Borsi, I

Farina, Angiolo

Fasano, Antonio

Primicerio, Mario

English

Florence Research

November 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p1BIOMASS GROWTH IN UNSATURATED POROUS MEDIA:HYDRAULIC PROPERTIES CHANGESI. BORSI, A. FARINA, A. FASANO, M. PRIMICERIODipartimento di Matematica - Universita` di FirenzeViale G. Morgagni, 67/A50134 Firenze, ItalyE-mail: borsi@math.unifi.itWe present a model to describe the biomass growth process taking place in anunsaturated porous medium during a bioremediation process. We focus on theso-called column experiment. At the initial time biomass and polluted water isinoculated in the column. The subsequent changes of hydraulic properties areanalyzed. We also show some preliminary simulations.Keywords: Microbial growth; Porous media; Bioremediation.1. IntroductionThe effect of the microbial growth on hydraulic properties of porous mediais a topic studied in the framework of many applications, e.g. oil recovery,wastewater treatment, bioremediation, etc. (see [1]).Studies on flow through porous media in presence of biomass growth arepresented in the papers by Rockhold et al. [2–4]. As stated there, additionalwork is needed for modelling unsaturated condition.The objective of this study is thus to analyze the flow through a contami-nated unsaturated porous medium in presence of biomass growth processeswhich induce changes in the hydraulic properties of the medium itself.In this paper we focus only on anaerobic processes, namely the model wedevelop does not account for the O2 consumption and diffusion.2. Problem description and physical assumptionsWe consider a vertical column (whose high is L ∼ 1m) of an unsaturatedcontaminated soil which represents a “laboratory scale” of a real vadosezone (the so-called “column experiment”, see Fig. 1). The physical model isdeveloped considering a 1-D approximation, so that x denotes the verticalNovember 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p2Fig. 1. A schematic of the column experiment.coordinate of the column, pointing upwards.At the initial time (t = 0) the saturation degree of the medium correspondsto the steady state. Then we inoculate biomass and (possibly) pollutedwater through the top surface. Our goal is to model the evolution of thebiomass, pollutant concentration and hydraulic properties of the soil aswell.Hereafter we list the most significant physical assumptions (see also [5]):A.1 The soil is a homogeneous, unsaturated, rigid porous medium.A.2 The liquid phase which shares the empty space with air is composed bywater (main component). We shall neglect the liquid density variation.A.3 The pollutant is dissolved in water and adsorbed onto the soil grains.Moreover, the dissolved pollutant (below a certain concentration) actsas nutrient for the bacteria (bio-reduction), but above a certain thresh-old may become a toxic agent (see [6]).A.4 The biomass is distributed either in water as suspension (“freebiomass”) or attached on the soil grains (“attached biomass”). In par-ticular, there is no cluster formation in the free biomass. The mass ofthe single bacterium is known and denoted by mb.A.5 The attached biomass forms porous clusters so that the liquid can dif-fuse through them. The clusters porosity is a known constant denotedby εb. The pores are saturated at all times. Moreover, the concentrationof the pollutant in the “entrapped water” equals the one of the “freewater” (see [3], for instance). The number of bacteria that forms theunit mass of the attached biomass is a known constant denoted by N?,[N?] = Kg−1. Of course, N?, εb and mb depend on the type of bacteriawhich are present inside the column.A.6 We consider the attachment of free biomass on the clusters, but weneglect the inverse process (i.e. we neglect detachment). Indeed theNovember 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p3experiments show that detachment is mainly due to the mechanicalaction caused by the “fast” water flux, [10].A.7 The concentration of pollutant and bacteria in the liquid phase is low(few ppm).Free biomass and attached biomass are responsible for different effectschanging hydraulic properties. More precisely:E.1 The free biomass causes essentially viscosity and surface tension varia-tions.E.2 The attached biomass growth causes medium porosity variations andaffects the contact angle.E.3 The above variations induce, in turn, changes in the permeability andin the relative saturation of the medium.3. Notations and basic equationsWe introduce the following quantities:• ε0, [ε0] = [−], initial porosity of the column (known parameter).