Upscaling of the advection–diffusion–reaction equation with Monod reaction

Advances in Water Resources 32 (2009) 1336–1351

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Advances in Water Resources

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Upscaling of the advection–diffusion–reaction equation with Monod reaction

F. Heße a, F.A. Radu a,b,*, M. Thullner c, S. Attinger a,b

a Department of Computational Hydrosystems, UFZ-Helmholtz Center for Environmental Research, Permoserstr. 15, D-04318 Leipzig, Germanyb Institute of Geoscience, University of Jena, Wöllnitzerstr. 7, D-07749 Jena, Germanyc Department of Environmental Microbiology, UFZ-Helmholtz Center for Environmental Research, Permoserstr. 15, D-04318 Leipzig, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 September 2008Received in revised form 27 May 2009Accepted 28 May 2009Available online 6 June 2009

Keywords:UpscalingPore-scale processesMonod reactionEffective parametersBioavailability

0309-1708/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.advwatres.2009.05.009

* Corresponding author. Address: Department of CUFZ-Helmholtz Center for Environmental ResearchLeipzig, Germany.

E-mail address: [email protected] (F.A. Radu).

The need for reliable models for the reactive transport of contaminants in the subsurface is well recog-nized. The predictive power of these models is determined by the accurate description of bioavailabilityof contaminants to microorganisms in porous media. Among many other factors influencing bioavailabil-ity, diffusive mass transfer processes may limit the substrate availability at the pore scale and hencereduce the effective degradation rate considerably. In this study we used a combination of analyticaland numerical methods to upscale surface catalyzed Monod-type reaction rates within a single pore,to obtain effective rate expression at a larger scale. Results show that in the upscaled description Monodkinetics lead to a concentration dependent transition between a reaction and diffusion-limited regime.Strictly, the effective rate repression does not follow Monod-type kinetics. However, we can presentappropriate effective parameters relations, which provide an acceptable approximation of degradationdynamics using an effective Monod-type reaction rate.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Anthropogenic groundwater contamination is a severe problemin many industrialized countries. Ex situ remediation means, suchas pump-and-treat systems, are often neither technically nor finan-cially feasible due to the size of the contaminated sites. For manyorganic carbon compounds in situ bioremediation, either passiveor enhanced, has shown to be a cost-effective alternative. En-hanced bioremediation uses the ability of subsurface microorgan-isms to degrade organic contaminants [55].

The biodegradation of groundwater contaminants has beenextensively investigated, both in the field and in the laboratory.However, due to the complex interplay of microbial, chemicaland physical processes occurring in groundwater, a direct quantifi-cation of in situ biodegradation is often hard to achieve. In order tojudge the effectiveness of biodegradation on contaminated sitesthe experimental characterization is often combined with numer-ical simulations using reactive transport models (e.g.[36,4,10,40]). Yet, their predictive power is restricted by the accu-racy of the implemented process descriptions.

The extrapolation of laboratory results on microbial degrada-tion processes to in situ biodegradation processes in the field andthe incorporation of these processes in reactive transport simula-

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omputational Hydrosystems,, Permoserstr. 15, D-04318

tions are – among many other aspects – challenged by finding anadequate description of the bioavailability of the substrate[20,51]. Factors controlling the bioavailability include the phys-ico-chemical structure of the substrate [8,22], physical occlusionby small pores [32,58,27] or mineral coatings [45], and macro-scopic mixing processes [12,50]. Most importantly, the bioavail-ability of a dissolved contaminant in porous media is highlyaffected by mass transfer processes at the pore or sub-pore scale.The activity of microorganisms is controlled by substrate concen-trations in their immediate vicinity [19,47,23]. In porous mediamicroorganisms primarily reside on the surface of the solid matrix(Fig. 1, right part). Microscopic transport processes within eachpore must provide the supply of the contaminant from the bulkpore water to the location of the microbial cells. This transport lim-its bioavailability, besides any of the other processes mentionedabove, which might impose an additional restriction to bioavaila-bilty. As a consequence, the bioavailable concentration, to whichmicroorganisms are exposed to, may differ considerably from theaverage concentration measured at the macroscale [43,33].

To understand the limitations of macroscopic degradation ratesby such pore-scale mass fluxes, research has focused on simplerepresentations of the pore space [3,28,30]. Looking at the porescale it can be shown that the effective reaction rate can be signif-icantly reduced when pore-scale diffusion becomes a limiting fac-tor for bioavailability [3,9,16,25,34]. However, the reaction rate inmost of these studies was assumed to follow first-order kineticswith respect to the concentration of the degraded species. In caseof microbially catalyzed reaction first-order kinetics valid for low

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Nomenclature

LatinA amplitude of Wc concentration of the solutec0 concentration at Ci

fcbio bioavailable concentrationcref reference concentrationC macroscale concentrationD microscale diffusion coefficientKm half-saturation constantLref reference length scaleL microscale differential operatorn unit vector normal to Cs

Pe Péclet numberqmax maximum conversion rateR general reaction ratev microscale fluid velocityV pore-scale average velocity

Greekg scaling coefficient

Cf boundary of X with the fluidCi

f inlet of Xp

Cof outlet of Xp

Cs boundary of X with the solidU2 Thiele modulusW transversal part of solution ck eigenvalue of Wsij coupling constant between mode i and jX whole domainXp pore domain

Miscellaneoush�i averaged quantitye� deviation quantity� scaled quantity�eff upscaled parameter�eqv constant approximation for �eff�i ith mode

F. Heße et al. / Advances in Water Resources 32 (2009) 1336–1351 1337

concentrations only [6]. In reality, the biodegradation of organiccontaminants often follows Monod-type kinetics [35]. Recently,Wood et al. [56] have performed investigations assuming aMonod-type reaction rate within a single pore. They derivedupscaling rules in the cases of either very low or very high sub-strate concentration. However, by comparing their upscaled equa-tion with numerical simulations in a complex array of pores theygot a mismatch for concentrations in the range of the Monod-half-saturation constant.

In this work we use a channel geometry comparable to [25,3]but consider a Monod-type reaction rate for the reactive surfaceof the pore. We chose this simple geometry in order to be able tomake use of analytical tools in the upscaling process. With our ap-proach we aim to verify: (i) whether the upscaled reaction ratelaws can be sufficiently described by Monod-type kinetics, andwhether the problems reported by Wood et al. can be resolvedand if yes: (ii) how the parameters of such macroscopic Monodkinetics can be linked to microscopic reaction rate parameters thatare valid at the local scale. The results obtained in this study forpore-scale systems may provide the base for interpreting resultsfrom laboratory column experiments. With further upscaling stepsand additionally taking into account large scale heterogeneities,our results can be applied for describing biodegradation efficiencyat the field scale.

In the following sections of this paper we will first introduce theconceptual approach used in this study including the underlyingequations, the geometric representation of the pore system, and

Continuum-Scale Pore-S

Fig. 1. Schematic of the complexity of the subsurfac

the applied numerical schemes and upscaling concepts (Section2). This is followed by the description of the analytical tools usedto obtain explicit solutions for the microscale problems and effec-tive equations for the macroscale continuum (Section 3). Analyticaland numerical results are presented and discussed in Section 4 andfinal conclusions for the scaling behavior of bioavailabilty con-trolled Monod-type reactions are given in Section 5.

2. Conceptual model

This section describes the conceptual approach used in thisstudy. This includes the governing equations describing transportand degradation of a reactive species as well as the pore geometrythese equations are applied to. Furthermore, the applied upscalingconcepts and numerical schemes are introduced.

2.1. Mathematical description

In the following we will derive the mathematical model forreactive transport at the pore scale. Starting with a general descrip-tion we will introduce appropriate scaling units and apply a fewsimplifications before stating the definite mathematicaldescription.

The scale of interest is that of a single pore (Fig. 2). All flow andtransport processes are taking place in the fluid phase X, only. For asingle pore, the boundaries of the fluid phase domain can be sepa-rated into a fluid–solid interface Cs and a fluid–fluid interface Cf .

