PEGASE Project
Developments in MARTHE code
Dominique THIERY
Jun 2001
1
1 Langmuir and Freundlich sorption
1.1
Implementation in the MARTHE code
Sorption with Langmuir and Freundlich isotherms has been implemented in MARTHE (Thiéry 1990,
1993, 1994, 1995).
a) Langmuir isotherm
The classical Langmuir sorption isotherm notation for the relation between solid phase
concentration SM and liquid phase concentration C is :
SM  Q 
KL  C
1 KL  C
With : SM = Solid phase mass concentration [kg/kg]
C = Liquid phase volumetric concentration [kg/ m3]
Q = Total sites for solid phase concentration [kg/kg]
KL = Langmuir constant [m3/kg]
As a matter of fact it is more physical and more practical to write this relation in term of solid
phase volumetric concentration SV (mass of product per terrain volume). SV = b . SM . It has
then the same dimension than the liquid phase concentration C (and is expressed in the same
unit).
b = Dry soil density [kg/m3] (about 1600 kg/m3 for a sol classical soil).
With these variables, Langmuir isotherm writes :
SV   b  Q 
KL  C
1 KL  C
or simpler, in dimensional form, as used in MARTHE :
S V  S VMAX 
C / C REF
1  C / C REF
With :

CREF = 1/KL = Reference concentration [kg/m3]. It is the liquid phase concentration above
which the effect of the solid phase is halved.

SVMAX = b . Q = Maximal solid phase volumetric concentration [kg/m3]. It is the solid
phase volumetric concentration, which corresponds to an infinite liquid phase
concentration.
b) Freundlich isotherm
The classical Freundlich sorption isotherm notation for the relation between solid phase concentration
SM and liquid phase concentration C is :
SM = KF . CB
2
With : B = Exponent (generally < 1)
KF = Freundlich constant in unit [m3/kg]B
It should be noted that KF unit is generally complex because it depends on exponent B.
Using solid phase volumetric concentration SV, Freundlich isotherm writes:
SV = KFV . CB
With : KFV = b . KF = Volumetric Freundlich constant in [kg/m3](1-B)
In MARTHE, Freundlich isotherm is described by 2 parameters:

KFV = b . KF = Volumetric Freundlich constant in [concentration unit] (1-B).

A2 = B (Dimensionless exponent)
Remarks :

when B=1 this isotherm becomes linear with KFV identical to a volumetric Rho_kd
(dimensionless)

Setting K FV  C REF 
C = CREF

Except when B=1, when C = 0 the ratio SV / C -> infinite, which corresponds to a
retardation factor equal to infinity.
1.2
1B
It appears that SV = C (which corresponds to Rho_kd = 1) when
Verification of the implementation
a) Description of the tests problems
Two sets of 1D simulations have been realized which are described below :

A set of simulations described by Zheng and Wang (1998) referring to a problem simulated by
Grove and Stollenwerk (1984).

A set of simulations described by Huang et al. (1998) referring to benchmarks problems of Goode
and Konikow (1989).
Both cases refer to a constant flow through a saturated column of soil. The common physical
parameters for all cases are :

Porosity
= 0.37

Seepage velocity
= 0.1 cm/s (10-3 m/s)

Bulk density b
= 1587 kg/m3

Concentration of source fluid Co
= 50 10-6 kg/m3

Distance of breakthrough results
= 8 cm
The differences between the 2 sets are :
3
Zheng & Wang
Huang et al.
0
0 or 10-2 or 10-3 s-1
Dispersivity
1 cm
0.1 cm
Duration of the
pulse
160 s
80 s
Decay
The interactions are as follows :
Langmuir interactions :
Zheng & Wang :
SM  Q 
KL  C
1 KL  C

C = volumetric concentration in the liquid phase

SM = mass concentration in the solid phase

Q =3 10-3 g/g
= 3 10-9 kg/kg

KL = 100 l/mg
= 105 m3/kg
The corresponding MARTHE parameters are :
S V  S VMAX 
C / C REF
1  C / C REF

SV = b . SM

CREF = 1/KL = Reference concentration [kg/m3]. It is the liquid concentration above which
the sorption of the solid phase is divided by 2.
CREF =1/KL = 10-5 kg/m3

