A fully coupled numerical model for two-phase flow with contaminant transport and biodegradation...

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2001; 17:325–336

A fully coupled numerical model for two-phase ,owwith contaminant transport and biodegradation kinetics

Claudio Gallo1 and Gianmarco Manzini2;∗

1CRS4; Zona Industriale Macchiareddu; Uta; Cagliari; Italy2IAN - CNR; via Ferrata 1; 27100 Pavia; Italy

SUMMARY

A fully coupled numerical model is presented which describes biodegradation kinetics and NAPL-aqueous two-phase ,ow in porous media. The set of governing partial di6erential equations is split intwo subsystems, the former one in terms of phase pressure and saturations, and the latter one in termsof contaminant concentration and bacterial population distribution. Non-linear saturation dependence inBrooks–Corey relative permeability functions and capillary pressure e6ects are incorporated in a mixed-hybrid :nite element model. The non-linear degradation kinetics %a la Monod is taken into account asa source term in the :nite-volume discretization of the equation modelling contaminant transport. Theglobal coupling is performed by using a nested block-iteration technique. A set of numerical experimentsdemonstrates the e6ectiveness of the method. Sensitivity analysis results are also presented. Copyright? 2001 John Wiley & Sons, Ltd.

KEY WORDS: bioremediation; two-phase ,ow; mixed :nite elements; :nite volumes

1. INTRODUCTION

An organic contaminant leaking in groundwater is generally advected as a separate non-aqueous phase liquid (NAPL) one, but may also be partially dissolved in the aqueous phase,namely water. This dissolved part of NAPL is liable to be degraded by the bacterial pop-ulation naturally present in soil. A major simpli:cation in the mathematical and numericalmodelling of such a process accounts for the contaminant as a solute passively transportedin the aqueous phase [1–3], no taking care of the presence of di6erent phases. However, abetter understanding of the underlying phenomena needs an explicit description in terms ofa multi-phase ,ow model coupled with a transport equation for the contaminant dissolvedin water [4; 5]. The dissolved NAPL contaminant is degraded according with a Monod-typekinetics, see for instance Reference [6].

∗Correspondence to: G. Manzini, IAN - CNR, via Ferrata 1, 27100 Pavia, Italy

Contract=grant sponsor: Faculty of Civil Engineering and Geosciences in Delft, The NetherlandsContract=grant sponsor: Italian C.N.R.Contract=grant sponsor: Sardinian Regional Authorities

Received 6 March 2000Copyright ? 2001 John Wiley & Sons, Ltd. Accepted 4 December 2000

326 G. GALLO AND G. MANZINI

It is worth mentioning that transport of contaminants may often take place in rather de-formable porous media. Hence, the multi-phase ,ow model should also be completed byincluding soil displacement e6ects, as in References [7–9]. In this work we explore a nu-merical coupling of contaminant transport and biodegradation in the framework of a simpletwo-phase ,ow model and its preliminary application to 1-D sensitivity analysis. Extensionto more complex scenarios, which may deserve for special numerical treatments, will be thesubject of future work. The discretization relies on a block-iterative splitting algorithm, andseparates the mathematical model in two sets of equations, the former for phase pressures andsaturations and the latter for contaminant transport and biodegradation. Pressure=saturationequations are re-formulated in terms of total pressure and velocity :elds following the ap-proach of Chavent and Ja6rIe, see Reference [10]. The main reason for this approach is thateJcient numerical methods can be devised to take advantage of the physical properties in-herent in the ,ow equations. This set of equations is approximated by the discontinuouslowest-order mixed-hybrid Raviart–Thomas :nite element method [11]. The saturation equa-tions are then approximated by cell-centre :nite volumes and advanced in time by an explicittwo-stage Runge–Kutta scheme. In the second set of equations, a predictor–corrector strategydecouples the advective transport of the dissolved contaminant concentration from the sourcereaction term which describes the consumption by soil bacterial population. The di6erentialequation that modellizes growth and decay of bacteria is analytically solved at any iterativestep of the computation after updating the dissolved contaminant concentration :eld [3]. Theoutline of the paper is the following. In Section 2 we introduce the mathematical model,in Section 3 we focus on the iterative splitting approach and shortly discuss the numericaltechniques, and in Section 4 we present some preliminary 1-D numerical results.

