000;MahandO’Toole,2001;Wingenderetal.,1999).

The study of biofilm growth in porous media is currently at-tracting interest forenvironmental applications such as bioreme-diation of contaminated sites (Al-Bader etal., 2013;Singh et al., 2006) ordevelopmentof bio-barriers forgroundwater protection (Huangetal., 2011;Seoetal., 2009).Oneofthemajorchallenges common to these decontamination methods is due to excessive

∗ _{Corresponding author.}

E-mail address: benioug@gmail.com (M. Benioug).

accumulation ofbiofilm in some pores (particularly inthe vicin-ityof the pollutant source or within the bio-barrier). This accu-mulationofbiofilm,indeed,leadstoadecreaseoftheporespace whichcausesareductioninporosityandpermeabilityofthe sys-tem while increasing the hydrodynamic dispersion (Kone et al., 2014). Eventually,theclogging mechanismmaychange the trans-portofdissolvedorganiccompoundsandthereforestrongly influ-encethe success ofthe application of cleaning methods for pol-lutedgroundwaters.Accurateknowledgeoftheformationand evo-lution of the biofilm over time is crucial to control its develop-ment.

Inrecentdecades,alargeexperimentalefforthasbeendevoted to study the biofilm structure and its change during growth. In parallel,therehasbeena growinginterestinthenumerical mod-eling ofthe involvedprocesses toreproduce the observed exper-imentalresults andconfirmorrefute existinghypotheses. Histor-ically, many models have been developed since the 90s. If the first numerical models of bacterial growth were limited to con-sidering the availability of the nutrient, they become more so-phisticatedto include theinfluence of shearforces (Alpkvist and Klapper, 2007;Morgenroth andWilderer, 2000), bacterial extinc-tion(HornandHempel, 1997) andadhesionmechanisms (Ebigbo etal.,2010;Kapellosetal., 2007).Recently,they succeededto de-scribemorecomplexbio-physico-chemicalprocessessuchas quo-rumsensing(Fredericketal.,2011),physiologicalstateofthecells

http://dx.doi.org/10.1016/j.advwatres.2017.06.009 0309-1708/© 2017 Elsevier Ltd. All rights reserved.

(Kapellos et al., 2007) and the effects of competition between bacterial populations (Ebigbo et al., 2013; Lackner et al., 2008; Nogueraetal.,1999;Pfeifferetal.,2001;Picioreanuetal.,2004a). Theinterest forbiofilm-related issuesin porousmedia has moti-vatedtheapplicationofthesemodelstosuchcomplexstructures. Avariety ofmodels havethus beenderived sincethe pioneering works of Suchomel et al. (1998) and Dupin et al. (2001) based onporenetworks. Ifsophisticated 2D and3D pore-scale descrip-tionshavebeenachieved(Ebigboetal.,2013;2010;Kapellosetal., 2007;Knutsonetal., 2005;Peszynska etal.,2016;von der Schu-lenburg et al., 2009) and have given additional insight on the bio-chemicalandphysicalmechanisms that drivethe biofilm dy-namics,importantquestionsonhowthemorphologyand distribu-tionofbiomassmayaffectthehydrodynamicpropertiesofporous mediumremaintobeaddressed.

The aim ofthis work isto presenta mathematical model ca-pable of predicting the temporal evolution of biofilm growth in porousmedia.Themodelwasvalidatedbyqualitativecomparison withbenchmarkcasesfromtheliterature.First,weconsideredthe caseof abiofilm developingon asolid flat surfaceinhydrostatic conditionsforthelimitinggrowthregimes,i.e.,transportor kinet-icslimitedregime.Then,wehaveimposedalateralflowtoverify theinfluence ofhydrodynamicconditions onthe biofilm pattern. Finally,wehaveapplied ourmodeltoa complexporousmedium inorder toobserve therole of bacterialgrowth onthe hydrody-namicpropertiesofthemedium.Inparticular,therelationship be-tweenporosityandpermeabilityhasbeendiscussedasafunction ofthebiofilmgrowthregime.

2. Literaturesurveyonbiofilmgrowthmodels

Overall,therearetwo possibleapproachestomodelingbiofilm growthinporousmedia(WangandZhang,2010),namelythe dis-crete (or individual-based) approach (including multi-agent and cellularautomata models)andcontinuousapproach.Inthe multi-agentmodel,thefundamentalentityistheagentrepresentingthe individual bacteria (Kreft et al., 1998) or bacterial colony. Each agent grows by consuming the substrate and divides when the volume reaches a critical threshold. The newly-formed agent re-mains in contact with the parent agent in a directionrandomly chosen.Then,biomasspropagationoccursbypushingagentssoon asthey become too close (Alpkvist et al., 2006; 2006; Batstone etal.,2006; Kreft etal., 2001;Picioreanuetal., 2004b). The first multi-agentmodelforthegrowthofbacterialcolonies,called Bac-Sim,wasdevelopedbyKreftetal.(1998)andextendedtoa multi-speciessimulationbyKreftetal.(2001).Multi-agent modelshave been successfully applied to describe and optimize various ap-plications on biofilm reactors (Batstone et al., 2006; Matsumoto et al., 2010; 2007; Picioreanu et al., 2005; Xavier et al., 2007; 2005a). Control strategies based on biofilm disruption were also studied(Xavieretal.,2005b). Usingthemulti-agentmodelis rel-evant if the interest is focused on interactions between individ-uals (Hellweger and Bucci, 2009; Laspidou et al., 2010). Indeed, thismodeloffersthe possibilityto workatthebacteriascaleand facilitatestaking intoaccount thecellular mechanisms. Empirical knowledgeofthe systemandofitsbiologymay beused directly to assign basic mechanisms of multi-agent model (see a recent review onadvantages andlimitations ofindividual-based models in Hellweger et al., 2016). However, it requires very long com-putation time particularly in a 3D configuration (von der Schu-lenburget al.,2009). Conversely, cellularautomata modeldivides the space into grid cells, each cell being considered as individ-ualentities that can take severalpossible states (e.g.,solid, fluid or biofilm). The state of a grid cell is determined by the previ-ousstate of its neighbors.Inthisapproach, biofilm isconsidered asaunit cellcontainingEPS,waterandbacteria.Thefirst

applica-tion ofcellularautomata modelto biofilm growthwaspresented by Wimpenny and Colasanti (1997). This model wasbased both on the work ofColasanti (1992)and the diffusion-limited aggre-gation(DLA)modelusedbySchindlerandRataj(1992);Schindler andRovensky(1994)andFujikawa(1994).Eversince,several mod-elshavebeendevelopedusingthisapproach(Hermanowicz,1998; 2001; Khassehkhan et al., 2009; Laspidou et al., 2012; Laspidou and Rittmann, 2004; Laspidou et al., 2005; Liao etal., 2012; Pi-cioreanuetal.,1998b).Theywereabletoproducecomplexgrowth patterns. Two cellular automata algorithms are particularly used intheliterature.Picioreanu etal.(1998b);1998c) divideintotwo equal parts the biomass of the cell when the thresholdvalue is reachedtomimic celldivision. Thisbiomassisthen redistributed amongthefouradjacentcells(fora2Dconfiguration)until reach-ingtheclosestunoccupiedcell.Theseso-calledpushingalgorithms canleadtoaspatialdiscontinuityofthebiomassdistribution,due tothedivision performed.Tomakethemcontinuous, another ap-proach hasbeen further developed by Noguera etal. (1999)and