= 10
[10-6 kg/m3]
SVMAX = b . Q = Maximal volumetric concentration in solid phase. It is the volumetric
concentration in the solid phase [kg/m3] that corresponds to an infinite liquid phase
concentration.
SVMAX = b . Q
= 4.761 10-6 kg/m3
= 4.761 [10-6 kg/m3]
Huang et al. (1998) :

Q =3 10-3 g/g
= 3 10-3 kg/kg

KL = 100 cm3/g
= 10-1 m3/kg
These parameters would correspond to MARTHE parameters :

C REF =1/KL = 10 kg/m3
= 107

SVMAX = b . Q = 4.761 kg/m3
= 4.761 106 [10-6 kg/m3]
[10-6 kg/m3]
However, probably due to some misspelling, these parameters turn out to be wrong (the
reference concentration being quasi infinite, the interactions should be linear which is not the
case). As a matter of fact the true parameters corresponding to the simulations by Huang et al.
(1998) are the same as those used by Zheng & Wang, namely :
4

CREF =1/KL = 10-5 kg/m3
= 10

SVMAX = b . Q
= 4.761 [10-6 kg/m3]
= 4.761 10-6 kg/m3
[10-6 kg/m3]
Freundlich interactions :
Zheng & Wang :
SM = KF . CB

SM = mass concentration in the solid phase

B = 0.7

K F = 0.3 g/g . (l/mg)B
= 0.3 10-6 . (1000 m3/kg)0.7
= 3.7768 10-5 (kg/m3)0.7
The corresponding MARTHE parameters are :
SV = KFV . CB

SV = b . SM

B = Freundlich exponent = 0.7

K FV = b . KF
= Volumetric Freundlich constant expressed in [kg/m3](1-B)
=1587 kg/m3 . 3.7768 10-5 (m3/kg)0.7
=5.9937 10-2 (kg/m3)0.3
As the concentrations are expressed in : 10-6 kg/m3
= 5.9937 10-2 /[(10-6)0.3] . [10-6 kg/m3]0.3
= 3.7818 . [10-6 kg/m3]0.3
= 3.7818 in the model
Huang et al. (1998) :

B = 0.7

K F = 0.3 cm3/g However, due to some misspelling, these units are incorrect and should
take into account the exponent B. The true value of KF is :

K F = 0.3 g/g . (l/mg)B
= 0.3 10-6 . (1000 m3/kg)0.7
= 3.7768 10-5 (kg/m3)0.7
This leads to the same parameters as those used by Zheng & Wang
b) Results obtained with MARTHE
Eleven (11) simulations have been realized with the 3D MARTHE code in order to verify the
implementations of the non-linear sorption schemes. The simulations have been realized with or
without decay, using all of the transport schemes available in MARTHE : the Finite differences
method (F.D.), the flux limiter method with Total Variation Diminishing (TVD), and the mass Method
Of Characteristics (MOC). The description of the different simulations appears in table 1.
For comparison purpose, the figures from the simulations by Zheng and Wang (1998), and those by
Huang et al. (1998), which are believed to be very accurate, have been scanned in order to be used as
5
reference. The times are in seconds for the simulations by Zheng and Wang (1998); the times are in
pore volumes for the simulations by Huang et al. (1998). As the pore velocity is 0.1 cm/s and the
distance is 8 cm, 1 pore volume corresponds to 80 seconds.
Figures 1 to 4 compare MARTHE simulations with the references simulations. These figures show
that :

The simulations are very accurate for every transport scheme (F.D., TVD and MOC).

The TVD scheme is optimal and enables to use larger grids than the F.D. scheme.

The MOC scheme is correct for Langmuir sorption, however for Freundlich sorption it leads to
some oscillations for very low concentrations before the breakthrough. This is due to the fact that,
with a Freundlich sorption using an exponent lower than 1, the equivalent retardation coefficient is
infinite for a concentration equal to zero.

The decay factors are taken into account accurately both with Langmuir and Freundlich isotherms.

