COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING
Commun. Numer. Meth. Engng
2001;
17
:325–336
A fully coupled numerical model for twophase owwith contaminant transport and biodegradation kinetics
Claudio Gallo
1
and Gianmarco Manzini
2
;
∗
1
CRS4
;
Zona Industriale Macchiareddu
;
Uta
;
Cagliari
;
Italy
2
IAN  CNR
;
via Ferrata 1
;
27100 Pavia
;
Italy
SUMMARYA fully coupled numerical model is presented which describes biodegradation kinetics and NAPLaqueous twophase ow in porous media. The set of governing partial dierential 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. Nonlinear saturation dependence inBrooks–Corey relative permeability functions and capillary pressure eects are incorporated in a mixedhybrid nite element model. The nonlinear degradation kinetics
a la
Monod is taken into account asa source term in the nitevolume discretization of the equation modelling contaminant transport. Theglobal coupling is performed by using a nested blockiteration technique. A set of numerical experimentsdemonstrates the eectiveness of the method. Sensitivity analysis results are also presented. Copyright
?
2001 John Wiley & Sons, Ltd.
KEY WORDS
: bioremediation; twophase ow; mixed nite elements; nite volumes
1. INTRODUCTIONAn organic contaminant leaking in groundwater is generally advected as a separate
nonaqueous 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 population naturally present in soil. A major simplication 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 dierent phases. However, a better understanding of the underlying phenomena needs an explicit description in terms of a multiphase ow model coupled with a transport equation for the contaminant dissolvedin water [4
;
5]. The dissolved NAPL contaminant is degraded according with a Monodtypekinetics, see for instance Reference [6].
∗
Correspondence to: G. Manzini, IAN  CNR, via Ferrata 1, 27100 Pavia, ItalyContract
=
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 2000
Copyright
?
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 deformable porous media. Hence, the multiphase ow model should also be completed byincluding soil displacement eects, as in References [7–9]. In this work we explore a numerical coupling of contaminant transport and biodegradation in the framework of a simpletwophase ow model and its preliminary application to 1D sensitivity analysis. Extensionto more complex scenarios, which may deserve for special numerical treatments, will be thesubject of future work. The discretization relies on a blockiterative 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 reformulated in terms of total pressure and velocity elds following the ap proach of Chavent and Jare, see Reference [10]. The main reason for this approach is thatecient numerical methods can be devised to take advantage of the physical properties inherent in the ow equations. This set of equations is approximated by the discontinuouslowestorder mixedhybrid Raviart–Thomas nite element method [11]. The saturation equations are then approximated by cellcentre nite volumes and advanced in time by an explicittwostage 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 dierentialequation 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 1D numerical results.2. MATHEMATICAL MODELThe twophase ow is governed by the couple of phase conservation equations
@@t
(
‘
s
‘
) +
∇ ·
F
‘
=
q
‘
+
q
‘
n
(1)
@@t
(
n
s
n
) +
∇ ·
F
n
=
q
n
−
q
‘
n
(2)where the subscripts
‘
and n refer to aqueous and nonaqueous phases,
t
is the time,
isthe porosity,
‘
and
n
are the constant phase densities,
s
‘
and
s
n
are the saturations,
q
‘
and
q
n
are the mass source
=
sink terms and
q
‘
n
is a mass exchange term between the two phases.The advective uxes
F
‘
and
F
n
take the usual Darcy’s form
F
‘
=
−
‘
‘
K
· ∇
(
P
‘
+
‘
gh
)=
‘
v
‘
(3)
F
n
=
−
n
n
K
· ∇
(
P
n
+
n
gh
)=
n
v
n
(4)where
P
‘
and
P
n
are the pressure elds,
K
is the absolute permeability tensor,
‘
and
n
are the phase mobilities,
v
‘
and
v
n
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 TWOPHASE FLOW
327head above a given datum. Phase mobilities are dened as
‘
=
k
rel
;‘
‘
;
n
=
k
rel
;
n
n
(5)where
k
rel
;‘
and
k
rel
;
n
are the soil relative permeabilities and
‘
and
n
are the viscosities.Let us assume that densities and viscosities are constant (isothermal system assumption), andthat, as usual, saturations sum to unity. Following the ideas of Chavent [10], we can restate(1)–(2) in terms of the aqueous phase pressure
P
‘
and the total velocity
v
T
=
v
‘
+
v
n
whichsatisfy
∇ ·
v
T
=
Q
T
v
T
=
−
(
‘
+
n
)
K
· ∇
P
‘
+
Q
S
(6)The global source terms
Q
S
and
Q
T
are given by
Q
S
=
K
·
q
S
; Q
T
=
q
‘
‘
+
q
n
n
(7)with
q
S
=
−
(
‘
‘
+
n
n
)
g
−
n
P
c
∇
s
‘
(8)where
g
=
∇
(
gh
) is the gravity vector. Assuming water as the wetting uid, we introducedin (8) the capillary pressure dened as
P
c
=
P
n
−
P
‘
.Phase velocities are then given back by
v
‘
=
‘
‘
+
n
(
v
T
−
Q
S
)
−
‘
‘
K
·
g
(9)
v
n
=
v
T
−
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
−
q
bio
(11)The source term
q
‘
n
takes into account the mass transfer process from NAPL to aqueous phase. This term is linearly proportional to the dierence between the maximum concentrationof NAPL that can be dissolved in water at equilibrium,
C
eq
, and its actual concentration,
C
,i.e.
q
‘
n
=
s
n
k
do
(
C
eq
−
C
) (12)The mass transfer proportionality coecient is denoted by
k
do
.The sink term
q
bio
is responsible for biodegradation and takes the form
q
bio
=
N
c
m
c
Y
bio
C K
1
=
2
+
C
(13)
Copyright
?
2001 John Wiley & Sons, Ltd.
Commun. Numer. Meth. Engng
2001;
17
:325–336
328
G. GALLO AND G. MANZINI
where
bio
is the maximum degradation rate,
K
1
=
2
the halfsaturation constant,
N
c
the number of bacterial colonies per unit volume,
m
c
the mass of a colony and
Y
a yield coecient [6].Finally, the rate of microbial growth
=
decay is obtained by balancing biomass reproductionand decay1
N
c
@N
c
@t
=
bio
C K
1
=
2
+
C
−
k
d
(14)where the parameter
k
d
is the bacterial decay constant [2
;
6].Finally, some constitutive relations must be specied to get closure of the model. Saturationsdepend on
P
c
via the Brooks–Corey relation [12],
P
c
=
P
d
∗
s
−
1
=‘
e
(15)In Equation (15),
P
d
is the entry pressure,
a tting parameter related to the poresizedistribution,
s
‘
e
the eective liquid saturation, dened as
s
‘
e
=(
s
‘
−
s
‘
r
)
=
(1
−
s
‘
r
−
s
nr
), and
s
‘
r
and
s
nr
, respectively, the wetting and nonwettingphase residual saturations. The relative permeabilities are given by
k
rel
;‘
=(
s
‘
e
)
; k
rel
;
n
=(1
−
s
‘
e
)
(16)the value of the parameter
being tted via regression on experimental data.3. NUMERICAL MODELBasically, our discretization approach consists in an external loop in time, namely
[Global]
,split in two internal loops, namely
[PS]
and
[Cbio]
. The loop
[PS]
updates the phase pressures
P
‘
and
P
n
and the saturations
s
‘
and
s
n
, by solving iteratively system (6) andEquations (1) and (2). The loop
[Cbio]
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 convergence is reached:
while
t¡t
max
loop
[Global]
until
convergence
loop
[PS]
until
convergence
loop
[Cbio]
until
convergence
check
global convergence
end loop
t
=
t
+
t
end while
Copyright
?
2001 John Wiley & Sons, Ltd.
Commun. Numer. Meth. Engng
2001;
17
:325–336
NUMERICAL MODEL FOR TWOPHASE FLOW
329The loop
[PS]
takes the form
loop
[PS]
until
convergence
estimate
∇
S
t
+
t;k
from
S
t
+
t;k
solve for
P
t
+
t;k
+1
‘
and v
t
+
t;k
+1T
[system (6)]
estimate v
t
+
t;k
+1
‘
and v
t
+
t;k
+1n
by using
S
t
+
t;k
[Equations (9)–(10)]
solve for
S
t
+
t;k
+1
[Equations (1)–(2)]
check
S
t
+
t;k
+1
−
S
t
+
t;k
6
[P
−
S]
end loop
where
S
denotes the vector of unknowns (
s
‘
; s
n
)
T
,
k
is the iteration index,
[P
−
S]
is a userspecied 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,
v
T
·
n
say, by the discontinuous lowestorder mixedhybridRaviart–Thomas nite elements and the aqueous phase pressure
P
‘
by piecewise constant elements over the computational cells and at cell interfaces [11]. Let us denote these approximateelds by, respectively,
q
h
,
h
and
h
. The symbol
h
denotes the maximum diameter of all thecells
K
in a given mesh
T
h
;

