1.
MODELING SOLUTE TRANSPORT AND
MASS‐TRANSFER PROCESSES AT THE
MADE‐SITE AQUIFER, MISSISSIPPI
Por
Luca Trevisan
Tesina presentada en el
DEPARTAMENTO DE INGENIERÍA DEL TERRENO Y CARTOGRÁFICA DE LA
ESCUELA TÉCNICA SUPERIOR DE INGENIEROS DE CAMINOS, CANALES Y PUERTOS
Para la obtención del título de
MÁSTER EN HIDROLOGÍA SUBTERRÁNEA
Expedido por la
UNIVERSITAT POLITÉCNICA DE CATALUNYA
con la colaboración de la
FUNDACIÓN CENTRO INTERNACIONAL DE HIDROLOGÍA SUBTERRÁNEA

2. 2

3. 3
INDEX
ABSTRACT ......................................................................................................................... 7
1. INTRODUCTION ...................................................................................................... 8
1.1 Description of the Macrodispersion Experiment (MADE) ................................... 8
1.2 Literature review of previous models at the MADE site .................................... 12
1.3 Objective of the study........................................................................................................ 15
2. METHODOLOGY ................................................................................................... 16
3. CONCEPTUAL MODEL ....................................................................................... 17
3.1 Boundary and Initial Conditions .................................................................................. 17
3.2 Description of Heterogeneity ........................................................................................ 18
4. MODEL DESCRIPTION ....................................................................................... 22
4.1 Flow Model ............................................................................................................................ 22
4.2 Transport Model I : Advective‐Dispersive Model ................................................. 22
4.3 Transport Model II : Dual‐Domain Mass Transfer Model .................................. 24
5. NUMERICAL MODEL FEATURES ................................................................... 26
5.1 Basic Flow and Transport Parameters ...................................................................... 26
5.2 Determination of Mass Transfer Coefficients ......................................................... 27
5.3 The Random Walk Particle Tracking Method ......................................................... 29
6. SIMULATION RESULTS ..................................................................................... 32
6.1 Particle Cloud Evolution .................................................................................................. 32
6.2 Integrated Mass Distribution with Longitudinal Distance................................ 35
6.2.1 Mass transfer VS No Mass transfer .................................................................... 36
6.2.2 Comparison between literature values of mass transfer rate
coefficients ..................................................................................................................................... 37

4. 4
6.2.3 Comparison between homogeneity and heterogeneity of mass
transfer rate coefficients .......................................................................................................... 38
6.2.4 Change of slope in the Bajracharya & Barry (1997) model .................... 39
6.2.5 Zonification of the capacity coefficient ............................................................ 40
7. CONCLUSIONS ....................................................................................................... 43
8. REFERENCES ......................................................................................................... 44

5. 5
LIST OF FIGURES
Figure 1 Macrodispersion Experiment (MADE) site location (after Boggs et al., 1992). ....... 8
Figure 2 Geological model of the MADE site interpreted by geophysical surveys (after
Bowling et al., 2005). ........................................................................................................................................... 9
Figure 3 Three‐dimensional view of the monitoring network at the MADE site. Vertical
lines indicate the locations of the flowmeters. Circles indicate multilevel sampler locations
(after Feehley et al., 2000). ................................................................................................................................ 9
Figure 4 a) Multilevel sampling network with tracer injection point at the origin of
coordinates (after Boggs et al., 1992); b) Locations of the 62 fully penetrating wells for
flowmeter tests (after Salamon et al., 2007). .......................................................................................... 10
Figure 5 Observed tritium plume at 328 days after injection (after Bowling et al., 2006). 11
Figure 6 Longitudinal concentration profiles of the Tritium plume and predictions using
the macrodispersion model and the mass transfer model (after Harvey & Gorelick, 2000).
.................................................................................................................................................................................... 12
Figure 7 Results of simulations using the mass transfer transport model and three
different K representations (after Bowling et al., 2006). .................................................................... 14
Figure 8 Mass transfer conceptual model for describing solute transport at large scales. 15
Figure 9 Scheme of the conceptual model with the aerial view of the initial position of the
tracer plume 27 days after injection. .......................................................................................................... 17
Figure 10 Hydraulic conductivity profile derived from borehole measurements at well K‐
12 (after Boggs et al., 1992). ........................................................................................................................... 18
Figure 11 Directional horizontal variogram and fitted model with hole effect for the ln K
flowmeter data (after Salamon et al., 2007). ........................................................................................... 19
Figure 12 Frequency distribution and univariate statistics of the 2495 flowmeter
measurements (after Salamon et al., 2007). ............................................................................................ 20
Figure 13 Realization # 26 of the hydraulic conductivity field. .................................................... 21
Figure 14 Stochastic distribution of the hydraulic conductivity values for the realization #
26. .............................................................................................................................................................................. 21
Figure 15 Results of simulations using the advection‐dispersion transport model and
three different K representations (after Bowling et al., 2006). ........................................................ 23
Figure 16 Cross – section visualization of the tracer plume 328 days after injection using
hydraulic conductivity field realization # 26: a) with homogeneous α; b) with
heterogeneous α. ................................................................................................................................................. 33

