Upscaling of transport processes in porous media with biofilms in equilibrium and non-equilibrium conditions
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Upscaling of transport processesin porous media with biofilms inequilibrium and non-equilibriumconditionsL. Orgogozo a , F. Golfier a & M.A. Bus aa Laboratoire Environnement , Gomcanique et Ouvrages, Nancy-Universit , Rue du Doyen Roubault-BP 40 F-54501, Vanduvre-ls-Nancy, FrancePublished online: 01 Oct 2009.
To cite this article: L. Orgogozo , F. Golfier & M.A. Bus (2009) Upscaling of transport processes inporous media with biofilms in equilibrium and non-equilibrium conditions, Applicable Analysis: AnInternational Journal, 88:10-11, 1579-1588, DOI: 10.1080/00036810902913862
To link to this article: http://dx.doi.org/10.1080/00036810902913862
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Applicable AnalysisVol. 88, Nos. 1011, OctoberNovember 2009, 15791588
Upscaling of transport processes in porous media with biofilmsin equilibrium and non-equilibrium conditions
L. Orgogozo*, F. Golfier and M.A. Bues
Laboratoire Environnement, Geomecanique et Ouvrages, Nancy-Universite, Rue du DoyenRoubault-BP 40 F-54501, Vanduvre-les-Nancy, France
Communicated by R.P. Gilbert
(Received 16 January 2009; final version received 11 March 2009)
Transport of biologically reactive dissolved solutes in a saturated porousmedium including a biofilm-phase occurs in various technologicalapplications such as in biochemical or environmental engineering. It isa complex process involving a wide variety of scales (from the bacteria-scaleto the aquifer-heterogeneities-scale in the case of groundwater remediation)and processes (hydrodynamic, physicochemical and biochemical). Thiswork is devoted to the upscaling of the pore-scale description of suchprocesses. Firstly, one-equation macroscopic models for bio-reactivetransport at the Darcy-scale have been developed by using the volumeaveraging method; they will be presented below. These one-equationmodels are valid for different limit cases of transport; their validity domainsin terms of hydrodynamic and biochemical conditions will also bediscussed. Finally, in order to illustrate such a theoretical development,an example of application to the operation of a packed bed reactor will bestudied.
Keywords: biofilm; porous media; transport; upscaling; volume averaging
AMS Subject Classifications: 74Q15; 76S05; 92C45
Modelling transport in saturated porous media of organic chemical species inpresence of a bacterial population growing in the form of biofilms is an importantarea of research to environmental and industrial applications, e.g. treatmentand remediation of groundwater contaminated by organic pollutants (bio-sparging,bio-barriers, ) or industrial processes (waste water treatment, bio-foulingelimination, ). Biofilms, which are composed of bacterial populations andextracellular organic substances, grow on the grains of a porous medium, in whichthree phases are present: fluid, solid and biofilm. In the biofilm-phase, bacterialmetabolism converts the dissolved organic chemical species into biomass or otherorganic compounds. In order to optimize various applications involving such
*Corresponding author. Email: Laurent.Orgogozo@ensg.inpl-nancy.fr
ISSN 00036811 print/ISSN 1563504X online
2009 Taylor & FrancisDOI: 10.1080/00036810902913862
phenomena, we need to carry out accurate numerical models of transport of organicsolute in a porous medium including a biofilm-phase. The pursuit of this objectivehas led to an extensive literature (see, e.g. ). One of the difficulties in thedevelopment of such models is the multi-scale aspect of these problems. Figure 1shows the different scales under consideration in this work. The upscaling oftransport phenomena from the bacteria-scale (I) to the pore-and-biofilm-scale (II)has been already studied (e.g. ). In this work we focus on the second level ofupscaling, i.e. from the pore-scale (II) to the Darcy-scale (III). We consider anorganic solute A, which is transported by convection and diffusion in the fluid-phase(the -phase) and by diffusion within the biofilm-phase (the !-phase) where it isconsumed by the bacterial metabolism. A classical Monod kinetics will be adopted todescribe the consumption reaction (e.g. ). In the general case, biodegradationkinetics involves two chemical species: the source of carbon our organic solute A and the electron acceptor, e.g. dioxygen or nitrate, denoted B. For the simplicity ofthe exposure, we will assume in this study that the electron acceptor is in large excessso that its concentration can be considered to be constant and thus the consumptionof solute A is governed by a simple Monod reaction kinetics. The fluid- and biofilm-phases are assumed each to be continuous and homogeneous except at the phaseboundaries whereas the solid-phase (the -phase) is considered to be passive
100 m to 1 cm
(e.g. aquifer or reactor)
1 dm to 10s of m
L >> R
R >> ro
10s of m to 100s of m
Figure 1. The different scales involved in transport phenomena in a porous medium witha biofilm-phase.
1580 L. Orgogozo et al.
(no reaction) relative to transport. We will consider also that there is noconcentration jump at the fluid-biofilm interface: there is continuity between theconcentration fields in each phase. Finally, we uncouple the solute transport
and bacterial growth processes by considering that the characteristic time-scaleof the first one is very small compared to the characteristic time-scale of thesecond one. Under these assumptions, the pore-scale transport problem for speciesA takes the following form (all the presented equations are dimensionless):
r DA rcA! DamcA!
1 cA!KA!in the !-phase, 1
r2cA in the -phase 2
n! DA rcA! 0 at A! 3
n rcA 0; at A 4
n! DA rcA! n! rcA at A! 5
cA cA!; at A!: 6
Here, cA and cA! represent the concentration of the organic solute A in the - and!-phases, respectively; DA is the effective diffusion tensor of species A in the biofilm;v is the fluid velocity; KA! is the effective half-saturation of the solute A. We haveused the terminology A to indicate the interface between - and -phases, and A!and A! to indicate the interface between - and !-phases and between - and!-phases, respectively. The term n indicates the unit normal pointing outwardfrom the -phase towards the -phase; n! and n! are similarly defined. Pem andDam are, respectively, the Peclet number and the Damkohler number associated to
the pore-scale.The upscaling of transport equations from the pore-scale to the Darcy-scale is
done by using the volume averaging method (e.g. ). Briefly, microscale equationsare averaged over a volume V of the medium which is representative of itsmicroscopic structure (Figure 1) and which satisfies the assumption of separation ofscales: its characteristic length must be large compared to the characteristic lengthsof the microscale and small compared to the characteristic lengths of the macroscale.The upscaling process leads to equations of transport at macroscale, which govern
averaged concentrations. These avera