2. MATHEMATICAL MODEL

The two-phase ,ow is governed by the couple of phase conservation equations

@@t

(��‘s‘) +∇ · F‘ = q‘ + q‘n (1)

@@t

(��nsn) +∇ · Fn = qn − q‘n (2)

where the subscripts ‘ and n refer to aqueous and non-aqueous phases, t is the time, � isthe porosity, �‘ and �n are the constant phase densities, s‘ and sn are the saturations, q‘ andqn are the mass source=sink terms and q‘n is a mass exchange term between the two phases.The advective ,uxes F‘ and Fn take the usual Darcy’s form

F‘ =−�‘‘K · ∇(P‘ + �‘gh)=�‘v‘ (3)

Fn =−�nnK · ∇(Pn + �ngh)=�nvn (4)

where P‘ and Pn are the pressure :elds, K is the absolute permeability tensor, ‘ and n are thephase mobilities, v‘ and vn are the phase velocities, g is the gravitational constant and h is the

Copyright ? 2001 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2001; 17:325–336

NUMERICAL MODEL FOR TWO-PHASE FLOW 327

head above a given datum. Phase mobilities are de:ned as

‘ =krel; ‘�‘

; n =krel;n�n

(5)

where krel; ‘ and krel;n are the soil relative permeabilities and �‘ and �n are the viscosities.Let us assume that densities and viscosities are constant (isothermal system assumption), and

that, as usual, saturations sum to unity. Following the ideas of Chavent [10], we can re-state(1)–(2) in terms of the aqueous phase pressure P‘ and the total velocity vT = v‘ + vn whichsatisfy

∇ · vT = QT

vT = −(‘ + n)K · ∇P‘ +QS(6)

The global source terms QS and QT are given by

QS =K · qS; QT =q‘

�‘+

qn

�n(7)

with

qS =−(‘�‘ + n�n)g− nP′c∇s‘ (8)

where g=∇(g h) is the gravity vector. Assuming water as the wetting ,uid, we introducedin (8) the capillary pressure de:ned as Pc =Pn − P‘.

Phase velocities are then given back by

v‘ =‘

‘ + n(vT −QS)− ‘�‘K · g (9)

vn = vT − v‘ (10)

The transport of the dissolved NAPL contaminant concentration C by the water bulk motionis described by

@(�s‘C)@t

+∇ · (v‘C)= q‘n − qbio (11)

The source term q‘n takes into account the mass transfer process from NAPL to aqueousphase. This term is linearly proportional to the di6erence between the maximum concentrationof NAPL that can be dissolved in water at equilibrium, Ceq, and its actual concentration, C,i.e.

q‘n =�snkdo(Ceq − C) (12)

The mass transfer proportionality coeJcient is denoted by kdo.The sink term qbio is responsible for biodegradation and takes the form

qbio =NcmcY�bio

[C

K1=2 + C

](13)



where �bio is the maximum degradation rate, K1=2 the half-saturation constant, Nc the numberof bacterial colonies per unit volume, mc the mass of a colony and Y a yield coeJcient [6].

Finally, the rate of microbial growth=decay is obtained by balancing bio-mass reproductionand decay

1Nc

@Nc

@t=

(�bio

[C

K1=2 + C

]− kd

)(14)

where the parameter kd is the bacterial decay constant [2; 6].Finally, some constitutive relations must be speci:ed to get closure of the model. Saturations

depend on Pc via the Brooks–Corey relation [12],

Pc =Pd ∗ s−1=‘e (15)

In Equation (15), Pd is the entry pressure, a :tting parameter related to the pore-sizedistribution, s‘e the e6ective liquid saturation, de:ned as s‘e = (s‘ − s‘r)=(1 − s‘r − snr), ands‘r and snr, respectively, the wetting- and non-wetting-phase residual saturations. The relativepermeabilities are given by

krel; ‘ =(s‘e)�; krel;n = (1− s‘e)� (16)

the value of the parameter � being :tted via regression on experimental data.

