A dual-porosity theory for solute transport in biofilm ...

Advances in Water Resources 62 (2013) 266–279

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A dual-porosity theory for solute transport in biofilm-coated porous media Laurent Orgogozo a,⇑, Fabrice Golfier b, Michel A. Buès b, Michel Quintard c,d, Tiangoua Koné b a

Géosciences Environnement Toulouse, 14 Avenue Edouard Belin, 31400 Toulouse, France Université de Lorraine, ENSG, CNRS, CREGU, GeoResources UMR 7359, 54518 Vandœuvre-lès-Nancy Cedex, France c Université de Toulouse, INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Allée Camille Soula, F-31400 Toulouse, France d CNRS, IMFT, F-31400 Toulouse, France b

a r t i c l e

i n f o

Article history: Available online 27 September 2013 Keywords: Porous media Biofilm Biodegradation Reactive transport Non-equilibrium Upscaling

a b s t r a c t In this work, we derive a Darcy-scale model for solute transport in porous media colonized by biofilms. Biofilms refer to structured communities of microbial cells that grow attached to an aqueous interface, e.g., the pore walls of a water-saturated porous medium. First, we present the pore-scale description of mass transport within and between the phases (water phase and biofilm phase) including the biologically-mediated reactions. Then, a two-equation Darcy-scale mass balance model (classically referred to as dual-porosity model), suitable both for local equilibrium and non-equilibrium conditions, is obtained from the method of volume averaging. Application of this macroscopic model to complex pore geometry of a two-dimensional porous medium supporting biofilms is presented. Comparisons with direct numerical simulation results confirm the large range of validity of the model. Finally, an illustrative example based on a scenario of reactive transport experimental tests is used to point out conditions for when the use of the proposed model is required. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Modelling of groundwater flow and species transport involves geochemical and microbiological processes that interact in a complex manner. Usually, these couplings induce non-equilibrium or mass transport limitations [1]. This is particularly true for biological processes in aquifers where bacteria are mainly present at the surface of the grains as biofilms. Such a biofilm phase is an active phase, i.e., it exchanges mass with the flowing phase, but with a low permeability and possibly low other transport properties [2], thus raising the potential for the need of non-equilibrium models. Historically, mathematical models of biofilm are classified along three different approaches in the literature [3]: the so-called ‘‘macroscopic model’’ (hereafter referred to as ‘‘local equilibrium model’’), the ‘‘biofilm model’’ and the ‘‘micro-colony model’’. The first category assumes that there are low concentration gradients between phases. In other words, the model is based on the local equilibrium assumption and the biodegradation rate may be included directly into the classical advection–dispersion equation (e.g. [4–6]). The simplicity of this approach has helped to maintain its popularity although it disregards most of the complex features of biofilm formation. In addition, data analysis from the literature indicates that microbial groundwater processes are usually characterized ⇑ Corresponding author. Tel.: +33 (0) 5 61 33 25 74; fax: +33 (0) 5 61 33 25 60. E-mail address: [email protected] (L. Orgogozo). 0309-1708/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.advwatres.2013.09.011

by moderate non-equilibrium conditions. Indeed, for the generally accepted orders of magnitude of the pore-scale Péclet number, in the range 0.01–100, and of the Damköhler number, lower than 10, the local-equilibrium model fails (see the comparison with experimental data of [7] or [8]. As an alternative, limitations by diffusion are considered between the microbial and bulk aqueous phase in microcolony models [9,10], while, in biofilm models [11–14], both external and internal mass transfer limitations are accounted for. These non-equilibrium models differ mainly by the assumption adopted for the geometrical description of the biomass. Bacteria are assumed to form either a thin continuous film (with increasing thickness as the biofilm layer grows) or patchy biofilm communities (with increasing the number or size of the colonies with growth). An improved description of mass transport within biofilm-coated porous media was proposed by Seifert and Engesgaard [15] through the use of a dual-porosity or dual-domain model. Nevertheless, while the above-mentioned approaches extend the validity range of biofilm models to non-equilibrium conditions, their applicability remains limited due to physical simplifications (biofilm of uniform thickness, spherical colonies) or some assumptions (value of the exchange rate parameter or the effectiveness factor). Most of the parameters are empirically-based and need to be fitted or determined a priori. From this literature survey, two major objectives motivate the present work:

L. Orgogozo et al. / Advances in Water Resources 62 (2013) 266–279

267

Nomenclature cA c cA x cBc cBx hcAcic hcAxix ðDAc DAx vc

lAx lAc lBx lBc KAx

pore-scale concentration of species A in the c-phase [mol m3] pore-scale concentration A in the x-phase [mol m3] pore-scale concentration of species B in the c-phase [mol m3] pore-scale concentration of species B in the x-phase [mol m3] averaged concentration of species A in the c-phase [mol m3] averaged concentration of species A in the x-phase [mol m3] diffusion coefficient of species A in the c-phase [m2 s1] effective diffusion tensor for species A in the x-phase [m2 s1] fluid velocity [m s1] specific degradation rate for species A in the x-phase [s1] specific degradation rate for species A in the c-phase [s1] specific degradation rate for species B in the x-phase [s1] specific degradation rate for species A in the c-phase [s1] half-saturation constants for species A in the x-phase [mol m3]

(i) The determination of a macroscopic model from an upscaling theoretical framework and suitable for a large range of operating conditions, (ii) A rigorous validation based on a comparative study with pore-scale numerical experiments. For this study purposes, we will focus on the development of a generalized macroscopic transport model, somewhat in the spirit of dual-porosity models (e.g. [16–19]). Physically, convective and dispersive transport occurs in the more permeable zone, i.e., the fluid phase, while water inside the biofilm phase, is assumed to be immobile. This behavior is reminiscent of ‘‘mobile-immobile’’ models (e.g. [17,20,21]) classically used in solute transport for heterogeneous porous media exhibiting a non-Fickian behavior. A dual-porosity medium description – hereafter referred to as twoequation or dual-porosity model – involving a separate mass balance equation for each phase is thus suggested to better describe mass transport within the system. The mass exchange between the two regions will be controlled by a diffusive process and described by first-order kinetics. To our knowledge, only a few contributions have applied this idea to biological systems (e.g. [3,7,14,15]) and the models were not developed from a comprehensive upscaling procedure. In this manuscript, the volume averaging approach is used to derive the averaged equations from the micro-scale physics and to obtain effective coefficients. The resulting macro-scale model is more complex but with the advantage of inducing fewer limitations and constraints. 2. Theoretical development of the dual porosity model 2.1. Background on upscaling theory applied to mass transport in biofilm-affected porous media One of the challenging features in the development of macroscopic models coupling flow and transport in porous media with

KBx KAc KBc

qx qc Aij V Vi nij

ex ec kdec Y

half-saturation constants for species B in the x-phase [mol m3] half-saturation constants for species A in the c-phase [mol m3] half-saturation constants for species B in the c-phase [mol m3] microbial density in the x-phase [mol m3] microbial density in the c-phase [mol m3] interface between the i-and j-phases (i, j:x, c, j) averaging volume volume of i-phase (i:x, c, j) in the averaging volume represents the unit normal pointing outward from the iphase toward the j-phase volume fraction of x-phase in the averaging volume [–] volume fraction of c-phase in the averaging volume [–] rate of decay of bacterial population [s1] stoichiometric coefficient of production of microbial biomass by unit of mass of biodegraded species [–]

