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Abstract

A sequential procedure of hydraulic tomographical inversion is applied to characterize at high resolution the spatial heterogeneity of hydraulic conductivity and specific storage at the field test site Stegemühle, Germany. The shallow aquifer at this site is examined by five short-term multilevel pumping tests with 30 pumping-observation pairs between two wells. Utilizing travel time diagnostics of the recorded pressure response curves, fast eikonal-based inversion is shown to deliver insight into the sedimentary structures. Thus, the structural information from the generated travel time tomogram is exploited to constrain full calibration of the pressure response curves. Based on lateral extrapolation from the measured inter-well profile, a three-dimensional reconstruction of the aquifer is obtained. It is demonstrated that calibration of spatially variable specific storage in addition to hydraulic conductivity can improve the fitting of the model while the structural features are only slightly changed. At the field site, two tracer tests with uranine and sodium-naphthionate were also performed and their concentrations were monitored for 2 months. The measured tracer breakthrough curves are employed for independent validation of the hydraulic tomographical reconstruction. It is demonstrated that major features of the observed solute transport can be reproduced, and structures relevant for macrodispersive tracer spreading could be resolved. However, for the mildly heterogeneous aquifer, the tracer breakthrough curves can also be approximated by a simplified homogeneous model with higher dispersivity. Therefore, improved validation results that capture specific characteristics of the breakthrough curves would require additional hydraulic measurements.
RESEARCH ARTICLE
10.1002/2014WR016402
Prediction of solute transport in a heterogeneous aquifer
utilizing hydraulic conductivity and specific storage
tomograms
S. Jim
enez
1
, R. Brauchler
1,2
,R.Hu
3
,L.Hu
1
, S. Schmidt
3
, T. Ptak
3
, and P. Bayer
1
1
Department of Earth Sciences, ETH Zurich, Zurich, Switzerland,
2
Now at AF-Consult Switzerland Ltd., Baden, Switzerland,
3
Geoscientific Centre, University of G
ottingen, G
ottingen, Germany
Abstract A sequential procedure of hydraulic tomographical inversion is applied to characterize at high
resolution the spatial heterogeneity of hydraulic conductivity and specific storage at the field test site
Stegem
uhle, Germany. The shallow aquifer at this site is examined by five short-term multilevel pumping
tests with 30 pumping-observation pairs between two wells. Utilizing travel time diagnostics of the
recorded pressure response curves, fast eikonal-based inversion is shown to deliver insight into the sedi-
mentary structures. Thus, the structural information from the generated travel time tomogram is exploited
to constrain full calibration of the pressure response curves. Based on lateral extrapolation from the meas-
ured inter-well profile, a three-dimensional reconstruction of the aquifer is obtained. It is demonstrated that
calibration of spatially variable specific storage in addition to hydraulic conductivity can improve the fitting
of the model while the structural features are only slightly changed. At the field site, two tracer tests with
uranine and sodium-naphthionate were also performed and their concentrations were monitored for 2
months. The measured tracer breakthrough curves are employed for independent validation of the hydrau-
lic tomographical reconstruction. It is demonstrated that major features of the observed solute transport
can be reproduced, and structures relevant for macrodispersive tracer spreading could be resolved. How-
ever, for the mildly heterogeneous aquifer, the tracer breakthrough curves can also be approximated by a
simplified homogeneous model with higher dispersivity. Therefore, improved validation results that capture
specific characteristics of the breakthrough curves would require additional hydraulic measurements.
1. Introduction
Nearly all hydraulic tomographic field studies are driven by the need to provide spatial high-resolution
parameter fields for solute transport predictions. In fact, tomographic approaches, among others [e.g.,
Mariethoz et al., 2010], can resolve sedimentary structures or fractures that control preferential flow. Their
potential and superiority to traditional field investigation techniques was demonstrated in several previous
studies [Gottlieb and Dietrich, 1995; Yeh and Liu, 2000; Illman et al., 2010; Berg and Illman, 2011a, 2015]. How-
ever, the effort of data collection and data evaluation is higher for tomographic investigation methods in
comparison to conventional methods that avoid spatial assignment of estimated hydraulic parameters. This
motivates a strong interest for enhanced tomographic field and inversion techniques [Bohling et al., 2002;
Zhu and Yeh, 2006; Lochb
uhler et al., 2013]. Naturally, the development of new field technologies and field
data collection strategies is delayed in time in comparison to the computer-based development of inversion
schemes. Numerical studies with virtual aquifers are essential means for motivating, developing, and testing
new schemes, but their viability can only be approved by often laborious field experiments. Therefore, espe-
cially during the last few years, the number of field studies has been catching up. These started from simpli-
fied two-dimensional, depth-integrated characterizations [e.g., Straface et al., 2007b] to arrive at full three-
dimensional reconstructions [e.g., Illman et al., 2009; Berg and Illman, 2011b; Cardiff et al., 2013] based on a
large number of interference tests.
