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ORIGINAL PAPER

A general model for bioremediation processes of contaminated soils

I. Borsi · A. Fasano

c© Indian Institute of Technology, Madras

Abstract We review the mathematical models that have been proposed for bioremediation processes, confining our- selves to the case of homogeneous soils with only one species of bacteria and one species of pollutant. We focus mainly on the case of unsaturated soil, trying to clarify some aspects, since most of existing studies refer to saturated soils. The unsaturated case is of great interest, because it is pre- cisely the situation of the vadose zone. Therefore we will try to present a sufficiently complete model for the vadose zone, including some original feature. Finally, a dimensional analysis will lead to a simplified formulation.

Keywords Flow in porous media · Groundwater pollution · Bioremediation

1 Introduction

Soil and groundwater pollution as well as their remedia- tion are challenging subjects in the general framework of environmental risk assessment. There are several sources of soils pollutants, such as spreading of wastes, atmospheric deposition, incinerator or metal smelting emissions, spillage of oil or industrial solvents, irrigation with polluted water, etc. A large number of remediation technologies have been developed in order to reduce the risk of contamination. An exhaustive review of different type of contaminants and relative remediation techniques can be found in [1], [2] and the references therein.

One possible restoration treatment is bioremediation, namely the process in which indigenous or inoculated micro- organisms (i.e., fungi, bacteria and other microbes) degrade organic contaminants found in soil or ground water. There

I. Borsi · A. Fasano Dipartimento di Matematica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy E-mail: [email protected]; [email protected]

are two main types of bioremediation the ex situ implemen- tation, when the contaminated plume can be excavated and remediated in a contained system, and the in situ remedia- tion process, in which microbes are injected into the conta- minated subsurface via a grid of wells (see [3]).

Such techniques has been extensively studied, especially for what concerns the saturated subsurface soils. Less efforts have been done for the so-called vadose zone, namely the unsaturated zone extending from the soil surface to the groundwater table. Nevertheless, the microbial processes taking place in the vadose zone are equally important to understand pollution related phenomena occurring in soils (see [6], [7], for instance).

Much work has been done also concerning mathemati- cal aspects, and simulation softwares based on mathematical models have been developed (e.g., [4] and [5]). In this paper we present a review of different approaches used to model such complex phenomena. In the course of our presentation we will try to build up a model which includes some origi- nal feature. In particular, we may classify the cited works according to the different methods in describing the follow- ing tasks, which play an important role in giving a coherent and exhaustive description of the process, i.e.,

• Flow of water and contaminants – Section 2. • Description of the biomass behaviour (growth kinetics,

attachment/detachment process, etc.) – Section 3. • Influence of the biomass on the hydraulic properties of

the medium – Section 4.

As a matter of fact, a model for a pollutant biodegradation has to consider

1. A mass balance equation for the water flowing through subsurface.

2. Equations for advection, diffusion and reaction of con- taminants, also considering possible sorption/desorption kinetics.

3. Description of biomass growth and consequent contami- nants degradation processes.

34 I. Borsi et al.

4. Relationship between the biomass volume fraction and the hydraulic properties of the soil.

How deeply these aspects have to be investigated depends essentially on the specific target of the model. A mathemati- cal system considering all of them together is pretty com- plex to be managed: indeed, even if it is possible to define a global theoretical model, it can prove to be unsuitable for real applications, due both to the lack of experimental data and to the difficulty of the involved numerics (see [8], [9] for a discussion on the numerical problems). In the sequel the following notation will be used

Nomenclature

ε Porosity, [ε] = [−] φ Volume fraction, [φ ] = [−] S Saturation, [S] = [−] p Pressure, [p] = [N/m2] θ Moisture content, [θ ] = [−] C f Concentration of contaminant dissolved in the fluid,

[C] = Kg/m3

Ca Concentration of pollutant adsorbed on the soil grains, [C] = Kg/m3

b f Volume of the free biomass per unit volume of the medium, [b f ] = [−]

ba Volume of the attached biomass per unit volume of the medium, [ba] = [−]

K Saturated hydraulic conductivity tensor, [K] = m/s k Permeability tensor, [k] = m2

kr Relative permeability, [kr] = [−] q Darcy’s velocity or specific discharge, [q] = m/s Dh Hydrodynamic dispersion tensor of contaminant in

water, [Dh] = m2/s Db Mechanical dispersion tensor of bacteria in water,

[Db] = m2/s B f Biodegradation rate due to the free biomass, [B f ] =

Kg/(s m3) Ba Biodegradation rate due to the attached ++ bio-

mass, [B f ] = Kg/(s m3) ρ Density, [ρ ] = Kg/m3 μ Viscosity, [μ] = Pa s g Gravity acceleration, [g] = m/s2