Indices

c x j A B

fluid phase biofilm phase solid phase substrate compound electron acceptor compound

biochemical processes is the variety of spatial scales involved in these phenomena. As discussed for example in Sturmann et al. [22], the natural attenuation of organic pollutants in the subsurface ranges from the cell scale (hundreds of nanometers–tens of micrometers) to the field scale characterized by geological heterogeneities that comprise the aquifer (tens of meters and more). One method to address this issue is to apply a form of upscaling, i.e., to develop some formal averaged macroscopic models involving effective parameters which describe at the macro-scale the main transport processes features (convection, dispersion, etc.). The formalism of the volume averaging method, adopted in this work, keeps an explicit coupling between the micro-scale information and the macroscopic parameters (through so-called closure problems), thus allowing predicting macro-scale parameters from the micro-scale structure. Mathematical models have been developed both for local mass equilibrium [6,23] or non-equilibrium conditions [1,24,25] but still for restrictive conditions (reaction limited by external mass transfer or kinetics; time asymptotic conditions). In this paper, we propose to consider the development of a two-equation or dualporosity model. If the mass transfer theory in two-region media has been explored in detail for Darcy-scale heterogeneous systems from an upscaling perspective, the contribution of this work is twofold: (i) biologically-mediated reactions are included both in the high-conductivity (bulk aqueous phase) and low-conductivity (biofilm phase) regions and (ii) we apply this theory to biofilm-colonized porous media and provide guidelines for when the use of a dual-porosity model is appropriate. Derivation of this upscaled model, based on a volume averaging approach, and its validity domain will be discussed in the following sections. As the volume averaging method involves significant algebraic effort which may discourage non-specialists, we have decided to present within the text only the main steps of the mathematical development, all details being provided in the Appendices.

268


A last point needs to be clarified. Bacterial populations grow and decay lead to spatial evolution of the biofilm [26]. However, this process is usually slow – with a characteristic time of the order of the day – compared to transport processes characteristic times. A quasi-steady approach is thus classically considered in biofilm models to describe this coupling (e.g. [27–30]; recent experimental evidences can also be found in Davit et al. [31]). Moreover, under usual subsurface conditions, biofilm supported by aquifer grains is not thick enough to disturb groundwater flow and growth processes may be neglected. We will follow this approach by considering a biofilm occupied-pore space, denoted by ex, that does not vary across time. The coupling with bacterial population dynamics – which could be performed through a sequential procedure – will not be investigated here except for the application case study.

ideal mixing of highly diluted solute; constraints associated to this approximation are discussed in Quintard et al. [38]. Thus we can write the dispersion equation in terms of molar concentrations, or any other definition of concentration. Under these assumptions, the mass balance equations for the fluid-biofilm-solid system can be written in the following way:

x-phase

@cAx cAx ¼ r ðDAx rcAx Þ lAx qx cBx @t c Ax þ K Ax

c-phase

@cAc c Ac þ r ðv c cAc Þ ¼ r ðDAc rcAc Þ lAc qc cBc @t cAc þ K Ac

ð1Þ

ð2Þ B:C:

nxj DAx rcAx ¼ 0 at Axj ;

ncj DAc rcAc ¼ 0 at Acj ;

and ncx DAx rcAx ¼ ncx DAc rcAc

c Ac ¼ c Ax

at Acx ;

2.2. Pore-scale problem This investigation starts with the set of mass balance equations which are used to describe the mass conservation at the pore scale (Fig. 1) in the fluid (water; c-phase) and the biofilm (polymeric gel containing bacterial populations which develops on the pore walls of the considered porous medium; denoted as the x-phase). For the sake of generality, both sessile and suspended bacteria are considered so that biodegradation may occur in both phases. The biofilm phase is assumed to be homogeneous, and such a description can be rigorously done following previous upscaling work (e.g. [32,33]). Classical assumptions are adopted here - consistent with a number of previous studies on biofilms (e.g. [1,34]) – and more details about the development of this set of equations can be found in Golfier et al. [6]. We do not here consider the effect of other physical parameters such as osmotic pressure or temperature variations on the metabolism of the bacterial population, although these have been shown to be important parameters for the intensity of bacterial activity (e.g. [35]) In this work, we consider systems with a single substrate (carbon and energy source – species A) and a single electron acceptor (such as dioxygen – species B). For the simplicity of the exposure, we will focus on the case of a large excess of the species B, for which the concentration cBx could be considered as temporally and spatially constant. Anyway, such a peculiar situation, i.e., microbial growth limitation by a single compound (electron acceptor or donor), is frequently encountered in groundwater systems [36]; for instance, in the case where organic matter is abundant in contaminated aquifers and electron acceptor availability is usually the limiting step (e.g. [37]). We consider an

ð3Þ with :

cBx ¼

cBx c Bx þ K Bx

and cBc ¼

c Bc cBc þ K Bc

ð4Þ

DAx, describing the effective diffusion of species A in the biofilm phase, is a tensor since the biofilm phase is not necessarily homogeneous and isotropic. One can see Wood and Whitaker [32] for more theoretical details on this tensor. Information on the experimental measurement of these quantities may be found in the review of Stewart [39]. Note that in the pore-scale equations given by Eqs. refspseqn1 (2)–(4), three dimensionless numbers – governing species behavior – can be introduced, namely the Péclet number PeA and the Damköhler numbers DaA, defined as follows

PeA ¼

kv c k‘c ; DAc

DaA;x ¼

lAx qx ‘2c DAc K Ax

;

DaA;c ¼

lAc qc ‘2c

ð5Þ

DAc K Ac

with kv c k the magnitude of the characteristic velocity and ‘c a characteristic length of the fluid phase (see Fig. 1). The Péclet number characterizes the ratio between convective and diffusive effects, while the Damköhler numbers expresses the ratio of reaction kinetics to diffusion in each phase. 2.3. Unclosed averaged equations Referring to the averaging volume, V, illustrated in Fig. 1, we define the intrinsic average concentrations of species A in the x-phase (biofilm) and the c-phase (fluid) as:

hcAx ix ¼

hcAx i

ex

¼

1 Vx

Z

cAx dV;

hcAc ic ¼

Vx

hcAc i

ec

¼

1 Vc

Z

cAc dV

Vc

ð6Þ in which Vx (resp., Vc is the volume of the x-phase (resp., c-phase) contained in the averaging volume. In the following development, we will make use of the classical decomposition of microscale concentration fields as the sum of their average and a deviation field (as in Whitaker [40]), we have

cAx ¼ hcAx ix þ ~cAx ;

Fig. 1. Representative elementary volume of pore-scale processes associated with subsurface bioremediation.

cAc ¼ hcAc ic þ ~cAc

ð7Þ

The scalar fields ~cAx and ~cAc are referred to as spatial deviation concentrations. The process of volume averaging starts by forming the superficial average of the conservation equations (1)–(3). The general transport theorem is used to interchange time derivative and averaging operators, while the spatial averaging theorem is used for the spatial derivatives [41–44]. These averaging operations applied to Eqs. (1)–(3) are fully developed in the appendix A in Golfier et al. [6] and in Wood and Whitaker [32] regarding the reactive terms. The result is:


269

Averaged equation for the biofilm (x-phase)

" !# Z Z @ðex hcAx ix Þ 1 1 x ~ ~ ¼ r ex DAx rhcAx i þ nxc cAx dA þ nxj cAx dA V x Axc ðtÞ V x Axj ðtÞ |fflfflfflfflfflfflfflfflffl@t {zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} Accumulation Diffusion

Z 1 hcAx ix þ nxc ðDAx rcAx ÞdAex lAx qx cBx V Axc ðtÞ hcAx ix þ K Ax |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð8Þ

Reaction

Interfacial Flux

Averaged equation for the fluid (c-phase)

" !# Z Z @ðec hcAc ic Þ 1 1 þ r ec hv c ic hcAc ic ¼ r ec DAc rhcAc ic þ ncx ~cAc dA þ ncj ~cAc dA V c Axc ðtÞ V c Acj ðtÞ @t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Convection Accumulation Diffusion

Z hcAc ic 1 þ ncx ðDAc rcAc ÞdA r hv~ c ~cAc i ec lAc qc cBc V Axc ðtÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} hcAc ic þ K Ac |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Dispersive Transport |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Interfacial Flux

2.4. Closure of the dual porosity model At this point, averaged concentrations and deviations are the solutions of coupled macro-scale and micro-scale equations. This coupling is quite complex and may involve different characteristic times, especially for describing mass exchange between the different phases. For some limiting conditions, such as local mass equilibrium condition, time-asymptotic regime, mass-transfer- or reaction-rate-limited effective biodegradation rate, simplifications can be made to obtain a single equation describing the macroscopic balance for the concentration. Most models are based on such assumptions (among others, [1,6,24,25]). If we wish to extend the applicability of these models, a more accurate description of the complex coupling process between the phases is required. By nature, mass exchange phenomena are transient and theoretically need a spatial-time convolution to be rigorously described. However, approximate solutions may still be found under different constraints and this leads to different models as described in Debenest and Quintard [45], Golfier et al. [46] or in Davit et al. [47], namely we may use mixed micro-scale and macro-scale models, multi-rate models, two equation models with different kinds of exchange terms, etc. In this paper, we develop a two-equation model, or dual-porosity model, corresponding to a quasi-steady linear closure. In the process of closure, unknown pore-scale quantities, i.e., the deviation fields which still occur in the unclosed macro-scale transport equations, are represented in terms of known effective parameters and volume-averaged variables. This process finally leads to a fully closed system of macroscopic mass balance equations. A way of closing our problem is to obtain an approximate solution of the governing equations for the deviation fields. Actually, by subtracting unclosed equations (Eqs. (8), (9)) from pore-scale equations (Eqs. (1)–(3)), we obtain a set of equations governing the deviation concentrations (Appendix A). These

ð9Þ

Reaction

deviation equations involve macroscopic quantities which can be identified as source terms for the deviation fields. Mapping of the deviations as a function of these source terms are then used to establish the required closure relations. This mapping is determined by combining closure equations in the x- and c-phases, which relate the deviation fields to the associated macroscopic source terms, with the set of equations governing the deviation concentrations. This approach leads to a set of three closure problems which govern the closure fields (Appendix B). It should be emphasized that an additional assumption is required to force the uncoupling between these solutions, due to the non linearity of the reaction kinetics. Thus, we assume hereafter that the reactive phenomena do not prevail over the convective and diffusive processes, i.e., DaA 6 PeA , so that we can ignore any coupling between the different closure fields through hcAx ix and hcAc ic due to the non-linearity of the kinetics term (cf. Appendix B). Apart from the non-linear reaction terms, the problem under concern is analogous to those studied by Ahmadi et al. [18], Cherblanc et al. [48] or Davit et al. [47] in some cases of transport in heterogeneous porous media. Following the same development, it can be shown that, under a quasi-steady linear approximation, the closure relations take the form (cf. Appendix B)

~cAc ¼ bAcc rhcAc ic þ bAcx rhcAx ix þ r Ac hcAx ix hcAc ic

ð10Þ

~cAx ¼ bAxc rhcAc ic þ bAxx rhcAx ix þ r Ax hcAx ix hcAc ic

ð11Þ

Here, bAcc, bAxc, bAcx and bAxx are closure vector fields ([L1]); rAc and rAx are closure scalar fields ([–]). These closure fields relate the micro-scale variables ~cAc and ~cAx to the macro-scale variables rhcAc ic ; rhcAx ix ; and ðhcAx ix hcAc ic Þ. Now, injecting the closure assumptions (10) and (11) in the unclosed averaged equations (8) and (9) leads to the following final set of macroscopic transport equations.

270


Closed averaged equation for transport in the x-phase: x

ex

@hcAx i ¼ r DAxx rhcAx ix aA ðhcAx ix hcAc ic Þ þ r @t ðDAxc rhcAc ic Þ þ r ðdAx ðhcAx ix hcAc ic ÞÞ þ uAx rhcAx ix ðuAc þ DAc rec Þ rhcAc ic ex lAx qx cBx

hcAx ix hcAx ix þ K Ax

ð12Þ

Closed averaged equation for transport in the c-phase:

3. Model assessment from comparison with pore-scale simulations

@hc ic ec Ac þ r ðec hv c ic hcAc ic Þ ¼ r ðDAcc rhcAc ic Þ @t þ aA ðhcAx ix hcAc ic Þ þ r ðDAcx rhcAx ix Þ þ r ðdAc ðhcAx ix hcAc ic ÞÞ

þ uAc rhcAc ic ðuAx þ rex DAx Þ rhcAx ix ec lAc qc cBc

hcAc ic hcAc ic þ K Ac

ð13Þ

DAcc ; DAcx , DAxc and DAxx represent the effective dispersion tensors associated to the macroscopic concentrations hcAc ic and hcAx ix in the c-phase and in the x-phase, aA is the mass exchange coefficient between the c-phase and the x-phase and dAc , uAc , dAx and uAx are non-classical additional convective terms. These effective coefficients are functions of the mapping variables through the closure relations, Eqs. (10) and (11). As an example, one can find below the expression of the dispersion tensor DAcc . Similar results apply to the other effective coefficients and are given in Appendix B.