Knutsonetal.(2005).Inthisapproach,onlytheexcessofbiomass isredistributed.Cellularautomatamodelsyieldedgoodagreement withexperimental results.Theyhavetheadvantage ofeasily tak-ing into account the different cellular mechanisms and facilitate implementingboundaryconditionsandcomplexsystemgeometry. However, they are less suitable formulti-bacterialbiofilms while preferentialorientationoftheCartesiangridmaycauseunrealistic growth.Unlikebothpreviousformulations,continuousmodelsare purely deterministic. Thisapproachrelies onthe idea that bacte-rial growthcausesthedevelopmentofapressure field insidethe biofilm(Eberletal.,2001).Thispressurefieldconstitutesthe driv-ing force ofavery viscous flowandcauses slowdisplacementof thebiofilm(WannerandGujer,1986).Asimilarideahasbeenused forthefirst timeby WannerandGujer (1986)toexpressthe ve-locityofthefluid/biofilminterfaceduetobiofilmgrowthina one-dimensionalmodel.Theaveragevelocityofbiomassisthus deter-minedbymechanicallaws,andmorepreciselybytheEuler equa-tionsoffluiddynamics.Continuousmodelsareabletopredictthe changes in biofilm thickness, the transientdynamics of bacterial communityaswell asthespatialdistributionofmicrobialspecies andchemicalcompoundsinthebiofilm.Variousnumerical meth-odscanbeusedtodescribethedisplacementofinterfaces,suchas the levelset method (Alpkvist andI.Klapper, 2007) and Volume-Of-Fluid (VOF) method (Ebigbo et al., 2013). Continuous models havethe advantageof beingpurelydeterministic, rigorous, based onconservationlawsandwell-knownPDEs.However, thistypeof modelgivesonlyamacroscopicinsightintothesimulateddomain. They are therefore particularly suited to large scale systems but theyarenot validwhenthesizeoftherepresentativeelementary volumeisclosetothebacteriaormicro-colonyscale.

Basedonthisin-depthreviewofbiofilmmodels,we optedfor themostflexibleapproachi.e.,thecellular-automatonmethod.We mustnotethatamajorshortcomingofthisapproachisrelatedto theelementary sizeof theunit cellused by thecellular automa-ton.Thiscellsize constrainstheminimal changeinbiofilm thick-ness duringbiomass growthandmay dramaticallyimpact model predictions at the first stage of colonisation or when the pores are nearlyplugged. Hereafter,thisdrawback willbe offsetby the useofefficient non-conformingnumericalmethods. Werefer the readertoBeniougetal.(2015)foracomparative analysisand ac-curacy assessment ofsuch methods applied to flow and reactive transport inporous media. In thepresentstudy, we will usethe LatticeBoltzmann Method coupled withthe Immersed Boundary Method(IB-LBM)forcalculatinggroundwater flowandaVOF-like methodforsolvingsolutetransport(Beniougetal.,2015).The cel-lularautomatonmodelwilltake advantageofthesenumerical ef-fortstoworkonentitiessmallerthanthemeshsize.

3. Modeldescription

Simplifying assumptions are often required to simulate the main mechanisms describing the complexity of biological and physico-chemicalphenomenainvolvedinthedevelopmentof bac-terial populationsin porousmedia (e.g.,nutrient availability, me-chanical strength of the biofilm and influence of shear forces on the detachment of sessile bacteria, competition for the sub-strate and symbiotic effect between different bacterial popula-tions,quorumsensing,evolutionofthephysiologicalstateofcells) (Brackmanetal.,2009;Daviesetal.,1998;Vegaetal.,2014;Zhang etal., 2009). Wedo notpretend heretodescribe all thefeatures of this complexityand consequently, we will focus on a specific class ofproblem wherewe will moreparticularly investigatethe biocloggingissuesinporousmedia. Inourcase,the following as-sumptionswillbemade:

• The solid matrix is supposed non deformable andthe porous mediumisfullysaturated.

• A singlebacterial species is presentin the medium andonly undersessileform(mono-speciesbiofilm).

• Biofilm is considered as a continuous and homogeneous medium (WoodandWhitaker, 1998), althoughthis one is in-herently characterized by a structural heterogeneity (bacterial cells,EPS,nutrients)andanintrinsicbiodiversity(bacteria, pro-tozoa).

• Metabolic reactions that are involved in the bacterial growth process only depend on a single solute, i.e., the organic molecule servingasacarbonandenergysourceandactingas theelectrondonorintheredoxprocess.Othersubstances (elec-tron acceptor,mineralsalts,aminoacids) which arenecessary forthesurvival ofbacteriaare assumedinlarge excessinthe medium.Inthosecircumstances,thebiodegradation kineticsis describedbyMonodkinetics(MeeGeeetal.,1970).

• The nutrient is transported by convection - diffusion in the fluidphaseandonlybydiffusioninthebiofilmphase.Inother words, macro-porosities within the biofilm where convection could occurare assumedsufficientlylargeto beexplicitly dis-cretizedandarethereforeattachedtothefluidphase.

• Theconcentrationofthechemicalspeciesdissolvedinthefluid phase issufficiently low so that solute transportin the water doesnot affecttheinitial physicalpropertiesofthe fluid,e.g., its dynamicviscosityanditsdensity. Inthismanner, theflow andtransportequationscanbeuncoupled.

Finally, we will use the time-scale separation assumption for the different processes involved (flow, nutrient transport and biomassgrowth) byadoptingaquasi-stationaryapproach.Indeed, thecharacteristicrelaxationtimeofthevelocityfieldisverysmall comparedwiththecharacteristictime oftransport,whichitselfis generallyverysmallcomparedtothecharacteristictimeofbiofilm growth (Picioreanu et al., 1998a). This assumption is valid in mostcases(Skowlund,1990).Consequently,wesolvetheflowand speciestransportequationsinsteady-stateconditions.The tempo-ral evolution is performedthroughthe bacterialgrowth equation (andtheassociatedcellularautomatonmodel).

Based on the assumptions above, the problem at the micro-scopicscalecanfinallybestatedasfollows.

3.1. Liquidflow

In thefluid phase (f), weconsider thesteady-state flow ofan incompressibleNewtonian fluid,at low Reynoldsnumber so that the massbalanceandmomentum conservationequationsare de-scribedusingthefollowingequations:

∇

· u=0, (1)

ρ

f



∂

u

∂

t +u·

∇

u



=−

∇

p+

μ

f

∇

2u, (2) with non-slip boundary conditions for the fluid-solid and fluid-biofilminterfaces,respectivelyA_fs andA_fb,

C.L.1 u=0, atAf b (3)

C.L.2 u=0, atAf s (4) where u, p,

μ

_f,

ρ

_f are respectively the velocity vector, pressure, fluid dynamic viscosity and fluid density. Aij represents the in-terfacebetween the i andj phases(i, j = f (fluid), b (biofilm), s

(solid)),andn_ij istheexternalnormalvectortotheinterfaces be-tweenphasesiandj.Inthefollowing,thephaseiwillbedenoted by



i.WemustkeepinmindthattheinterfaceAib betweenboth regions is assumed to evolve versus time but sufficiently slowly so that a non-slip conditioncan be adopted. Note also that the viscoelasticbehaviourofbiomassthatmaycausedeformationand slipflowatthebiofilmsurface(AlpkvistandKlapper,2007;Shaw etal.,1996)isneglectedhere.