6. 6
Figure 17 Aerial visualization of the tracer plume 328 days after injection for each
hydraulic conductivity field realization: A) # 7; B) # 19; C) # 26; D) # 47; E) # 56; F) # 80.
.................................................................................................................................................................................... 34
Figure 18 Comparison chart between observed mass distribution profile of the Tritium
plume and simulations using dual – domain mass transfer or advective – dispersive model.
.................................................................................................................................................................................... 36
Figure 19 Comparison chart between observed mass distribution profile of the Tritium
plume and simulations using mass transfer model with different mass transfer rate
coefficients. ............................................................................................................................................................ 37
Figure 20 Comparison chart between observed mass distribution profile of the Tritium
plume and simulations using mass transfer model with heterogeneous values or the same
value of mass transfer rate coefficient. ...................................................................................................... 38
Figure 21 Comparison chart between observed mass distribution profile of the Tritium
plume and simulations using mass transfer models with different terms in the correlation
model between average pore – water velocity and mass transfer rate coefficients. .............. 39
Figure 22 Comparison chart between observed mass distribution profile of the Tritium
plume and simulations using mass transfer model with different capacity coefficients
along downgradient distance. ....................................................................................................................... 40
Figure 23 Longitudinal mass distribution profile of the Tritium plume 328 after injection
and six predictions using classical advection‐dispersion model for each hydraulic
conductivity field realization. ........................................................................................................................ 41
Figure 24 Longitudinal mass distribution profile of the Tritium plume 328 after injection
and six predictions using dual – domain mass transfer model for each hydraulic
conductivity field realization. ........................................................................................................................ 42

7. 7
ABSTRACT
Several natural gradient field tracer tests have been conducted in the last
decades to gain understanding on the mechanisms governing subsurface
solute transport. Among them, the Macro‐Dispersion Experiment (MADE)
site, which is distinguished for being representative of a highly
heterogeneous system, has been the most controversial one for revealing an
important anomalous behavior in the shape of the tracer plume, i.e., highly
asymmetric spreading of the plume with high concentrations maintained
near the source and a far and dilute reaching front of the plume. Many
conceptually different transport models have been developed that can
partially explain this behavior. In particular, transport models with rate‐
limited mass transfer processes have been claimed to better reproduce this
behavior as opposed to macrodispersion caused by heterogeneity in
hydraulic conductivity. Recently, a new stochastic model of the MADE aquifer
(Salamon et al., 2007) has shown that, when small – scale variability of
hydraulic conductivity is correctly modeled at the flowmeter measurement
support scale, the advection – dispersion simple model is capable of
reproducing much of the anomalous tracer spreading at the MADE site. The
objective of this Master thesis is to assess the relative importance of mass
transfer processes at the MADE site. To achieve this, we extend Salamon’s
model to include mass transfer processes. Sensitivity analyses of the mass
transfer model parameters show the interplay between Darcy – scale mass
transfer and (lnK) heterogeneity. Remarkably, significant improvement in the
plume shape characteristics is obtained when a direct relationship between
the characteristic times of mass transfer and advective processes is included
in the model.

8. 8
1. INTRODUCTION
1.1 Description of the Macrodispersion Experiment (MADE)
The field data used for this study are from the MADE site at the Columbus Air
Force Base in northeastern Mississippi (Figure 1). The shallow aquifer which
immediately underlies the site consists of an alluvial terrace deposit with an
average thickness of 11 m and is characterized by a high degree of
heterogeneity (variance of the natural logarithm on hydraulic conductivity of
about 4).
Based on resistivity data and additional information obtained from other
geophysical methods such as ground penetrating radar, Bowling et al. (2005)
divided the MADE site into four stratigraphic units: 1) an upper layer of
meandering fluvial sediments, 2) an upper‐middle layer with gravelly sands
deposited by a braided fluvial system, 3) a layer associated with fine sands
and silts of the Eutaw Formation (Eutaw Sand), and 4) a layer associated with
the high clay content of the Eutaw Formation (Eutaw Clay).
The top two units were deposited by a meandering fluvial system and
braided fluvial system respectively, while the lower two units are part of the
marine Eutaw Formation (Figure 2). According to Bowling et al. (2005),
stratigraphy of the second unit consists of multiple lenticular bodies on the
Figure 1 Macrodispersion Experiment (MADE) site
location (after Boggs et al., 1992).

9. 9
order of 5 – 10 m in length; this unit represents the main aquifer at the MADE
site and accounts for the considerable heterogeneities in hydraulic
conductivities that have been reported.
An array of 258 multilevel samplers consisting of a network of approximately
6000 sampling points (Figure 4a) was built for monitoring the tracer plume
in three dimensions. Whereas the spatial distribution of hydraulic
conductivity at the test site was determined from 2495 measurements of
conductivity obtained from borehole flowmeter tests performed in 62 fully
penetrating wells (Figure 4b). A comprehensive three‐dimensional view of
the whole sampling network is given in Figure 3.
Figure 2 Geological model of the MADE site interpreted by geophysical surveys
(after Bowling et al., 2005).
Figure 3 Three‐dimensional view of the monitoring network at the MADE site.
Vertical lines indicate the locations of the flowmeters. Circles indicate
multilevel sampler locations (after Feehley et al., 2000).

10. 10
Two field‐scale macrodispersion experiments were conducted to carry out a
complete hydraulic characterization of the aquifer and to execute a tracer
injection and plume tracking program.
MADE‐2 represents the second large‐scale natural‐gradient tracer
experiment conducted at Columbus Air Force Base. The field study started in
June 1990 and was carried out during the following 15 months performing
detailed field observations of the transport of tritiated water.
The method of injecting the tracer solution into the aquifer was designed for
producing a uniform pulse release of tracers into approximately the middle
of the saturated zone with the minimal amount of disturbance of the natural
flow field. In order to do that, a total volume of 9.7 m3 of tracer solution was
injected through a linear – five – well array during approximately 2 days
registering a maximum increase in hydraulic head within the injection wells
Figure 4 a) Multilevel sampling network with tracer injection point at the origin
of coordinates (after Boggs et al., 1992); b) Locations of the 62 fully
penetrating wells for flowmeter tests (after Salamon et al., 2007).