Subpore-Scalecale

e and the variety the different scales involved.

s

ps

fo

fi

Fig. 2. Schematic of the computational domain.

p

x

y

&Contaminant

Ly,refReaction

Fig. 3. Schematic sketch of the semi-infinite channel used to describe processes in asingle pore.

1338 F. Heße et al. / Advances in Water Resources 32 (2009) 1336–1351

The latter can be further separated into the inlet boundary Cif and

the outlet boundary Cof , each of them described by different bound-

ary conditions. In a single pore, the fate of a single species withconcentration c is described by (a) the advective diffusive transportin the fluid phase, (b) microbial degradation following Monod-typekinetics at the fluid–solid interface, (c) a constant concentrationalong the inlet boundary, and (d) a zero-concentration gradientat the outlet boundary

@

@tc þ Vr � ~vc ¼ DDc in Xp; ð1aÞ

Drc � n ¼ � qmaxcKm þ c

on Cs; ð1bÞ

c ¼ c0 on Cif ; ð1cÞ

rc � n ¼ 0 on Cof : ð1dÞ

Here, the water flux v is given as v ¼ V ~v, with V being the pore-scale average velocity and ~v the rescaled velocity, D is the moleculardiffusivity, n is the outer unit normal, qmax is the maximum conver-sion rate and Km is the half-saturation constant. The implementa-tion of the reaction rate in Eq. (1b) assumes the microorganismsto be localized at the solid liquid interface in a thin biofilm beingconstant in space and time.

Eqs. (1a)–(1c) were transferred into a non-dimensional formusing reference lengths Lx;ref and Ly;ref as well as a reference concen-tration cref (values for Lx;ref ; Ly; ref and cref are addressed in Section2.2). This allows for the definition of the following dimensionlessvariables

x¼ xLx;ref

; y¼ yLy;ref

; c¼ ccref

; bK m¼Km

cref; and t¼ Dt

L2y;ref

: ð2Þ

Furthermore, two-dimensionless quantities are used: the Pécletnumber and the Thiele modulus. The Péclet number

Pe ¼VL2

y;ref

DLx;refð3Þ

indicates whether the advective or the diffusive transport is domi-nant at the scale of interest. High Péclet numbers mean advectiondominates diffusion and vice versa. At the pore scale of groundwa-ter systems, typical values of Ly;ref < 1 mm and V < 1 m=d result invalues of Pe � 10 or below. This is in contrast to the continuumscale were Péclet numbers can be considerably higher. The Thielemodulus [49]

U2 ¼ qmaxLy;ref

DKmð4Þ

compares the dynamics of the reactive consumption and the diffu-sive flux. This dimensionless quantity is related to the Damköhlernumbers Da [14], commonly used in chemical engineering to relatethe kinetics of reactions to mass transfer processes [18]. The Thielemodulus can be used to describe the bioavailabilty of a substrate(e.g. [37,11]). In pore-scale systems U2 as well as Km can vary overseveral orders of magnitude. In our work we will focus on valuescomprising the transition from the reaction-limited to the diffu-sion-limited as well as from a first-order to a zeroth-order regime.

Using the definitions given by Eqs. (2)–(4) we write Eqs. (1a)and (1b)

@

@tc þ Per � ~vc ¼

L2y;ref

L2x;ref

@2

@x2 c þ @2

@y2 c in Xp; ð5aÞ

rc � n ¼ � U2c

1þ c=bK m

on Cs: ð5bÞ

Eqs. (1c) and (1d) exhibit no significant changes using non-dimen-sional variables. In the remainder of this publication, we will usethe same symbols for dimensional as well as for non-dimensionalvariables. The occurrence of the Péclet number and the Thiele mod-ulus in the equations will be the indicator whether dimensional ornon-dimensional variables are considered.

Before stating the definite system of equations we will applythree simplifications justified by the scope of the study. First, wewill drop the time derivative, since we are mainly interested inthe steady state solution. Second, we restrict our analysis to travelpaths of the contaminant with L2

y;ref � L2x;ref . This is corresponding

to a flow path of the contaminant being effectively longer alongthan perpendicular to the flow field. Rephrasing this constraintas L2

y;ref=L2x;ref � 1 shows that we can neglect the longitudinal diffu-

sion in Eq. (5a). The assumption is for example supported by thefindings of Liedl et al. [31] who showed that the longitudinal dis-persivity has practically no impact on the steady state plumelength. The last simplification regards the velocity field, whichhas only a component in the direction along the flow path. Withthese simplifications and using the inlet boundary concentrationas a reference (cref ¼ c0) the pore system is described by

Pef ðyÞ @@x

c ¼ @2

@y2 c in Xp; ð6aÞ

c ¼ 1 on Cif ; ð6bÞ

rc � n ¼ � U2 c1þ c=Km

on Cs: ð6cÞ

Eqs. 6a, 6b and 6c were used to perform all analysis presented in thefollowing sections. The coefficient function f ðyÞ in Eq. (6a) is aplaceholder for an arbitrary velocity profile. Note that for first-orderreaction rates or c � Km Eq. (6c) reads rc � n ¼ �U2c.

2.2. Geometrical description

The pore system to which we apply Eqs. 6a, 6b and 6c is repre-sented by a channel extending in x- and y-direction (Fig. 3). A sin-gle pore with such a geometry will lead to a porous mediumconsisting of a compound of capillary tubes [15]. Compared toother two-dimensional arrays of single pores (e.g. [28,30,17]) thisrepresents a simplification of the pore geometry which is necessaryto obtain analytical solutions in closed form expressions. Such asingle pore system, although simple, has been proven to giveappropriate indications on the relation between pore geometry,diffusion and reaction in general ([26,38]) and the geometry used


here has previously been used by other authors to describe reactiveprocesses in porous media (e.g. [25,34,53]). Results obtained forsingle pores can be transferred to more realistic porous media rep-resentations using the ratio of the reactive surface and the free vol-ume as a shape factor [56].

The reference length Ly;ref is chosen to be half the width of thepore resulting in a pore space Xp given by the dimensionless coordi-nate ranges of 0 < x <1 and�1 < y < 1. The reference length Lx;ref

is the characteristic length for which the concentration should bedetermined. As described in Eq. (6b), the fixed concentration con-sidered at the inlet boundary is used as reference concentration cref .

The pore space, the boundaries and thus all obtained solutionsof Eqs. 6a, 6b and 6c are symmetric with respect to the x-axis(Fig. 3). For this reason the domain was split along the y-axis. Allanalytical and numerical solutions were calculated for 0 < y < 1,considering rc � n ¼ 0 as boundary condition at y ¼ 0.

2.3. Scenarios considered for calculations

In Eq. (6a) the form of the coefficient function regarding thevelocity profile was not further specified. For the given pore geom-etry a parabolic profile is the most realistic velocity distribution[48,1]. The focus of this study is on the scaling behavior ofMonod-type reaction kinetics. However, in order to verify the re-sults of our approach with those presented and discussed in the lit-erature for first-order reaction rates and uniform velocity profiles[25], we here consider the same in form of scenario I. Next to themost simple (scenario I) and the most realistic scenario (scenarioIV), two further scenarios (II and III) of intermediate complexitywere considered (Fig. 4). In scenarios II and III the remaining com-binations of reaction kinetics and velocity profile were addressedto investigate the influence of each individual feature on the ob-tained results.

2.4. Upscaling of the pore-scale processes

The focus of this study is to use upscaling methods to obtain aneffective one-dimensional representation of the system describedby Eq. (6). Generally the purpose of upscaling is to find an effectivedescription of the process of interest on a coarse level by startingwith a well defined representation of the process on a fine level.The most common methods for upscaling in subsurface hydrology[44,57] are homogenization [2,39] and volume averaging [54].

To arrive at an effective representation we have to average theprocess over the y-axis (Fig. 3). The scheme of the upscaling pro-cess used in this study is outlined in Fig. 5.