For field investigations, one of the important and at the same time most challenging tasks is the evaluation
of the significance and reliability of the reconstructed hydraulic parameter fields. Independent from the
inversion technique, all field studies use the residual error from data fitting as a first measure for the quality
of their inversion results. Unfortunately, this information is not sufficient because a large number of
Key Points:
Sequential hydraulic tomographical
inversion
Combination of structural
information and full signal inversion
Tracer tests used as independent
validation
Correspondence to:
S. Jimenez,
santos.jimenez@erdw.ethz.ch
Citation:
Jim
enez, S., R. Brauchler, R. Hu, L. Hu,
S. Schmidt, T. Ptak, and P. Bayer (2015),
Prediction of solute transport in a
heterogeneous aquifer utilizing
hydraulic conductivity and specific
storage tomograms, Water Resour. Res.,
51, 5504–5520, doi:10.1002/
2014WR016402.
Received 18 SEP 2014
Accepted 13 JUN 2015
Accepted article online 17 JUN 2015
Published online 19 JUL 2015
V
C2015. American Geophysical Union.
All Rights Reserved.
JIM
ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5504
Water Resources Research
PUBLICATIONS
parameter distributions might exist that equally honor the measured data. Hence, independent information
and measures have to be exploited to evaluate the quality of the reconstructed parameter fields. Geological
information such as deposition information or fault information based on a detailed structural geological
study is utilized by Straface et al. [2007a] and Illman et al. [2009] to support reconstructed parameter fields.
Berg and Illman [2012] used a large number of permeameter and grain size tests in combination with multi-
level slug tests to interpret the estimated hydraulic conductivity (K) fields. The tests were performed at the
North Campus Research Site (NCRS), Waterloo, Canada, in a heterogeneous confined aquifer built up by tills
and glaciofluvial deposits. The high vertical resolution of multilevel slug tests and direct-push injection-logs
were exploited by Brauchler et al. [2010, 2013] to interpret reconstructed diffusivity fields estimated at the
Stegem
uhle Site, G
ottingen, Germany. This site is characterized by a shallow confined aquifer consisting of
fluviatile sediments. In comparison to the conditions at the NCRS [variance of log conductivity, r
K2
51.72,
Alexander et al., 2011], the aquifer at the Stegem
uhle Site is less heterogeneous (r
K2
50.2).
Cardiff et al. [2012, 2013] compared porosity logs and multilevel slug tests with reconstructed parameter
fields. They performed a three-dimensional (3-D) transient hydraulic tomographic field experiment utilizing
highly flexible packer systems at the Boise Hydrogeophysical Research Site (BHRS), USA. The BHRS site [e.g.,
Straface et al., 2011; Cardiff et al., 2013] is characterized by a mixed sand/gravel/cobble facies and in com-
parison to the other two test sites it shows unconfined conditions.
Hydraulic tomographical measurements were performed by Vasco and Karasaki [2001, 2006] to reconstruct
preferential flow paths at the Raymond Field Site, California, USA. Intensive geological experiments allowed
for a comparison of identified preferential flow paths, imaged in the hydraulic tomograms, with borehole
conductivity logs and seismic tomograms. Such utilization of independently collected data for comparison
with the reconstructed hydraulic tomograms can be a challenge due to different observation scale and mis-
match in resolution [e.g., Brauchler et al., 2012]. Huang et al. [2011] successfully applied independent valida-
tion pumping tests to field data recorded at the test site of the National Yunlin University of Science and
Technology in Taiwan. The site consists of fluviatile sediments with mean hydraulic conductivity values of
around 10
24
m/s, which are comparable to those found at the Stegem
uhle Site. A more direct way to vali-
date tomograms is to use direct visual comparisons between inverted hydraulic conductivity and laboratory
experiments [e.g., Illman et al., 2007, 2010].