11. 11
of 0.45 m. The injection fluid had a tritium concentration of 55’610 pCi/mL
and the total injected mass (activity) was 0.5387 Ci. Each injection well was
screened over a 0.6 m interval between elevations 57.5 m and 58.1 m a.m.s.l.
(Boggs et al., 1992).
After 328 days since the injection took place, the two clearest features
showed by the tracer plume observations, were:
• the strong asymmetry of the concentration distribution in the
longitudinal dimension;
• a highly variable mass balance during the experiment.
In fact, from studies conducted at more homogeneous aquifers, like the
Borden or the Cape Cod sites, it would be predicted that the plume would
have traveled a considerable distance from the injection area 328 days after
the start of injection. However, the location with the maximum observed
concentration of approximately 3800 pCi/mL was less than 6 m away from
the injection location (Figure 5). This may be explained by the fact that the
aquifer is extremely heterogeneous and the injection wells are located in a
zone with relatively low hydraulic conductivity acting as a tracer trap
(Feehley et al., 2000).
Figure 5 Observed tritium plume at 328 days after injection (after Bowling et al., 2006).

12. 12
1.2 Literature review of previous models at the MADE site
Till now many numerical models attempted to explain the extreme non‐
Gaussian behavior of the tracer plume at the MADE site. Amongst all the
following ones deserve a special attention.
Harvey & Gorelick (2000) stated in their paper that a combination of physical
nonequilibrium mass transfer (molecular diffusion into and out of low
permeability areas) and chemical sorption could better explain the behavior
of the plumes at the Columbus Air Force Base than a macrodispersion model
alone (Figure 6).
Moreover, they asserted that declining mobile mass and extreme asymmetric
spreading, were both evidences of transport subject to rate‐limited mass
transfer, in the case when the capacity coefficient (ratio of immobile to
mobile mass at equilibrium) was large and the timescale of mass transfer was
similar to the timescale of the experiment. In fact, they suggested that
diffusional mass transfer might have been significant in the Columbus aquifer
because three main criteria were met: 1) the plume was observed over the
timescale similar to the timescale of diffusion in and out of the low
conductivity zones, 2) diffusion was dominating advection within the low
Figure 6 Longitudinal concentration profiles of the Tritium plume and predictions
using the macrodispersion model and the mass transfer model (after
Harvey & Gorelick, 2000).

13. 13
conductivity zones, and 3) advection was dominating diffusion through the
high conductivity zones.
A complementary study was led by Feehley et al. (2000) where a dual –
domain mass transfer model was compared with the Advection – Dispersion
Equation (ADE) accounting two methods of hydraulic conductivity
interpolation, ordinary kriging and conditional geostatistical simulation
based on fractional Brownian motion. The results showed that tracer
spreading using ADE was not modeled correctly with neither of the two
heterogeneous fields. The authors attribute this to the fact that preferential
flow pathways, which strongly influenced asymmetric tracer spreading,
might exist at a scale smaller than the grid spacing. To overcome this inability
they suggested the use of a dual‐domain approach which, after calibrating the
immobile porosity and the mass transfer rate, was able to recreate the non‐
Gaussian shape of the tracer plume.
Berkowitz and Scher (1998) also analyzed the MADE site experiment using
the continuous time random walk formalism. They compared the field
experiment with the dominant aspects of anomalous solute transport in
fracture networks. Characterizing key features of fracture properties and
mapping them on probability distributions, the spatial distribution of the
plume concentration as well as breakthrough curves can be calculated
analytically using the continuous time random walk approach. Comparing the
results of the fracture network to the MADE site Berkowitz and Scher (1998)
demonstrate that time dependent anomalous as, i.e., non‐Gaussian transport
also exists in other geological formations than rock fractures. They conclude
that when mapping preferential flow paths and high flow variability of the
heterogeneous aquifer at the Columbus Air Force Base to a series of channels
or “fractures” tracer transport can be reproduced using the continuous time
random walk method.
Another explanation of the failure to model correctly the tracer experiment
was given by Barlebo et al. (2004). Their paper introduced the possibility that
the small‐scale flowmeter measurements were too noisy and possibly too
biased to use so directly in site‐scale models and that the hydraulic head and
transport measurements were more suitable for site‐scale characterization.

14. 14
Therefore they suggested using a finite element three‐dimensional inverse
flow and transport model to overcome the inaccuracy of flowmeter
measurements, concluding that the macrodispersion model is still able to
reproduce the extensive plume spreading.
A more recent attempt to model the MADE site tracer plume using the mass
transfer model was presented by Bowling et al. (2006). The intent of these
authors was to demonstrate that a highly heterogeneous aquifer could be
modeled properly with minimal hydrological data supplemented with
geophysical data. Thus, in order to test the validity of their approach, they
used many geophysically derived hydraulic conductivity representations to
compare the macrodispersion model (Figure 15) to the mass transfer model
(Figure 7).
To conclude this issue, the last work to be cited is the one of Salamon et al.
(2007), whose end was to highlight the importance of generating high –
resolution hydraulic conductivity fields on the basis of a detailed
geostatistical analysis. They concluded that, when small – scale variability of
hydraulic conductivity is correctly modeled at the flowmeter measurement
support scale, the ADE is capable of reproducing the tracer spreading at the
MADE site and that the heterogeneity at that scale is the main contributor to
the anomalous plume behavior.
Figure 7 Results of simulations using the mass transfer transport model and three
different K representations (after Bowling et al., 2006).