Subscripts

w Water s Soil

2 Flow regimes and contaminant transport

The liquid flow through the subsurface is frequently mod- elled by the well-known Richards’ equation (see [10]),

∂ ∂ t

(ρ θ ) + ∇ ·(θ v) = 0, (1)

where

• θ = ε S is the liquid volume fraction (or moisture con- tent), • ρ is the liquid density, • v is the liquid velocity.

The latter is given by Darcy’s law in terms of the specific discharge q, i.e.,

q = θ v = −k kr μ

∇ ·(pw + ρ gez) , (2)

where

• pw is the liquid pressure, • k is the permeability tensor, • kr is the relative permeability (which typically is a func-

tion of the saturation), • μ is the liquid viscosity.

In the framework of the application we have in mind, by liquid we mean the mixture of water and water transported bacteria (the pollutant is considered to occupy a negligible volume).

When the liquid density ρ is constant, then (2) can be rewritten in the form

q = −K∇ ·(ψ + ez), (3) where K = (krkρ g)/μ is the hydraulic conductivity tensor and ψ = pw/(ρ g) is the suction (or pressure head). Note that [K] = L T−1 and [ψ] = L.

Moreover, a constitutive relationship linking θ and ψ is assumed, the so-called saturation (or retention) curve. We note that prescribing such a relationship means that the equilibrium between pressure and water content is reached instantaneously. In the presence of sufficiently large varia- tion rates of q, this is by no means the situation, due to the occurrence of relaxation phenomena. For such cases a dif- ferent approach has been proposed (see [11] for instance) which includes dynamical effects expressed by a differen- tial equation linking θ and ψ . If we confine to the case of slowly varying flows we may use equations (1), (3).

Moreover, we note that the flow model (1), (3) disregards thermal effects. Non-isothermal flows require of course more complicated systems (see [12] and [13]).

Even in the simplified framework of Richards’ equation an accurate description of the flow regime should consider the variation of the hydraulic properties of the medium (such as saturation curve, hydraulic conductivity and permeabil- ity). This subject will be discussed in Section 4.

When the flow takes place in a saturated porous medium, then Richards’ equation reduces to

∇ ·q = 0.

A general model for bioremediation processes of contaminated soils 35

If no influence of the biomass on the medium is considered, then the Darcy’s velocity enters the biochemical section of the model only as a prescribed function and the mathemati- cal system to be studied is only the one describing the bio- mass evolution and the pollutant transport/reaction (see [14]–[16]). The same method is true when the characteris- tic time of the flow is much shorter than the time scale of the biomass growth, so that the flow field may be assumed quasi-steady (see [17] and [18]).

However it must be said that the flow is generally affected by the presence of the biomass. The latter consists of

(a) bacteria freely transported by the flow. (b) bacteria attached to the soil and forming a gel.

The bacteria in the first class may occasionally form clusters and they may also enter the class (b). The converse transition is less important. Both classes have their evolution laws that will be considered later. The free biomass can be considered to form a mixture with water. The hydraulic conductivity tensor in (3) is modified for two reasons:

(i) the viscosity of the mixture depends on the concentra- tion of the biomass.

(ii) the medium permeability is modified due to the reduc- tion of porosity produced by the growth of the attached biomass.

An excessive growth of the biomass may lead to clogging, a phenomenon which will not be considered in the sequel.

Since the density of bacteria is very close to the den- sity of water, the biomass growth at the expenses of water does not imply any significant volume change. For the same reason gravity acts in the same way on both elements. This makes it possible to consider a constant density in equation (3), irrespectively of the composition of the mixture. The tensor K instead will be variable.

In writing equation (1) for the mixture we must consider that in our case θ is the sum

θ = θw + b f ,

where θw is the moisture content (volume of water per unit volume of the system) anf b f is the analogous quantity for the free bacteria.