Although the main motivation of hydraulic tomographic field studies is to provide high-resolution informa-
tion for solute transport predictions, only a small number of field studies were published that employ tracer
test data to interpret or validate reconstructed tomograms. Bohling et al. [2007] utilized a solute tracer test
to evaluate the capability of steady-shape tomography. They show that the tracer test could support the
existence of a highly conductive layer, which was reconstructed by hydraulic tomography. For further verifi-
cation, a large number of small-scale hydraulic tests were performed at the Geohydrologic Experimental
and Monitoring Site (GEMS), USA, but none of these revealed the presence of this high-conductivity zone.
The GEMS site is a heavily studied alluvial confined aquifer that consists of 11 m of sand and gravel overlain
by silt and clay. Another field example utilizing tracer test data and flow data for inversion was presented
by Vasco and Finsterle [2004] at the Grimsel Rock Laboratory in Switzerland. In contrast to the work of Boh-
ling et al. [2007], the tracer test data were not used for validation. Instead, the different data types were
inverted together to improve the significance of the reconstructed hydraulic tomograms. Illman et al. [2012]
showed that estimated hydraulic conductivity tomograms predicted better tracer distribution patterns dur-
ing a dipole tracer test than other traditional methodologies (i.e., effective parameter/macrodispersion
approach or heterogeneous approach using ordinary kriging based on core samples). They also emphasize
the difficulties of capturing details of the tracer breakthrough due to intrinsic methodological limitations,
such as effects of noise in head measurements and ‘‘the less diffusive nature of the tracer which demands a
much higher resolution mapping of the K-field.’’ Ni et al. [2009] compared the predictive capabilities of a 2-
D tomographic reconstruction to that of a homogeneous model. In their theoretical study, they showed
that the tomographic variant was capable of reproducing tracer breakthrough curves (BTCs) independently
of the transport distance. In contrast, a homogeneous advection-dispersion model with empirically esti-
mated dispersivity could not match the BTC. When calibrated to the BTC, the homogeneous model would
properly reproduce the BTC, but obtained dispersivity would need to be raised with transport distance.
In this paper, our main objective is to validate subsurface reconstructions from hydraulic tomographic inver-
sion for predicting solute transport in the field. Therefore, we choose the hydraulic tomography procedure
Water Resources Research 10.1002/2014WR016402
JIM
ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5505
developed by Jim
enez et al. [2013], which combines eikonal and pilot point-based inversion approaches.
The procedure was originally presented for the reconstruction of a K-field and theoretically assessed by
application to a virtual aquifer. Here we further develop and adapt it to the requirements of the simultane-
ous 3-D reconstruction of specific storage and hydraulic conductivity. We apply it to the inversion of short-
term pumping tests at the Stegem
uhle site. Between the same wells originally used for the hydraulic tomo-
graphic investigations by Brauchler et al. [2013], a forced gradient tracer test is performed. Two fluorescent
tracers are injected in two different depths of the aquifer and the BTCs are recorded in the observation well
over the entire thickness of the aquifer. The tracer test results are contrasted with the new findings from
hydraulic inversion, and we reveal, to what extent and at which accuracy it is feasible to reconstruct struc-
tures relevant for subsurface transport. Similarly to Ni et al. [2009], we also compare the BTC prediction by
reconstructed model to that of a homogenous one.