Starting point of the analysis is the system of two-dimensionalpartial differential equations as given by Eqs. 6a, 6b and 6c. Themost ‘straight forward’ analysis is first solving these equations

If irst − order reaction

uni f orm velocity f ield

IIIMonod − type reactionuni f orm velocity f ield

Fig. 4. Schematic of the different scenar

either numerically or analytically (left side of Fig. 5). The resultingtwo-dimensional concentration distribution is then averaged overthe width of the pore (i.e. the y-axis; Fig. 3) providing a one-dimen-sional concentration profile along the length of the pore. The de-rived concentration profiles were used as references for analternative approach where the steps of analytical solution andaveraging are permuted (right side of Fig. 5). In the latter approach,first the averaging over the y-axis results in a new one-dimensionaleffective differential operator Leff with a reduced complexity butnew effective parameters (see Section 3). The evaluation of theseparameters is the main part of the analytical upscaling process.After that, the upscaled parameters are used to calculate an effec-tive solution (one-dimensional concentration profile along thelength of the pore). A comparison between the solutions is a goodmeasure for the accuracy of the effective parameters (Fig. 5).

2.5. Numerical scheme

To support the analytically derived results numerical solutionsfor Eqs. 6a, 6b and 6c were calculated (see Fig. 5). These numericalsimulations were performed using the software platform UG(‘Unstructured Grids’, [5]). Steady state results were obtained bysimulating a transient problem with arbitrary initial conditions un-til steady state was reached. The time derivative is discretized bythe one-step implicit Euler method. In order to ensure the localmass conservation, the mixed finite element method is appliedfor the spatial discretization. More precisely, the lowest order finiteelements of Raviart–Thomas type are used for the approximationof the fluxes and piecewise constants for the concentrations. Theresulting algebraic system of equations is hybridized by adding La-grange multipliers on the edges according to Radu et al. [41,42].Then the nonlinear problem is linearized by a damped Newtonmethod and the resulting linear systems are solved by a multigridalgorithm.

3. Upscaling and analytical methods

In this section we present (i) analytical solutions for the coupledtransport degradation problem in two-dimensions and the subse-quent averaging over the y-axes, as well as (ii) one-dimensionaleffective equations obtained by the upscaling theory. Both ap-proaches are first applied for first-order reaction rates and after-wards modified to solve the case of a Monod-type reaction rate.

3.1. First-order reaction rate

Assuming a first-order reaction rate, Eq. (6c) can be written as

rc � n ¼ �U2cbio ð7Þ

IIf irst −

−

order reactionparabolic velocity f ield

IVMonod type reactionparabolic velocity f ield

ios (I–IV) investigated in this study.

Fig. 5. Schematic of the upscaling process.


introducing cbio as cðx; yÞjy¼1 or the concentration available to thesurface bound microorganisms. To solve the resulting system ofequations we assume the concentration to be given as an infinite se-quence of modes

cðx; yÞ ¼X1i¼1

CiðxÞWiðyÞ: ð8Þ

This ansatz separates every mode into a longitudinal and a transver-sal component (relative to the flow direction), under the assump-tion that the velocity field is constant along the longitudinaldirection. Thus both sides of Eq. (8) are not coupled by the coeffi-cient function f ðyÞ from Eq. (6a). A comprehensive discussion onthe solution of Eq. (8) can be found in the Appendix A.1. As a resultwe get the following expression

T@

@xC ¼ KC ð9Þ

which can be rearranged to

@

@xC ¼ T�1KC ¼ CC: ð10Þ

The entries of the unknown vector C are the longitudinal modes Ci

of the concentration c. The entries of the system matrix C depend onthe velocity field f ðyÞ. In this form Eq. (10) represents a system oflinear ordinary differential equations. In the following subsectionswe will solve this system for the cases of a uniform and a parabolicvelocity field.

3.1.1. Uniform velocity fieldFor a uniform velocity field the coefficient function in Eq. (6a) is

given by f ðyÞ ¼ 1.

3.1.1.1. Analytical solution. For this velocity field the system matrixC is diagonal so the single longitudinal modes are decoupled

CiðxÞ ¼ AisinðkiÞ

kie�

k2i

Pex: ð11Þ

For a comprehensive derivation of this solution and the calculationof the coefficients Ai and ki see Appendix A.2. The y-averaged solu-tion can be written as

CðxÞ ¼XN

i¼1

e�k2

iPex 4 sin2ðkiÞ

kiðsinð2kiÞ þ 2kiÞ: ð12Þ

From Eq. (11) it can be concluded, that only the first few modes arerequired to obtain a good approximation of CðxÞ. Since the elementsof fkigiP1 are monotonously increasing (see Fig. 18) the respectivemodes exhibit a steeper exponential decay. Furthermore, the coeffi-cients Ai in Eq. (11) are decreasing with increasing i (see Eq. (36)).Consequently, the contribution of higher modes to CðxÞ isinsignificant.

3.1.1.2. Effective equation. Details on the direct upscaling of the sys-tem given by Eqs. 6a, 6b and 6c can be found in the Appendix A.2.As a result of this procedure we get the following differentialequation:

@

@xCðxÞ ¼ �U2

eff

PeCðxÞ ð13Þ

which exhibits a first-order dependency on the y-averaged concen-tration. The new effective coefficient U2

eff will be discussed in Sec-tion 4.1 in more detail.

3.1.2. Parabolic velocity fieldFor a parabolic velocity field the coefficient function in Eq. (6a)

is now given by f ðyÞ ¼ 1:5ð1� y2Þ. The details of the determinationof the analytical solution as well as the effective equation are givenin Appendix A.3.

3.1.2.1. Analytical solution. For this velocity field the differentmodes of the unknown vector C are now coupled and have to bediagonalized before they can be solved in analogy to Eq. (11). Con-sequently we get

wiðxÞ ¼ wið0Þe�diix; ð14Þ


where dii are the entries of the diagonalized Matrix C from Eq. (10)and the vector of the initial conditions is wð0Þ ¼ G�1Cð0Þ. The re-quired solution is then found by re-transforming the solution ofEq. (14).

3.1.2.2. Effective equation. For a parabolic velocity field we get anordinary, second order differential equation for the first mode ofthe concentration

veff@

@xC1ðxÞ ¼ Deff

@2

@x2 C1ðxÞ þ Reff C1ðxÞ ð15Þ

In comparison to Eq. (13) new transport parameters veff and Deff areintroduced. Since all quantities in Eq. (15) are non-dimensionalizedthese effective parameters represent the ratio between the micro-scale and the physically effective values. The transport parameterscan be determined by solving Eq. (46)

veff ¼ s11 þ s22k2

1

k22

; ð16aÞ

Deff ¼Pe

k22

ðs21s12 � s11s22Þ and ð16bÞ

Reff ¼ �k2

1

Pe: ð16cÞ

Here sij are the entries of the matrix T from Eq. (9). Note that therepresentation of the effective parameters is arbitrary. The presentform has been chosen such that Reff is a good approximation of thereaction rate in the former scenario (see Eq. (47)).

3.2. Monod-type reaction rate

For Monod-type reaction rates given by Eq. (6c) the coefficientskiðxÞ are now x-dependent so Eq. (10) must be modified

@

@xC ¼ CðxÞC: ð17Þ

Further details on the calculations are again given in the AppendixA.4. The solution of Eq. (17) depends on the velocity field f ðyÞ and isin the following discussed in analogy to Section 3.1.

3.2.1. Uniform velocity field3.2.1.1. Analytical solution. As mentioned above in the case of a uni-form velocity field all modes in Eq. (10) are decoupled. Therefore,each single mode CiðxÞ is given by the differential equation

@

@xCiðxÞ ¼ �ciiCiðxÞ: ð18Þ

Here cii are the respective entries of CðxÞ from Eq. (17). Due to the x-dependency of the coefficient function we have to modify Eq. (11)to

CiðxÞ ¼ Aið0Þsinðkið0ÞÞ

kið0Þe�R x

0ciiðx0Þ dx0

: ð19Þ

This leads to the y-averaged solution for the concentration

CðxÞ ¼XN

i¼1

4 sinðkið0ÞÞ sinðkiðxÞÞe�R x

0ciiðx0 Þ dx0

kið0ÞkiðxÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisinð2kið0ÞÞ þ 2kið0Þ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisinð2kiðxÞÞ þ 2kiðxÞ

p :

ð20Þ

Because of the nonlinear Monod term in Eq. (6c) the coefficients ki

depend on the solution c. Therefore, Eq. (19) has to be solved itera-tively. Using the solution for first-order reaction rates as an initialguess c0, we solve the system to get the new approximation c1

and iterate the proceedings. The fixed point cI of the iterative loopis then the required solution. Although, we have not proven the

convergence of the iteration scheme, we see it numerically. More-over, for all investigated parameter settings we see a good agree-ment between the semi-analytical solution and the numericallycalculated solution (see Section 4).