DAcc ¼

DAc V

Z

ncx bAcc dA þ Axc

DAc V

Z

we will work with periodic unit cells and, therefore, use periodicity conditions which give the more general conditions for the deviations. In addition, it is necessary to introduce a condition that sets the levels of bAcc , bAxc, bAcx, bAxx , rAc and rAx to complete the closure problem and this condition is specified by a constraint of zero average per phase, which ensures that averages of deviations are equal to zero. The three resulting closure problems are detailed in Appendix B. The validity domain of this dual-porosity model will be investigated in the following section.

ncj bAcc dA hv~ c bAcc i þ ec DAc I

Acj

ð14Þ Note that, apart from the non conventional convective terms, we recover a classical mobile-immobile formulation with an interphase mass exchange flux described by first-order kinetics. The limitations associated to this approximation and the conditions for which such a formulation may not be acceptable are discussed in Cherblanc et al. [49] and Golfier et al. [50]. It is worth noting that non-classical terms obtained from the upscaling process can be neglected in certain situations (see some examples of calculation of such coefficients in Orgogozo et al. [24]). At this point, two procedures are available: 1. If pore-scale data are unknown, the macroscopic formulation can be directly applied and effective coefficients should be fitted by some inverse method from experimental results. 2. If the microscopic geometry has been characterized (cf. [31,51– 54]) or at least pore-scale information is available, macroscopic parameters can be mathematically determined by solving the closure problems. The closure problems need to be solved on a unit cell representative of the micro-scale medium geometry in order to calculate with accuracy the effective parameters involved in Eqs. (12) and (13). The amount of pore-scale information required cannot be determined a priori. However, one should remark that the deviation fields ~cAc and ~cAx appear only in integral terms in the averaged equations. Consequently, it should be possible to capture the effective behavior if the size of the unit cell, over which closure problems are solved, is significantly larger than the spatial correlation length for the geometry. Boundary conditions must be applied to the closure variables over the unit cell boundaries. In this paper

If, in essence, the two-equation model is expected to encompass the different one-equation limit behaviors (for interested readers, a comparison of these different models is detailed in the additional material available online), we need to investigate whether it may operate over a larger range of hydrodynamic conditions. The proposed dual-porosity formulation, as expressed by Eqs. (8) and (9), was tested by comparison of the elution curve predicted by the macroscopic model with the one determined by direct pore-scale numerical simulation. A realistic two-dimensional pore geometry was constructed for performing this analysis, see Fig. 2. This 2D configuration qualitatively agrees with some experimental observations, which can be found for instance in Davit et al. [31] but for a more mature biofilm. An initial porous medium of dimension L L = 8.8 8.8 mm was used as representative volume and duplicated three times to obtain the final domain. Biofilm was generated randomly using a simple cellular automaton model (see Golfier et al. [6], section 5.2). The obtained biofilm volume fraction is 0.046, and the fluid phase volume fraction is 0.625. The pore throat characteristic length, ‘c, is estimated to be 0.54 mm so that the constraint of separation of scales is verified. The comparison proceeded in three steps described below: 1. We first solved numerically the micro-scale conservation equations, Eqs. (1)–(4), over the domain illustrated in Fig. 2. The velocity field was obtained by solving Stokes equations. The diffusivity ratio, DAx =DAc , was fixed at 0.5I. As indicated in the figure, concentration and pressure were specified at the inlet. A specified pressure was also applied at the outlet and the boundary condition for concentration was of the convective flux type. Symmetry conditions were applied for concentration and pressure on the other faces. Pore-scale simulations were computed with COMSOL MultiphysicsÒ for different values of Pe and Da. 2. Effective coefficients given by the closure problems detailed in Appendix B (in supplemental online material) were computed over the representative unit cell of dimension L L. Such a unit cell is probably more complex than required to capture the pore-scale information, but we need to make sure that the complete correlation structure of the field is kept for a rigorous validation of the theory. Indeed, from a correlation length analysis [55] it was found that the medium characteristic length is about 0.12L, which can be used as a rough estimate for the minimum REV length. This figure is not dramatically changed by the introduction of the biofilm structure, given its generation characteristics. This makes the ratio length of the domain over REV length than one order of magnitude, which is enough in general to satisfy the separation of scale condition for such a comparison between direct numerical and macro-scale simulation. The calculation of the effective properties is done over the domain of length L used to generate our large-scale domain. Smaller windows could be used, but it is more secure to use the true periodic cell in order to avoid artifacts due to the closure problem boundary conditions when the unit cell is not truly periodic.


271

Fig. 2. Complex 2D geometry considered for additional numerical validation.

Fig. 3. Concentration fields for three representative hydrodynamical and biochemical conditions: PeA = 0.6 – DaA = 0.04; PeA = 60 – DaA = 4 and PeA = 60 – DaA = 20.

3. Darcy-scale simulations, given by Eqs. (8) and (9), were conducted over a one-dimensional equivalent domain of length equal to 3L and the resulting elution curves were compared to direct numerical simulations. The result for the concentration fields is represented in Fig. 3 for three typical numerical conditions. To assess the convergence of the solution, steady-state simulations have been initially performed for ensuring that we have a valid solution. The problem was solved with direct solver (PARDISO) with a relative tolerance and absolute tolerance of 0.1 and 0.01 respectively. Then, a local mesh refinement has been used to perform the mesh independence study. The final mesh is composed of 104,472 elements connecting 57,660 nodal points. For PeA = 0.6–DaA = 0.04, the solute

front propagates uniformly through the domain. Concentration is quasi-homogeneous along the cross-sections and we are close to local equilibrium conditions. As the Damköhler and Péclet numbers increase, concentrations in the fluid and biofilm phases change differently and channeling appears. We would qualify the results for PeA = 60–DaA = 4 as moderate non-equilibrium conditions. These results are consistent with the recent experimental evidences described in Koné [8] and Golfier et al. [56]. At last, for very high non-equilibrium situations (PeA = 60–DaA = 20), some low-concentration zones remain trapped behind the front due to the high biofilm reactivity. Comparison of the resulting breakthrough curves is illustrated in Fig. 4. We note a good agreement between direct numerical simulations and dual-porosity model simulations, even in the cases with significant ‘‘channeling effects’’.

272


Fig. 4. Comparison of breakthrough curves obtained by direct numerical simulations (DNS) and by the double-porosity (DP) model for a complex 2D geometry.

A moment analysis was conducted on the basis of the 0th, 1st and 2nd order moments of the distribution associated to the time derivative of the breakthrough curves. For local equilibrium (DaA = 0.04 and PeA = 0.6) and moderate non-equilibrium conditions (DaA = 4 and PeA = 60), less than 2% of relative error was observed for all the considered moments, and even for high nonequilibrium conditions (DaA = 20 and PeA = 60), the relative error over the moments increased up to 4% only. As expected, for a high value of the Damköhler number, strong concentration gradients appear within the averaging volume and the conditions of a weak deviation field is not respected anymore, i.e., ~cAx OðhcAx ix Þ. Under these conditions, the reactive term cannot be linearised, which causes the observed discrepancies between the dual-porosity model and the direct numerical simulations (see Wood et al. [55] for the impact of the concentration variance over the effective Michaelis–Menten kinetics). In a similar way, for too high Péclet numbers (e.g., PeA 60 in our tests, data not shown), the time scales associated with the diffusive (within the biofilm phase) and convective (within the fluid phase) processes become strongly different. The quasi-steady assumption for the closure fields is not valid anymore and thus a first-order model describing the mass exchange between the phases is not able to describe correctly the transport phenomena. A similar issue is met in the case of tracer transport in heterogeneous porous media for strong ratios of heterogeneity [50,57]. A more refined description of transport phenomena could probably be obtained by the use of a multi-rate mass transfer approach [58–60], in which more than two different regions (or more than two time scales) are considered. For linear problems, indeed, a dual-porosity formulation has been shown to be linked to multirate models by the fact that the first-order mass transfer coefficient value obtained from the method of volume averaging is an harmonic average of the eigenvalues (i.e., the different time-scales) of the diffusion problem for the low diffusivity phase [61]. However, in our case, the non linearity of the problem raises additional complexities which prevent a direct use of classical multi-rate ap-