2. Material and Methods
2.1. Field Site and Experiments
2.1.1. Stegem
uhle Site
The experimental field test site Stegem
uhle is located south of the city of G
ottingen, Lower Saxony,
Germany (Figure 1). In order to carry out hydrogeological and hydrochemical field research under
controlled natural conditions, five 100-observation, twenty-one 200 -observation (five of them are multi-
chamber wells), and three 600-observation wells were installed during the period of 2006–2011. The
composition of the shallow subsurface was determined by a variety of methods such as inspection
of sediment cores, grain size analysis, direct-push electrical conductivity logging, borehole gamma-ray
logging, electrical resistivity tomography, and seismic travel time inversion [e.g., Hu, 2007; Vogt, 2007;
Meyer et al., 2014; Hu, 2011]. The aquifer is composed of unconsolidated fluviatile sediments (sand
and gravel) of Quaternary age (Weichsel Glaciation). These sediments have a varying thickness of 1.0–
3.3 m and are overlain by alluvial clay. The aquifer bottom is at a depth of 1.9–7.0 m below land surface with
erosional contact to the underlying clay stone formation of Middle Keuper Age. In the middle of the field site,
which is the focus area of this study, the aquifer exhibits confined conditions. Here Hu [2011] and Brauchler
et al. [2013] applied multilevel slug tests and observed vertically varying hydraulic conductivities with higher
values at the bottom of the aquifer.
2.1.2. Field Implementation of Hydraulic Tests
A series of cross-well multilevel pumping tests were performed at the test site Stegem
uhle, implementing a
tomographic array along a straight line between a pumping well (P0/M25) and an observation well (P5/
M17.5) (Table 1). The distance between these two wells is 9 m (Figure 1). During each pumping test, the
water was partially pumped out of the pumping well P0/M25 by employing double packer systems with a
screened interval of 0.25 m. The tube connected to the pump has an internal diameter (ID) of 0.031 m. The
observation well P5/M17.5 (Figure 2) is a multichamber well constructed with the Continuous Multichannel
Tubing (CMT) System [Einarson and Cherry, 2002]. This well consists of a pipe with six continuous separate
Figure 1. Monitoring well network at the Stegem
uhle test site. The 200 wells are colored black, and multichamber wells are colored in blue.
The plane between P5/M17.5 and P0/M25 corresponds to the eikonal inversion domain.
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JIM
ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5506
channels (ID 50.014 m), which are
arranged in a honeycomb shape and lead
to different depths. This design allows for
the measurement of water level changes at
different depths of the aquifer.
For the profile between the pumping well
and the observation well, five short-term
pumping tests were carried out. For every
short-term pumping test and every pump-
ing interval, the pressure changes in the six
different depths of the multichamber wells
were recorded at a frequency of 50 Hz with the pressure transducer (PDCR 35/D-8070) connected to a data
logger (Campbell ScientificV
RCR 3000). By varying the pumping interval, a total number of 30 (5 36) draw-
down curves for the profile were recorded [Brauchler et al., 2013]. The pumping tests in series produced a
pattern of crossing trajectories between test and observation well, similar to the paths of a radar or seismic
experiment. The travel times and hydraulic attenuations between the wells P0/M25 and P5/M17.5 thus can
be utilized for eikonal-based cross-sectional reconstruction of hydraulic parameter distributions.
2.1.3. Field Implementation of Tracer Tests
Tracer test data are used for independent validation of the derived aquifer model. Nonreactive tracer tests
are effective means to identify preferential flow paths or integral transport parameters of the subsurface,
such as porosity and dispersivity. They can be conducted under natural gradient or forced gradient condi-
tions. Compared to the natural gradient tracer test, the forced gradient tracer test is hydraulically well
Table 1. Basic Informati on of the Two Wells P0/M25 and P5/M17.5 Used
for Hydraulic Tomography Inversion and Tracer Testing
P0/M25 P5/M17.5
Type Single screen Multichamber
Aquifer thickness (m) 1.98 1.99
Well height (m) 0.87 0.65
Elevation of the well top (m.a.s.l.) 152.23 151.54
Surface elevation (m.a.s.l.) 151.36 150.89
Well bottom (m.a.s.l.) 145.28 145.3
Aquifer top (m.a.s.l) 147.382 147.415
Aquifer bottom (m.a.s.l) 145.401 145.419
02040
60 80 100
6
4
2
0
151.36
149.36
147.36
145.36
m.a.s.l.
6
4
2
0
30 40 50 60 70 80
EC (mS/m)
EC (mS/m)
depth (m)
depth (m)
P0/M25 P5/M17.5
sodium-naphthionate
uranine
9 m
fluorometer
pump
sampling
bottle
Aquifer
Figure 2. Setup of the tracer experiment at Stegem
uhle site with well configuration and illustration of injection levels for uranine and
sodium-naphthionate. Additionally, electrical conductivity (EC) measurements are depicted (in red) for both wells, which delineate the
aquifer boundaries.