It has been verified that the mass balance with MARTHE numerical scheme is satisfactory (within
much less than 1%).
1.3
Simulation in Unsaturated Zone
In order to show the implementation of Langmuir or Freundlich sorption isotherms in Unsaturated
Zone flows, a test adapted from Vredepeel site, Netherlands, see Boesten and Van Der Pas (2000), has
been run. This test concerns the 1D simulation of the leaching of Bentazone in a soil 2 meters thick
overlying an aquifer, as described by Golaz (2001). A mass of 8 10-5 kg/m2 (0.8 kg/ha) of Bentazone is
laid (on day 0 = 22/Nov/1990) at the surface of the soil which is exposed to rainfall and evaporation.
As an illustration a Langmuir isotherm has been assumed with the following values :

CREF
= 9 . 10-5 kg/m3

SVMAX
= 8 . 10-5 kg/m3
The other parameters are :
First 30 cm
Lower than 30 cm
Dispersivity
0.02 m
0.10 m
Degradation
100 days
100 days
Half Life
Fig. 5 compares the concentrations profile after 103 days (i.e. on 05/Mar/1991), with Langmuir
sorption and without sorption. It appears that the sorption has a large effect because the concentrations
are nearly halved and the lowest concentrations are more sorbed than highest due to the non-linear
effect of the isotherm.
1.4
2D Simulations
A reference test adapted from the "Rocky Mountain Arsenal" test described by Voss (1984) has been
modeled with MARTHE in steady state (transport is also in steady state). The description of this test is
as follows (see fig. 6) :

Study area approximately 16 km X 20 km.

Flow approximately from North to South.

Injection of water at a concentration of 1000 units (ppm) in a lake situated in the northern part of
the domain.
6

Some recovery wells are situated in the southern part of the domain.
This test, which integrates a decay process, has been modeled without sorption and with a Langmuir
sorption isotherm.
MARTHE parameters controlling the transport are :

Decay Half Life
= 20 years

Grid size
= 1 km

Dispersivity :
Longitudinal
= 500 m
Transversal
= 100 m

Porosity
= 20 %

CREF
= Langmuir Reference Concentration

SVMAX
= Maximal volumetric concentration in solid phase = 120 concentration units
= 100 concentration units
The steady state concentration plume without and with Langmuir sorption are displayed in Fig 6. This
figure shows clearly that in presence of sorption the concentrations are lower and less spread. This is
due to the retardation effect of the sorption, which enables the decay to take place longer. The aim of
this test was to verify the implementation in 2D or 3D, and also to demonstrate the capability of
MARTHE in steady state.
The same simulation in transient state (not displayed here) during 60 years shows that the break
through in the recover wells is slower and more progressive in presence of Langmuir sorption.
2 Sequential First-Order Decay Reactions
This functionality takes into account the follow up of the degraded products when decay takes place.
Product 1 is degraded into product 2, which in turn is degraded into product 3 etc. Each product has its
own decay factor and its own retardation coefficient or kd factor. In MARTHE the number of products
is not limited. It should noted that, in the present release, 1 product is degraded into not more than 1
"son" product (hence the name of "chain" degradation).
In MARTHE, the chain degradation is available when transport is computed by Finite Differences
(F.D.) or by TVD. With MOC transport scheme, the chain degradation is only available when the
retardation coefficient or the kd factor is identical for all products. This is due to the fact that in
MARTHE the particles transport schemes use a single kind of particles containing all the chemical
components. The coupling method is sequential iterative, hence a transport in steady state can not be
calculated directly but should be calculated as a transient state during a long period.
The code has been verified with a 1D analytical solution from Cho (1971) described and used by Van
Genuchten (1984). A simulation in 2D has also been realized in order to verify the implementation of
the code in 2D or 3D.
2.1
1D Simulations
The test used is the test described by Van Genuchten (1984). It applies to the three-species nitrification
chain :
NH4+  NO2-  NO3-
7
It is a horizontal column receiving through its left limit a solution of NH4+ that degrades into NO2-.
Simultaneously NO2- degrades into NO3-. The initial concentrations are all equal to 0. The parameters
of the test are :

Pore velocity
= 1 cm/h

Dispersion coefficient
= 5 . 10-9 m2/s

Retardation factors for each product = 2 for NH4+ ; 1 for NO2- and NO3-

Decay factors for each product

(Decay Half Life for each product = 138.63 h for NH4+ ; 10 h for NO2- and infinite NO3-)
= 0.005 h-1 for NH4+ ; 0.1 h-1 for NO2- and 0 for NO3-
Three simulations have been realized :
Method
Grid
Time Step
Peclet
(cm)
(h)
Number
Simulation 1
F.D.
0.4
0.4
2.2
Simulation 1
TVD
0.4
4
2.2
Simulation 1
TVD
2
4
11.1
Figures 7 and 8 compare MARTHE simulations with the analytical calculations resulting from
CHAIN code from Van Genuchten (1984).
Figure 7 shows that the simulation is very accurate with a fine grid. The simulation with TVD scheme
is so close that the 2 curves are superimposed. Figure 8 with a much larger grid shows a very accurate
simulation.