TheseequationsaresolvedbyusingtheIB-LBmodelpresented inBeniougetal.(2015).Adirect-forcing ImmersedBoundary (IB) model is coupled with the Lattice Boltzmann Method (LBM) to keepthe fluid-solidinterface sharpandmaintaina directcontrol onnumericalaccuracy.Moredetails aboutthesenumerical meth-ods can be found in Beniouget al.(2015) and they will be only briefly remindedbelow. We usehere a Lattice–Boltzmann model withasinglerelaxationtime(SRT) andaBGK (Bhatnagar–Gross– Krook)collision operator. Athree-dimensional latticewith 19 ve-locityvectors(D3Q19 model)isconsidered.Thediscretized Boltz-mannequationisclassicallyexpressedasthecollisionpropagation equation: fk

(

x+ek



t,t+



t

)

= fk

(

x,t

)

−



t

τ



fk

(

x,t

)

− fk(eq)

(

x,t

)



(5)

where

τ

representsthe relaxationtime and f_k(eq) theequilibrium distributionfunctionassociatedwiththevelocityvectorek.As de-tailedinBeniougetal.(2015),thecouplingwithIBMisperformed byaddingaboundaryforcedensityintothemomentumequation. Consequently,Eq.(2)ismodifiedasfollows:

∂

u

∂

t +u·

∇

u=−

∇

p

ρ

f +

ν∇

2_u+ g

ρ

f, (6)

where

ν

is the kinematic viscosity and g represents the forcing termusedtosatisfytheno-slipconditionontheimmersed bound-aries.Thecomponentg_koftheforcingtermalongthevectore_kis givenbyKangandHassan(2011):

gk

(

x,t

)

=



1− 1 2

τ



ω

k



3ek− u

(

x,t

)

c2 +9 ek· u

(

x,t

)

c4 ek



· g2

(

x,t

)

(7) with: g2

(

x,t

)

=2

ρ

(

x,t

)

¯ U

(

x,t

)

− u∗

₍

_x,t

₎



t (8)

andu the velocity in the bulk fluid and U¯ the interpolated ve-locity at the boundary of the immersed cell. As illustrated in

Fig.1,thelinearinterpolationofthecomponentU¯jalongthe direc-tionnormalto theimmersedboundary andlocatedata distance

(

1₋

φ

(

x_f,t

))



xfromtheinterface,leadsto:

¯

Uj=

(

1−

φ

(

xf,t

))

xu j



x+

(

1−

φ

(

xf,t

))

x

Fig. 1. Illustration of the interpolation procedure in the x direction with the immersed boundary method (IBM).

where

φ

(x,t)describesthebiofilmvolumefractionforthegridcell

xattimet.

Note that LBM equations are solved in a transient form and steadystate solutionisassumedtohavebeenachievedwhenthe pressureconvergencecriterionisverifiedbetweentwoconsecutive timesteps.

3.2.Solutetransport

Thetransport ofchemicalspeciesbothinthefluidandbiofilm phasesisgovernedbythefollowingconservationequations:

∇

·

(

ucf

)

=

∇

·

(

Df

∇

cf

)

, inthefluidphase (10)

0=

∇

·

(

Db

∇

cb

)

− Rb, in thebiofilmphase (11)

Df

∇

cf· nfb=Db

∇

cb· nfb, atAf b (12)

cf=Kbfcb, at Afb (13)

0=Df

∇

cf· nfs, atAf s (14)

0=Db

∇

cb· nbs, atAbs (15) where cf (resp. cb) is the concentration of the solute within



f (resp.



b),uisthevelocityvectorfieldaspredictedfromEqs.(2)–

(4)andDfandDbarethemoleculardiffusioncoefficientsin



fand



b.Kbfisthepartitioncoefficientbetweenthetwophases



fand



b (itsvalueisusuallycloseto1forafluid/biofilmsystem).Rb is thereactiontermwhichisdescribedbyMonodkinetics:

Rb=

ρ

bio μmax cb

cb+K

(16)

where

ρ

bio,

μ

max, K are, respectively, the biomass

concentra-tion,themaximumreactionrateandthehalf-saturationconstant. DirichletandNeumann conditionsare classically imposed on ex-ternalboundariesofthedomain.Aspreviouslyfortheflow calcu-lation,non-boundary conforming method is required to simulate concentrationfieldsatthepore-scaleduetogrowth(orreduction) of



_b. We consider here a volume-of-fluid type model basedon theassumption that the systembehavior is close to the equilib-riumconditions.Thisimpliesthatthecharacteristictimefor trans-porthasthesameorderofmagnitudeinbothphasesandthe char-acteristictimeofthereactionkineticsislarge(Golfieretal.,2009). In other words, (i) the concentration gradients in the vicinity of theinterface are sufficientlysmall and(ii) the concentrations do notvarysignificantlyatthescaleoftheimmersedcells.Therefore, asingle concentration Cboth in



f and



b can be defined. This

approximation leadsto a one-equationtransport model valid ev-erywhereexpressedas:

∇

·



u_φC

=

∇

·



D_φ

∇

C

− R_φ, in



=



f∪



b (17) where

φ

(x,t)describesthevolumefractionof



_b over



.D_φ,R_φ

andu_φ represent respectivelythe molecular diffusion coefficient, thereaction termandthevelocity vector within



. Theyare ex-pressed as a function of

φ

so that, for the two limit cases, i.e.,

φ

=0 (fluid cell)and

φ

=1 (biofilm cell), Eqs. (10) and(11) are recovered.Thus,wehave:

D_φ

(

x,t

)

=

φ

(

x,t

)

Dr+

(

1−

φ

(

x,t

)

)

Df (18)

R_φ

(

x,t

)

=

φ

(

x,t

)

Rb

(

x,t

)

(19)

u_φ

(

x,t

)

=

(

1−

φ

(

x,t

)

)

u

(

x,t

)

(20)

3.3. Biomassgrowthanddecay

The time evolution of the biomass concentration

ρ

_bio is gov-ernedbythefollowingmassbalanceequation:

∂ρ

bio

∂

t =Fλ

ρ

bio μmax cb

cb+K− kd ρbio

(21)

F_λ is the stoichiometric coefficient ofthe biological reaction and

kd istheextinctioncoefficient ofthebacteria.Thissimplifiedlaw considersonlythebacterialgrowthproducedbythesubstrate con-sumptionandextinctionofbacteria.

The biomass concentration

ρ

_bio calculated at each time step fromEq.(21) isthen used todetermine thevalue ofthe volume fraction

φ

bycell(thebiomassdensityissupposedtobeconstant inthe whole domain).Note that thenet changein biomass con-centrationiscalculated overa referencetimestep ofgrowth, de-notedtgrowth andcorrespondingto anincrease in

φ

of0.01based onthemaximumreactionrate

μ

max.Assoonasthe

ρ

bio value ex-ceedslocally, within abiofilm cell,an arbitraryvalue denoted by

ρ

biomax (corresponding toa value of

φ

=1), the biomassdensity isdistributed spatiallyby usingthe cellularautomata model (de-scribedinmoredetailsinthenextsection).

3.4. Biomassspreading

Thespatialdistributionofbiofilmismodeledviaacellular au-tomaton. Themodel isactivatedwhen thebiomassconcentration exceeds locally the imposed threshold value. As for any biofilm model,the cellular-automata rules assume that bacteriagrow to-wardtheareaswherethesubstrateconcentrationisthemost im-portant.Basically, foreach gridcell (x,y),assoon as

φ

(x,y) >1 (i.e.,

ρ

bio(x, y) >

ρ

biomax), the excess of biomass



ρ

bio =

ρ

bio(x,

repeatedforeachcellinturnuntilalltheexcessbiomassis redis-tributed.