K

is the cell volume and
@K
the cell boundary. The variationalformulation reads as
K
−
1
(
‘
+
n
)
q
h
·
w
h
−
K
K
h
∇ ·
w
h
+
K
@K
h
w
h
·
n
K
=
Q
S
·
w
h
K
K
v
h
∇ ·
q
h
=
Q
T
v
h
(17)
K
@K
q
h
·
n
K
h
=0where the test functions
w
h
; v
h
and
h
are taken in the same functional spaces of the corresponding 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 nonconforming nite element methodguarantees that a zerodivergence constraint is well satised by the total velocity eld.This issue is required to fairly simulate the nonlinear degradation kinetics, as pointed out inReference [5].The semidiscrete nite volume formulation that approximates Equations (1) and (2) isobtained by integrating separately in a cellwise 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 velocities
v
‘
and
v
n
. This method takes the form

K

d
S
K
d
t
+
k
∈
(
K
)
l
k
H
(
S
int
k
;S
ext
k
;
n
k
) +
k
∈
(
K
)
l
k
H
(bc)
k
(
S
int
k
) =

K

Q
(
S
K
)
;
∀
K
∈
T
h
(18)
Copyright
?
2001 John Wiley & Sons, Ltd.
Commun. Numer. Meth. Engng
2001;
17
:325–336