2.2
It has been verified that the mass balance with MARTHE numerical scheme is very satisfactory
(within much less than 1/100 %).
2D Simulation
The same reference test adapted from the "Rocky Mountain Arsenal" test described by Voss (1984)
has been modeled with MARTHE in transient state. The parameters are the same as those used for the
"Rocky Mountain test" with Langmuir sorption except for the following :

Transfer of 3 products with first-order chain reactions

Simulation in transient state

Retardation factors for each product = 2 for product 1 ; 1 for products 2 and 3

Decay Half Life for each product
product 3
= 20 years for product 1 ; 1 year for products 2 and Infinite for
Figure 9 shows the effects of the chain degradation : product 2 that degrades quickly (1 year) has a
much lower concentration than the others have.
8
3 Degradation function of Saturation and Temperature
The decay factor (or degradation factor) may depend on temperature and/or on water content
(saturation). This functionality is also available when the decay factor has spatial variations as the
influence of the temperature and/or of the water content appears as correction factors applying to the
model values. Following a review by Golaz (2001) of the formulations available in the literature five
(5) corrections formulations have been included in MARTHE for the temperature effect and six (6) for
the water content effect :
3.1
1 LEACHP
2 AGRIFLUX (aerobic)
3 MACRO
4 WAVE
5 PELMO
6 AGRIFLUX (anaerobic)
Temperature effects
For each formulation the notations are as follows :
T
= Water Temperature in °C.
TREF
= Reference Temperature (MARTHE parameter) in °C.
TK, TKREF
= °K equivalent of T and TREF.
A1
= Correction Parameter

LEACHP Formulation :
F(T) = A1(T - TREF) / 10
PELMO formulation is identical
The reference values for A1 are on the order of 2 to 6 for Ethoprophos and Bentazone.

AGRIFLUX Formulation
F(T) = Exp[(1/TKREF - 1/TK) . A1 / 8.31]
A1 is the activation energy (vaporization heat) in J/mol.


MACRO Formulation

T > 5 °C

0 < T < 5 °K

T < 0 °C
=> F(T) = Exp[ A1 . (T - TREF) ]
=> F(T) = (T/5) .Exp[ A1 . (5 - TREF) ]
=> F(T) = 0
WAVE Formulation
F(T) = A1 . (T - TREF) / 10.
(We add : F(T) = 0 when T < TREF)
9

PELMO Formulation
PELMO formulation is identical to LEACHP formulation.
3.2
Saturation effects
For each formulation the notations are as follows :

R
S
= Water content
= Residual Water content
= Water content at Saturation
A1, A2, A3 = parameters. As a matter of fact each formulation depends of only one single parameter
except for LEACHP formulation which depends on 3 parameters.

LEACHP Formulation :
T_min = max (A1 , R)
T_max = min (A2 , S)
F_sat
= A3
F() = (R) / (T_min - R)
if < T_min
F() = F_sat + [(1 - F_sat) .(S)] / (S- T_max …
if > T_max
F() = 1
if T_min < < T_max
This formulation depends on 3 parameters.
Reference values for F_sat parameter (which is < 1) are in the range 0.6 to 1
N.B. When this formulation is used with MARTHE code, parameters A1 and A2 which correspond to
water contents are in water content unit (i.e. in % if the water content is expressed in %).

Aerobic AGRIFLUX Formulation
T_cap = min(A1 , S) = Field capacity.
F() =  / T_cap
if  < T_cap
F() = T_cap / 
if  > T_cap
Degradation is maximal at field capacity. It is reduced at both side of this value.
N.B. When this formulation is used with MARTHE code, parameter A1 which corresponds to water
contents is in water content unit (i.e. in % if the water content is expressed in %).

MACRO Formulation
F() = ( / S)A1
As a matter of fact in MACRO formulation S refers to the porosity, excluding the possible
macroporosity.

10
WAVE Formulation
WAVE formulation is quasi-identical to MACRO formulation
F() = ( / REF)A1 . We assume : REF = S
Which leads to : F() = ( / S)A1 as in MACRO formulation.

PELMO Formulation
PELMO formulation is identical to WAVE formulation.