It is important to remark that even if the free bacteria proliferate, the increase of b f occurs at the expense of θw, so that no source or sink term has to be introduced for prolifer- ation. The same is true for the bacteria which die, provided that they are degraded to water in a sufficiently short time. This may not always be a good approximation, but we will keep it to avoid the necessity of introducing a population of dead cells degrading according to some kinetics. The main change to equation (1), when it is interpreted as the mass balance of the mixture is the presence of a free term

∂ ∂ t

(θw + b f ) + ∇ ·q = −Λ , (4)

where Λ is the net mass loss or gain rate due to the biomass attachment/detachment process (see [19]).

It seems necessary to deal with the mixture rather than the two components separately, because the tensor K can be defined just for the mixture. The mass balance for b f will be written later.

In this section we present a rather general model. Let us start by describing the processes of contaminant transport and consumption. For simplicity we refer to the case of a single species.

We start by defining the following concentrations:

• C f is the concentration of pollutant in the flowing water, so that θwC f is the mass of movable pollutant per unit volume of the system. • Ca is the mass of pollutant per unit volume of the system,

which is either adsorbed on the soil grains or distributed within the attached biomass.

We ignore the processes taking place at the microscale and which are responsible for transporting the pollutant from the soil to the biomass and we adopt only one macroscopic quantity, namely Ca, to describe the content of pollutant not transported by the flow.

In this framework the most delicate choice in the evolu- tion model for the pair C f , Ca is the one concerning

(I) the way each species is consumed by the bacteria,

(II) the possible exchange C f ←→Ca. In the water flow there will be a direct consumption by the free bacteria proportional to the product θwC f b f . We denote by B f the corresponding rate constant. Sometimes the degra- dation rate B f is multiplied by a yield coefficient, represent- ing the mass of pollutant consumed per mass of bacteria generated (see [14] and [15], for instance).

Also the attached biomass (ba is the volume of attached biomass per unit volume of the system) is likely to con- sume the free pollutant at a rate which is proportional to the interface area between the fluid and the attached biomass. The interface area per unit volume will be proportional to ba up to some threshold (not considered in the previous litera- ture) ba and will stabilize beyond that value. Therefore we can introduce a “covering index”

σ = max (

ba ba

, 1

)

and write the contribution of the attached biomass to the consumption of the free pollutant in the form

Baσ θwC f ,

where Ba is the rate constant for the action of the attached biomass on the pollutant.

Concerning the immobile pollutant, as we just said we have a consumption due to the attached biomass whose rate

36 I. Borsi et al.

can be controlled as BaCaba. The action of the free biomass on Ca could be described as (1−σ )B f Caθwb f . However it seems reasonable to neglect this quantity and suppose that the free bacteria only consume the free pollutant.

According to the scheme just described we can write the equations

∂ ∂ t

( θw θ

C f

) + ∇ ·(C f q)

= ∇ · ( θwDh∇C f

) −R−B f θwC f b f −Baσ θwC f ba, (5)

∂Ca ∂ t

= R−BaCaba, (6) where

• Dh is the hydrodynamic dispersion tensor, given by the sum of molecular diffusion tensor D and of the mechan- ical dispersion Dd (see [21] for more details). • R is the adsorption/desorption term, whose definition

ranges in a large variety of models (see [10] and [21]). In general, R is defined as the difference between a desorption and an adsorption term, depending on Ca and C f , respectively, i.e.,

R = F1(Ca)−F2(C f ), where F1 and F2 have to be specified.

A simple way of defining R is

R = h(σ )(Ca −θwC f ), (7) where the mass transfer rate coefficient h decreases from a maximum hmax when σ = 0 to a minimum hmin when σ = 1.

Normally, biodegradable contaminants are not soluble in water (e.g., the so-called DNAPL, Dense Non-Aqueous Phase Liquids) and Ca > θwC f , so equation (7) describes a desorption process, which can be very slow (in the sense that hmax � Ba). Sometimes it makes sense to take h ≡ 0, which makes C f = 0 an equilibrium solution of equation (5).

3 Biomass evolution

Biodegradation can completely destroy, transform, or immo- bilize hydrocarbon species, which are substrate for micro- bial growth and an energy source for biodegradation processes that can occur under aerobic or anaerobic con- ditions. As we said, the biomass may stay either in water or attached to the soil grains. The evolution of the two quan- tities b f , ba is driven by several phenomena. A sufficiently descriptive system is the following

∂ b f ∂ t

+ ∇ · (

b f θ

q )

= ∇ · (

θ Db∇ (

b f θ

)) −Λatt + Λdet + Ff ,

(8)

∂ ba ∂ t

= Λatt −Λdet + Fa, (9)

where

• Db is the mechanical dispersion tensor referred to the free biomass (in general pure diffusive processes are neg- ligible). • Ff and Fa are the bacteria productions rates in the two

states (see Section 3.1). • Λatt and Λdet are the attachment and detachment rate,

respectively (see Section 3.2).