15. 15
1.3 Objective of the study
After revising the previously cited works we found that the normalized
Tritium mass profile still has not been well reproduced, in particular
regarding two aspects:
• the steepness of the tail slope;
• the presence of an unexpected second peak at the front of the plume.
The goal of this study consisted of studying the relative importance of Darcy‐
scale mass transfer model as an alternative upscaled model (Figure 8) at the
macrodispersion experiment, in order to better reproduce the non‐Gaussian
behavior of the Tritium plume.
To achieve this objective we compared the advection‐dispersion model with
a single‐rate mass transfer model and, according to the study of Salamon et
al. (2007), we took into account 6 of the 80 sequential realizations of the
hydraulic conductivity field in order to depict several possible scenarios.
These realizations were chosen because they seemed to be the ones best
reproducing the plume mass distribution.
Figure 8 Mass transfer conceptual model for describing solute
transport at large scales.

16. 16
2. METHODOLOGY
The study started with the conceptualization of the flow model, fixing the
boundary conditions of the domain system and the initial concentration of
the tracer mass injected, as well as selecting 6 different realizations of the
hydraulic conductivity field among the 80 already existing in the paper of
Salamon et al. (2007). These hydraulic conductivity fields were useful to
observe the variation of the simulation results depending on each realization
and independently of the choice of the model grid or the potential incapacity
of the ADE to simulate anomalous transport.
Afterward, through an additional FORTRAN program, the binary file
originated by the MODFLOW model containing the intracell flow rates for the
whole domain’s grid was converted to a GSLIB format file where flow rates
were transformed into pore‐water velocity. Subsequently, through the
Bajracharya & Barry (1997) model, the pore‐water velocity was correlated to
the mass transfer coefficient α. This task allowed studying the velocity –
dependence of the mass transfer rate coefficient.
Then a comparison between a constant – α model and a heterogeneous – α
model was performed to demonstrate that a heterogeneous field of mass
transfer rate coefficients could explain much better the observed tracer
plume behavior than a model using the same coefficient on the whole domain
(Figure 20).
Furthermore we proceeded with a trial and error estimation of the values for
the intercept and slope terms of the model by Bajracharya & Barry (1997), so
that we could achieve a better adjust to the observed tracer mass
distribution.
The last section of the work consisted of a comparative process between
many simulation results through a semi – logarithmic graphic of normalized
mass distribution with distance from the injection point. As it will be showed
in the following chapters, that class of graphic represented the best way to
highlight the extreme asymmetric spreading of the observed tracer plume.

17. 17
3. CONCEPTUAL MODEL
3.1 Boundary and Initial Conditions
Dirichlet condition of specified head was fixed at x = 0 m and x = 280 m since
a natural gradient flow system at steady state regime was considered,
whereas Neumann conditions of no flow were set at y = 0 m, y = 100 m (y‐
axis parallel to flow lines), z = 0 m and z = 10.5 m. Specified heads at the
northern and southern boundaries were interpolated from average heads of
1‐year observations using ordinary kriging.
For the transport model, boundary conditions of zero – mass – flux were
fixed at all boundaries except for the northern and southern boundaries,
where the solute could exit freely out of the model domain.
As initial concentration for the tracer plume, the mass distribution for the
27th day was employed (Figure 9), because of the complexity of reproducing
specific local conditions as, e.g., well characteristics, hydraulic head cone
Figure 9 Scheme of the conceptual model with the aerial view of the initial position
of the tracer plume 27 days after injection.

18. 18
during injections, etc., which the tracer injection is generally subject to. Then,
with the purpose of using the mentioned snapshot as starting point within
the transport model, the tracer concentration was spatially interpolated in
the aquifer and a total of 50’000 particles were proportionally assigned to
each cell (Salamon et al., 2007).
3.2 Description of Heterogeneity
The most important feature that differentiates the MADE site aquifer from
previous natural gradient experiments, like the Cape Cod or the Borden
aquifers, is its important degree of heterogeneity in the hydraulic
conductivity field. Rehfeldt et al. (1992) reported that the overall variance of
the natural logarithm of hydraulic conductivity estimated from the borehole
flowmeter measurements was 4.5 (Figure 12). This result indicates that the
alluvial aquifer at the Columbus site is approximately 1 order of magnitude
more heterogeneous than the aquifers at the Borden and Cape Cod sites,
where the variances measured were 0.29 and 0.26 respectively.
The flowmeter derived hydraulic conductivity profile in Figure 10 illustrates
the extreme heterogeneity of the aquifer.
Figure 10 Hydraulic conductivity profile derived from
borehole measurements at well K‐12 (after
Boggs et al., 1992).

19. 19
Although the statistical section has not been treated extensively in this work,
the study of the plume evolution requires a good knowledge of the hydraulic
conductivity field through which the solute is moving. However, as full
knowledge of aquifer properties is not possible, many different equally likely
aquifer realizations were realized. Among 80 realizations, we analyzed 6 of
them, corresponding on the best performance based on the advective and
dispersive transport mechanisms (Salamon et al., 2007).
Based on the work of Salamon et al. (2007), these 6 realizations of the
hydraulic conductivity spatial distribution were obtained through stochastic
simulation of random fields as a variogram model consisting of a nested
structure and a so called “hole effect” model (Figure 11). Hole effect
structures are used to indicate a form of periodicity, e.g., lenses of high/low
conductivity, which is a common spatial characteristic in sedimentary
geology.
Figure 11 Directional horizontal variogram and fitted model with hole effect
for the ln K flowmeter data (after Salamon et al., 2007).

20. 20
Table 1 Univariate statistics of the hydraulic conductivity fields.
Model mean ln K (cm/s) variance ln K
Flowmeter Measurements ­5.35 4.245
Realization # 7 ‐5.1589 5.3814
Realization # 19 ‐4.914 4.754
Realization # 26 ‐4.8234 4.6195
Realization # 47 ‐5.229 4.54
Realization # 56 ‐5.343 4.969
Realization # 80 ‐5.2897 5.166
Figure 12 Frequency distribution and univariate statistics of the 2495
flowmeter measurements (after Salamon et al., 2007).