Anaerobic AGRIFLUX Formulation
T_cap = min(A1 , S) = Field capacity
F() = 0
if  < T_cap
F() = ( - T_cap) / (S- T_cap)
if  > T_cap
Microbial activity takes place only when the water content is close to saturation (i.e. above
T_cap)
N.B. When this formulation is used with MARTHE code, parameter A1 which corresponds to water
contents is in water content unit (i.e. in % if the water content is expressed in %).
3.3
1D simulations in Unsaturated Zone
In order to check the implementation of the degradation as a function of temperature and saturation, a
test adapted from Vredepeel site (Netherlands) described by Boesten and Van Der Pas (2000), has
been run. This test concerns the 1D simulation of the leaching of Bentazone in a soil 2 meters thick
overlying an aquifer, as described by Golaz (2001). A mass of 8 10-5 kg/m2 (0.8 kg/ha) of Bentazone is
laid (on day 0 = 22/Nov./1990) at the surface of the soil which is exposed to rainfall and evaporation.
Four (4) simulation have been performed :

A reference simulation with a constant decay half life equal to 100 days.

A simulation with the influence of the temperature

A simulation with the influence of the saturation

A simulation with both influence : temperature and saturation
a) Reference simulation
The parameters for this simulation are :
First 30 cm
Lower than 30 cm
Dispersivity
0.02 m
0.10 m
Rho*Kd (%)
11
0.55
Degradation
100 days
100 days
Half Life
The transport scheme is TVD.
The simulation period extends from 22/Nov/1990 (day 0) to 10/Mar/1992 (day 474).
11
b) Simulation with temperature effects on the degradation
The air temperature has been monitored during the period November 1990 to March 1992. The initial
temperature in the 2 meters soil layer has been set to the "average" air temperature that is equal to
8.33 °C. At each model time step (equal to 1 day), the monitored air temperature has been prescribed
in the first cell of the model, at a depth of 3.75 cm. During the whole simulation, the temperature at
bottom of the layer has been kept equal to the average temperature (8.33 °C).
The thermal parameters used are :