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JIM
ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5507
controlled and not constrained to a given
natural flow field. Moreover, higher mass
recovery rates can be achieved [e.g., Ptak
et al., 2004]. Consequently, forced gradient
conditions were also favored at the field
site.
The experiment was conducted between
the same wells previously used for the
pumping tests (P0/m25, P5/m17.5). We
selected two different tracers with similar
transport properties, the fluorescent dye
tracers uranine and sodium-naphthionate [Leibundgut et al., 2009], in order to allow for a robust analysis
and to mitigate the effects of possible measurement errors, data noise, or tracer-specific transport behavior.
Prior to tracer injection, a steady radial flow field was established by continuous water extraction from the sin-
gle screen well at a constant rate of 0.3 L/s. A mass of 15 g uranine and 150 g sodium-naphthionate dissolved
in water were injected into the multichamber well, each in a different depth (Figure 2). The injection periods of
uranine and sodium-naphthionate were 5 min and 3 min, respectively, and by this a pulse-like injection was
realized. The different injection periods were employed due to technical reasons. They have negligible influ-
ence on the results as they are very small compared to the time of tracer arrival. At the monitoring well depth-
integrated concentrations were measured every 30 s by a flow-through field fluorometer (type GGUN-FL30),
calibrated to the specific tracers and local groundwater conditions. For quality control, at some timepoints the
pumped-out water was also manually sampled and subsequently analyzed in the laboratory. The entire tracer
experiment lasted 2 months, and the water levels at the two wells were measured every few days to ensure
the flow field was steady. Detailed information on experimental settings is summarized in Table 2.
2.2. Tomographic Inversion
The inversion procedure we use here encompasses the sequential application of an eikonal-based and pilot
point-based inversion scheme. It was presented in detail by Jim
enez et al. [2013] and makes use of the
strengths of both inversions and minimizes drawbacks. On the one side, eikonal-based inversion is a fast,
well-tested methodology capable of providing insight into hydraulic parameters and aquifer structure.
Strictly speaking it represents an approximation, since it treats the parabolic flow equation as a wave equa-
tion. In addition, the eikonal approach, as presented in this paper, only uses a time diagnostic from the
whole pressure signal leaving the rest of the information unused. As a consequence, the eikonal-based
inversion is highly efficient for fast detection of structures, but as an approximation it is less accurate in esti-
mation of hydraulic parameter values. On the other side, pilot point-based inversion commonly works by fit-
ting the flow equation, using the full recorded pressure signal and thus making full use of the measured
information. Therefore, hydraulic parameter values can be determined, but this is computationally demand-
ing, especially for 3-D reconstructions.
Jim
enez et al. [2013] showed how to link both schemes in a synergetic way (Figure 3). Eikonal-based inver-
sion is utilized to extract structures from reconstructed diffusivity fields (D-tomograms) but not parameter
values. Since we are interested mainly in K, the original procedure is refined here by utilizing eikonal-
based estimates of K-distribution rather than diffusivity fields. As a bridging step, specific storage (S
s
)-
tomograms are developed by attenuation tomography [Brauchler et al., 2013]. Given D5K/S
s
, a fully
eikonal-based K-tomogram can be derived from the D-tomograms and S
s
-tomograms. The reliability of
the tomographic models is assessed by means of null-space energy maps. A null-space energy map repre-
sents a measure of the reliability of a tomogram. It relates the trajectory distribution to the mesh used for
the discretization of the investigated area [e.g., B
ohm and Vesnaver, 1996] and comprises a singular value
decomposition of the tomographic matrix.
We consider the eikonal-based K-distribution as a proxy, but carrying valuable insight into subsurface struc-
tures. For extracting this information, it is converted into a zonal image (conceptual map) using a k-means
clustering algorithm. Pilot points serve as auxiliary variables that are often combined with regularization
Table 2. Experimental Design of Two Tracer Tests at the Stegem
uhle
Field Site
Uranine
Sodium-
Naphthionate
Pumping rate (L/s) 0.301–0.305
Injection start
date and time
18 Oct 2011, 13:45 18 Oct 2011, 13:53
Injection mass (g) 15 150
Injection chamber 6 1
Height of chamber (m.a.s.l) 145.761 147.261
Injection duration (s) 300 180
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ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5508
techniques to fill a model parameter field. During the pilot point-based inversion procedure here, the infor-
mation content of the conceptual maps is exploited as follows [Jim
enez et al., 2013]:
Initial Values. Each individual pilot point is assigned an initial value equal to that of the corresponding clus-
ter centroid.