• εb, [εb] = [−], clusters porosity (considered a known constant).• φf , volume fraction occupied by the liquid and the gaseous (air) phase.• φc, volume fraction occupied by the clusters.• σ, [σ] = [−] liquid phase saturation.• θlm =volume of “mobile” liquidporous medium volume, [θlm] = [−]. In particular, θlm = σφf .• θlb = εbφc =volume of “clusters− stored” liquidporous medium volume, [θlb] = [−].We thus have thatφf + φc = ε0.So, the volume fraction occupied by the liquid (accounting for mobile andstored) isθlm + θlb = σφf + εbφc = (σ − εb)φf + εbε0. (1)The dependent variables which have to be determined are:• σ(x, t), the liquid phase saturation.• φc (x, t), or alternatively φf = ε0 − φc.• Nl(x, t) =number of free bacteria in the liquid phaseunit mass of free liquid, [Nl] = Kg−1.• wA(x, t) =mass of pollutant dissolved in the liquid phaseunit mass of free liquid.November 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p4• ws(x, t) =mass of pollutant adsorbed onto soil grainsunit mass of solid matrix.Darcy’s law and Richards equationWe define the water pressure P and introduce the capillary pressure Pc andthe pressure head ψ, settingPc = Pair − P = −P, ψ =−Pρg=Pcρg,since, as usual, Pair has been rescaled to 0.The well-known Darcy’s law describes the specific discharge q,q = −ρgKsatµkrel(∂ψ∂x+ 1),where ρ water density, g gravity acceleration and• Ksat = Ksat(φf ), saturated permeability, [Ksat] = m2. For instance, wemention the Kozeny-Carman formula [11].• krel = krel(ψ), [krel] = [−], relative permeability (see [14]).• µ is the liquid phase viscosity. We assume the following law (based onexperimental observations [2])µ = µ˜(Nl) = µ0 (1 + b3Nl) , (2)with b3 empirical parameter and µ0 water viscosity (i.e. the liquid vis-cosity in absence of biomass).Next, the Richards’ equation [14] reads as∂∂t[θlm + θlb] = −∂q∂x. (3)Now, introducing the saturation curve σ = σ˜ (ψ), we have θlm = φf σ˜(ψ)so that, exploting (1), the mass conservation (3) rewrites as∂∂t[(σ − εb)φf]= −∂q∂x. (4)Evolution equation for the attached biomass phaseAccording to the literature (see for instance, [7] § 2.3 and [12]) we set∂φc∂t= (c1N?) [ε0f (wAεbφc)− φc]φc︸ ︷︷ ︸biomass “bulk growth”+λH (ws − wA)Nlθlm︸ ︷︷ ︸,attachment(5)November 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p5where:• [c1] = Kg s−1.• wAεbφc is the amount of nutrient available for the attached biomass.• ε0f (wAεbφc), is a modified form of the carrying capacity (see [13] § 1.2).Actually, ε0 is the maximum volume fraction allowed for the attachedbiomass and it is “modulated” by the function f , 0 ≤ f ≤ 1, whichaccounts of both amount of nutrient and toxic effects (see assumptionA.3).• H(·) is the Heaviside function. We notice that the attachment termcould be multiplied also by a function of φc, i.e. an “effective surface”term modeling the so-called collector and collision (or sticking) effi-ciencies.Evolution equation for the free biomassThe free biomass is a component of the liquid phase. Therefore, following[8,9], the evolution equation for Nl is∂∂t(Nlθlm) = −∂∂x[q Nl] +∂∂x[aL |q |∂Nl∂x]︸ ︷︷ ︸advection/dispersion+ c1 [Nmaxf(wA)−Nl]Nlθlm︸ ︷︷ ︸free biomass growth−λH (ws − wA)Nlθlm︸ ︷︷ ︸ .attachment(6)The first term in (6) is the divergence of the advective flux Jadv = −qNl.The second term represents the dispersive flux Jdisp, which in 1–D settinghas the following formJdisp = −σφfD, D =aL |q|σφf,with aL longitudinal dispersion coefficient (see [14]).The “bulk growth” of the free biomass is modeled as in the case of theattached biomass, i.e. by means of a logistic–type dynamics. Here Nmax isthe equilibrium value. We setNmax = nN?, 0 < n < 1.November 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p6Equation for the adsorbed pollutant.We describe the dynamics of ws by the following equation∂wS∂t= hAwAεbφc(w? − wS)︸ ︷︷ ︸adsorption term− hDB N?φc wS︸ ︷︷ ︸ ,bio-reduction term(7)stipulating, essentially, that only two effects are important: adsorption (des-orption) and bio-reduction. In particular,• wAεbφc = εbθlb is the amount of pollutant dissolved inside the biomassclusters.