Water specific heat = 4185 J/kg/°K

Mineral (solid matrix) volumetric specific heat = 2 106 J/m3/°K

Water thermal conductivity = 0.6 W/m/°K

Mineral thermal conductivity = 1.5 W/m/°K
The bulk specific heat used in the code is an average weighted by water and matrix volumes.
Figure 10 displays the computed temperature during the simulated period (474 days) at 4 depths :
3.75 cm (prescribed), 41.25 cm, 101 cm and 191.25 cm. That figure shows that, with the chosen
hypothesizes, the temperature variations are important even at the base of the modeled layer. At middepth, the temperature is in the range 2 °C - 17°C.
The degradation versus temperature formulation has been chosen as a LEACHP scheme with a
parameter (A1) equal to 2. The reference temperature (TREF) has been chosen as 20°C.
Figure 11 compares the profile of concentrations on day 103 (i.e. on 05/Mar/1991) with and without
thermal effects on degradation.
c) Simulation with saturation effects on the degradation
This simulation is identical to the reference simulation but the saturation effect formulation has been
chosen as a LEACHP scheme with :
T_min = 3 %
T_max = 8 %
F_sat
= 0.6
Figure 11 compares the profile of concentrations on day 103 (i.e. on 05/Mar/1991) with and without
saturation effects on degradation.
d) Simulation with both temperature and saturation effects on the degradation
This simulation takes into account both thermal and saturation effects with the same formulations and
parameters than above. See figure 11.
From these simulations it appears that, with the used parameters, the thermal effect is large (maximal
concentration 0.30 mg/l instead of 0.20 mg/l), the saturation effect being smaller (maximal
concentration 0.22 mg/l instead of 0.20 mg/l). The simulation with both effects shows an additive
effect.
12
Table 1 : 1D Simulations with Langmuir or Freundlich Interactions
1
2
3
4
5
6
7
8
9
10
11
12
13
Order
Cref
Total sites
Kfv
Expon.
Refer.
Decay
Decay
Disp-
dx
dt
Courant
Peclet
Number
Langmuir
Langmuir
Freundl.
Freund.
Half
Const.
pers.
Number
Numb.
10-6 kg/m3
-
15
Maxim. Scheme
Coupl.
Iterat.
Time
10-6
14
(s)
(s-1)
(cm)
(cm)
(s)
(-)
(-)
3
kg/m
1
3.7818
0.7
Zheng
-
-
1
0.16
1
0.625
0.16
30
DF
2
3.7818
0.7
Zheng
-
-
1
0.25
10
1*
0.25
8
TVD
3
3.7818
0.7
Zheng
-
-
1
0.4
4
1
0.4
1
MOC
4
10
4.761
Zheng
-
-
1
0.16
1
0.625
0.16
30
DF
5
10
4.761
Zheng
-
-
1
0.16
10
1*
0.16
2
TVD
6
10
4.761
Zheng
-
-
1
0.4
4
1
0.4
1
MOC
7
3.7818
0.7
Huang
-
-
0.1
0.1
4
1*
1
6
TVD
8
3.7818
0.7
Huang
-
-
0.1
0.1
2
1*
2
6
MOC
9
3.7818
0.7
Huang
693.15
10-3
0.1
0.1
4
1*
1
6
TVD
Huang
-
-
0.1
0.1
2
1*
1
6
TVD
0.1
0.1
2
1*
1
6
TVD
10
11
10
10
4.761
4.761
Huang
69.315
10
13
-2
MF_DF
Refer
0.4
C/Co
0.3
0.2
0.1
0
0
500
1000
1500
Time (s)
a) MARTHE : Finite Differences
MF_TVD
Refer
0.4
C/Co
0.3
0.2
0.1
0
0
500
1000
1500
Time (s)
b) MARTHE : TVD
MF_MOC
Refer
0.4
C/Co
0.3
0.2
0.1
0
0
500
1000
1500
Time (s)
c) MARTHE : MOC
Fig. 1 Freundlich sorption. Test Zheng et al. 1998 : Thick line = MARTHE ; Slim line = Reference
14
ML_DF
Refer
1
C/Co
0.8
0.6
0.4
0.2
0
0
100
200
300
400
500
Time (s)
a) MARTHE : Finite Differences
ML_TVD
Refer
1
C/Co
0.8
0.6
0.4
0.2
0
0
100
200
300
400
500
Time (s)
b) MARTHE : TVD
ML_MOC
Refer
1
C/Co
0.8
0.6
0.4
0.2
0
0
100
200
300
400
500
Time (s)
c) MARTHE : MOC
Fig. 2 Langmuir sorption. Test Zheng et al. 1998 : Thick line = MARTHE ; Slim line = Reference
15
HF_TVD
Refer
1
C/Co
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
Pore Volume s
a) MARTHE : TVD (No decay)
HF_MOC
Refer
1
C/Co
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
Pore Volume s
b) MARTHE : MOC (No decay)
HF_DEC
Refer
1
C/Co
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
Pore Volume s
c) MARTHE : TVD (With decay)
Fig. 3 Freundlich sorption. Test Huang et al. 1998 : Thick line = MARTHE ; Slim line = Reference
16
HL_TVD
Refer
1
C/Co
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Pore Volumes
a) Langmuir sorption : No decay
HL_DECTVD
Refer
1
C/Co
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Pore Volumes
b) Langmuir sorption : With decay
Fig. 4 Langmuir sorption without and with decay. Test Huang et al. 1998
Thick line = MARTHE TVD ; Slim line = Reference
17
No Sorption nor Kd
Bentazone
Langmuir
0
-20
-40
Depth (cm)
-60
-80
-100
-120
-140
-160
-180
-200
0
0.05
0.1
0.15
0.2
0.25
Concentrat (mg/l)
Fig. 5 Transfer of Bentazone in an unsaturated soil with Langmuir sorption and decay (adapted from