21. 21
The conductivity field realizations were saved in the GSLIB format so that it
was possible to visualize them in SGeMS software (Figure 13) and to
represent the whole values by a histogram summarized by the mean and the
variance (Figure 14).
Figure 13 Realization # 26 of the hydraulic conductivity field.
Figure 14 Stochastic distribution of the hydraulic conductivity values for the
realization # 26.

22. 22
4. MODEL DESCRIPTION
4.1 Flow Model
Groundwater flow within the aquifer was modeled using a three‐dimensional
block‐centered finite difference code, using a model domain with 280
columns, 110 rows and 70 layers (2’156’000 nodes covering a total volume of
280 x 110 x 10.5 m3). The grid spacing is then 1 x 1 x 0.15 m.
Since the attention was focused on the influence of the heterogeneity of
hydraulic conductivity in the longitudinal mass distribution profile, we
considered sufficient to simulate the flow at steady state conditions in a
confined system using the MODFLOW code. Yet, it is recognized the
possibility that a transient regime of flow may had been affecting the three –
dimensional picture of the tracer cloud.
4.2 Transport Model I : Advective­Dispersive Model
The classical, Fickian advection‐dispersion transport equation for
conservative tracers is:
( )ij i s s
i j i
C C
D q C q C
t x x x
θ θ
⎛ ⎞∂ ∂ ∂ ∂
= − +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ [1]
where C [ML‐3] is the solute concentration, θ is the porosity, t is time, Dij is the
hydrodynamic dispersion tensor [L2T‐1], qi is the Darcy flux, while qs and Cs
are the Darcy flux and concentration respectively of the fluid sink/source.
The advection – dispersion equation describes the transport of a
conservative solute (non – reactive). Therefore simple interaction
phenomena like retardation coefficients or coupling with geochemical
models are not considered.
This equation expresses that the dispersion of a tracer plume around its
center of mass can be compared to a diffusion mechanism; thus we refer to
Fickian, diffusive or even Gaussian dispersion. Such dispersion is
characterized by some coefficients that are measured both on laboratory and

23. 23
field. Several tracer experiments and theoretical studies (e.g., Matheron and
de Marsily, 1980; Gelhar, 1993) show that:
• depending on the tracer experiment scale, dispersion coefficients are
different;
• at a given scale, dispersion coefficients increase with the distance from
injection point.
This scale‐dependent behavior (also sometimes referred to as “pre‐
asymptotic” or “non‐Gaussian”) is what we shall refer to as “non‐Fickian”
transport. Other evidence of non‐Fickian behavior lies in the often observed
“unusual” early breakthrough, and “unusual” long late time tails, in measured
breakthrough curves.
Due to this deficiency, advection‐dispersion simulations fail to reproduce the
low concentrations of tritium observed at distances much greater than 100 m
from the injection point, although they generally explain the observed peak
concentrations (Figure 15) (Bowling et al., 2006).
Figure 15 Results of simulations using the advection‐dispersion transport model
and three different K representations (after Bowling et al., 2006).

24. 24
4.3 Transport Model II : Dual­Domain Mass Transfer Model
In order to overcome these deficiencies in representing the tail of
breakthrough curves, alternative transport models have been promoted. For
instance, an alternative to the single – porosity advection – dispersion
equation is the dual – porosity or dual – domain mass transfer model, as
given below for conservative tracers:
( )m im m
m im m ij i m s s
i j i
C C C
D q C q C
t t x x x
θ θ θ
⎛ ⎞∂ ∂ ∂∂ ∂
+ = − +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ [2]
where θm is the porosity of the “mobile” domain, denoting pore spaces filled
with mobile water, θim is the porosity of the “immobile” domain, denoting
pore spaces filled with immobile water, Cm [ML‐3] is the solute concentration
in the mobile domain, and Cim [ML‐3] is the solute concentration in the
immobile domain.
With the dual – domain model the aquifer is conceptualized as consisting of
two overlapping domains, a mobile domain in which transport is dominated
by advection and an immobile domain in which transport is mostly driven by
molecular diffusion. For this reason the dual – domain approach gives a
plausible explanation of the mass overestimation at early times and
underestimation at late times (Feehley et al., 2000).
Typically this model defines the rate of change of solute concentration within
the immobile zone as:
( )im
im m im
C
C C
t
θ α
∂
= −
∂
[3]
where α is a first‐order rate coefficient which controls the rate at which
solute moves between immobile and mobile domains. The larger the rate
constant α is, the faster transport occurs between the mobile and immobile
zones for a given concentration difference.
Another important term for treating the dual – domain model is the term β
[dimensionless], which is the “capacity coefficient”, equal to the ratio

25. 25
between tracer mass in the immobile zone and the mass in the mobile zone at
equilibrium:
[4]
The capacity coefficient relates to the total porosity as follows:
[5]
This dual – domain mass transfer model can be generalized to account for
multiple mass transfer processes occurring simultaneously. In this case, the
transport equation is known as the multi – rate mass transfer model. The
model describes mass transfer between a mobile zone and any number of
immobile zones, and is coupled with the advective – dispersive transport
model for solute transport.
im
m
θ
β
θ
=
1
m
φ
θ
β
=
+