3.2.1.2. Effective equation. The direct upscaling described above forfirst-order reaction rates is now applied using Eq. (6b) instead of(6a). This leads to

Pe@

@xC ¼ � U2 cbio

1þ cbio=Km: ð21Þ

Furthermore, we introduce a new coefficient function

Km;eff ¼C

cbioKm: ð22Þ

With this new effective half-saturation constant and the effectiveThiele modulus U2

eff , given in analogy to Eq. (52), we obtain

Pe@

@xC ¼ � U2

eff C1þ C=Km;eff

: ð23Þ

Both effective coefficient functions U2eff and Km;eff are scaled by the

same scaling factor

g ¼ cbio

C; ð24Þ

which is the ratio of the bioavailable concentration cbio and the y-averaged or upscaled concentration C. Using Eq. (24), we can re-write Eq. (23) to obtain an analytical expression of the governingdifferential equation for the upscaled concentration C

Pe@

@xC ¼ � U2 C

1=gþ C=Km: ð25Þ

If the coefficients U2eff and Km;eff are constant and if Cð0Þ ¼ 1 is used

as boundary condition, the analytical solution of Eq. (23) is given by

C ¼ Km;eff LambertWe

1�U2

effPeKm;eff

x

Km;eff

0B@1CA: ð26Þ

The function LambertWðzÞ is the solution of z ¼ wew (see [13])which has already been used in the context of microbial reactionkinetics (e.g. [46,21]). Comparing Eq. (26) to the analytical ornumerical solutions of the two-dimensional problem allows to ob-tain direct estimates for the effective parameters Ueff and Km;eff .

3.2.2. Parabolic velocity field3.2.2.1. Analytical solution. Because of the x-dependency of thecoefficients kiðxÞ, Eq. (19) has to be modified in analogy to Eq. (11):

C0 ¼ CðxÞC ¼ GðxÞDðxÞG�1ðxÞCG�1ðxÞC0 ¼ DðxÞG�1ðxÞCw0 ¼ DðxÞw:

By decoupling the system we have arrived at a form comparable toEq. (17). The analytical solution in analogy to Eq. (19) is now

wiðxÞ ¼ wið0Þe�R x

0diiðx0 Þdx0

: ð27Þ

The required solution is found by re-transforming the solution ofEq. (27).

3.2.2.2. Effective equation. As in the case of a first-order reactionrate with a parabolic velocity field, no closed solution for the directupscaling exists. However, by applying a similar scheme as usedfor first-order reaction rates we can derive an effective equationfor the first longitudinal mode C1ðxÞ

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

x

C

numericalanalytical

Fig. 6. Examples for first-order reaction rate with uniform velocity field. The localparameters are U2 ¼ 10 and Pe ¼ 2. Top: simulated two-dimensional results.Bottom: comparison of one-dimensional analytical and numerical solutions.


veffðxÞ@

@xC1ðxÞ ¼ DeffðxÞ

@2

@x2 C1ðxÞ þ Reff ðxÞC1ðxÞ: ð28Þ

All coefficients of the effective Eq. (28) are now x-dependent func-tions. Their evaluation has therefore, become cumbersome for prac-tical applications compared to the case of a first-order reaction rate.Nonetheless, these coefficient functions are useful for theoreticalconsiderations

veff ¼ 1þ c11

c22� c12

c22

@

@x1c12

; ð29aÞ

Deff ¼ �1c22

and ð29bÞ

Reff ¼ ðc11 �c12c21

c22� c12

c22

@

@xc11

c12Þ: ð29cÞ

Here the coefficients cij are the entries of the system matrix C of Eq.(17). The increase in complexity is attributed to the new mixingterms for the effective velocity veff and the effective reaction termReff . Though the effective dispersion Deff contains no additionalterms all coefficient functions are x-dependent (see also Eq. (23)).

3.3. Synopsis of analytical methods

In Table 1 a summary of the analytical solutions and effectiveequations derived from Eqs. 6a, 6b and 6c for the different porevelocity profiles and reaction rates is given. For first-order reactionrates our results are comparable to those found in the literature[25,3]. In the case of the parabolic velocity field the effective equa-tion only provides results for the first mode. The introduced errorby neglecting higher modes is confined to small values of x.

4. Results and discussion

In this section we first present and discuss the analytical andnumerical results obtained by applying the different approachesoutlined in Fig. 5 for different combinations of velocity fields andreaction kinetics (see Fig. 4). Calculated values of the effectiveparameters used in the one-dimensional upscaled equation, andthe dependency of these parameters on local parameters are eval-uated. Finally, the applicability of an effective Monod-type reactionrate is discussed.

4.1. First-order reaction rate with uniform velocity field

The case of a first-order reaction rate with a uniform velocityfield is well studied in the literature [25]. We briefly review andcompare those results vis-à-vis our numerical findings. Calculatedconcentration profiles exhibit an exponential decrease along the x-direction and a cosine-like profile along the y-direction (see Fig. 6for an arbitrary example). The strong gradient in the y-directionshows that the transversal diffusion is not fully able to transportthe contaminant from the bulk of the domain to the reactiveboundary at y ¼ �1. The y-averaged profile of the numerical andthe analytical solution match very well in all investigated scenar-ios, which indicates the soundness of the used numerical scheme.

Calculated values for U2eff , using Eq. (13), show a hyperbolic

behavior with respect to U2 (Fig. 7). Consequently, we can identify

Table 1Summary of cases and corresponding equations.

First-order reaction rate Monod-type reaction rate

Analytical Effective Analytical Effective

Uniform vel. profile (12) (13) (20) (23)Parabolic vel. profile (14) (15) (27) (28)

three different regimes. The first one is termed reaction-limited andis valid for low U2 values. Here U2

eff shows a nearly linear depen-dency and the scaling unit g is accordingly close to 1. This indicatesa strong coupling between the local and global behavior. Thus, thereaction in this regime is sufficiently slow for the transversal diffu-sion to provide the reactive boundary with enough substrate. Thebioavailable concentration cbio is therefore nearly the same as they-averaged concentration C. As a result the upscaled reaction rateis mostly governed by the small scale reaction henceforth thename. The second regime is called diffusion-limited and is validfor high values of U2. Here U2

eff asymptotically approaches p2=4,corresponding to a linear decrease of g (Fig. 7). In this regime thereaction is too fast for the transversal diffusion to transport suffi-cient amounts of substrate to the reactive boundary. As a result,strong concentration gradients occur along the width of the poreand the bioavailable concentration is much smaller than the y-averaged concentration, i.e. cbio � C. The third regime is the transi-tion zone between the two other regimes and is characterized byreaction as well as diffusion. Both are limiting processes and con-trol the upscaled behavior. These findings agree with results fromthe literature [25,56], which in case of [56] also shows that, byapplying appropriate scaling steps, the results from a simple geom-etry can be extended to more realistic scenarios.

4.2. First-order reaction rate with parabolic velocity field

All other boundary conditions assuming the same as in Section4.1, we discuss the case of a first-order reaction rate with a para-bolic velocity field. As noted in Section 2, this case has been dis-cussed in the literature [3], but for different conditions asconsidered here. Nevertheless, our results are similar to those pre-viously reported. Calculated concentration profiles for a parabolicvelocity field (Fig. 8) show only minor differences to profiles ob-tained for a uniform velocity field. For small values of x, i.e. closeto the inlet, y-averaged concentrations are slightly smaller in thecase of a parabolic velocity field. For increasing x however, higherconcentrations are observed for the parabolic velocity field.