proaches (see Silva et al. [62], or Willmann et al. [63], for an application of this method to non-linear phenomena). 4. Illustrative scenarios of equilibrium and non equilibrium conditions In this section, our goal is to investigate how non-equilibrium affects the biodegradation during contaminant transport and, also, the suitability of the dual-porosity model for describing such conditions. A scenario of nutrient transport and removal in a twodimensional packed bed was designed to assess the model capability. For the sake of simplicity, the influence of planktonic biomass is neglected (i.e., qc 0) and there is no inhibition of biodegradation (single substrate limitation, i.e., cBx 1Þ. In order to illustrate the coupling between transport processes and microbial dynamics, bacterial growth due to substrate uptake is considered. However, due to the relative short duration of the experiment (12 h), the biofilm volume is kept constant, only the microbial density change is taken into account (i.e., qx = f(t) but ex = constant). Thus we have an additional macroscopic mass balance equation for the attached biomass. For this equation, we use a simple heuristic model of bacteria growth and decay inspired from the literature [3]. Additionally, we consider a dense and uniform biofilm all over the domain and, thus, rex is equal to zero. Finally, Eqs. (12) and (13) can be simplified and the resulting set of macroscopic equations is the following: Solute transport in the fluid phase:

ex

@hcAx ix ¼ r ðDAxx rhcAx ix Þ aA ðhcAx ix hcAc ic Þ @t þ r ðDAxc rhcAc ic Þ þ r ðdAx ðhcAx ix hcAc ic ÞÞ

þ uAx rhcAx ix uAc rhcAc ic ex lAx qx

hcAx ix hcAx ix þ K Ax

ð15Þ


Solute transport in the biofilm phase: c

ec

@hcAc i þ r ðec hv c ic hcAc ic Þ ¼ r ðDAcc rhcAc ic Þ þ aA ðhcAx ix @t hcAc ic Þ þ r ðDAcx rhcAx ix Þ þ r ðdAc ðhcAx ix hcAc ic ÞÞ

þ uAc rhcAc ic uAx rhcAx ix ð16Þ Microbial density evolution equation:

@ qx hcAx ix ¼ Y lAx q kdec qx @t hcAx ix þ K Ax x

ð17Þ

In the latest equation, Y is the stoichiometric coefficient and kdec is the rate of decay of the bacterial population. It should be noted that Monod-type kinetics is employed in Eq. (17) to represent microbial growth processes, as it is widely used to describe bacterial growth in cultures [64] and by extension, in various experimental systems of biofilm-coated porous media (for example: [65–67]). Phenol biodegradation by P. putida F1 biofilm is used to provide an example of realistic conditions for which the dual-porosity and local equilibrium models differ significantly. The kinetics parameters for phenol removal are based upon the work of Seker et al. [68] and Reardon et al. [69]. The concentration injected is consistent with the experiments of Abuhamed et al. [70]. Only the microbial decay rate and the initial density of attached bacteria have been chosen arbitrarily. The hypothetical flow-cell used for the simulations measures 30 cm wide and 50 cm long. The cell is filled up with a regularly packed bimodal distribution of spherical grains coated with biofilm. We consider a uniform flow of the fluid phase along the longest dimension of the flow cell. The resulting volume fractions are 0.3 for the fluid phase and 0.06 for the biofilm phase. Simulation experiments are conducted over a 12-h period during which nutrient is continuously released. Injection is performed only through a part of the inflow side to keep a 2D configuration. Fig. 5 shows the two-dimensional domain, boundary conditions, and the microstructure used for the closure calculations. Two scenarios will be considered to emphasize the impact of non-equilibrium effects. In the first case (PeA = 2.1 and DaA = 0.1), we simulate phenol transport and biodegradation under low flow rate in a very fine sand medium with grain diameters of 20 and 85 lm, respectively (i.e., LREV = 100 lm). For the second one (PeA ¼ 25:7 and DaA ¼ 7:6), a higher flow rate is applied in a coarse glass bead medium. A similar arrangement of grains is used but with much larger diameters, similar to the ones used by Vayenas

273

Table 1 Parameter values used for simulations of biodegradation under equilibrium and nonequilibrium conditions.

cinj vDarcy vpore D Ac

lAx intial qx K Ax Y kdec PeA DaA LREV

(phenol biodegraded by P.putida F1) 102 mg cm3 5.79 105 cm s1/1.16 104 cm s1 1.93 104 cm s1/3.86 104 cm s1 9.1 106 cm s1 3.06 105 s1 0.1 mg cm3 3.2 102 mg cm3 0.8 g g1 5 107 s1 2.1/25.7 0.1/7.6 0,1/15 mm

et al. [71] for instance, such as LREV is about 1.5 cm. Values of kinetics and hydrodynamic parameters have been gathered for the two test cases in Table 1. As in Section 3, the effective coefficients involved in Eqs (15) and (16) are previously computed – based on the volume averaging approach detailed in this paper – on the representative pore-scale geometry (cf. Fig. 5) and for different values of hcAx ix and qx. Then, the macroscale equations are solved with Comsol MultiphysicsÒ on the two-dimensional domain of interest. The spatial distribution of substrate concentrations within the bulk fluid and biofilm and of bacterial density after 12 h is given in Fig. 6 for both cases. For the first scenario (PeA = 2.1 and DaA = 0.1), observations of the concentration fields show practically no differences between the two phases. Concentration is almost identical in the bulk fluid and the biofilm (Fig. 6a and c), which is characteristic of a behavior close to local equilibrium. In the second case (PeA = 25.7 and DaA = 7.6), due to relatively high reaction rates and high flow rates, non-equilibrium effects appear which lead to a significant difference between the phase concentration fields (Fig. 6b and d). The increase with time of bacteria density (Fig. 6f), close to the inlet, enhances this situation with a local increase of the Damköhler number. A comparison of the results of the dual-porosity model with concentrations simulated by the local equilibrium model (corresponding to the equilibrium case with aA ! 1 in Eqs (15) and (16); see [6]) is illustrated in Fig. 7. If predictions of both models are similar for the first scenario (Fig. 7a), the mobile concentration of phenol is largely underestimated by the local equilibrium model for the test case 2 (Fig. 7b). Clearly, a local equilibrium model cannot predict with accuracy the phenol degradation for such conditions and a non-equilibrium model must be used with the proper values of the effective parameters.

Fig. 5. Domain and boundary conditions for the 2D transport problem.

274


Fig. 6. Results simulated for a 12-h phenol transport experiment in equilibrium (a–c–e) and non equilibrium (b–d–f) conditions: concentration field in the fluid (a–b) and the biofilm phase (c–d); bacterial density field (e–f).

Fig. 7. Comparison of longitudinal profiles of concentration and bacterial density predicted by the double-porosity model and the local equilibrium model for (a) Test case 1 and (b) Test case 2.