During biomass spreading,depending onthe nature of neigh-boringcells,threedifferentcasescanbeidentified:

1. If there is one fluid cell or more among the neighboring cells,



ρ

bio isplaced inone ofthem, randomly chosen withequal probability,andthisone becomesa partially-filledcell witha volumefraction

φ

= ρbio

ρbiomax.

2. If alltheneighboring cellsarebiofilmsbut, atleast one ofthem containsabiofilmvolumefractionlessthan1,



ρ

_bio israndomly distributedinthesecellswithequalprobability.Ifthevolume fractionofthe lattercellsbecomeshigher than1, thecellular automatonintervenesagaintomovethesurplusofbiomass to-wardsadjacentcells.

3. If all the neighboring cells are biofilms and have a biofilm vol-umefractionequal orgreater than 1,



ρ

_bio isplacedin one of them,randomlychosenwithequalprobability andtheprocess ofbiomass spreading continues until a fluid orpartially-filled cellisencountered.

Unlike classical models in which the fluid cell becomes a biofilm cell when it receives biomass, the main improvementof this algorithmis based on themanagement ofthe cellspartially occupiedby biofilm. Thismodificationis,ofcourse, motivatedby theuseofIB–LBmethodsfortheflowcalculationandtheVOF ap-proachemployedforsolutetransport.However,wemustnotethat this algorithm doesnot allow to specify the shape of the fluid-biofilminterfacewithinthecell,necessaryforusingtheImmersed Boundary Method in computingthe velocity andpressure fields. Tofacilitatetheresolutionoftheflow,we assumedinthe follow-ing that thesefictitious interfacesare parallel tothe meshfacets andperpendiculartothedirectionofgrowth.

3.5. Biomassdetachment

Onlythedetachmentby erosion,i.e.,undertheeffectofshear stresses acting on the superficial biofilm layers (i.e., cells with fluid/biofilminterface)istakenintoaccountinthemodel.We as-sumethat thepartial ortotaldetachmentofthebiomass present in a cell occurs when the shear stress exceeds locally a fixed thresholdvalue,denoted

τ

max(ChoiandMorgenroth,2003; Stood-ley etal., 2001;Sudarsan etal., 2005). Althoughthisapproachis simplifiedwithregardtothecomplexityoftheprocessesinvolved (e.g.,changesinmechanicalpropertiesofbiofilmdependingonthe quantityofEPSproduced)(Ruppetal.,2005;Stoodleyetal.,2002; Telgmann etal.,2004)it presentstheadvantageof beingeasyto calibrate incomparison withlaboratory data sets e.g., Paul et al. (2012) andWalter etal.(2013).It alsoallowsustoconsiderboth surface erosion(for low valuesof

φ

det) andvolume erosion (the maximumsizeofparticles thatcanbe detachedislimitedbythe cellsize)whichseemstoplayamajorroleintheevolutionofthe biofilmthickness(Derlonetal.,2013).

Todeterminethequantityoferodedbiomass,wecalculateina firststeptheshearstressexertedforeachbiofilmcellonthefacets incontactwiththefluid.Themaximumstressexertedoneachcell is takenasreferencevalue forthecalculation ofthe detachment. Thisvalueisobviouslyupdatedateachtimestepaccordingtothe changesofthemediumgeometryandofthevelocityfield.Forthe biofilmcell(i,j)representedinFig.2,wehavein2Dforinstance:

τ

=max

(

τ

north,

τ

south,

τ

east,

τ

west

)

(22)

τ

=

μ

f× max

∂

uy

∂

x,

∂

ux

∂

y



(23)

Fig. 2. Cell ( i, j ) with its four neighbors and their velocity vectors for calculating the shear stress in 2 D .

wherethe cellseast, west, southand northrepresentthe neigh-boringfluidcells.Theparietalvelocitygradientsarediscretizedby finitedifferencesandcalculatedbetweenthefluid/biofilminterface -wherethevelocity iszero-andthecentreoftheadjacentfluid celldistantoflengthlstress.Thus,fortheshearstressontheupper face,i.e.,thenorthface,weobtain:

τ

north=

μ

f×

ux

(

i,j+1

)

− 0

lstress

(24)

IntheparticularinstanceillustratedinFig.2,lstressisgivenby:

lstress=



₃

2−

φ

(

i,j

)





y (25)

Notethat ifthecell(i,j) wouldbefullyoccupiedby biofilm, i.e., withthefluid/biofilminterface located at(i, j+1

2) we would

re-coverlstress=2y.

Tomodeltheprocess ofdetachment,we arebased onthe ex-perimental results of Paul et al. (2012) which indicate that the thickness of the biofilm exponentially decreases with increasing shearstress.Fromtheseobservations,wehaveestablishedan em-piricalformulation relating the detached biofilm volume fraction

φ

dettotheshearstress

τ

:

φ

det =0 for

τ

<

τ

max (26)

φ

det =eCdet

(τ−2×τmax)

τmax for

τ

_max≤

τ

≤ 2×

τ

_max (27)

φ

det =1 for

τ

≥ 2×

τ

max (28)

withCdet an empiricalconstant.Theremainingvolumefractionof biomassafterdetachmentcanbederivedeasilyasfollows(we as-sumethatthebiomasskeepsaconstantdensity):

φ

t+1₌

₍

_φ

t₋

_φ

det

)

(29)

suchas

φ

t_{isthebiofilmvolumefractionattheprevioustimestep,} i.e.,beforedetachment.

The last step consists in describing the fate of the detached biofilmparticles,i.e.,theprocessofbiomassattachment.

Fig. 3. Schematic representation of stages of bacterial attachment.

3.6.Biomassattachment

As the transport of planktonic bacteria is neglected in this model,wesupposethatthedetachedbiofilmparticlesareonly en-trainedbytheflow.Thus,iftheshearstressesarefavourable,i.e.,

τ

≤

τ

attachthebiomasscan potentiallyreattachto thegrainsurface ortothesurfaceofanotherbiofilmencounteredalongitspathway. Itshould benoted thatdespite its impacton thebio-plugging of groundwatermedia(Buntetal.,1993;Cowanetal.,1991; Scheuer-manetal.,1998),thismechanismisrarelyaccountedforinbiofilm growthmodels(Kapellosetal.,2007).

Thedetachedbiofilm particlesareassimilatedtobacterialflocs advected along streamlines. They are assumed to be sufficiently smallnot to be submitted to inertialeffects (Stokesnumber less than 1) and have the same density as the fluid (a biofilm con-sistsof morethan 90 % water)so that the sedimentationeffects are negligible. Moreover, we take into account the assumption thatattachmentconditionsdepend onlyonenvironmental hydro-dynamicstresses.Consequently, thealgorithm is designedas fol-lows:foreachfluidcellcontaininganon-zerobiomass concentra-tion (i.e.,biomass previously detached), bacteria concentration is transportedalong thestreamline (fromcell centre tocell centre) duringthecalculationtimesteptgrowth.Foreachcellcrossedalong thepathway,themodelcheckstheprobabilityofbiomass attach-ment(Fig.3a).Ifoneormoreoftheneighboringcellsarebiofilm orsolidcellsandifthevalueofthelocalshearrateislowenough (

τ

≤

τ

attach),then thevirtual biofilm particleisattached, i.e.,the fluidcellisconvertedintopartially-filledcell(Fig.3b).Otherwise, thebiofilmparticlefollowsitspathwayuntilitleavesthedomain. Finally,thegeneralalgorithmofourmodelincludingfluidflow, chemical species transport, biofilm growth, biomass attachment anddetachmentisillustratedinFig.4.