The mass balance law for the water can be deduced from the system (4), (8).

3.1 Microbial growth kinetics models

A lot of growth kinetics models has been defined in order to described the microbial evolution for both free and attached biomass. In [6], [14], [20] and [22]–[24] many examples are reported. The simplest choice is a first-order model, i.e.,

Ff = γ1b f C f , (10)

where γ1 is a constant growth coefficient. However, it is often expected that as the pollutant concentration increases the growth rate stabilizes. The linear model (10) does not exhibit this behaviour, so that the majority of authors use the so-called Monod’s kinetics (see [25]), which corresponds to the Michaelis-Menten model used in the context of enzyme reactions, namely

Ff = γ2b f C f

k2 + C f , (11)

with γ2 and k2 constant. If the process is aerobic, then the previous expression

rewrites considering also the contribution of the oxygen, i.e.,

Ff = γ2b f (

C f k2 + C f

) ( CO2

k3 + CO2

) , (12)

and at the same time we have to add to the system an equa- tion for the oxygen diffusion-consumption.

Some author considers also a toxicity factor (see [15], for instance). Indeed, the pollutant above a certain thresh- old, say Cmaxf , may become a toxic agent for the microbial population. Such effects may be taken into account rewriting (11) in the following way

Ff = γ2b f (

C f k2 + C f

) ( 1− C f

Cmaxf

) . (13)

An alternative approach in describing the microbial growth is the one based on a logistic-type dynamics, which is a typi- cal method used in populations dynamics (see [26]).

For instance, in [27] and [28] we exploited this approach defining the growth rate as

Ff = γ3[bmaxf g f (C f )−b f ]b f , (14)

A general model for bioremediation processes of contaminated soils 37

where γ3 is a constant. The so-called carrying capacity bmaxf is modulated by the function g f , whose definition allows to take into account several phenomena, as the already men- tioned toxicity effect.

Many different kinetics have been proposed for the evo- lution of cells populations, but apparently only kinetics of logistic type have been used in the present context.

The same arguments apply to Fa, but in that case it is realistic to change the value of the constant parameters involved, as well as the definition of g f in case of model (14).

3.2 Attachment/detachment models

Another important feature in system (8)–(9) is the choice of the attachment coefficient. This process is greatly influenced by the microscopic properties of the medium and the type of biomass as well, since both factors play an important role in determining the attachment capability. Microorganisms have a strong tendency to become associated with surfaces. Once they are attached to a substratum surface, a multistep process starts forming a complex, adhering microbial com- munity that may lead to the aggregation in clusters and/or biofilms which are consolidated by the production of an extracellular matrix. As pointed out in [20], the possibil- ity of sorption can be distinguished in three different types: reversible sorption at solid-liquid interfaces (and/or on other attached cells); irreversible sorption at solid-liquid interfaces (e.g., on iron-oxide coatings); and, in case of an unsaturated porous medium, irreversible sorption at gas-liquid interfaces. Moreover, in general detachment processes are driven by the water flux erosion. Therefore they are often neglected since the typical flow rate in bioremediation is fairly low. An exhaustive review of various mechanism for bacterial adhesion is given in [29].

Normally the bacteria have a strong tendency to attach to the immobile biofilm and the flow is too weak to detach them. Therefore often it makes sense to put Λdet = 0, while a linear attachment kinetics looks reasonable (see [32] and [33], for instance),

Λatt = λ b f . (15)

Other authors, as in [18] and [34], use Λdet = const. and

Λatt ∝ (bmaxa −ba)b f , (16) where bmaxa is the maximum value admitted for the adsorbed cells.

Some models involve the microscopic properties, as in [30], where Λatt is assumed proportional to a power of the shear stress of biomass from soil particles, which in turn is given as a function of porosity, pore diameter, and fluid viscosity and velocity.

In [20] the authors mention the following model for a reversible attachment process taking place in a variably sat- urated porous medium, i.e.,

Λatt ∝ |q|

θwd p

( 1−θw

θw

) b f , (17)

Λdet ∝ ba exp(−τ), (18) where d p is the mean pore diameter and τ is a constant related to the time of exposure of the biomass to the sub- strate.