5. Conclusions and perspectives In this paper, we have developed, using a volume averaging method, a dual-porosity model for reactive transport in porous media including a biofilm phase, which can be used under various flow and transport conditions. A first-order mass transfer coefficient is used to describe inter-phase diffusive fluxes and it can be computed from pore-scale characteristic features by solving a stea-

dy-state closure problem. This work extends and generalizes earlier results obtained for local mass equilibrium conditions [6], specific limit cases of non-equilibrium conditions (reaction-rate limited reaction and transport limited reaction, [24]; analytical works on 1D configurations that deal with such non equilibrium limiting cases have also been the subject of recent work, see for instance [72,73]) and asymptotic non-equilibrium conditions [25]. This model extends also previous non-equilibrium models that

275


were proposed on a more heuristic base. This model can be easily extended to a mobile-mobile formulation if we wish to introduce flow with a non-zero permeability value in the biofilm phase. The assumptions on which this model is based are reminded below: – We consider a porous medium colonized both by planktonic bacteria and a biofilm phase with a sufficient separation of scales. Biodegradation reaction is described by single Michaelis–Menten kinetics (a large excess of electron acceptor is assumed). – We assume an ideal mixing of highly diluted solute. – Biofilm growth and solute transport phenomena are uncoupled due to the separation of time scales of both processes. – Averaged concentrations must be large compared to the porescale deviations to handle the non-linearity of the reaction term, i.e., ð~cAi Þ ¼ OðhcAi ii Þ; for i ¼ c or x. The main contribution of this work is twofold: (i) the conditions for which a dual-porosity model can be applied to biofilm coatedporous media have been assessed, and, (ii) the capacity of the volume averaging method to give a fair estimate of the mass transfer coefficient and the other effective parameters such as dispersion has been confirmed through a comparison with direct pore-scale numerical simulations. Overall, we have investigated the ability of this dual-porosity model to predict substrate biodegradability from pore-scale simulations. The results emphasize that, even for highly complex microstructures, the theory captures the main features of the solute behavior when pore-scale information is available so effective properties can be estimated properly. As a conclusion, this model is suitable both for equilibrium conditions and for moderate to high non-equilibrium conditions, under the following restrictions: – Low to moderate values of the Péclet number. Beyond a certain critical value (between 100 and 1000 depending on the porescale geometry and the accuracy required by the user), a more accurate description of the mass transfer process should be considered. – The transport phenomena must not be dominated by reaction processes, i.e., the Damkhöhler number must be inferior to 1 or lower than the Péclet number. This limitation is due to the non linearity of the considered biodegradation kinetics. However, the resulting range of operating conditions in terms of Péclet and Damköhler numbers encompasses most situations encountered in chemical and environmental engineering applications. Overall, the use of this model should help both to interpret experimental data and to design engineered porous systems involving microbial processes. The next step will be to confront our theoretical predictions with experimental observations. To this end, an experimental study has been recently carried out. A flowcell containing quartz sand colonized with biofilm (Shewanella oneidensis MR-1 species) was used for monitoring both biofilm growth and solute transport [8,56] and results will be published soon. Multi-species biofilm (including various electron acceptors or donors) and dual-limitation of biofilm kinetics will be examined in further studies.

Acknowledgments This work was partially supported by the French National Research Agency (ANR) through the MOBIOPOR project, with the reference ANR-10-BLAN-0908 and by the French Scientific Interest Group-Industrial Wasteland (GISFI) program.

Appendix A. Development of the governing problem of the concentration deviations First, we subtract the unclosed averaged equations of transport, Eqs. (10) and (11), to the pore scale transport equations, Eqs. (1) and (2) and the result takes the following form: x-phase @ ~cAc c @ ec þ e1 ¼ r ðDAc r~cAc Þ v c r~cAc v~ c rhcAc ic c hcAc i @t @t c þ ðe1 c rec Þ ðDAc rhcAc i Þ " !# Z Z 1 1 1 ~ ~ ncx cAc dA þ ncj cAc dA ec r ec DAc V c Axc V c Acj

e1 c e1 c

1 V 1 V

Z

ncx ðDAc rcAc ÞdA Axc

Z Ajc

~ c ~cAc i ncj ðDAc rcAc ÞdA e1 c r hv

c e1 c ðRAc hRAc i Þ

ðA:1Þ

c-phase @ ~cAx x @ ex þ e1 ¼ r ðDAx r~cAx Þ x hcAx i @t @t x þ ðe1 x rex Þ ðDAx rhcAx i Þ " !# Z Z 1 1 e1 nxc ~cAx dA þ nxj ~cAx dA x r ex D A x V c Axc V c Acj

e1 x

1 V

e1 x

1 V

Z

nxc ðDAx rcAx ÞdA

Axc

Z Axj

x nxj ðDAx rcAx ÞdA e1 x ðRAx hRAx i Þ

ðA:2Þ

with:

B:C:1

nxj DAx r~cAx ¼ nxj DAx rhcAx ix ;

B:C:2

ncj DAc r~cAc ¼ ncj DAc rhcAc ic ;

B:C:3 hcAx ix þ ~cAx ¼ ~cAc þ hcAc ic ;

at Axj ;

at Acj

at Acx

ðA:3Þ ðA:4Þ ðA:5Þ

B:C:4 ncx DAx r~cAx þ ncx DAx rhcAx ix ¼ ncx DAc r~cAc þ ncx DAc rhcAc ic ;

at Acx

ðA:6Þ

where:

RAx ¼ lAx qx cBx

c Ax cAx þ K Ax

and RAc ¼ lAc qc cBc

cAc c Ac þ K Ac

ðA:7Þ

and:

hRAx ix ¼ lAx qx cBx

hcAc ic hcAx ix and hRAc ic ¼ lAc qc cBc x hcAx i þ K Ax hcAc ic þ K Ac ðA:8Þ

The averaging of the reactive terms presented in equation (A.8) has been previously developed in Wood and Whitaker [32]. It is important to keep in mind that this averaging requires the following assumption:

hcAx ix >> ~cAx

and hcAc ic >> ~cAc

ðA:9Þ

At this point, simplifications can be made from order-of-magnitude analysis of the various terms and using the constraint of separation of scales [74]. This investigation leads to the following relations:

276


c ~ ~ c ~cAc i; r ðDAc r~cAc Þ >> ðe1 c rec Þ ðDAc rhcAc i Þ; v c rcAc >> r hv

"

r ðDAc r~cAc Þ >> e1 c r ðDAc r~cAc Þ >> e1 c

Z

1 Vc

r ðDAc r~cAc Þ >> e1 c r ec DAc

1 V 1 V

ncx ~cAc dA þ

Axc ðtÞ

Z

1 Vc

Using a quasi-steady assumption, we finally obtain the set of equation governing the concentration deviations: Deviation equation in the x-phase

!#

Z

ncj ~cAc dA

;

Acj ðtÞ

lAx qx cBx

ncx DAc rhcAc ic dA;

Z

ncj ðDAc rhcAc ic ÞdA;

Ajc ðtÞ

1 Vc

r ðDAx r~cAx Þ >> e1 x 1 V

r ðDAx r~cAx Þ >> e1 x

1 V

Z

Z

nxc ~cAx dA þ

Axc

Z

1 Vc

Z

lAc qc cBc !# ;

A cj

nxc ðDAx rhcAx ix ÞdA;