4. Resultsanddiscussion

To illustrate and evaluate the predictive capability of our nu-merical model, two numerical benchmarks taken from the liter-ature(Eberl et al., 2001; Picioreanu et al., 1998a; 2000; Wanner et al., 2006) have been initially performed. The two tests were carriedout ina 2D vertical configurationbut forhydrostaticand hydrodynamicconditions.Forthe first one, thebiofilm growson a solid flat surfaceand the nutrient is conveyed downwardonly by diffusionfrom theupper boundary. From thisbenchmark, we investigate the ability of the model to properly reproduce the biofilm growth limitingregimes, i.e., under masstransfer-limited andreaction-rate-limited conditions. In the second test case, we imposealateralflowtoexaminetheimpactofhydrodynamic

con-Fig. 4. General algorithm of model.

ditionsandverifythatthemodelcapturesthedynamicsofthe de-velopingbiofilmpattern.

Finally,we apply ourmodeltoa 2D porousmedium. Theaim of thisstudy isto assess the influence ofbiofilm growthon the changes of the geometrical (bioclogging) and physical properties (porosity, permeability) of the porous medium. A peculiar atten-tion is paid to the porosity-permeability relationship andits de-pendencytobiofilmgrowthregimes.

Tofacilitate the analysisandthe interpretation ofresults, nu-mericaldatawillbereportedinadimensionlessformasdescribed below: x= x l, z ₌ z l, t _=t Df l2, ci= ci C0,

(

i= f,b

)

(30) p= pl

μ

fumoy, u= u umoy,

ρ

 bio=

ρ

bio

ρ

biomax ,

τ

₌

τ

l

μ

fumoy (31) K= K C0,

μ

 max=

μ

max l2 Df , k_d=kd l2 Df ,D_i= Di Df (32)

Fig. 5. Schematic representation of the computational domain. Bacterial colonies are green, the solid substrate is shown in red. (a) biofilm growth in hydrostatic conditions. (b) biofilm growth in hydrodynamic conditions. The thick arrow indicates the direction of nutrient transport. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

ρ

biomax, C0, umoy and l respectively represent the maximum biomassconcentration,thesubstrateconcentrationimposedatthe inlet, the fluid averaged velocity and the characteristic length of themedium.Wealsointroducethefollowingdimensionless num-bers,PeandDathatcharacterizemasstransportandbacterial reac-tivityandthedimensionlessrelativeshearstress,denotedby

τ

∗:

Péclet number: Pe=umoy· l

Df

(33)

Damköhler number: Da=

μ

max· l2·

ρ

biomax

DfK =

μ

 max K



with



=

ρ

biomax C0 (34)

Dimensionless shear stress:

τ

∗=

τ

∞

τ

max (35)

where

τ

_∞representsthemaximumshearstressvalueencountered inthewholedomain.

4.1. Case1.Biofilmgrowthunderhydrostaticconditions

The computational domain



=[0,lx]× [0,lz] is discretized with a uniform mesh size

(

Nx_{− 1}

)

_×

(

Nz_{− 1}

)

and a resolution



x= lx

Nx−1=



z=Nzlz−1 whereNxandNzarethenumberofnodes

alongthelongitudinal(x)andvertical(z)directions.Weconsider

n0 biofilm coloniesrandomly distributedonthesolid flatsurface,

locatedatz=0(Fig.5a).Eachcolonyoccupiesinitiallyonecellin thex-directionandtwocellsinthez-direction.

The initial biomass concentration is the same for all the colonies ofbiofilm and is

ρ

_bio

(

t=0

)

=1. A constant concentra-tion C0 innutrient isimposed on theupperlimit ofthedomain,

Table 1

Simulation data values. Parameter value lx 3 lz 1 Nx 151 Nz 51 n0 70 ρ bio 1 D_f 1 D_b 0.25 Fλ 0.7 k_d 5 × 10 −3  300

i.e.,atz=1. Periodicboundaryconditionsare imposedin thex -directioninordertoavoidsideeffects.Finally,awallconditionis imposedonthesolidborderatz=0.

Dimensionless values of simulation data are given below (Table1).Notethat

μ

max andKvaluesdependonthefixedvalue

ofDa.

Timeevolutionofthebiofilmstructure

In this simulation, an intermediate value of the Damköhler number corresponding to Da=10 is considered. During the first time steps of simulation, thestratification of nutrient concentra-tionfieldisobservedundertheeffectofthediffusion.The isocon-centrationsarepracticallyhorizontalalongthex-direction.We ob-serveahomogeneousdevelopmentofbiofilmwhichoccupies prac-ticallytheentiresolidsurface(Fig.6a).Thebiofilmgrowthis pro-gressivelycausingachangeintheavailabilityofnutrient(Fig.6b).

Fig. 6. Temporal evolution of the biofilm geometry for Da = 10 . Biofilm is shown in green, fluid in blue and solid substratum in red. The black lines indicate the lines of equal concentration. These lines show the decrease of substrate concentration from the upper fluid surface (blue region) to the biofilm (green), with a variation of 10% between the lines. a) t  _{= 10 , b) t}  _{= 50 , c) t}  _{= 100 , d) t}  _{= 160 . (For interpretation}

of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Thisdisturbanceoftheconcentrationfieldintensifiesprogressively anddestabilizesthebiomass development(Fig.6bandc).Indeed, thesubstrate loading rate beingthe limiting factor, the bacterial colonies nearthe source continue to grow producing biomass at thedetrimentofthosewhicharemostremote.Thisheterogeneous distribution,thatfavorsthelargebacterialcolonies(Fig.6d),leads totheformationoffinger-likestructures(Fig.6candd)whichhave alsobeen reportedin theliterature (Costerton etal., 1994; Eberl etal.,2001;Picioreanuetal.,1998c).

InfluenceofDanumberonbiofilmpattern

In thissection,we assesstheDamköhlernumberinfluenceon thebiofilm growth.The numerical simulations are performedfor a large range ofDa numbers varying from5· 10−2 _{to 10}3 _to

in-vestigatebiofilmgrowthregimes,bothundermasstransfer-limited and/orreaction rate-limited conditions.Forlow Da,theinfluence ofmass transfer ratewithin the biofilm is predominant. Indeed, thelow nutrient consumption by bacteria results ina large pen-etrationof substrate into the biofilm (Fig. 7a -right). This obser-vation iscorroborated by Fig.7a-left, where the average concen-trationC¯along thex-direction remainsabovezeroovertheentire thicknessofthebiofilm.Suchhydrostaticconditionsleadtoa com-pactandsmooth biofilm(Fig.7a),corresponding toagrowth lim-ited by the kinetics.In contrast,when Daincreases, thereaction kineticsbecomesmoreimportantrelativetothediffusionrate.The biofilmgrowthismoreandmorelimitedby thesolutetransport. Consequently,theoutsidesurfaceofthebiofilm becomesmore ir-regular andthe biofilm evolvestowards a more porousstructure withmanychannelsandvoidsbetweenthecoloniesthat growin the formof fingers (Fig. 7b andd). The concentration no longer

penetrates inside the biofilm and theunstable growth process is amplified (the fingers become thinner and narrower). The aver-agebacterialconcentration

ρ

¯ isalsomorespreadout overthe z -directionanddecreaseswhenapproachingthesolidsurfacewhere themostmaturebacterialayersareaffectedbythecellular extinc-tionduetonutrientdepletion.