Finally, in [27] we used

Λatt ∝ H (Ca −C f )b f , (19) H being the Heaviside function. It means that the attach- ment process is driven by the difference between the amount of nutrient (i.e., the pollutant) in the liquid and the soil. A similar approach is used in [31]. Moreover, we notice that the attachment term could be multiplied also by a function of ba, representing an “effective surface” term modelling the so-called collector and collision (or sticking) efficiencies (see [30]).

4 Changes in hydraulic properties

As we noted above, bacteria tend to migrate towards interfaces (either liquid-solid or liquid-air) giving rise to agglomerations in form of clusters or biofilms. Therefore, in many situations the biomass growth leads to considerable variations of the hydraulic properties of the medium: the mathematical system describing the bacteria evolution and the pollutant degradation becomes strongly coupled with the water flow equation. Bacteria affect the hydraulic properties in different ways, e.g.,

• The free biomass may cause density, viscosity and sur- face tension variations. • The attached biomass growth may change the medium

porosity and affects the contact angle, thus affecting cap- illarity. • The above variations induce, in turn, changes in the sat-

urated permeability and in the relative permeability and saturation curves as well.

Such phenomena are well known and many experimentalists claim the observation of permeability reduction, increasing of pressure head, etc. Moreover, a very similar problem has been studied also in the field of surfactants (SURface ACTive AgeNTS) and their use in soil remediation (SEAR: Surfactant Enhanced Aquifer Remediation), see [35]–[38], for instance. SEAR is a promising technique for remediat- ing aquifers contaminated by NAPLs (non-aqueous phase liquids). Indeed, surfactants decrease the interfacial tension of the NAPL, eventually increasing its mobility.

38 I. Borsi et al.

As pointed out in [39] most of the research on this topic is devoted to liquid-saturated porous media (see [40]–[46], for instance), while minor efforts have been done in case of unsaturated or variably saturated systems (see [47]–[49]).

In case of saturated system, many authors address the problem from a microscopic point of view, in order to derive the effective properties of the medium by upscaling the influ- ence of biofilms on the flow regime into the pores (see [40], [44], [45] and [46]). Many works approaching the problem at the macroscopic scale can be found in the references of [39].

In the unsaturated case, the micro scale description is very involved from a theoretical point of view. Moreover, one should consider also the processes taking place at the air/liquid interface, which might be not negligible due to the tendency of some biophase to concentrate at gas/liquid interfaces (see [34] and [39]).

In [47] a numerical and experimental work is presented, giving a rich description of many phenomena linked to this topic. In particular, Richards’ equation is coupled with the system for biomass and pollutant. The influence on the cap- illary pressure was included introducing the so–called scal- ing factor approach (see also [50]). Even if this method is useful in the framework of experimental works, it seems to suffer of a physical drawback. In [28] we discussed this point and pointed out the limitations of the scaling factor approach by means of a simple example.

In [49] an accurate description of the changes in porosity and structural properties of the medium is given. The authors propose a model in a 1-D setting with the following features:

• The medium is unsaturated. • The unit volume of the medium is divided into: soil

matrix, air, mobile water, water stored in the biomass and biomass substrate. • The total water content comprises a mobile part (i.e., free

water) and an immobile fraction stored in the microbial cells. • The microbial mass is described only as attached to the

solid matrix, namely the free biomass and their advection- dispersion in water is neglected. In our notation, b f ≡ 0.

Moreover, the biomass affects the hydraulic properties of the medium and in particular

• The soil porosity decreases due to the microbial growth, i.e.,

ε = ε(ba) = ε0 −ba, ε0 = ε(0) being the soil porosity in the biomass-free stage. • The so–called air entry potential increases. This para-

meter is related to the soil capillary characteristics and corresponds to the flow potential at which the air enters the medium in a drainage process.

• The average pore–size decreases. • The water surface tension is decreased by the presence

of bacteria at the air-water interface.

The above changes induce, in turn, an alteration of the sat- uration curve and the hydraulic conductivity, so that they enter the Richards’ equation also as functions of ba. There- fore, the flow problem is coupled with the equation for the biomass growth, which is described by means of a logistic- type dynamics.