ðA:10Þ

@ ~cAc ¼ r ðDAc r~cAc Þ v c r~cAc v~ c rhcAc ic @t ! Z Z 1 1 DAc ncx r~cAc dA þ ncj r~cAc dA V c Axc V c Ac j ec ðRAc hRAc i Þ ðA:11Þ

x-phase @ ~cAx ¼ r ðDAx r~cAx Þ @t ! Z Z 1 1 nxc r~cAx dA þ nxj r~cAx dA DAx V x Axc V x Ac j x 1 ex RAx hRAx i ðA:12Þ Here, we use the assumption (A.9) so that the deviation reactive term may be expressed as:

lAx q

x cBx

cAx hcAx ix cAx þ K Ax hcAx ix þ K Ax

~cAx K Ax hcAx ix2 þ ~cAx hcAx ix þ K Ax hcAx ix þ K Ax hcAx ix þ K Ax ~cAx þ K 2Ax ~cAx K Ax x2

hcAx i

x

þ K Ax hcAx i þ K Ax hcAx i

~cAx K Ax ðhcAx ix þ K Ax Þ

Z

nxj ðDAx r~cAx ÞdA

Axj ðtÞ

ðhcAc ic þ K Ac Þ

B:C:1

1 Vc

Z

2

þ v~ c rhcAc ic þ v c r~cAc |fflfflfflffl{zfflfflfflffl}

1 Vc

Source term

Z

ncx ðDAc r~cAc ÞdA

Axc ðtÞ

ncj ðDAc r~cAc ÞdA

ðA:15Þ

Ajc ðtÞ

nxj DAx r~cAx ¼ nxj DAx rhcAx ix |fflfflfflfflffl{zfflfflfflfflffl}

at Axj ;

ðA:16Þ

2

x

B:C:2

ncj DAc r~cAc ¼ ncj DAc rhcAc ic ; |fflfflfflffl{zfflfflfflffl}

at Acj

ðA:17Þ

Source term

B:C:3

hcAx ix hcAc ic þ~cAx ¼ ~cAc |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

at Acx

ðA:18Þ

Source term

0

0

1

1

B C B:C:4 ncx DAx @rhcAx i þr~cAx A ¼ ncx DAc @ rhcAc ic þr~cAc A; |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} x

Sourceterm

Source term

at Acx

ðA:19Þ

Appendix B. Closure problems associated to the dual porosity model

c

lAx qx cBx

nxc ðDAx r~cAx ÞdA

Axc ðtÞ

Source term

yield to neglect them. A similar analysis can be conducted for terms involving a time derivative [74]. Under these assumptions, the deviation equations can be stated as: c-phase

¼ lAx qx cBx

Axj

while the local terms have an order of magnitude of Oðl1c Þ, which

RAx hRAx ix ¼ lAx qx cBx

K Ac ~cAc

¼ r ðDAc r~cAc Þ

Axc ðtÞ

Non-local terms are here clearly identified, i.e., terms which are evaluated not only at the centroid of the volume of averaging. These terms are averaged quantities, and their order of magnitude is Oð1L Þ

1

1 Vx

Z

Deviation equation in the c-phase

@ ~cAx x @ ex ; r ðDAx r~cAx Þ >> e1 x hcAx i @t @t

r ðDAx r~cAx Þ >>

1 Vx

ðA:14Þ

nxj ~cAx dA

nxj ðDAx rhcAx ix ÞdA;

¼ r ðDAx r~cAx Þ

r ðDAx r~cAx Þ >> ðe1 x rex Þ ðDAx rhcAx i Þ; "

2

x

r ðDAx r~cAx Þ >> e1 x r e x DA x

ðhcAx ix þ K Ax Þ

Axc ðtÞ

@ ~cAc c @ ec ; r ðDAc r~cAc Þ >> e1 ; c hcAc i @t @t

r ðDAc r~cAc Þ >>

K Ax ~cAx

þ K 2Ax

ðA:13Þ

The system of governing equations of the deviation fields ~cAc and ~cAx is presented in Appendix A. Closure relations which relate the deviation fields to their macroscale source terms, identified in the previous set of equations, are required for upscaling the porescale transport processes. Linearizing the averaged reaction term requires the following assumption, already used for the development of the unclosed macroscopic equations of transport (Eqs. (10) and (11), see [6]):

hcAx ix >> ~cAx and hcAc ic >> ~cAc

ðB:1Þ

Apart from the non-linear reaction term, this problem is analogous to those studied by Ahmadi et al. [18] and by Cherblanc [48] in some cases of transport in heterogeneous porous media. First, we consider a classical closure assumption, by decomposing the deviation terms according to their source terms, which appear in the boundary conditions (A.16)–(A.19) (see a discussion in the body of the present paper for a brief presentation of other approaches):

~cAc ¼ bAcc rhcAc ic þ bAcx rhcAx ix þ rAc ðhcAx ix hcAc ic Þ

ðB:2Þ

~cAx ¼ bAxc rhcAc ic þ bAxx rhcAx ix þ r Ax ðhcAx ix hcAc ic Þ

ðB:3Þ

Upon substituting these closure relations into the deviation problem (A.14)–(A.19), we obtain the three following closure problems:

277


Vector closure problem I: c-phase 0

PeA ðv~ c þ v c rbAcc Þ þ

DaA;c cBc

bAcc ðK 0eff ;Ac

ðB:22Þ

B:C:8 hbAxx ix ¼ 0

ðB:23Þ

Scalar closure problem: c-phase

2

1

B:C:7 hbAcx ic ¼ 0

þ 1Þ Z Z 1 1 ¼ r2 bAcc ncx rbAcc dA ncj V c Axc ðtÞ V c Ajc ðtÞ

PeA ðv c rr 0Ac Þ þ DaA;c cBc

rbAcc dA

ðB:4Þ ¼ r2 r 0Ac

x-phase 0 bAxc

r2 b0Axc DaA;x cBx 1 Vx

Z

ðK 0eff ;Ax

1

þ 1Þ

0

ðB:5Þ

B:C:1 nxj rbAxc ¼ 0; at Axj ;

ðB:6Þ

Axc ðtÞ

ncj ¼ ncj rbAcc ;

at Acj

B:C:3 ncx þ ncx rbAcc ¼ ncx Dc r 0 bAxc ;

B:C:4 bAcc ¼

Axc ðtÞ

1

þ 1Þ

2

ncx rr0Ac dA

ðB:24Þ

r2 sAx DaA;x cBx

ðsAx 1Þ ðK 0eff ;Ax

1

þ 1Þ

2

¼

Z

1 Vx

ncx rsAx dA

Axc ðtÞ

ðB:25Þ

0

B:C:2

Z

ðK 0eff ;Ac

x-phase

2

ncx rbAxc dA

¼

1 Vc

ðr 0Ac 1Þ

0 bAxc ;

at Acx

at Acx 0

0

B:C:5ðPeriodicityÞ bAxc ðrÞ ¼ bAxc ðr þ li Þ;

i ¼ 1; 2; 3;

B:C:1 nxj rsAx ¼ 0;

at Axj

ðB:7Þ

B:C:2 ncj rr 0Ac ¼ 0;

ðB:8Þ

B:C:3 ncx Dc rsAx ¼ ncx rr 0Ac ;

ðB:9Þ

B:C:4 sAx ¼ r 0Ac ;

at Axe

ðB:26Þ

at Acj ;