The differences between the two velocity fields can be attrib-uted to the occurrence of the effective dispersion coefficient Deff

and the effective velocity veff (see Eq. (15)), which result in a fastertransport of substrate along the length of the pore. The relation

100 10210−1

100

Φ2

Φ2 ef

f

(a) Development of Φ 2eff.

100 10210−3

10−2

10−1

100

Φ2

η

(b) Development ofη.

Fig. 7. Dependency of U2eff and g on U2 in case of a first-order reaction rate with a uniform velocity field. Together with the linear and constant asymptotes representing the

reaction-limited (reached for low values of U2) and the diffusion-limited regime (reached for high values of U2). The value of U2eff has been evaluated using Eq. (52).


between the effective velocity veff and U2 depends on the regimegoverning the overall consumption of the substrate (see Fig. 9a).In the reaction-limited regime the effective velocity is close to 1(i.e. equal to the average flow velocity) and both, uniform and par-

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

x

C

uniform velocity fieldparabolic velocity fieldnumericalparabolic velocity fieldanalitical

Fig. 8. Examples of a first-order reaction rate with a parabolic velocity field. Thelocal parameters are U2 ¼ 10 and Pe ¼ 2. Top: simulated two-dimensional results.Bottom: comparison of one-dimensional analytical and numerical solution withresults for a uniform velocity field.

10−2 100 1021

1.1

1.2

1.3

1.4

1.5

Φ2

v eff

(a) Development of veff.

Fig. 9. Dependency on the effective parameters veff and Deff from U2 for the case of a fisolving Eqs. (16a) and (16b).

abolic velocity fields, yield almost identical results. The effectivevelocity increases with increasing U2 and eventually saturates forhigh U2-values in the diffusion-limited regime. In the latter regimestrong transversal concentration gradients exist and highest con-centrations correlate with highest flow velocities. As a result thebulk of the substrate mass is transported faster downstream. Theeffective dispersion coefficient Deff remains small compared tothe molecular diffusion coefficient (i.e. Deff < 1) and shows a re-verse dependency on U2 than observed for veff (see Fig. 9b). Thesteep gradient at the immediate vicinity of pore inlet (Fig. 8) iscaused by the uniform constant concentration distribution usedas boundary condition along the entire inlet. This results in highsubstrate concentrations at the reactive pore wall, leading to reac-tion rates that are not limited by any transversal mass transfer atthe vicinity of the inlet.

4.3. Monod-type reaction rate with uniform velocity field

For the combination of a Monod-type reaction rate and uniformvelocity field analytical and numerical results agree well (Fig. 10)which confirms the semi-analytical scheme used for the analysis.The two-dimensional concentration distribution obtained for theMonod-type reaction rate (Fig. 10 top) is qualitatively similar tothe one presented above for a first-order reaction rate (Fig. 6top). The one-dimensional concentration profile, however, showsa clear contrast between the two cases (Fig. 10 bottom). At highy-averaged concentrations C, the concentration decrease is muchweaker for the Monod-type reaction rate because at these concen-trations the upscaled reaction rate approximately follows zeroth-order kinetics. The slope of the concentration profile is therefore

10−2 100 1020.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

Φ2

Def

f

(b) Development of Deff.

rst-order reaction rate with a uniform velocity field. The values were evaluated by

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

x

C

First−order reaction rateMonod type reaction ratenumericalMonod type reaction rateanalytical

Fig. 10. Examples for the Monod-type reaction rate with uniform velocity field. Thelocal parameters are U2 ¼ 10; Pe ¼ 2 and Km ¼ 0:1. Top: simulated two-dimen-sional solution. Bottom: comparison of analytical and numerical one-dimensionalsolutions with results from a first-order reaction rate.


nearly linear in contrast to the exponential decrease observed for afirst-order reaction rate. Only when C drops to small values (i.e.C 6 Km), the upscaled reaction rate approaches a first-order kinet-ics. This qualitative analysis shows, that in the upscaled equationsthe characteristics of a Monod-type reaction rate is preserved.

As in case of a first-order reaction rate the scaling parameter gdescribes the coupling between local and global parameters (Eq.(25)). Compared to the former case, where g does not depend onC (the slight variations for C � 1 are attributed to the inlet bound-

10−310−210−110−3

10−2

10−1

100

C

η

5

10

50

100

200

500

1000

20

(a) Development of η forthefirst-orderreactionrate.

10−310−210−110−3

10−2

10−1

100

C

η

1000

500

200

100

50

20

510

(c) Development ofη for Km = 0.1.

Fig. 11. Development of g for different U2 and Km for the case of a Monod-type reactiocurves. The results were evaluated by using Eq. (24).

ary conditions), the behavior of g is more complex for a Monod-type reaction rate. The parameter now depends on the y-averagedconcentration C (Fig. 11a) approaching a constant value only forsufficiently small concentrations (C � Km; Fig. 11b–d). Besidesthe sensitivity of g towards Km, results also vary with U2. Lowervalues of U2 extend the concentration range where g depends on C.

This characteristic allows to distinguish between three differentregimes: an effective zeroth-order, an effective first-order and atransition regime. For an effective zeroth-order regime withKm � C, g is nearly constant and close to 1. This regime is character-ized by a combination of high concentration values and low valuesof U2 (Fig. 11d in the upper left part). For Km � C we have a transi-tion regime were g, and therefore the correlation between the localand the global parameters, shows a strong dependency on C. In thethird regime, characterized by low concentrations and/or high U2; gis well approximated by constant values representing an effectivefirst-order regime, i.e. Km > C. A comparison of Fig. 11a–d showsthat the behavior of g for high values of U2 is very similar for allKm. Therefore, the results for first-order reaction rates can beapplied to Monod-type reactions rates with high values of U2.

4.4. Monod-type reaction rate with parabolic velocity field

This case is the most complex of the scenarios investigated andthe obtained results (Fig. 12) represent a combination of the effectsdiscussed in Sections 4.3 and 4.2.

As for a first-order reaction rate, a reaction-limited and adiffusion-limited regime can be distinguished. The behavior ofthe upscaled equation in the reaction-limited case (see Fig. 13a)can qualitatively be understood as a superposition of the cases de-scribed in Sections 4.2 and 4.3. Compared to the uniform velocityfield, the concentration decreases relatively sharp in the vicinityof the inlet but exhibits weaker gradients further downstream ofthe pore. These effects have already been discussed in Section

10−310−210−110−3

10−2

10−1

100

C

η

5

20

500

1000

10

200

100

50

(b) Development of η for Km = 1.

10−310−210−110−3

10−2

10−1

100

C

η

10 5

1000

500

200

100

50

20

(d) Development ofη for Km = 0.01.

n rate with uniform velocity field. The respective value of U2 are tagged along the

100 10210−3

10−2

10−1

100

Φ2

ηeq

v

First−order reaction rateKm = 1

Km = 0.1

Km = 0.01

Fig. 14. Dependency of the approximated equivalent scaling parameter geqv on U2

in case of a Monod-type reaction rate with uniform velocity field.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

x

C

numericalanalytical

Fig. 12. Examples for the Monod-type reaction rate with a parabolic velocity field.The local parameters are U2 ¼ 10;Km ¼ 0:1, and Pe ¼ 2. Top: simulated two-dimensional solution. Bottom: comparison of analytical and numerical one-dimensional solutions.


4.2. Furthermore, for the Monod-type reaction rate, a zeroth-orderbehavior is observed for high concentrations and a first-orderbehavior for low concentrations (see Section 4.3). In contrast, forthe diffusion-limited regime the upscaled concentration profilesshow a similar dependency on the velocity field. The results forthe first-order and Monod-type reaction rate are almost identical(see Fig. 13b). This again emphasizes that the relation between dif-fusion and reaction rate determines the upscaled behavior.