In [27] we defined a model for a macroscopic description of the problem and accounting for:

• Variably saturated flow in the medium (Richards equa- tion), with:

– Porosity depending on the volume fraction occupied by attached biomass

– Variable saturated permeability (since it depends on porosity)

– Moisture content description based on mixture the- ory (i.e., considering mobile water and water stored in the attached biomass)

• Advection, diffusion and reaction equations for pollutant and biomass in water. • Reaction equations for attached biomass and pollutant

adsorbed on soil.

The model was presented along with some numerical simu- lations showing its qualitative consistency.

In [28] we introduced a slightly different version with more attention to the modification of the porosity accom- panying the evolution of the biomass. More in detail, we defined a one-dimensional system based on equations (3), (5), (8) and (9), basically neglecting the concentration of pollutant dissolved in the fluid. Moreover, the hydraulic properties changes were considered including:

• An explicit dependence on the varying porosity ε = ε0− ba of: saturated permeability, saturation curve and rela- tive permeability curve. • An explicit dependence on b f of the liquid viscosity.

The growth kinetics was selected of the type (14), while the attachment process was described by the linear model (15).

For such a model we proved the existence of a solution to the corresponding mathematical problem and we criti- cized also the consistency of the so-called fluid media scal- ing approach.

5 Scaling

The complete system describing a bioremediation process can be simplified by means of a scaling procedure.

In particular, hereafter we consider the system of equa- tions (4), (5), (6), (8), (9), with:

A general model for bioremediation processes of contaminated soils 39

• expression (7) for R; • the growth rate Ff given by (14); • the growth rate Fa = γ4[bmaxf ga(C f )−ba]ba; • Λatt as in (15). • Λdet = 0. • θ = Θ (ψ), being Θ a selected saturation function (see

[10], for instance).

Then we rescale the quantities Ca, C f , ba, b f , h, Dh, Db, ψ and q selecting some reference values C0a , C

0 f , b

0 a, b

0 f , h

0,

D0h, D 0 b, ψ

0 and q0, respectively. Moreover, we consider a characteristic length L and the following characteristic time

t0 = 1

b0a Ba . (20)

The choice (20) corresponds to selecting the bioreduction of adsorbed pollutant in (9) as the most meaningful phenome- non of the whole process.

Next, setting

ψ 0 = L and q0 = |K|, after the scaling procedure, the system as1

1

b0f

∂ θw ∂ t

+ ∂ b f ∂ t

+ 1

b0f A1∇ ·q = −A2b f , (21)

q = − K|K|∇ ·(ψ + ez), (22)

∂Ca ∂ t

= h(σ ) ( A3Ca −A4θwC f

) −Caba, (23)

∂ ( θwC f

) ∂ t

+ A1∇ · (

θw θ

C f q )

= A5∇ · ( θwDh∇C f

) −A6C f b f −σ θwC f ba,

−h(σ )(A7Ca −A3θwC f ) (24)

∂ b f ∂ t

+ A1∇ · (

b f θ

q )

= A8∇ · (

θ Db∇ (

b f θ

)) −A2b f + A9[g f (C f )−b f ]b f ,

(25)

∂ ba ∂ t

= A10b f + A11[ga(Ca)−ba]ba, (26)

where Ai, i = 1, . . . , 11 are the dimensionless quantities defined in Table 2.

Using the data set reported in Table 1, we get

t0 ≈ 38, 6 day

1 For the sake of simplicity we denote the dimensionless variable with the same letters.

Table 1 Data used for the scaling procedure

Parameter Value Reference

C0a 4.8×10−4 Kg/m3 [30] C0f 3.3×10−4 Kg/m3 [30] b0a 3.53×10−2 Kg/m3 [15] b0f 4×10−5 [47] λ 10−4 s−1 [33] B0a 10

−5 s−1 [15] B0f 10

−5 s−1 [15] q0 10−5 m/s [47] γ3 1.37×10−4 s−1 [47] γ4 3.6×10−4 m3/(Kg s) [49] D0h 10

−9 m2/s [47] D0b 10

−8 m2/s [47] L 5×10−2 m [47] h0 7×10−11 s−1 [15] ρb 1.1×103 Kg/m3 [47]

Table 2 List of dimensionless quantities and their order of magni- tude on the basis of data reported in Table 1

Coefficient Magnitude Process

A1 = (t0q0)/L O(10 2) Darcy flux

A2 = t0λ O(102) Biomass attachment A3 = t0h0 O(10−4) Pollutant adsorption A4 = (t0h0C0f )/C