3. NUMERICAL MODEL

Basically, our discretization approach consists in an external loop in time, namely [Global],split in two internal loops, namely [P-S] and [C-bio]. The loop [P-S] updates the phasepressures P‘ and Pn and the saturations s‘ and sn, by solving iteratively system (6) andEquations (1) and (2). The loop [C-bio] updates the concentration of the NAPL contaminantdissolved in water and the microbial population distribution.

At any time step, the outer loop alternatively iterates on the two inner loops until conver-gence is reached:

while t¡tmax

loop [Global] until convergenceloop [P-S] until convergenceloop [C-bio] until convergencecheck global convergence

end loopt= t +Ot

end while



The loop [P-S] takes the form

loop [P-S] until convergenceestimate ∇St+Ot; k from St+Ot; k

solve for Pt+Ot; k+1‘ and vt+Ot; k+1

T [system (6)]estimate vt+Ot; k+1

‘ and vt+Ot; k+1n by using St+Ot; k [Equations (9)–(10)]

solve for St+Ot; k+1 [Equations (1)–(2)]check ‖St+Ot; k+1 − St+Ot; k‖6 �[P−S]

end loop

where S denotes the vector of unknowns (s‘; sn)T, k is the iteration index, �[P−S] is a user-speci:ed tolerance, and ‖ · ‖ is the Euclidean norm.

At each internal step, system (6) is solved by approximating the normal component of thetotal velocity :eld at cell interfaces, vT ·n say, by the discontinuous lowest-order mixed-hybridRaviart–Thomas :nite elements and the aqueous phase pressure P‘ by piecewise constant ele-ments over the computational cells and at cell interfaces [11]. Let us denote these approximate:elds by, respectively, qh, h and h. The symbol h denotes the maximum diameter of all thecells K in a given mesh Th; |K | is the cell volume and @K the cell boundary. The variationalformulation reads as

∫P

K−1

(‘ + n)qh · wh −

∑K

∫K h∇ · wh +

∑K

∫@K

h wh · nK =∫PQS · wh

∑K

∫Kvh∇ · qh =

∫PQTvh (17)

∑K

∫@K

qh · nK �h =0

where the test functions wh; vh and �h are taken in the same functional spaces of the corre-sponding unknowns. Standard algebraic manipulations yields a linear problem which is solvedfor the set of Lagrangian multipliers h by using the static condensation technique. Backwardsubstitution closes the solution algorithm [11]. This non-conforming :nite element methodguarantees that a zero-divergence constraint is well satis:ed by the total velocity :eld.This issue is required to fairly simulate the non-linear degradation kinetics, as pointed out inReference [5].

The semi-discrete :nite volume formulation that approximates Equations (1) and (2) isobtained by integrating separately in a cell-wise fashion each phase saturation equation, ap-plying the divergence theorem, approximating the interface integrals with the midpoint rule,and :nally computing the advective ,uxes by an upwind estimation of the phase velocitiesv‘ and vn. This method takes the form

|K |dSK

dt+

∑k∈�(K)

lkH (S intk ; Sext

k ; nk) +∑

k′∈�′(K)lk′H

(bc)k′ (S int

k ) = |K |Q(SK); ∀K ∈Th

(18)



where SK indicates the K th cell-averaged value for both s‘ and sn; �(K) is the set of cellsadjacent to K; �′(K) is the set of boundary edges on @K; H (S int

k ; Sextk ; nk) and H (bc)

k′ (S intk )

are the numerical ,uxes at internal and boundary edges, Q(SK) stands for the source termsin (1) and (2), and lk ; lk′ are the edge lengths. The numerical ,uxes depend on S int

k andSextk , which are the traces at each cell interface of the linearly reconstructed saturation. The

boundary ,ux H (bc)k′ (S int

k ) may also depend on a set of suitable external data. A speciallinear reconstruction procedure interpolates the cell-averaged saturation values {SK}K∈Th andensures a TVD stability condition by limiting the reconstructed slopes. This avoids order oneoscillations to appear in sharp gradient solution regions [5]. An explicit two-stage second-order Runge–Kutta scheme advances in time the approximate solutions SK . The method isformally second-order accurate [3].