4.5. Equivalent Monod-type parameters

Results presented in Section 4.3 show that the coupling betweenthe local and global coefficients is concentration dependent in thecase of a Monod-type reaction rate (Fig. 11). However, a qualitativeanalysis of the results in Sections 4.3 and 4.4 demonstrates aMonod-like behavior of the reaction rate in the upscaled equations,too. This suggests that approximations for concentration indepen-dent parameters for the upscaled rate expression can be found. Inthe following we investigate these equivalent parameters for thecase of (i) a uniform and (ii) a parabolic velocity field.

4.5.1. Uniform velocity fieldIn case of a Monod-type reaction rate and a uniform velocity

field Eq. (25) shows the importance of the ratio g for the behaviorof the upscaled reaction rate. Although a rigorous analysis revealed

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

x

C

First−order reaction rateUniform velocity fieldFirst−order reaction rateParabolic velocity fieldMonod reaction rateUniform velocity fieldMonod reaction rateParabolic velocity field

(a) Reaction-limited regime Φ2 = 10.

Fig. 13. Examples for a Monod-type reaction rate with a parabolic

that g is not constant but a function of concentration (Fig. 11a–d),we attempt to find a simple constant approximation of g which wecall geqv. To estimate geqv we fitted Eq. (25) to the exact one-dimen-sional concentration profiles derived from the two-dimensionalsolutions.

The results of the fitting procedure reveal that geqv depends onU2 and Km (Fig. 14). For higher Km results for the Monod-type reac-tion rate are comparable to those obtained for a first-order reactionrate. Both reaction kinetics show a similar decrease of geqv withincreasing U2. This indicates a transient shift in the regime fromreaction-limited (geqv � 1) to diffusion-limited (geqv � 1). In turn,for lower Km results for Monod-type reaction rates differ, withthe reaction-limited regime apparently prevailing longer withincreasing U2. This leads to higher values for geqv compared tothose obtained for first-order reaction rates. At sufficiently highU2 results for Monod-type and first-order reaction rates again con-verge asymptotically to values of geqv � p2=ð4U2Þ supporting thestatements made in Section 4.2. The accuracy of the estimated con-centration profiles obtained using geqv is given by the differencesbetween fitted and exact solutions (Fig. 15). In general, a goodaccuracy (errors 6 1%) is only found in the extreme cases of eitherlow or high values of U2. The much higher errors found in the tran-sition regime (errors > 10%) correspond to the values of U2 whereg exhibits the strongest dependency on C (Fig. 11). Furthermore, itwas noticed that errors increase with decreasing Km (results notshown).

To improve the quality of the estimates obtained by fittingeffective Monod-type reaction rates to the exact solutions weintroduced an additional degree of freedom and fitted Eq. (26).Considering now both parameters, U2 and Km, to be independentof each other. The improvement resulted in two new equivalentparameters, U2

eqv and Km;eqv. This procedure gives us significantlysmaller errors in the transition zone between reaction- and diffu-

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

x

C

First−order reaction rateUniform velocity fieldFirst−order reaction rateParabolic velocity fieldMonod reaction rateUniform velocity fieldMonod reaction rateParabolic velocity field

(b) Diffusion-limited regime Φ2 = 100.

velocity field. The local parameters are Km ¼ 0:1 and Pe ¼ 2.

100 101 102 103 1040

2

4

6

8

10

12

Φ2

Δ C

[%]

Monod 2Monod 1

Fig. 15. Concentration error DC by applying geqv (dashed line) compared to fittingwith two independent parameters (dotted line) in case of a Monod-type reactionrate with uniform velocity field, Km ¼ 0:1.


sion-limited regimes (Fig. 15), e.g. for Km ¼ 0:1 errors remain be-low 1–2 %. In all investigated cases, i.e. 0:01 6 Km 6 100, the errorof the improved fitting procedure was always less than 3 %, whichis in the lower range of the experimental accuracy for concentra-tion measurements indicating the applicability of the approach.

4.5.1.1. Parabolic velocity field. As explained in Section 4.4 the scal-ing behavior in this case can qualitatively be seen as a superposi-tion of the cases described in Sections 4.3 and 4.2. Thecalculation of the coefficient functions in Eq. (28) shows theappearance of complex mixing terms prohibiting the derivationof the parameter for the effective equation. Thus, numerical solu-tions were used as a reference to obtain constant rate parametersapplying again a fitting procedure using Eq. (28) but with constant

100 101 102 1030

1

2

3

4

5

6

Φ2

Φ2 eq

v

first−orderKm = 1

Km = 0.1

Km = 0.01

(a) Φ2eqv.

Fig. 16. Dependency of the approximated equivalent parameters U2eqv and Km;eqv on U2

100 10210−2

10−1

100

101

Φ2

K m

diffusionlimited

reactionlimited

transition

(a) Dominating local process (diffusion- or reaction-limited).

Fig. 17. General survey of the upscaling behavior in case of a Monod-type reaction rate.using arbitrary threshold values.

coefficients. The effective transport and reaction expressions resultin four unknown parameters in this case. However, to avoid over-parametrization we consider the transport parameters determinedfor a first-order reaction rate to be applicable for Monod-type reac-tions, and fit only U2

eqv and Km;eqv, the two parameters of the reac-tion term.

For Km P 1 the behavior of U2eqv with respect to U2 is similar to

the results of a first-order reaction rate (Figs. 16 and 17, see alsoFig. 7). In contrast, for Km < 1 significant differences can be ob-served. In the latter case, the linear dependency between U2

eqv

and U2 proceeds till larger values. This shows that the reaction-limited regime is extended towards higher values of U2, which cor-responds to the behavior of geqv (Fig. 14). Nevertheless for high val-ues of U2;U2

eqv converges towards p2=4 regardless of the value ofKm. The dependency of Km;eqv on U2 supports these statements(Fig. 16b). In the reaction-limited regime, i.e. for low values ofU2, the local and the global half-saturation constants are approxi-mately identical, i.e. Km � Km; eqv. Again, with increasing U2 the lo-cal and global behavior diverge with higher values of Km showingearlier divergence. Eventually, for high values of U2 all global half-saturation constants Km;eqv increase to high values (100 or above).For such values the global behavior is always well approximated bya first-order reaction rate regardless of the value of Km at the locallevel.

The concentration independent parameters and their depen-dency on local reaction rate parameters determined by this proce-dure could be used for larger scale simulations, e.g. in the form oflook-up tables, in pore network models of porous media([51,7,29]). Such simulations would allow investigations of furthereffects, caused for example by the tortuosity and pore connectivityof the medium.

100 102 10410−2

10−1

100

101

102

Φ2

K m,e

qv

Km = 1Km = 0.1Km = 0.01

(b) K m eqv.

for several Km in case of a Monod-type reaction rate with a parabolic velocity field.

100 10210−2

10−1

100

101

Φ2

K m

first−orderbehavior

zeroth−orderbehavior

transition

(b) Behavior of the upscaled equations (zeroth- orfirst-order).

Borders of the zones with different regimes are drawn for demonstration purposes


4.6. Synopsis

In case of a Monod-type reaction rate and a uniform velocityfield the assumption of a single constant scaling parameter, link-ing local and global reaction parameters, leads to significant er-rors for the transition between reaction-limited and diffusion-limited regimes. Specifically, the upscaled rate expression doesnot follow a Monod-type reaction rate. This can explain theproblems reported in studies, assuming a single scaling parame-ter for the reaction rate expression [56]. However, the errors ob-tained by assuming a Monod-type reaction rate can be clearlyreduced by using two independent scaling relations for U2 andKm. This results in two new, equivalent parameters U2

eqv andKm;eqv. Though we lack a rigorous analytical derivation of aneffective rate expression for a parabolic velocity field, the analy-sis of the numerical results support the extension of the abovestatements to parabolic flow fields as well. Consequently, our re-sults can be applied to Monod-type reactions with a parabolicvelocity field, that enable us to formulate general upscalingrules.