0 a O(10

−4) Pollutant desorption A5 = (t0D

0 h)/L

2 O(10) Pollutant dispersion A6 = t0b

0 f B f O(10

−3) Biodegradation of dissolved pollutant due to free biomass

A7 = (t0h0C0a )/C 0 f O(10

−4) Pollutant adsorption A8 = (t0D0b)/L

2 O(1) Biomass dispersion A9 = t0γ3b0f O(10

−2) Free biomass growth A10 = (λ t0b0f ρb)/b

0 a O(10

−1) Biomass attachment A11 = t0γ4b0a O(10) Attached biomass growth

while the values of the dimensionless constants are the ones reported in Table 2. The analysis of the orders of magni- tude implies that some processes may be neglected, e.g., the pollutant adsorption/desorption (see values of A3, A4 and A7), the biodegradation of the dissolved contaminant due to the free biomass (see A6), the free biomass growth (see A9) and the contribution of biomass attachment to the growth of ba (see A10). In particular, comparing the values (1/b0f ), (A1/b0f ) and A2 in (21) we see that

A1∇ ·q = O(1) ⇒ ∇ ·q ≈ A−11 = O(10−2) ≈ 0, (27)

40 I. Borsi et al.

which exploiting (22), entails

∇ · (

K |K|∇ ·(ψ + ez)

) = 0. (28)

The latter is an elliptic equation whose solution gives ψ and q once the initial and boundary conditions have been speci- fied.

In turn, using the definition of the retention curve θ = Θ (ψ), we determine also θ . It means that we can solve equa- tions (23)–(26) considering θ and q as known quantities.

Exploting the above considerations on the orders of mag- nitude, we get a reduced form of system (23)–(26), namely

∂Ca ∂ t

= −Caba, (29)

A1q∇ ((

1− b f θ

) C f

) = A5∇ ·(θwDh∇C f ) (30)

q ·∇ (

b f θ

) = −A2

A1 b f , (31)

∂ ba ∂ t

= A11[ga(Ca)−ba]ba, (32)

which describes the growth process of the attached biomass population and the corresponding pollutant bioreduction – equations (32) and (29) – both taking place in a stationary flow regime, described by equations (30) and (31).

Let us confine ourselves to a 1-D nondimensional case, i.e., considering the problem for x ∈ [0, 1], where x is the vertical coordinate pointing upward. Moreover, we close the model with the following boundary and initial conditions.

On the bottom boundary, x = 0, we set a saturation con- dition for the flow problem, namely

ψ = 0(⇔ θ = ε), (33) while for C f and b f we impose a no diffusive flux condition, corresponding to

∂C f ∂ x

∣∣∣∣ x=0

= 0, (34)

∂ b f ∂ x

∣∣∣∣ x=0

= 0, (35)

On the top boundary, x = 1, we assume to know the entering flux, i.e.,

q(1,t) = qin(t), (36)

and that the entering liquid is not polluted at all,

C f (1,t) = 0, (37)

with a known concentration of free bacteria,

b f (1,t) = b1(t). (38)

Finally, we assume the following initial conditions

q(x, 0) = q0(x), (39)

θw(x, 0) = θ0(x), (40)

ba(x, 0) = ba,0(x) > 0, Ca(x, 0) = Ca,0(x) > 0, (41)

b f (x, 0) = b f ,0(x), C f (x, 0) = C f ,0(x) (42)

Integrating equations (27) and (28) with conditions (33) and (36), we get

q(x,t) = qin(t), (43)

and

ψ(x,t) = − (

1 + qin(t)

K

) x, (44)

from which we have also

θ (x,t) = Θ (ψ(x,t)) . (45)

Moreover, integrating in (x, 1) equation (31) we have

b f (x,t) = θ (x,t) b1(t)

θ (1,t) exp

[ A2 A1

∫ 1 x

θ (y,t)dy ] , (46)

so that b f (x,t) is completely determined once θ (x,t) has been computed from (45). Moreover, equation (30) is a sec- ond order ODE w.r.t. the variable x, endowed with condition (34), (37) and whose solution gives C f (x,t). Finally, Ca(x,t) and ba(x,t) can be found solving (29), (32) which is a sys- tem of first order ODEs w.r.t. t, with the initial conditions (41).

Therefore, the problem is completely solvable in a classi- cal way.

Remark 1 Note that C f (x,t) ≡ 0 is a trivial solution for the problem (30), (34), (37). If the initial condition C f ,0(x) is strictly positive, then there will be a short time interval in which C f (x,t) goes to zero.

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