4.2. Case2.Biofilmgrowthunderhydrodynamicconditions ImpactofPécletnumber

Thissecondtestcaseisidenticalwiththeoneaboveexceptfor theboundaryconditionsandthevaluesofthedimensionless num-bersPeandDa.Allthe othernumericaldata giveninTable1 re-main unchanged. The condition imposed on the upper limit, i.e., atz₌1_,isnowa symmetryconditionandthesubstrate concen-trationC=1isinjectedatx=0withaparabolicvelocityprofile (Fig.5b).Theseconditionsareexpressedasfollows:

u_x

(

0,z

)

=

(

6z/lz− 6z2_/lz2

₎

_,z_{∈[0,1] (36)} uz

(

0,z 

)

=0,z∈[0,1] (37)

∂

ux

∂

z



x,1

=0,x∈[0,1] (38) uz

(

x  ,1

)

=0,x∈[0,1] (39)

∂

u_x

∂

x



1,z

=0,z∈[0,1] (40) u_z

(

1,z

)

=0, x∈[0,1] (41)

∂

C

∂

n





z=1 =0 (42)

A random distribution ofn0=25 colonies wasapplied at the

initialtime.Tolimittheinfluenceoftheboundaryconditions,the colonies were placed between x=0.26 and x=0.74. At t=0,

each colony are occupying one cell of width and two cells of height.

Theinfluenceofconvectiononsubstrateavailabilityandonthe changes in bacterial population is investigated by modeling the biofilm growth under different hydrodynamic regimes from the variationofthePécletnumber.Consequently,theDamköhler num-ber is kept constant for all the simulations, i.e., Da=1 and the values of the Péclet numberare Pe₌0_.1_,1 and 100. The results areillustratedinFig.8.Itisimportanttonotethatinthese simu-lationsthemechanisms ofattachment/detachment ofthebacteria havebeendesactivated.

• Forthelargest Penumber (Fig. 8c),the biofilmgrowth seems not tobe influenced by the changes inthe velocity field. The biofilmcoloniesshowrelativelyuniformgrowth.Thehighflow velocitiesfavor a constant supplyof nutrient everywhere, the isoconcentrationsarethusgloballyuniformaround thebiofilm patch.

• ThedecreaseinthePécletnumber(Fig.8aandb),causesa dif-ferentiatedbiofilmgrowthbyfavoringthecoloniesnearthe nu-trient source. Indeed, when Péclet number decreases and Da

remains constant, the convection characteristic time becomes verylargerelativetothereactioncharacteristictime.The sub-strate availability then decreases progressively away from the source (here, x=0). This phenomenon is amplified by the

Fig. 7. Right: Biofilm patterns observed for different Da . The black lines represent the isoconcentrations with a variation of 10% between each one. Left: profiles of the average substrate concentration ¯C in red, and the average biomass concentration ¯ρin green. (a) Da = 0 . 05 , t  _{= 600 , (b) Da = 10 , t}  _{= 160 , (c) Da = 10 0 0 , t}  _{= 46 . (For interpretation}

of the references to colour in this figure legend, the reader is referred to the web version of this article.)

growth of colonies closest to the source. These colonies cre-ate an obstacle to the flow and change the streamline distri-bution.These numericalresultsare inperfectagreement with thoseobtainedbyPicioreanuetal.(2000).

Impactofbacterialdetachment

In asecond step,ourattentionis focused on themechanisms of bacteria attachment/detachment and their impact on biofilm growth.However,asthismechanismimplicitlydependsonthe ve-locityfield, we firststudied themesh-dependency ofour numer-ical results. Twodifferent meshesare considered: Nx× Nz= 100 × 33 and 80 _{× 26.} The Péclet number is Pe₌50 with a criti-cal shearstress

τ

_max =103 _{andtheDamköhler numberisDa=1.}

The medium isinitiallyinseminated by avolume ofbacteria cor-responding to 3.8% of total volume. The bacteria were randomly distributedonthesolidsurfacebutbetweenthesamelimitsthan above, i.e., between x=0.26 andx=0.74. The resultsare illus-trated in Fig. 9. First, we observe that the biofilm patterns are very similar, independently of the mesh size (Figs. 9a and b). The analysisoftheincrease ofbiomass concentrationversus time

(Fig. 9d)supports thisfinding andconfirmsthecapability ofthis IB–LBbased modelto predictbiomass detachment.Second, com-parison with numerical simulations in the absence of bacterial detachment (Figs. 9c and d) highlights the role of this regula-torymechanism on biomass development. Impact of detachment becomes clearly visible after t=22 (Fig. 9d). After t=32, the biofilmheightreachesaplateauandstabilizesaround0.4(Figs.9a andb) whilethe biomass continuesto growinfinitely inthe ab-senceofbacterialdetachment,atleastifthe substrateavailability issufficient(Fig.9c).The progressivebiofilm growthcausesa re-ductionoftheflowcross-sectionandhence,anincreaseofthe lo-calshearstresses.Whentheshearstressreachesthecriticalvalue

τ

max,erosionofsuperficialbiofilm layersoccurs.Theonsetofthis

steady-stateplateaucorrespondstoabalancebetweengrowthand detachment.

As aconclusion, the numericalsimulations carried out inthis sectionindicatethatthemodelcapturesproperlythedynamicsof biofilmgrowthandresultsareinagreementwiththeliterature.In thenext section, we apply our modelto porousmedia colonised bybiofilms.

Fig. 8. Biofilm growth patterns for Da = 1 and different Pe values. The black lines represent the isoconcentrations with a variation of 10% between each one. The ve- locity vectors are also indicated. (a) Pe = 0 . 1 , t  _{= 1400 , (b) Pe = 1 , t}  _{= 1100 , (c)} Pe = 100 , t  _{= 450 .}

4.3. Applicationtoa2Dporousmedium

Weconsidernowarepresentativevolumeofacomplexporous medium (Fig.10). Such a geometrygives usan overview ofhow thepore-scalearchitecture (soliddistributionandpore connectiv-ity) may impact biofilm development. The domain has a size of

lx_{× lz=}12_{× 12}mm2 _{with an initial porosity}

0 of 0.6 and the

pore throat characteristic length, lf, is estimated to be 0.3 mm (Woodetal.,2007).Themediumisinitiallyinseminatedbya vol-umeofbacteriacorrespondingto5.5%oftotalvolume.Thebacteria arerandomlydistributedontheporewalls.Avelocityfield is im-posedattheinletofthedomain(i.e.,atx=0)whilethepressure iskeptconstantattheoutlet(i.e.,atx=1).Periodicityconditions are introduced atthe boundariesz=0 andz=1.The influence offluidflow,substrateconsumptionandtheimpactofdetachment andattachment mechanismsonthe structuralarchitecture ofthe biofilmaretheninvestigatedthroughdifferentvaluesattributedto thedimensionlessPécletandDamköhlernumbers.

Table2containsthedimensionlessparametersusedforthe nu-mericalsimulations andvaluesareintherangeofthosefoundin theliterature(Botteroetal.,2013;Knutsonetal.,2005;Picioreanu etal.,1998a).Notethatthevalueof



andF_λhavebeenchosento achieveafulluncouplingbetweenthebiofilmgrowthandnutrient transportandconsumption.Theextinctioncoefficientkd wasfixed atkd=3.4× 10−8s andthe half-saturation constant K varies be-tween0.0032mg/land0.32mg/l.Themaximumgrowthrate

μ

max

wasimposedviatheDamköhlernumber.