ðB:27Þ at Acx

ðB:28Þ

at Acx

ðB:29Þ

B:C:5ðPeriodicityÞ sAx ðrÞ ¼ sAx ðr þ li Þ;

i ¼ 1; 2; 3;

at Axe

ðB:10Þ B:C:6ðPeriodicityÞ bAcc ðrÞ ¼ bAcc ðr þ li Þ;

i ¼ 1; 2; 3;

at Ace ðB:11Þ

c

B:C:7 hbAcc i ¼ 0 B:C:8

0 hbAxc ix

¼0

0

1 Vc

Z

r bAxx ¼

1 Vx

B:C:1

ðB:13Þ

B:C:8 hsAx ix ¼ 1

ðB:33Þ

ðK 0eff ;Ac

1

0

bAcc

2

¼

þ 1Þ

ncx rbAcx dA

ðB:14Þ

DaA;x cBx Z

bAxx ðK 0eff ;Ax

1

þ 1Þ

ncx rbAxx dA

Axc ðtÞ

nxj ¼ nxj rbAxx ; 0

2

1 Vx

DaAx ¼

Z

ncj rbAxx dA

ðB:16Þ

at Acj ; 0

B:C:3 ncx Dc þ ncx Dc rbAxx ¼ ncx rbAcx ; B:C:4 bAxx ¼

0 bAcx ;

ðB:15Þ

Ajc ðtÞ

at Axj

at Acx

at Acx

B:C:5ðPeriodicityÞ bAxx ðrÞ ¼ bAxx ðr þ li Þ;

lAx qx l2c DAc K Ax

v~ c ; kv c k

K 0eff ;Ax ¼

;

v 0c ¼

K Ax x ; hcAx i

DaAc ¼

vc kv c k

kv c k‘c DAc

ðB:18Þ

1 kv c k ¼ hv c ic hv c ic 2

at Axe

bAxx lc

ðB:34Þ

r 0Ac ¼ 1 þ r Ac

PeA ¼

ðB:20Þ B:C:6ðPeriodicityÞ bAcx ðrÞ ¼ bAcx ðr þ li Þ;

v~ 0c ¼

K Ac c ; hcAc i

0

bAxx ¼

ðB:17Þ

ðB:19Þ i ¼ 1; 2; 3;

bAcc ; lc

SAx ¼ 1 þ r Ax ; K 0eff ;Ac ¼

B:C:2 ncj rbAcx ¼ 0;

at Ace

ðB:31Þ ðB:32Þ

x-phase 2

at Ace

B:C:7 hr 0Ac ic ¼ 1

bAcx

0

Axc ðtÞ

i ¼ 1; 2; 3;

with:

0

¼ r2 bAcx

B:C:6ðPeriodicityÞ r 0Ac ðrÞ ¼ r0Ac ðr þ li Þ;

ðB:12Þ

Vector closure problem II: c-phase

PeA ðv c rbAcx Þ þ DaA;c cBc

ðB:30Þ

lAc qc l2c DAc K Ac

ðB:35Þ DC ¼

DAx DAc

ðB:36Þ

ðB:37Þ

ðB:38Þ

ðB:39Þ

ðB:40Þ

Classical closure assumptions (periodicity conditions, zero average conditions) are imposed to ensure the unicity of the solutions. Here, PeA is the cell Péclet number, DaAc and DaAx are the cell Damköhler numbers for both phases, and K 0eff ;Ac and K 0eff ;Ax are effective dimensionless half-saturation coefficients. Since both K 0eff ;Ac and K 0eff ;Ax depends

i ¼ 1; 2; 3; ðB:21Þ

on a macroscopic variable (hcAc ic or hcAx ix ), note that a coupling still remains between the calculation of the microscopic closure fields

278


and the macro-scale mass transport equation. A tabulation of the effective parameters estimated on the basis of the closure variables as a function of K 0eff ;Ac and K 0eff ;Ax is a way to deal with this coupling. The linearized reaction terms of the deviation equation (A.14) 2 (e.g.: lAx qx cBx K Ax ~cAx =ðhcAx ix þ K Ax Þ ) leads to additional complexity. Actually, a non-trivial difficulty arises within the upscaling process in the sense that the decompositions stated by the closure assumptions are not linear anymore, since hcAx ix occurs explicitly in the coupled governing equations of the deviation fields. The deviation reaction term, written as a function of macroscopic source terms, takes the following form

lAx qx cBx

bAxc ðhcAx ix þ K Ax Þ

rhcAc ic 2

ðhcAx ix þ K Ax Þ

2

rhcAx ix ðB:41Þ

This hold also for the fluid phase. The assumption that the reactive phenomena do not prevail over the convective and diffusive processes, i.e., DaAc 6 PeA and DaAx 6 PeA ; is required to overcome this difficulty. These closure fields are used to evaluate the effective parameters of the macroscopic equation, according to the following expressions, which are derived by the injection of the closure assumptions (12) and (13) in the unclosed macroscopic equations (10) and (11).

Z

DAc V

ncx bAcc dA þ

Axc

DAc V

Z

ncj bAcc dA hv~ c bAcc i Acj

þ ec DAc I Z

DAc V

DAc V

uAc ¼

1 V

ðB:42Þ ncx bAcx dA þ

Axc

Z

ncx r Ac dA þ Axc

Z

Z

ncj bAcx dA hv~ c bAcx i ðB:43Þ

Acj

Z

DAc V

ncj r Ac dA hv~ c r Acx i

ðB:44Þ

Ac j

ncx :DAc rðbAcc ÞdA þ

1 V

Z

DAc V

Axc

¼

Z

1 V

ncj :DAc rðbAcc ÞdA

Acj

1 V

nxc DAx rðbAxc ÞdA þ

Axc

Z

! nxj DAx rðbAxc ÞdA

Axj

DAc rec ðB:45Þ

aA ¼

1 V

Z

ncx :DAc rðr Ac ÞdA þ

Axc

1 ¼ V

1 V

nxc r Ax dA þ

Axc

Z

DAx V

Z

1 V

Z

nxj DAc rðrAc ÞdA

Ac j

Z

!

Z

1 nxc DAx rðr Ax ÞdA þ V Axc

nxj DAx rðr Ax ÞdA

Axj

ðB:46Þ DAxc ¼

DAx V

DAxx ¼

DAx V

Z

nxc bAxc dA þ

Axc

Z Axc

DAx V

nxc bAxx dA þ

Z

DAx V

nxj bAxc dA

ðB:47Þ

Axj

Z

nxc r Ax dA

ðB:49Þ

Axj

nxc DAx rðbAxx ÞdA þ

Axc

1 V

Z

nxj

Axj

DAx rðbAxx ÞdA ! Z Z 1 1 ¼ ncx :DAc rðbAcx ÞdA þ ncj :DAc rðbAcx ÞdA V Axc V Ac j rex DAx

Appendix C. Supplementary data

bAxx

þ lAx qx cBx K Ax

dAc ¼

uAx ¼

Z

ðB:50Þ

r Ax ¼ lAx qx cBx K Ax 2 2 x x ðhcAx i þ K Ax Þ ðhcAx i þ K Ax Þ

þ lAx qx cBx K Ax

DAcx ¼

DAx V

K Ax ~cAx

ðhcAx ix hcAc ic Þ

DAcc ¼

dAx ¼

nxj bAxx dA þ ex DAx

Axj

ðB:48Þ

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