• w? is maximum concentration of pollutant (known parameter) whichcan be adsorbed by the soil.• hA, [hA] = s−1, is the adsorption/desorption rate per unit concentra-tion.• hDB, [hDB] = Kg s−1, bio–reduction rate per unit mass.Equation for wA.The total amount of pollutant (per unit volume of porous medium) dis-solved in the free and “entrapped” water is wA(θlm+θlb), i.e. wA(θlm+εbφc).According to [9], we write for it the following equation∂∂t[wA(θlm + εbφb)]= −∂∂x[q wA] +∂∂x[aL |q |∂wA∂x]︸ ︷︷ ︸advection/dispersion term− hAwAεbφc(w? − wS)︸ ︷︷ ︸adsorption term− hDB [θlmNl + φcN?]wA (θlm + εbφc)︸ ︷︷ ︸ .bio-reduction term(8)The bio–reduction term in (8) depends on the amount of bacteria whichare present either in the liquid and in the clusters.Summarizing, we have to solve the system of the governing equations (4),(5), (6), (7) and (8), which has to be endowed with initial and boundaryconditions.November 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p74. A simplified approach: biofilm and fluid media scalingWe now illustrate the basic idea of an approach to simplify the problem.The key point is to consider a porous medium constituted by a network ofcapillary tubes distributed uniformly in space. Next, we assume that theattached biomass phase forms a uniform layer (biofilm) completely coatingthe internal surfaces of the capillary tubes.Focusing now on a single capillary tube, we compare two scenarios: capillarytube partially filled with “pure” water and capillary tube whose walls arecoated by the biofilm and partially filled by a liquid whose components arewater, bacteria and pollutant.Denoting by pc and pc, bio the capillary pressures which refer to the abovescenarios, we may write the following Laplace formulaspc =2γ cosαR, pc,bio =2γbio cos(αbio)Rbio,where R and Rbio denote the capillary radii, γ and γbio are the surfacetensions and α, αbio are the contact angles. Therefore,pc,biopc=γbioγcosαbiocosαRRbio.We now assume that the above formula holds true also for the averagedquantities, i.e.Pc,bioPc=γbioγcosαbiocosα〈R〉〈Rbio〉, (9)where Pc,bio = 〈pc,bio〉, Pc = 〈pc〉, since 〈 · 〉 denotes the R.E.V. average.Now, selecting appropriate constitutive equationsa forcosαbiocosα,γbioγand〈R〉〈Rbio〉,we can define the parameter β = β(φf , Nl), (called scaling factor for thecapillary pressure), such thatPc,bioPc= β(φf , Nl), ⇒ ψ = β(φf , Nl)ψ0, (10)where ψ0 = −Pcρgis the pressure head in absence of biomass. Therefore,θlm = σ˜(β(φf , Nl)ψ0)(ε0 − φc) .aSuch equations strongly depend on the intrisic geometry of the medium, see [5].November 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p8Such an approach (often called fluid media scaling, see [15]), presents anevident advantage: once the flow problem is specified, we determine ψ0using the “classical” Richards’ equation, i.e. equation (4) where the termdue to porosity changes is absent. As second step we evaluate ψ exploiting(10) and q through (2). Therefore, the mathematical problem is stronglysimplified and its numerical solving turns out easier. Of course, the fluidmedia scaling suffers from an evident drawback: ψ and q do not fulfill theRichards’ equation, i.e. mass conservation. Hence, such a property needs tobe tested a posteriori.5. Numerical simulationsIn this section we present few numerical simulations we worked out consid-ering the simplified model. A deeper analysis of the results can be found inthe forthcoming paper [5]. Our main goals are: (i) to show that the foundsolution satisfies (within a suitable tolerance) the Richards’ equation; (ii)to show that the results obtained agree, at least qualitatively, with theexperimental data; (iii) to put in evidence that, in certain circumstances,variation of porosity and hydraulic properties is significant.5.1. Problem settingThe PDEs system was solved in a 1D domain. Concerning the soil, weused the well-known van-Genuchten and Mualem forms for the saturationand permeability curve, respectively (see [14]). Moreover, a Cozeny-Karmanfunction for the saturated permeability ksat(φf ) has been selected (see [11]).Finally, we run the simulation considering Tmax = 7 day as maximum timeof the process.Initial conditionsThe initial stage of the experiment is characterized by absence of attachedbiomass and a given pollutant concentrations. Hence we setwA(x, 0) = w0A, wS(x, 0) = w0S , Nl(x, 0) = 0, φc(x, 0) = 0,with w0A = 0.3 and w0S = 