In the case of a first-order reaction rate the global behavior ofthe y-averaged solution can be separated into a reaction-limitedand a diffusion-limited regime (Fig. 7) whereby the transition be-tween them is controlled by U2. In addition, for a Monod-type reac-tion rate we also have to distinguish between a first-order and azeroth-order regime, the transition of which is now controlled byU2 and Km. Furthermore, Km also has an impact on the transitionbetween reaction-limited and diffusion-limited regimes (Fig. 16).As a result the ratio between Km and the concentration C has astrong influence on the scaling behavior of Monod-type reactionrates making it far more complex as for first-order reaction rates.Such concentration dependent transitions between reaction- anddiffusion-limited systems have been reported before ([52,24]). Thisfurther supports that scaling rules obtained for first-order reactionrates can not easily be expanded to Monod-type reaction rates.Only for C � Km, or in the marginal cases of either high or lowU2, the different reaction rates scale similarly.

5. Summary and conclusion

We have presented an new upscaling approach from a two-dimensional system with transport and surface catalyzed degrada-tion of a single reactive species in a simple pore geometry to aneffective one-dimensional reactive transport equation. For theanalysis we neglected the longitudinal diffusion and motivatedthe decision in mathematical and physical terms. The validity ofthe developed model was tested with results from analytical andnumerical solutions to verify the soundness of the upscaling pro-cess and to evaluate the effective parameters of the upscaledequation.

The main focus was the scaling behavior of Monod-type reac-tion rates. Two cases have been considered regarding the velocityprofile within the pore: a simple uniform and a more realistic par-abolic velocity distribution. For both distributions, the results forMonod-type reaction rates have been compared with results ob-tained for first-order reaction rates.

The first two investigated scenarios of the analysis were simplecases of reactive transport with a first-order reaction at the reac-tive boundary of the medium (scenario I and II in Fig. 4). Solutionsfor the upscaled system are already known [3,25] and served as averification of the conceptual approach applied in this study andthe resulting effective reaction rates. Results show that the macro-scopic reaction rate can be strongly reduced when diffusion is thelimiting factor and that effective transport parameters must beconsidered for a parabolic velocity field.

For Monod-type reactions (scenarios III and IV in Fig. 4) theupscaling results showed a concentration dependent coupling be-tween local and global scales with highest sensitivities for localconcentrations in the same order of magnitude as the Monod-half-saturation constant (i.e. Km � cbio). For scenario III where Km

was either much higher or much smaller than the bioavailable con-centration cbio, the upscaled reaction rate could be well approxi-mated by a first-order or zeroth-order reaction rate, respectively.Coupling of local and global behaviors using a single parameterdemonstrated that scaling parameters either required concentra-tion dependent scaling or resulted in significant errors whenremaining constant. The use of this parameter is therefore eithercumbersome or inaccurate. However, by using two independent,constant scaling parameters the global behavior could be repro-duced reasonably well. Such independent scaling parameters couldbe derived for both types of velocity fields by fitting the effectiveone-dimensional profiles to explicit solutions of the two-dimen-sional problem. For scenario IV our study revealed that the upscal-ing in case of a parabolic velocity field is analytically as well asnumerically cumbersome, thus limiting the applicability of theanalytical upscaling approach. However, by using the effectivetransport parameters obtained for the first-order reaction and re-determining the upscaled reaction rate parameters through fittingthe numerical results, we could achieve acceptable results. Theseupscaled parameters, determined by fitting, now represent a goodtradeoff between accuracy and applicability.

Results of this work provide an effective upscaled reaction rateconsidering mass transfer limitations taking place at the scale of asingle pore. The use of a simplified representation of a pore al-lowed an analytical treatment and understanding of the physicalprocesses involved. By considering Monod-type reactions at thepore scale, the obtained effective equations comprise the restric-tions of substrate bioavailability caused by pore-scale diffusion.For such processes the obtained scaling behavior depends on thesubstrate concentration. This result is caused by the concentrationdependent transition between reaction-limited and diffusion-lim-ited regimes and is not observed for first-order reactions. The ap-proach presented in this study allows the determination ofconcentration independent scaling parameters, which provide glo-bal concentration estimates of an acceptable accuracy. The ob-tained relations between local and global reaction rateparameters can be transferred to larger scale models, e.g. by usingthem in pore network simulations. Future steps should include theexperimental validation of these theoretical results.

Acknowledgements

We would like to use the opportunity to acknowledge the con-tribution of Arne Nägel, Jan Friesen, Christoph Schneider, Anke Hil-debrandt, Rohini Kumar and one anonymous reviewer for thetechnical support, several proofreadings and important sugges-tions on this work.

Appendix A. Development of the upscaled solution

In this appendix we will provide the details for the upscaling ofthe pore-scale processes. This comprises (i) the analytical solutionas well as (ii) the effective differential equation. Eqs. (6a)–(6c)describing these processes can be found in the body of the paperand will not be listed again.

A.1. Separation of the variables

The scheme used to arrive at an analytical solution in case of afirst-order reaction rate can be found in [3]. It has been modified to


account for the mathematical and geometrical description used inthis study.

A.1.1. Transversal directionLet fkigiP1 be the set of eigenvalues and fWigiP1 the respective

set of orthonormal eigenfunctions of the following self-adjointeigenvalue problem

@2

@y2 WiðyÞ ¼ �k2i WiðyÞ; ð30aÞ

@

@yWijy¼0 ¼ 0; ð30bÞ

@

@yWijy¼1 ¼ �U2Wijy¼1: ð30cÞ

The boundary condition given by Eq. (30b) reflects the symmetry ofthe medium and Eq. (30c) is the reaction term in case of a first-orderreaction rate. The solution of Eq. (30a) is known to consist of thetrigonometric functions sine and cosine. Therefore we can write

WiðyÞ ¼ Ai cosðkiyÞ þ Bi sinðkiyÞ: ð31Þ

Here ki is the frequency and Ai and Bi are the respective amplitudes.These coefficients have to match the boundary conditions. First weuse the boundary condition given by Eq. (30b)

@

@yWijy¼0 ¼ 0 ¼ Bi cosð0Þ � Ai sinð0Þ ð32Þ

from which is clear that Bi ¼ 0. The boundary condition given by Eq.(30c) yields

@

@yWijy¼1 ¼ �U2 Ai cosðkiÞ ¼ �Aiki sinðkiÞ: ð33Þ

Rearranging this expression we get

ki tanðkiÞ ¼ U2: ð34Þ

The behavior of the left hand side of Eq. (34) is that of a strictlymonotonously increasing curve from �1 to þ1 within each inter-val i

2 p;i2þ 1� �

p� �

with ði ¼ 1;3;5; . . .Þ. The solutions ki are thendetermined by evaluate this expression within each of this intervals(Fig. 18).

For the calculation of the amplitudes Ai we refer to the ortho-normality of the eigenfunctions, i.e.Z 1

0WiWj dy ¼ di;j: ð35Þ

When we use this condition and insert Eq. (31), we get

Ai ¼Z 1

0cos2ðkiyÞ dy

� ��12

¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiki

sinð2kiÞ þ 2ki

s: ð36Þ

Now we can formulate the explicit solution for every transversalmode of Eq. (8) by introducing the values for Ai and Bi into Eq. (31)

Fig. 18. Solution of the eigenvalue problem described by Eq. (34) displayed as theinterfaces of the k tanðkÞ function (continuous line) and a constant reaction rate(dashed line, arbitrarily set to U2 ¼ 50 in this example).

WiðyÞ ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiki

sinð2kiÞ þ 2ki

scosðkiyÞ: ð37Þ

A.1.2. Longitudinal directionIn order to solve for the longitudinal part of every mode we in-

sert Eq. (8) into Eq. (6a)

Pef ðyÞX1i¼1

WiðyÞ@

@xCiðxÞ ¼

X1i¼1

CiðxÞ@2

@y2 WiðyÞ: ð38Þ

Multiplying this equation by Wj and integrating over y yields

PeX1i¼1

Z 1

0f ðyÞWiðyÞWjðyÞ dy

@

@xCiðxÞ

¼X1i¼1

CiðxÞZ 1

0

@2

@y2 WiðyÞWjðyÞ dy: ð39Þ

By definingZ 1

0f ðyÞWiWj dy ¼ sij ð40Þ

and inserting Eq. (30a), because of the ortho-normality of the Wi’s(35) we getX1i¼1

sij@

@xCiðxÞ ¼ �

k2j

PeCjðxÞ: ð41Þ

To get a good approximation of the complete solution a finite num-ber of modes will certainly be sufficient. By considering only Nmodes of the series we can rewrite Eq. (41) in a matrix notation

T@

@xC ¼ KC ð42Þ

with

T ¼s11 � � � s1N

..