26. 26
5. NUMERICAL MODEL FEATURES
5.1 Basic Flow and Transport Parameters
An average total porosity of 0.32 was used for the entire model area, which
corresponds to the value determined from the soil cores collected at the test
site (Boggs et al., 1990). A value of 0.1 m was assigned to the longitudinal
dispersivity and the ratios of the transverse and vertical dispersivities to the
longitudinal dispersivity were fixed at 0.1. Additionally an apparent diffusion
of 1.0 cm2/day was chosen for Tritium according to Gillham et al. (1984).
Table 2 Input parameters for solute transport model.
Parameter Simulation 1 Simulation 2 Simulation 3
Θm 0.04 0.04 0.04; 0.008; 0.22
Θim 0.28 0.28 0.28; 0.312; 0.122
β 7.0 7.0 7.0; 39; 0.62
ln K Realization 26 26 26
α(days­1) constant, 0.02
variable1,
A= ‐ 0.61; B= 1.5
variable1,
A= ‐ 0.61; B= 1.5
Longitudinal
Dispersivity (m)
0.1 0.1 0.1
Transverse
Dispersivity (m)
0.01 0.01 0.01
Total porosity 0.32 0.32 0.32
Courant number 0.0125 0.0125 0.0125
(1) log(tα) = A + B * log(tad)
(2) first, second and third zone

27. 27
5.2 Determination of Mass Transfer Coefficients
As it has been introduced in chapter 4.3, the dual – domain mass transfer
model expresses a solute mass interchange between two overlapping
domains, a mobile domain and an immobile one. Subsequently there is the
need to determine mass transfer rate coefficients that control this change of
solute concentration within both domains.
Many papers (i.e. Brusseau, 1992; Bajracharya and Barry, 1997; Griffioen et
al., 1998) found that the first‐order rate coefficient α [T‐1] changes
approximately linearly with average pore‐water velocity ν.
Two physical explanations underlie the mechanism that correlates α with ν.
The first process believed to control the relationship between α and ν is the
advection through low – K zones. Zinn and Harvey (2003) show that α varies
linearly with ν 1) if high – K regions are spatially connected, 2) if the variance
and spatial scale of K heterogeneity fall in a range where tailing occurs, and
3) if transport into low – K regions is not controlled by diffusion. Zinn and
Harvey conclude that advection into low – K areas may create tailing that
cannot be explained by a Fickian macrodispersion model in fields with
connected high – K paths, even though these fields have conventional
univariate lognormal histograms and covariance functions (Haggerty et al.,
2004).
The second, less accounted explanation for the relationship between α and ν
is that the mass transfer model does not adequately characterize the
underlying physics of mass transfer. An example of this theory is given by
Glueckauf (1955) and Rao et al. (1980), who show that when a simple linear
mass transfer model is used to approximate diffusion into spheres, the
effective α changes with the duration of the experiment.
For the purpose of this study we considered advection through low – K zones
as the predominant process within the Columbus aquifer, so we accounted
the Bajracharya and Barry (1997) model as the best one to reproduce the
correlation between mass transfer rate coefficient and average pore – water
velocity. This model is based on experiments conducted in two columns of

28. 28
different lengths with a design such that mass transfer was expected to be
dominated by advection.
The results from Bajracharya and Barry’s experiments showed that log tα is
better correlated to log tad than log v, and the relation of tα to residence time
is not related to column length. As a matter of fact it was demonstrated that
the strongest tailing in an experiment is always due to the mass transfer
process with approximately the same timescale as advection through he
system. Bajracharya and Barry (1997) model formulates this proportionality
between “mass transfer time” tα and “advection time” tad through the
following equation:
log log adt tα = Α + Β⋅ [6]
where the intercept term A = ‐ 0.61 and the slope term B = 1.10.
First of all we know that, for a first‐order single‐rate mass transfer model
(Haggerty et al., 2004), the characteristic residence time in the immobile
zone or “mass transfer time”, tα [T], is:
1
tα
α
=
[7]
We also know that the characteristic residence time in the mobile zone or
“advection time”, tad [T], is:
ad
x
t
v
∆
= [8]

29. 29
5.3 The Random Walk Particle Tracking Method
We consider the movement of water and chemical species (i.e., tracers)
moving through a geological formation. Clearly, heterogeneities occur on a
broad range of scales in most geological formations. These heterogeneities
can consist of fractures (joints and/or faults), variations in the rock matrix
(e.g., grain sizes, mineralogy, layering, and lithology), and/or large‐scale
geological structures. Under an external pressure gradient, the velocity and
flux distributions are determined by the liquid properties and by the
structure of the aquifer heterogeneities. Tracer particles (representing the
contaminant mass) transported within the water move through the
formation via different paths with spatially changing velocities. Different
paths are traversed by different numbers of particles. Typically,
heterogeneous systems show a broader distribution of velocities than
homogeneous systems. This reasoning represents the conceptual basis of the
random walk particle tracking method (RWPTM).
The RWPTM consists in moving a cloud of solute particles advectively
according to the flow path lines and adding a random displacement for each
time step to simulate dispersion. Its main advantages are the nonexistence of
numerical dispersion, computational efficiency, and local as well as global
mass conservation (Salamon et al., 2006). In fact, this modeling approach was
designed mainly to avoid the numerical dispersion that might be affecting
solute transport simulations owing to the addition of mass transfer
processes.
In the study of Salamon et al. (2006), the concept of transition probability
was drew on the context of continuous time Markov chains, denoting the
probability that a process presently in state i will be in state j a time t later.
Using this principle in particle tracking and provided that we know the
transition probability function, we can determine if a particle is in the mobile
phase after a time step ∆t by performing a simple Bernoulli trial on the
appropriate phase transition probability.