0.7.Boundary conditionsFollowing [4] we set on the column bottom, x = 0,∂Nl∂x∣∣∣∣x=0= 0,∂wA∂x∣∣∣∣x=0= 0.November 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p9On the column top, x = 1 water (with pollutant and bacteria) is inoculatedin the medium. We stipulate that pollutant and bacteria concentration inthe inflow water are known, hence,Nl(1, t) = N1(t), wA(1, t) = wA 1 (t) .5.2. Simulation resultsFirst, we mention the computed value of the quantitymax(x,t)∈[0,1]×[0,tc]∣∣∣∣∂(θlm + θlb)∂t +∂q∂x∣∣∣∣ ∼ 3.8× 10−7which definitively shows that the computed solution ψ = βψ0 satisfies theRichards’ equation.Moreover, in Fig. 5.2-5.2 we report the plot of the most sgnificant quanti-ties computed during the simulation. All the values are plotted at initial,intermedium and final time step.Further comments on the obtained results will be reported in the forthcom-ing work [5].References1. Holden P.A., Fierer N., Microbial processes in the vadose zone,Vadose ZoneJ. 4 (2005), 1-21.2. Rockhold M.L., Yarawood R.R., Niemet M.R., Bottomley P.J., Selker J.S.,Consideration for modelling bacterial-induced changes in hydraulic propertiesof variably saturated porous media, Adv. Wat. Res. 25 (2002), 477-495.3. Rockhold M.L., Yarwood R.R., Seker J.S., Coupled microbial and transportprocesses in soils, Vadose Zone J. 3 (2004), 368-383.4. Rockhold M.L., Yarawood R.R., Niemet M.R., Bottomley P.J., Selker J.S.,Experimental observations and numerical modeling of coupled microbial andtransport processes in variably saturated sand, Vadose Zone J. 4 (2005), 407-417.5. Borsi I., Farina A., Fasano A., Primicerio M., Modelling changes of hydraulicproperties induced by biomass growth in unsaturated porous media, to appear.6. Gallo C., Hassanizadeh S.M., Modeling NAPL dissolution and biodegradationinteractions: effect of toxicity and biomass growth limitations,ComputationalMethods in Water Resources, S.M. Hassanizadeh et al. (Eds.), vol. 1 (2002),859-866, Elsevier, Amsterdam.7. Rajagopal K.R., Tao L., Mechanics of Mixtures, World Scientific, Singapore,1995.8. Scha¨fer A. et al., Transport of bacteria in unsaturated porous media, J. Con-tam. Hydrol. 33 (1998), 149-169.November 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p109. Hassanizadeh S.M., Derivation of basic equations of mass transport in porousmedia, Part 2. Generalized Darcy’s and Fick’s laws, Adv. Water Resources,9, 207–222 (1986).10. MacDonald T.R., et al., Effects of shear detachment on biomass growth andin situ bioremediation, Ground Water, 37 (1999), 555–563.11. Carman, P.C, Fluid flow through granular beds, Transactions of ChemicalEngineering, 15, 150-166, London (1937).12. Kingsland, S.E., Modeling Nature, University of Chicago press, Chicago,1985.13. Brauer, F., Castillo–Chavez, C., Mathematical Models in Population Biologyand Epidemiology, Texts in Applied Mathematics, Vol. 40, Springer, NewYork, 2001.14. Bear J., Verruijt A., Modelling Groundwater Flow and Pollution, Theory andApplications of Transport in Porous Media, Bear J. (Ed.), vol. 1, Kluwer,Dordrecht, 1987.15. Miller, E.E., Miller R.D., Physical theory for capillary flow phenomena, J.Appl. Phys., 27 (1956), 324–332.16. Coirier J., Me´canique des Milieux Continus, Dunod, Paris 1997.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 111.051.11.151.21.251.31.35Depth, mScaling factor βt=0t=3 dayt=7 dayFig. 2. Plot of the scaling factor β at different time values.November 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p110 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.4−1.2−1−0.8−0.6−0.4−0.20Pressure head, mDepth, mt=0t=3 dayt=7 dayFig. 3. Pressure head ψ at different times.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.240.260.280.30.320.340.360.380.40.42Depth, mEvolution of Moisture Contentt=0t=3 dayt=7 dayFig. 4. Moisture content (θlm + θlb) at different times.November 28, 2006 16:21 WSPC - Proceedings Trim Size: 9in x 6in simai2006˙b˙f˙f˙p120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.40.50.60.70.80.91Depth, mRelative permeability, [−]t=0t=3 dayt=7 dayFig. 5. Relative permeability krel at different times.

BIOMASS GROWTH IN UNSATURATED POROUS MEDIA: HYDRAULIC PROPERTIES CHANGES

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BIOMASS GROWTH IN UNSATURATED POROUS MEDIA: HYDRAULIC PROPERTIES CHANGES

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