Vredepeel site).
18
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
0
2000
4000
6000
8000
10000
12000
14000
16000
14000
16000
a) Rocky No interactions
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
0
2000
4000
6000
8000
10000
12000
b) Rocky : Langmuir sorption
Fig 6 Rocky test problem. Transport in Steady State with decay.
19
NH4 -> NO2 -> NO3
Finite Differences
1.2
1
C / Co
0.8
0.6
0.4
0.2
0
0
50
100
150
200
150
200
Distance
a) MARTHE : Finite Differences
NH4 -> NO2 -> NO3
TVD Fine
1.2
1
C / Co
0.8
0.6
0.4
0.2
0
0
50
100
Distance
b) MARTHE : TVD (fine grid)
Fig 7 Chain First-Order reaction of NH4-. Thick line = MARTHE ; Slim line = Reference
20
NH4 -> NO2 -> NO3
TVD Large
1.2
1
C / Co
0.8
0.6
0.4
0.2
0
0
50
100
150
Distance
Fig. 8 Chain First-Order reaction of NH4-.
Thick line = MARTHE TVD with a large grid ; Slim line = Reference
21
200
20000
20000
18000
18000
16000
16000
14000
14000
12000
12000
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
0
2000
4000
6000
8000
10000
12000
14000
16000
0
2000
4000
Product 1
6000
8000
10000
Product 2
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
0
2000
4000
6000
8000
10000
Product 3
Fig 9 Chain First-Order reaction : Rocky test problem.
22
12000
14000
16000
12000
14000
16000
Simulated Temperature in the Soil at 4 depth
Bottom Temperature prescribed
3.8 cm
41 cm
101 cm
191 cm
25
20
Temper (°C)
15
10
5
0
-5
-10
0
100
200
300
400
500
Days
Fig 10 Vredepeel site : simulation of the temperature at different depths.
Bentazone
0
-20
-40
Refer
-60
Decay_Satu
Depth (cm)
-80
Decay_Temp
-100
Decay_Satu+Temp
-120
-140
-160
-180
-200
0
0.1
0.2
0.3
0.4
Concentrat (m g/l)
Fig. 11 Transfer of Bentazone in an unsaturated soil (adapted from Vredepeel site): effect of thermal
and saturation effects on degradation. Profile after 103 days.
23
.
24
References
BOESTEN & VAN DER PAS 2000. Movement of water, bromide ion and the pesticides
ethoprophos and bentazone in a sandy soil : description of the Vredepeel data set, Agricultural
water management, 44, 21-42.CHO, C. M., 1971 Convective transport of ammonium with
nitrification in soils. Can. Jour. Soil Sci. Vol. 51 no 3 pp 339-350.
GOODE, D.J. & KONIKOW, L.F. 1989 Modification of a method-of-characteristics solute
transport model to incoporate decay and equilibrium-controlled sorption or ion exchange. U.S.
Geological Survey Water-Resources Investigations Report 89-4030, 65 pp, Reston, Virginia.
GOLAZ, C. 2001. Comparison of two simulations made with LEACHP and MARTHE models on
the Vredepeel data set, report of PEGASE project, (in press).
GROVE, D.B. & STOLLENWERK, K.G. 1984 Computer model of one-dimensional equilibriumcontrolled sorption processes. U.S. Geological Survey Water-Resources Investigations Report
89-4059, 58 pp.
HUANG, K., MOHANTY B., LEIJ F. and Van GENUCHTEN M. Th. 1998 - Solution of the
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THIERY, D. 1990 - MARTHE. Modélisation d'Aquifères par maillage Rectangulaire en régime
Transitoire pour le calcul Hydrodynamique des Ecoulements. Version 4.3. Rapport BRGM 4S/EAU
n° R32210 (users guide).
THIERY, D. 1993 Modélisation des aquifères complexes - Prise en compte de la zone non saturée et
de la salinité. Calcul des intervalles de confiance. Revue Hydrogéologie, 1993, n° 4 pp. 325-336.
THIERY, D. 1994 - Modélisation 3D des écoulements en Zone Non Saturée avec le logiciel
MARTHE. Version 5.4, Rapport BRGM n° R38108 HYT/DR 94 (users guide).
THIERY, D. 1995 - Modélisation 3D du Transport de masse avec le logiciel MARTHE. Version 5.4,
Rapport BRGM n° R38149 HYT/DR 95 (user's guide).
VOSS C.I. 1984 - SUTRA - A Finite-Element simulation model for saturated-Unsaturated, FluidDensity-Dependent Groundwater Flow with Energy Transport or Chemically Reactive Single
species solute transport. U.S. Geological Survey National Center. Reston, Virginia.
25
ZHENG C. and WANG C. 1998 - MT3DMS A modular three-dimensional multispecies transport
model for simulation of advection, dispersion and chemical reactions of contaminant in
groundwater systems. Documentation and User's Guide. Waterways Experiment Station. US
Army Corps of Engineers Vicksburg, Mississippi 39187.
Van GENUCHTEN M. Th. 1985 - Convective-Dispersive transport of solutes involved in
sequential first-order decay reactions. Computer Geosciences Vol. 11 no 2 pp 129-147.
26