Fig. 9. Biofilm growth for Pe = 50 and Da = 1 . The two black lines represent the biomass concentration contour, solid line at ρ

bio = 0 . 1 , dashed line at ρbio = 0 . 9 (a) Biofilm

growth with bacteria detachment at t  _{= 35 (mesh 100 × 33), (b) Biofilm growth with bacteria detachment at t}  _{= 35 (mesh 80 × 26), (c) Biofilm growth without bacteria}

Fig. 10. Initial geometry (representative region of a porous medium extracted from Wood et al., 2007 ), the biofilm phase is illustrated in green, solid in red and fluid in blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 2

Physical and biological parameter values used in the simula- tions.

Parameter physical value dimensionless value

lx 12 mm lx = 1 lz 12 mm lz = 1 Nx 150 Nz 150 ρbio(t = 0) 10 4 mg / l ρbio (t = 0) = 1 C0 32 mg / l C  0 = 1 Df 10 −9 m 2 / s D  f = 1 Db 0 . 25 × 10 −9 m 2 / s D  b = 0.25 Fλ 0.7 kd 3 . 4 × 10 −8 s k  d = 5 × 10 −3 K 0 . 0032 − 0 . 32 mg / l K = 10 −4 − 10 −2  300 τmax 1.39 Pa Cdet 5

Note that the dimensionless shear stress value,

τ

, implicitly related to the Péclet number (through the pore velocity magni-tude) varyalsowiththenumericalcomputationsperformed(

τ

max

iskeptconstant).Fromallthesimulationscarriedout,threepairs of Pe− Davalueshavebeen chosen asparticularlyrepresentative ofthedifferentgrowthregimesobtained:Pe=1-Da=10−2 (low

Pe,lowDa,low

τ

),Pe=10−1 -Da=102_{(lowPe,highDa,low}

τ

₎ andPe=500 -Da=10−1 (highPe, low Da, high

τ

). The results arediscussedbelow.

4.3.1. LowPe,lowDa,low

τ

For low values of Pe (Pe=1), there is no influence of shear stress on the biofilm growth. Moreover, as the Damköhler num-ber(Da=0.01)issmallcomparedtothePécletnumber,transport mechanisms (diffusion in the biofilm phase and convection and diffusioninthefluidphase)prevailrelativetothe reaction kinet-ics. Consequently,the substrateis uniformlyavailable throughout theporousmedium (Fig.11a)andthebiomassgrowsinarelative uniformmanneraroundthesolidgrains(Fig.11c).

However,evenfortheselowvaluesofPe− Da,biomass accu-mulationisslightlymorepronouncednearthenutrientsource,i.e., atx₌0(Fig.11b)duetoalowbutnon-zeroconcentration gradi-entalong the flow direction. Thissmall discrepancyingrowth is amplifiedwithtime andleadstoa localdecreaseofthemedium porosity,andeventuallya clogging ofthepores inthevicinity of the inlet (Figs. 11c and d). Ultimately, the bacterial colonies far fromtheentrancewillreceivelesssubstratewithconsequenceson theirdevelopmentandfinally,causingtheirextinction.Weshould recoveratlarge timesthe observed behaviorfor low Pe,highDa

(seeSection4.3.2).

4.3.2. LowPe,highDa,low

τ

We now considerthe case ofa growing biofilm under condi-tions of large Damköhler number, i.e., Da=100 but still at low Pécletnumber (Pe₌0_.1). The characteristicreaction time is now significantly lower than the characteristic time associated with nutrient transport. In other words, the electron donor species is consumedbeforecompletelypassing throughthemedium. Aswe can see in Fig.12a, we have a very large concentration gradient on the first cells of domain and a concentration that drops

Fig. 11. Simulation results for Pe = 1 , Da = 0 . 01 . (a) substrate concentration field at t  _{= 100 , (b) biomass concentration field at t}  _{= 100 , (c) geometry of the medium at} t = 100 , (d) geometry of the medium at t  = 9800 .

downclosetozerojustaftertheentrance.Thissignificant reduc-tion of the concentration leads at the inlet, where the substrate concentrationismaximum,toarelativelylarge increasein bacte-rialconcentration(Fig.12bandc).Onthecontrary,neartheoutlet wheretheamountofavailablesubstrateisreducedorinsufficient, biomass growth is very limited or even non-existent (Fig. 12a). Bacteria extinction begins to appear far from the injection zone (Fig.12b).

It is for such conditions that the plugging of the medium may be fast. In our case, a full bio-clogging near the entrance ofthe porousmedia occursafter t=2 (Fig. 12c andd). The hy-drodynamicpropertiesofthemedium arestrongly impacted.The porosityofthemediumdecreases slightlyovertheentiredomain (althoughitdecreasesstronglylocally)whilethepermeabilityfalls ofseveralordersofmagnitude.

4.3.3. HighPe,lowDa,high

τ

We end this analysis by the case at high Péclet number,

Pe=500 and low Damköhler, Da=0.1. As in the case consid-ered in section 4.3.1 (Pe=1 – Da=0.01), the Péclet number is much largerthan theDamköhler number,whichimpliesthat the transport mechanisms are predominant relative to the reaction rate.Atshorttimes,thebehaviourisidenticalwiththatdescribed inthe above-mentioned paragraphcharacterized by a rather uni-formbiofilmgrowtharoundthegrainsanduniformdistributionof nutrientconcentrationwithinthemedium(Fig.13a).

However,withthedevelopmentofthebiofilm,thepores grad-ually shrink. We observe firstly a clogging of the finer pores, whereasonlyafewchannelsremainopen.Thesechannelsrapidly create preferential flow paths which lead to an increase of local flowvelocitiesandhence,ofshearstressesexertedonthebiofilm

Fig. 12. Simulation results for Pe = 0 . 001 , Da = 100 . (a) substrate concentration field at t  _{= 1 , (b) biomass concentration field at t}  _{= 1 , (c) medium geometry at t}  _{= 1 ,}

(d) medium geometry at t  _{= 2 .}

surface(Fig.13e).Assoonastheshearstressesexceedtheimposed critical value

τ

max (i.e.,

τ

> 1),the biomass issubjected to

ero-sion (Fig. 13c). In stagnant flow regions, located between several solid grains,theclogging becomesalmostcomplete (Fig.13d). Af-tersometime,t=140,thesystemseemstoreachanequilibrium betweenbacterialgrowthanddetachmentprocesses.Thesubstrate isonlyavailablewithinthechannelswherethegrowthisnolonger possibleduetotheintensityoftheshearstress(Fig.13b).The pref-erential flow mechanism encountered in this simulation is very similar to the one observed during dissolution in porous media withthedevelopmentofwormholes(Golfieretal.,2002).

4.4. GrowthregimediagramPevsDavs

τ

Based on the simulations carried out, three characteristics regimes of biofilm growth in porous media havebeen observed. Theseregimescanberelatedtothedimensionlessnumbersvalues ofPe,Daand

τ

:

Reactionrate-limited growth:biofilmdevelopsmainly closeto theinjection zone of the nutrient, leadingrapidly to a to-talbio-pluggingofthemedium(Fig.12).Thisregimeoccurs whenPe _Daand

τ

_<1.