. . .. ..

.

sN1 � � � sNN

0BB@1CCA; ð43Þ

K ¼� k2

1Pe 0

. ..

0 � k2N

Pe

0BBB@1CCCA ð44Þ

and

C ¼C1

..

.

CN

0BB@1CCA ð45Þ

for the unknown vector C, whose entries are the longitudinal modesof the solution c. Eq. (42) can be rearranged into

@

@xC ¼ T�1KC ¼ CC ð46Þ

In this form we have a simple system of homogeneous linear differ-ential equations of order one with constant coefficients.

A.2. Analysis of the case of a first-order reaction rate with a uniformvelocity field

In this part of the appendix we will give the details for the caseof a first-order reaction rate with a uniform velocity field. Thiscomprises the analytical solution as well as the effective equation(see Fig. 5).


A.2.1. Analytical solutionIn case of a uniform velocity field the matrix T is the identity

matrix according to Eq. (40). Therefore C is diagonal and Eq. (10)are decoupled. The governing ordinary differential equation forevery mode then reads

@

@xCiðxÞ ¼ �

k2i

PeCiðxÞ ð47Þ

The solution of this equation is given by

CiðxÞ ¼ Cið0Þe�k2i

Pex: ð48Þ

For the evaluation of the initial conditions Cið0Þ we have to refer toEq. (8) and insert the boundary condition given by Eq. (6b)

1 ¼ cð0; yÞ ¼XN

i¼1

Cið0ÞWiðyÞ ð49Þ

multiplying both sides with Wj and integrating over y yields

Cjð0Þ ¼Z 1

0cð0; yÞWjðyÞdy ð50Þ

because of the ortho-normality of the eigenfunctions WiðyÞ. Sincecð0; yÞ ¼ 1 we finally arrive at

Cjð0Þ ¼Z 1

0WjðyÞ dy ¼ Aj

sinðkjÞkj

: ð51Þ

With this equation we get the starting value for every transversalmode Cj of the solution.

A.2.2. Effective equationTo find the upscaled effective description of Eq. (6) we directly

apply the upscaling operatorR 1

0 dy to Eq. (6a) and insert the bound-ary conditions given by Eqs. (6b) and (6c). In case of the uniformvelocity field we get

Z 1

0Pe

@

@xc dy ¼

Z 1

0

@2

@y2 c dy

Pe@

@xC ¼ @

@ycbio �

@

@ycjy¼0

¼ �U2 cbio:

Here we introduce an effective Thiele modulus U2eff , as

U2eff ¼

cbio

CU2: ð52Þ

Since C is only zero at þ1 this equation is valid almost everywhere.It can be shown that U2

eff shows only variation for small values of xso it can be approximated as a constant.

Fig. 19. First five rows and columns of matrix T in graphical and numerical display, eval

A.3. Analysis of the case of a first-order reaction rate with a parabolicvelocity field

A.3.1. Analytical solutionThe matrix T is no longer a diagonal matrix in case of a parabolic

velocity field. As a result the entries of C in Eq. (10) are now cou-pled and must be solved in a closed form. Nevertheless, the matrixT is still symmetric, so sij ¼ sji (Fig. 19).

To find the solution of Eq. (10) we diagonalize the system ma-trix C. To that end we have to find a representation of the formC ¼ GDG�1. Here D is a diagonal matrix and G is the orthogonal ma-trix of the eigenvectors of C. Applying this transformation we canrewrite Eq. (10) into

c0 ¼ CC ¼ GDG�1C

G�1c0 ¼ DG�1Cw0 ¼ Dw

with w ¼ G�1C. In this form the modes are decoupled so we can fol-low the same proceedings as in case of a uniform velocity field.

A.3.2. Effective equationA direct analytical upscaling like in Appendix A.2 is not possible

in case of a parabolic velocity field. Applying the upscaling opera-tor

R 10 dy on Eq. (6a) we get

1:5@

@xC �

Z 1

0y2c dy

� �¼ �U2

eff

PeC: ð53Þ

Unfortunately no explicit solution is known for the remaining inte-gral in Eq. (53). Therefore we have to pursue an alternative proceed-ing to arrive at an effective equation. Using the linear system ofordinary differential Eq. (46) we can get an expression for the lead-ing first mode C1ðxÞ. In order to obtain the effective description wehave to rewrite the system of N differential equations of order 1 toan ordinary differential equation of order N. The general solution ofthis procedure isXN

n¼0

an@n

@xnC1 ¼ 0: ð54Þ

In Eq. (54) the numbers an are the respective coefficients of thecharacteristic polynomial of C in Eq. (46). To link the solution forthis mode with the upscaled solution C we have to multiply it withthe corresponding y-averaged transversal mode: hC1i ¼ C1

RW1 dy.

Since C ¼P

ihCii holds, we make an error by neglecting the highermodes of C. Nevertheless, these modes decrease very fast so the er-ror is confined to small values of x and even there it is comparablysmall. An important simplification can be made, regarding the num-ber of modes N, which has to be taken into account, to arrive at agood estimate for C1.

A numerical analysis shows that a good approximation is al-ready reached by using few modes (see Fig. 20). Only in the case

uated for U2 ¼ 10 in case of a first-order reaction rate and a parabolic velocity field.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

x

<C1>

12

34

5

(a) The y-averaged first mode C1 calculated usingdifferent number of modes.

1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

Number of Modes

squa

re s

um e

rror

(b) Square sum error for several modes.

Fig. 20. Error made by using only a limited number of longitudinal modes for the calculation of hC1i in case of a first-order reaction rate and a parabolic velocity field.


when one mode is considered, i.e. the first mode itself, we get a sig-nificant error. This error however decreases dramatically whenusing more modes, which justifies the use of only two. By restrict-ing therefore our analysis to N ¼ 2 we get a differential equation ofsecond order for C1.

A.4. Analysis of the case of a Monod-type reaction rate

For a Monod-type reaction rates given by

rc � n ¼ � U2cbio

1þ cbio=Kmð55Þ

the procedure presented in Appendix A.1 must to be modified.

A.4.1. Transversal directionThe coefficients of the transversal component in Eq. (8) now de-

pend on the concentration

@

@yWjy¼1 ¼ �

U2

1þ cbio=KmWjy¼1: ð56Þ

Applying the same procedure as for first-order reaction rates, theequation for the evaluation of the eigenvalues in Eq. (34) now reads

ki tanðkiÞ ¼U2

1þ cbio=Km: ð57Þ

Furthermore, the calculation of the transversal modes is modified to

Wiðx; yÞ ¼ AiðxÞ cosðkiðxÞyÞ: ð58Þ

Because of the nonlinearity of this problem the coefficients kiðxÞ hasto be found in an iterative scheme where the solution of each stepserves as a guess for their evaluation.

A.4.2. Longitudinal directionThe procedure to arrive at Eq. (46) has to be modified as well

when considering the case of Monod-type reaction rate. Now itreads

TðxÞ @@x

cðxÞ þ BðxÞcðxÞ ¼ KðxÞcðxÞ: ð59Þ

Here the entries of the matrix BðxÞ are given by

bijðxÞ ¼Z 1

0f ðyÞ @

@xWjðx; yÞ

� �Wiðx; yÞdy: ð60Þ

Rearranging yields

@

@xcðxÞ ¼ TðxÞ�1ðKðxÞ � BðxÞÞcðxÞ ¼ CðxÞcðxÞ; ð61Þ

so we get again a system of homogeneous linear differential equa-tions. In contrast to Eq. (46) the entries of the coefficient matrix CðxÞare not constants but x-dependent functions.

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Documents

Upscaling of the advection–diffusion–reaction equation with Monod reaction