Pilot Point Positioning. Higher pilot point density is favored at locations where parametric variability is sus-
pected, i.e., at cluster boundaries. This step is performed using a finite element mesh generator. For each
cluster a mesh is designed, and at each node a fixed pilot point is assigned. This procedure automatically
leads to a refinement at the cluster boundaries.
Regularization. Interpolation among the pilot points is based on the conceptual maps as well. Three condi-
tions are proposed, which have to be fulfilled so that two pilot points are correlated: (i) both pilot points
pertain to the same cluster; (ii) the distance between the pair of pilot points is smaller than the average
length of the cluster in horizontal direction; and (iii) there must be no other pilot point from a different clus-
ter within a given space of influence [Jim
enez et al., 2013]. For the regularization implementation, i.e., the
spatial relationships among the pilot points, a graph theoretical concept is adopted [e.g., Bhark et al., 2011].
Initially, prior to the calibration, each pair of pilot points is examined and an adjacency matrix is developed:
0 aij
..
.
aij  0
0
B
B
B
@
1
C
C
C
A
Cðpi;pjÞ5
1;conditions fulfilled
0;otherwise
((1)
where i5j51;...;number of pilot poins,aij is a boolean indicator, pdenotes a pilot point, and CðÞis
the adjacency matrix. The adjacency matrix dictates if two pilot points are connected in a graph or not,
based on the three conditions listed above, and it is used to calculate the regularization function, Ur.
Note that we want to arrive at a 3-D parameter field, but the conceptual map gives only insight into struc-
tures in a 2-D vertical slice between source and receiver well. For 3-D extrapolation, variograms along the
horizontal and vertical axes are constructed from the eikonal-based diffusivity tomogram. Along the tomo-
gram, pilot point values are considered as hard data, and the values for each cell of a given numerical
model grid are derived from 3-D kriging.
Transient
pressure
signal
Eikonal
inversion
Pilot point
inversion
Specific storage
tomogram
Reliability
Diffusivity
tomogram
Conceptual
map
Pilot point
distribution
Initial values
Tikhonov
regularization
Hydraulic
reconstruction
Travel time diagnosticFull signal
Tracer test
Validation
Hydraulic
conductivity
tomogram
Figure 3. Main elements of sequential inversion procedure: the transient pressure signal is inverted by an eikonal-based approach to
deliver a conceptual map. This structural information is used to constrain full pressure signal inversion based on pilot points.
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JIM
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The hydraulic parameters with spatial heterogeneity, which are addressed by the inversion procedure, are K
and S
s
. On one hand, the resulting K-field and S
s
-field must honor the recorded pressure response data. This
is evaluated by implementation in a flow model and comparison to the field data. On the other hand, the
parameter fields are constrained by regularization. This is described by an objective function, which is
solved based on Lagrange multipliers [Doherty, 2010; Jim
enez et al., 2013].
2.3. Numerical Modeling
2.3.1. Hydraulic Test Simulation
Inversion methods such as the pilot point-based approach use a large number of iterative flow model runs.
Therefore, any possibilities for minimizing simulation time are of interest. An appealing option is using
locally refined grids, and accordingly hydraulic cross-well simulations were performed with MODFLOW-LGR
[Mehl and Hill, 2005; Vilhelmsen et al., 2012], a transient, three-dimensional groundwater flow code.
MODFLOW-LGR allows local refinement of a finite difference grid, as an extension to the classical MOD-
FLOW code. It couples two or more finite difference grids called parent and child. A parent grid can cover a
large area in order to accommodate regional flow and boundaries. A much more refined child grid can be
used to study more local phenomena, for instance, the hydraulic effects in the vicinity of a pumping well.