30. 30
In more detail, the evolution in time of each representative particle is driven
by a drift term that relates to the advective movement and a superposed
Brownian motion responsible for dispersion. The displacement of a particle
in a given ∆t step is calculated as follows:
( ) ( ) ( ) ( ) ( )1 2, ,p p p pt t t t t t t t+ ∆ = + ∆ + ⋅ ∆Χ Χ Β Χ Β Χ ξ [9]
where Xp(t) is the particle position, B1 is the “drift” vector, B2 the
displacement matrix, and ξ(t) is a vector of independent, normally
distributed random variables with zero mean and unit variance. B1
corresponds to:
1
mR
+ ∇⋅
=
v D
Β [10]
and the displacement matrix B2 is related to the dispersion tensor according
to the following relationship:
2 2
2 T
mR
= ⋅
D
Β Β [11]
After having described the spatial evolution of a particle through equations
[9],[10], and [11], the random walk method can easily incorporate any mass ‐
transfer process by performing a simple Bernoulli trial on the appropriate
particle transition probability distribution between the mobile domain and
any immobile domain (Salamon et al., 2006). The transition probability
stochastic matrix was found as:
( )1
exp t−
⎡ ⎤= ∆⎣ ⎦Ρ Α Β [12]
where
11
11 1 1
00
00
0 0
N
j j Nj
N
N N
α β αα
αα β
αα β
=
−
⎛ ⎞−
⎜ ⎟
−⎜ ⎟
= ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
∑
Α Β [13]

31. 31
Having calculated the phase transition probabilities, numerical
implementation into particle tracking is done easily. For each time step a
uniform [0; 1] random number Y is drawn for each particle and is compared to
the corresponding probability.
The state of a particle being in the mobile phase is adjusted according to:
( )
mobile if
immobile if
m m
p
m m
Y P
t t
Y P
→
→
<⎧
+ ∆ = ⎨
≥⎩
Χ [14]
For a particle being located in the immobile phase the final state is adjusted as
follows:
( )
immobile if
mobile if
im im
p
im im
Y P
t t
Y P
→
→
<⎧
+ ∆ = ⎨
≥⎩
Χ [15]
Having finished the trial, a particle is only allowed to move when being in the
mobile phase.

32. 32
6. SIMULATION RESULTS
6.1 Particle Cloud Evolution
One of RW3D’s main output files is the rw3d_part_evol.dat, a TECPLOT – input
format which includes the spatial coordinates of the whole tracer plume after
a previously selected time step. Some simulations of the plume distribution
328 days after tracer injection for different conductivity field realizations are
presented in Figure 17.
It can be seen from these pictures that this kind of representation may help
visualizing the first impact of the hydraulic conductivity field on the
evolution of the contaminated plume. Nevertheless it’s not useful to observe
the results’ variation owing to the trial and error phase of the simulation
process. In fact, after having chosen one of the six conductivity realizations,
we proceeded testing the sensitivity of the mass transfer parameters;
however we didn’t detect such an important change in the plume’s shifting
(Figure 16), as we could see through the semi – logarithmic chart of the
longitudinal integrated mass distribution profile presented in the following
issue.

33. 33
b)
a)
Figure 16 Cross – section visualization of the tracer plume 328 days after
injection using hydraulic conductivity field realization # 26: a) with
homogeneous α; b) with heterogeneous α.

34. 34
A) B)
C) D)
E) F)
Figure 17 Aerial visualization of the tracer plume 328 days after injection for each hydraulic conductivity
field realization: A) # 7; B) # 19; C) # 26; D) # 47; E) # 56; F) # 80.

35. 35
6.2 Integrated Mass Distribution with Longitudinal Distance
For easier comparison between the simulated plumes and the observed
Tritium plume, we analyzed all the simulation’s results through the same
“Normalized Tracer Mass versus Distance from Injection Point” chart. In
order to do that, the model domain has been divided into 28 zones with a 10
– meter – interval along the general flow direction (the y – axis). Then, after
integrating concentrations laterally and vertically along the y – axis, the 1‐D
mass profile 328 days after tracer injection was normalized by the total mass
injected so that the resulting area under each model curve is equal to 1.
Through this visualization we could notice at a glance which simulation was
better reproducing the observed mass distribution and where were the most
significant discrepancies with the simulated mass distribution. Moreover,
after considering realization # 26 as the best performing for the simulation of
mass distribution, we decided calibrating mass transfer parameters
employing always this hydraulic conductivity field realization.

36. 36
6.2.1 Mass transfer VS No Mass transfer
The first attempt to model the Tritium mass distribution consisted of testing
the macrodispersion model using the traditional Advection – Dispersion
Equation in order to compare its resulting curve with the predictions
corresponding to the Dual – Domain Mass Transfer Model.
The transport and mass transfer parameters employed for this first
simulation are reported in Table 2.
The mass transfer coefficient α selected for the represented simulation was
0.02, corresponding to the harmonic mean of the whole values within the
heterogeneous field computed according to the Bajracharya & Barry (1997)
model.
The weak divergence of the DDMT curve from the ADE one is probably due to
the high value of α (0.02 days‐1) which implies high rates of mass transfer
between mobile and immobile zones and does not lead to a clear effect of the
solute retention by the immobile zone.
A more general view of this comparison for all six realizations is reported in
Figures 23 and 24.
Figure 18 Comparison chart between observed mass distribution profile of the Tritium
plume and simulations using dual – domain mass transfer or advective –
dispersive model.

37. 37
6.2.2 Comparison between literature values of mass transfer rate
coefficients
For this second step of the calibration process we took into account two
values of α which were previously used in the literature by Feehley et al.
(2000); subsequently we analyzed their effect on the simulation curve in
comparison with the mass transfer rate coefficient used in simulation 1.
From this chart we can appreciate the physical meaning of the mass transfer
coefficient α; in fact we can observe that, while the shape of the mass
distribution profile keeps the same, the mass concentration within the
central zone of the aquifer decreases proportionally with the value of α. This
is due to the fact that a decrease of α leads to an increase of mass transfer
time causing a slowdown of the solute mass flow.
Figure 19 Comparison chart between observed mass distribution profile of the Tritium
plume and simulations using mass transfer model with different mass
transfer rate coefficients.