A predictor–corrector strategy is deserved to treat the non-linear dependence on C in thesink source term qbio

loop [C-bio] until convergencecompute C̃t+Ot;m+1 [Equation (11) with qbio = 0]estimate q‘n by using C̃t+Ot;m+1 [Equation (13)]compute Ct+Ot;m+1 by using C̃t+Ot;m+1 and qbio [Equation (11)]compute Nt+Ot;m+1 [Equation (14)]check: ‖Ct+Ot;m+1 − Ct+Ot;m‖6 �[C−bio]

end loopwhere �[C–bio] is a user-de:ned tolerance. The :eld C̃t+Ot;m+1 is initially predicted by assumingqbio = 0 in (11). Then, the bacterial consumption source term is evaluated by using C̃t+Ot;m+1

in Equation (13). Finally, Equation (11) is re-iterated with the estimated value of qbio to cor-rect the dissolved contaminant concentration :eld at t +Ot. The new bacterial population iscalculated using this latter concentration value in Equation (14). The whole procedure is iter-ated until convergence. As for the phase saturations, the concentration equation is discretizedby using a second-order :nite volume scheme. Further details are given in Reference [3].

4. NUMERICAL EXPERIMENTS

4.1. Validation

The di6erent parts of the numerical simulator were separately validated on a set of suitablemodel problems. The validation of the contaminant transport solver and the biodegradationmodule are part of a previous work, see Reference [3], and will not be discussed herein.

The coupled two-phase ,ow solver in loop [P-S] has been extensively validated by com-paring numerical results with the exact solution of the classical 1-D Buckley–Leverett problemin a set of representative situations [13], an example of whose is given in Figure 1. This cal-culation was performed using 40 cells, and a good agreement is evident between the exactand the numerical solution. The combined rarefaction wave-shock discontinuity pattern is dueto the non-convexity of the physical ,ux function and is captured quite fairly. Discontinuitiesare approximated in at worst four cells and little dissipative smearings characterize rarefactionwaves.



Figure 1. Two-phase ,ow without bioremediation: (a) the initial solution of the aqueous saturation;(b) the exact solution (solid) and its numerical approximation (circles) at t=1 day.

4.2. 1-D column simulations

At t=0 a 10 m length homogeneous porous medium column is completely :lled in therange [0; 200 cm] by a NAPL contaminant with constant saturation sN =0:16. The waterphase saturation in the rest of the column is sl =1:0. The bacterial population is uniformlydistributed. At t¿0 the column is ,ushed by pure water and the NAPL contaminant begins todissolve and be degraded. The computational domain is discretized by using 250 1-D regularcells and a time-step Ot=1s. The value of the mass-transfer coeJcient kdo ranges throughout10−7 and 10−4 s−1, while the maximum degradation rate, �bio, ranges throughout 3:4× 10−8

and 3:4× 10−4 s−1. The values of the other parameters in the simulations are given in Table I.The column distributions of the water saturation s‘ at di6erent instants of the simulation

are shown with and without biodegradation in Figure 2. The condition sn = snr is earlier



Table I. Values of the parameters in 1-D column simulations.

� �n (kg=m−3) |K| (m2) kd (s−1) Ceq (g=l) K1=2 (g=l)

0:74 1:000 10−8 2:14× 10−7 0.02 0:02

Figure 2. Aqueous saturation at di6erent times (one curve every 0.5 days): (a) with and; (b) withoutbiodegradation. The arrow indicates the direction of increasing time.

veri:ed when biodegradation takes place, thus implying that NAPL trapping e6ects, namelyimmobilization, are enhanced.

4.2.1. Convergence of the non-linear iterative scheme. Table II illustrates the e6ectivenessof the split iterative scheme described in the previous section. We evaluate the performance



Table II. Performance of the non-linear iterative procedure.

kdo (s−1) NOtG NOt

PS NOtCbio NG

PS NGCbio

10−4 2.9 15.1 8.6 5.2 2.910−5 2.0 8.4 5.0 4.3 2.510−6 2.0 8.4 5.0 4.2 2.510−7 1.8 8.4 4.8 4.2 2.4

of the method by measuring the iterations required to get convergence in both internal loopsand in the external one.