Mass transfer-limited growth: biofilm grows uniformlyin the medium resultingina progressivebutuniformdecrease in porosity(Fig.11).ThisregimeoccurswhenPe_Daand

τ

<1.

Shear rate-limited growth: preferential flow paths develop within the colonised medium, under the effect of shear forces. The obstruction of the medium is never complete (Fig.13).Thisregimeoccurswhen

τ

>1.

4.5.Calculationofmacroscopicproperties

At this point, we can take advantage of these pore-scale simulations to investigate the impact of physical changes in the microstructureon theeffectivepropertiesof theporousmedium. To illustrate these biofilm-induced changes on the macroscopic

Fig. 13. Simulation results for Pe = 500 and Da = 0 . 1 . (a) Substrate concentration at t  _{= 70 , (b) substrate concentration at t}  _{= 140 . (c) medium geometry at t}  _{= 70 ,}

Fig. 14. Temporal variation of biofilm volume fraction and permeability. (a)-(b) Pe = 0 . 001 , Da = 100 , (c)-(d) Pe = 1 , Da = 0 . 01 , (e)-(f) Pe = 500 , Da = 0 . 1 .

behaviour ofthe medium, we willfocus on the relation existing between permeability and porosity. Indeed, as a consequence of biofilm growth, the relationship between the macroscopic quantities may be historical. It must be emphasized that this question is of a general interest, and similar problems were found ingeochemistry,orin dissolutioninporous media(Golfier et al., 2002). Anapproach, classically made, consistsin assuming a direct relationship between the permeabilityand the porosity, i.e.K(

)insteadofK(t)and

(t).Here,wewilladoptthispointof view.

In order to test this assumption, the biofilm growth patterns obtainedat thepore-scale inthe previous Section 4.3were used todeterminethemacroscopicparameters.Fromtheseriesof sim-ulationsperformedfordifferentvaluesofPeandDaonthe repre-sentativeelementaryvolume(REV)ofporousmedium,thebiomass volumefractioniscalculatedatthedifferenttimesteps byspatial integration. In a similar way, the permeability is calculated over

theREVbyDarcy’slaw:

K=umoy

μ

f

· L

dP , (43)

whereKisthepermeability,Lthelengthofthedomain(hereequal tolx)anddPthepressuredifference.

Fig. 14 shows the temporal evolution of both the biofilm volume fraction

b and the dimensionless permeability K= KK0, whereK0istheinitialpermeability.FordifferentvaluesofDaand

Pe,aquickdecayofthepermeabilityatthebeginning ofthe sim-ulationis firstobserved. Thissharpreduction ofK isdueto the choice of initial conditions.We start the simulations withC=1 everywhereandahighvalueoftheinitialbacterialconcentration (

ρ

_bio =1) that enhance a fast spreading of biomass inthe pores before reaching a stabilisation of theconcentration profile.Then, permeabilitygraduallydecreasesmoreorlessdrastically.For reac-tionrate-limitedregime,permeabilityfallsdowntozeroduetothe finalbio-cloggingofthedomain(Fig.13bandd)anddropsof

sev-Fig. 15. Relationships between permeability and biomass volume fraction.

eralordersofmagnitudeforthemasstransferlimitedregime.Such asignificantpermeabilitydecreaseisinagreementwiththe exper-imental observations ofCunningham etal. (1991) andKim et al. (2006) who have found a reduction in permeability up to three ordersofmagnitudedueto biofilmformation.Conversely, forthe shearrate-limitedgrowthregime,thepermeabilityconvergestoa plateau(Fig.13f)corresponding to adecrease ofaboutone order ofmagnitude. Thisphenomenon is dueto theformation of pref-erential flow channelsthat preserve partiallythe permeability of the porous medium. In the same time, the biofilm volume frac-tionincreasesquasilinearlyforallthesimulationsafteranabrupt increase atshort times (related to the sharpdecrease of perme-abilitydiscussed above). Forboth reaction rate-limited andmass transfer-limited growthregimes, the increase of

_b is limited by the bioclogging(Fig. 13a andb) while forthe shear rate-limited growthregime,itstabilizesaroundaconstantvalue(about0.4for

Da=0.1andPe=500)duetothequasi-equilibriumstatebetween biomassgrowthandbacterialdetachment(Fig.13e).

Finally,wehaverepresentedinFig.15thecorrelationsrelating the change in permeability with the increase in biofilm volume fraction for all the growth regimes. We observe significant dis-crepanciesbetweenthesedifferentrelationshipsduetothe differ-encesinbiomassdistribution forthebiofilm growthregimes.For comparisonpurpose,wehavealsorepresentedinthesamefigure thepermeabilitychangespredictedbythesemi-empiricalKozeny– Carman law classically used in macroscopic models of biofilm growth, K=CS

3

(

1−

)

2 =CS

(

0−

b

)

3

(

1−

0+

b

)

2 , (44)

where CS represents a geometrical shape factor,

0 the initial

porosityofthedomainand

bthebiomassvolumefraction. If thisformulationpredicts correctlythe permeabilitychanges forthe mass transfer limitedregime since the biofilm is coating gradually the solid grains, it fails to reproduce the behavior ob-served whenthe shear stress or thereaction rateis the limiting

stepforbiofilmgrowth.Thisresultemphasizestheneedoffurther research for predicting bio-cloggingand shows that an universal relationshipbetweenhydraulicconductivityandeffectiveporosity cannotbeformulated.Thisimportantpointshouldbekeptinmind forlarge-scalesimulationsinbiofilm-relatedapplications.

5. Conclusions

In this work, a two-dimensional pore-scale numerical model ofbiofilmgrowthwasdeveloped.Wehaveadopteda cellular au-tomatamodelfordescribingthespatialandtemporalevolutionof thebiomassundertheinfluenceofbacterialgrowthandthe mech-anismsinducedbyfluidflow,suchasdetachmentorattachmentof bacterialfloc.Theuseofimmersedboundarymethodsallowedus to overcome themain drawback encountered inthis type of ap-proachbecause ofthe sizeof theelementary cell thatconstrains the volumeof biomass producedat each time step.Fluidflow is simulated with an immersed boundary–lattice Boltzmann model whilesolutetransportisdescribedwithavolume-of-fluid-type ap-proach.First, this modelwas validatedfrom qualitative compari-sonwithbenchmarkcasesissuedfromtheliterature.Under hydro-staticconditions, the numerical solutions are strongly dependent of the Damköhler number. At low Da, the influenceof the mass transferratewithinthebiofilmispredominant.Thelarge penetra-tion of nutrient concentration inside the biofilm leads to an en-hanced microbial growth and a bacterial concentration relatively uniformacrossthebiofilm.AthighDa,thebiofilmgrowthis lim-itedbysubstratediffusion,generatinganunstabledevelopmentin theformoffingersforbacterialcolonies. Inhydrodynamic condi-tions, ourinterestis focusedon theinfluence,through thePéclet number,ofconvectiononsubstrateavailabilityandthechangesin bacterial population. When the detachment mechanisms are not considered, the bacterial growth is heterogeneous at low Péclet number and tends towards a homogeneous distribution at high Péclet number.In a second step,the model wasapplied to a 2D complexporousmedium,inordertoanalyzetheinfluenceof envi-ronmentalconditions(reaction kinetics,flow rate)onbiofilm pat-terns.Based onthenumericalresults obtained,a regime diagram representingthedifferentmodesofbiofilm growthwasproposed.