38. 38
6.2.3 Comparison between homogeneity and heterogeneity of mass
transfer rate coefficients
After comparing a range of values for the mass transfer rate coefficient α, we
evaluated the effect of a heterogeneous field of α’s compared to a constant
value over the whole domain.
The heterogeneity of α was reproduced using the already cited Bajracharya &
Barry (1997) model with its original values of intercept A and slope B.
This step gave not much positive results, for this reason we passed to the
next step of the calibration process.
Figure 20 Comparison chart between observed mass distribution profile of the Tritium
plume and simulations using mass transfer model with heterogeneous values
or the same value of mass transfer rate coefficient.

39. 39
6.2.4 Change of slope in the Bajracharya & Barry (1997) model
Here we proceeded calibrating the slope term within the Bajracharya & Barry
(1997) model and finally we selected the value of 1.5 instead of the original
value of 1.10 (Table 2).
We can notice from this chart the strong sensitivity of the slope value, in
comparison with the parameter α which does not cause such a variation in
the curve profile by changing from 0.0033 to 0.0016 day‐1. Moreover it is
evident the improvement of the model for what concerns the tailing
reproduction (clear appearance of two peaks) although the peak
corresponding to the 200 – meter distance from the injection point is not well
simulated yet, at least not in the right position.
Another positive result of this prediction is the better resolution of the slope
of the tail, quite more horizontal than the previous one.
As a further intent to reduce the tracer mass after the first peak we also
decreased the intercept term from – 0.61 to – 0.061 in order to reduce more
α and the solute concentration within the immobile zone.
Figure 21 Comparison chart between observed mass distribution profile of the Tritium
plume and simulations using mass transfer models with different terms in
the correlation model between average pore – water velocity and mass
transfer rate coefficients.

40. 40
6.2.5 Zonification of the capacity coefficient
A further attempt to improve the normalized tracer mass curve focused the
attention on the position of the anomalous peak at the end of the plume tail.
With the aim of reproducing this crest more shifted to the front of the plume
we subdivided the aquifer along the y – axis into three sections of different
capacity coefficient corresponding to 0 – 50 m, 50 – 160 m, and 160 – 280 m.
Thus, assigning a different multiplier to the mobile porosity at each zone, we
obtained three zones of porosity within the aquifer.
In the chart presented above we assigned a multiplier of 0.2 to the 2nd zone,
and a multiplier of 5 to the 3rd zone, while the porosity values of the first zone
were left the same (Table 2).
Figure 22 Comparison chart between observed mass distribution profile of the Tritium
plume and simulations using mass transfer model with different capacity
coefficients along downgradient distance.

41. 41
Figure 23 Longitudinal mass distribution profile of the Tritium plume 328 after injection and six predictions using classical advection‐dispersion model for
each hydraulic conductivity field realization.

42. 42
Figure 24 Longitudinal mass distribution profile of the Tritium plume 328 after injection and six predictions using dual – domain mass transfer model for
each hydraulic conductivity field realization.

43. 43
7. CONCLUSIONS
The Macro‐Dispersion Experiment (MADE) site has been highly controversial
for showing an important anomalous behavior in the shape of the tracer
plume. Many conceptually different transport models have been developed
that can partially explain this behavior. Among them, remarkably, Salamon’s
model [2007] has demonstrated that the main plume characteristics, i.e. non‐
Gaussianity of the concentration profile, should be attributed to aquifer
heterogeneity (spatial variation in hydraulic conductivity). This model did
not include Darcy‐scale mass transfer processes that are believed to occur at
the test site.
In this work we have evaluated the relative importance of Darcy‐scale mass
transfer processes at the MADE site. To do this, we modified Salamon’s model
to incorporate Darcy‐scale mass transfer processes represented through an
additional source/sink term in the classical advection‐dispersion equation.
For simplicity, the single‐rate mass transfer model was selected. Our study
only considers the six successful stochastic realizations of the hydraulic
conductivity field from Salamon et al. (2007).
We found that the new dual – domain mass transfer model seemed to offer a
more plausible interpretation of the anomalous behavior of the Tritium
plume as it improves the prediction of the tracer front profile. In particular,
the interplay between mass transfer processes and tracer velocity has been
shown to be significant. When a positive correlation between the solute
residence time in the immobile domain and mobile domain exist, the
longitudinal profile of the tracer mass in the aquifer exhibited the presence of
a double peak.

44. 44
8. REFERENCES
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Feehley, C. E., & Zheng, C. (2000). A dual‐domain mass transfer approach for
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47. 47
APPENDIX 1
The RW3D Code
The program is designed to calculate the following outputs according to the utility
of the study:
• Particle Breakthrough Curve;
• Cumulative Particle Breakthrough Curve;
• Particle Cloud Evolution;
• Particle Paths;
• Cartesian Spatial Moments;
• Spatial Moments of Particle Position at Control Planes;
• Particle Position at Control Planes;
• Dilution Index of Kitanidis;
• Radial Spatial Moments;
• Temporal Moments;
• Dispersivities from Control Planes.
The input file is divided into many packages accounting for the parameters in
brackets:
• Domain’s Geometry (grid’s spacing and boundary conditions);
• Time Discretization (constant move or constant time method);
• Advection (Eulerian or Exponential method, unitary flux, porosity);
• Dispersion (longitudinal, horizontal transverse and vertical transverse
dispersivity);
• Control Surfaces (wells or planes with respective coordinates);
• Reactive Package (e.g. Reactive Power Law Distribution of Mass Transfer,
Reactive Multirate Mass Transfer, Reactive Equilibrium Linear Sorption,
Reactive Multirate Mass Transfer, Reactive Spherical Diffusion).