From the left-most to the right-most column of the table, kdo, is the mass-transfer coeJcient;NOt

G ; NOtPS , and NOt

Cbio are the average number of iterations per time step required to achieveconvergence in, respectively, the outer loop [Global] and the inner loops [P-S] and [C-bio];NG

PS and NGCbio are the average number of iterations per global iteration to achieve convergence

in the inner loops.The performance table illustrates that convergence is achieved in [P-S] and [C-bio] in

a comparable number of iterations, even if the former loop demands for about three-halfthe number of iterations of the latter one. Decreasing the value of kdo below 10−4 s−1, thecomputational e6ort needed to solve the two inner loops does not change signi:cantly, andattains reasonable values, that is the average number of inner iterations per global one isalways between 2.5 and 5.5.

4.2.2. Sensitivity analysis. Cumulative e6ects are analyzed in the transient [0; T ] by consid-ering the mass fraction of the NAPL contaminant dissolved in water

M ∗diss(T )=

Mdiss

Mtot; t=0=

∫ T

0

∫L�snq‘n dx dt∫

L�sn|t=0�n dx

(19)

and the one degraded by bacteria

M ∗deg(T )=

Mdeg

Mtot; t=0=

∫ T

0

∫L�s‘qbio dx dt∫

L�sn|t=0�n dx

(20)

The two adimensional groups

V=(|K|OPmax)=(�lkdoL2)

R=(Nc;0mc�bio)=(�lkdo)(21)

have been recognized and used as independent parameters in our numerical sensitivity analysis.We stress that both parameters, as de:ned in (21), may be given a physical interpretation.

In fact, V is a sort of inverse ratio of the characteristic time of the ,ow to the characteristic



Figure 3. Sensitivity analysis: degraded mass curves, M∗deg(T ), and total dissolved mass curves, M∗

diss(T ),versus the adimensional group R for di6erent values of V .

time of the mass-transfer process, while R of the inverse ratio of the characteristic time ofdegradation kinetics to the characteristic time of the mass-transfer process.

In Equations (20) and (21) OPmax is the pressure gradient between the two ends of thecolumn, L is the column length, and Nc;0 is the average initial concentration of bacteria.Figure 3 shows the dependence on V and R of M ∗

diss and M ∗deg. Three distinct regimes can

be identi:ed from left to right: a mass-transfer limited one, a transition one, and a kinetics-limited one. For the lowest values of V , i.e. the left-most parametric curves in Figure 3,a little variation of the smallest value of R may produce signi:cant changes in the totaldissolved and degraded mass. Instead, the system behaviour is less dependent on the choiceof R for increasing values of V , that is the dissolved mass tends to stabilize to a constantvalue. It is informative to say that all the values of R were obtained by using the samevalue of �bio, except in the case of the minimum value of V . This latter situation is theone more strongly in,uenced by a variation in R. In facts, when small values of �bio occurs,



the contaminant concentration gradient, which is roughly proportional to (Ceq − C), is verysmall. Thus, equilibrium conditions are nearly attained, and we may expect that the NAPLcontaminant be very poorly degraded. Since in this case the dissolution process is likely tobe almost depressed, only incoming pure water can bu6er the dissolved NAPL contaminantand only a variation in the ,ow condition can modify the dissolution rate.

5. CONCLUSION

A numerical model was investigated which couples phase pressure and saturations in two-phase ,ow and NAPL concentration transport and biodegradation kinetics. A mixed-hybrid:nite element model incorporates the non-linear saturation dependence in Brooks–Corey rel-ative permeabilities and capillary pressure e6ects. A :nite volume method takes into accountthe contaminant concentration transport which also incorporates the non-linear degradationkinetics as a suitable source term. One-dimensional numerical simulations were performed fora homogeneous porous medium and illustrate the e6ectiveness of the approach in predictingsigni:cant e6ects in the behaviour of the contaminant distribution. The approach addressedin the paper has satisfactorily been tested against a set of relevant situations in 1-D column,ow simulations.

ACKNOWLEDGEMENTS

Part of this work was performed by the :rst author (C. G.) at the Faculty of Civil Engineering andGeosciences in Delft, The Netherlands, whose hospitality and :nancial support are gratefully acknowl-edged.

The work was also supported by the Italian C.N.R. and the Sardinian Regional Authorities.The authors appreciated the constructive criticism of the anonymous reviewers, whose many sugges-

tions were helpful in improving the paper.

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