2.3.2. Tracer Test Simulation
By numerical simulation of the tracer test with the reconstructed aquifer, the suitability of the hydraulic
tomography approach for predicting solute transport can be evaluated. For this purpose, the reconstructed
3-D aquifer is implemented in a flow and transport model, and the simulated results are contrasted with
those by a homogeneous case. The heterogeneous model represents the full heterogeneity in Kand S
s
obtained from the tomographical inversion procedure. In the homogeneous model, Kand S
s
are set con-
stant, taking the mean of the estimated values. For flow modeling, the forward model used for pilot point-
based full signal inversion is selected. The code MT3DMS [Zheng and Wang, 1999] is chosen for solving sol-
ute transport.
By comparing measured and, by these models, simulated tracer BTCs, we can assess the gain from resolving
aquifer heterogeneity and also validate the inverted model. Still, additional parameters need to be specified
before the transport models can be run. Crucial unknowns are dispersivity and effective porosity. Since
these parameters cannot be determined separately with sufficient accuracy, we consider their possible
value ranges, and estimate the most likely parameter ranges for homogeneous and heterogeneous models
through a Bayesian approach, a Markov Chain Monte Carlo (MCMC) sampling procedure. It utilizes the
Metropolis-Hastings algorithm [Metropolis et al., 1953; Hastings, 1970] to sample realizations of longitudinal
dispersivity and effective porosity, separately for heterogeneous and homogeneous models. For simplifica-
tion, transversal dispersivity is set 1/10 the value of the longitudinal one, which is a rough but common
assumption in related work [Molina-Giraldo et al., 2011]. For generating new realizations within the MCMC
framework, (i) one of the parameters is selected randomly, (ii) a new parameter value is proposed using a
Gaussian random walk, and (iii) the acceptance ratio ais computed:
a5min 1;LðmnewÞ
LðmoldÞ
gm
new !mold

gm
old !mnew
ðÞ
(2)
where Lis the likelihood function, gis the proposed distribution, and mdenotes a model parameterization.
Finally, a random number uis drawn from a uniform distribution on [0, 1] and the realization is accepted if
a>u, and rejected otherwise. As search criterion, the RMSE between measured and modeled tracer BTCs is
selected. The function that maps from RMSE to likelihood is L5102RMSE
2r2with requal to 0.2.
3. Results
3.1. Eikonal-Based Inversion of Hydraulic Tests
Brauchler et al. [2013] reconstructed a diffusivity (D) and specific storage tomogram (S
s
) utilizing the eikonal-
based inversion approach. The derived tomograms displayed in Figures 4a–4e are shortly discussed in the
following; however, for more details we refer to Brauchler et al. [2013].
For the eikonal-based inversion, a starting 2-D model domain of 45 cells was applied. Utilizing the method
of staggered grid, the mesh was shifted four times in the horizontal direction and three times in vertical
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direction, which led to the final resolution of 540 pixels imaged in Figure 4. For the diffusivity reconstruction
displayed in Figure 4a, the 50% travel time diagnostic was employed for the inversion. The travel time and
attenuation inversion, including the staggered grid calculation, took less than 1 min on a conventional note-
book. For quantifying the reliability of the D-tomograms and S
s
-tomograms (Figures 4a and 4c), null-space
energy maps are provided in Figures 4b and 4d. These maps illustrate the uncertainty associated with the
eikonal-based inversion. A value of 1 of the null-space (black color) means the lowest possible confidence
on the values obtained for a given cell in a tomogram, and a value of 0 (white color) indicates highest
confidence.
The D-tomograms and S
s
-tomograms indicate horizontal layering (Figures 4a and 4c), with values of D
between 2 and 20 m
2
/s, and of S
s
between 3 310
25
and 10
24
m
21
. The K-tomogram (Figure 4e) is
obtained by D5K/S
s
. It shows a similar structure, with moderate heterogeneity and values ranging from
K510
24
to 10
23
m/s, which are considered typical for sand and gravel aquifers.
3.2. Conceptual Map and Pilot Points Configuration
The conceptual map to support the subsequent pilot point inversion was developed from the eikonal-
based results and is displayed in Figure 4f. The number of clusters was set to four, which, after visual inspec-
tion, was considered the maximum possible to keep the main structures of the tomograms. Following
Jim
enez et al. [2013], cells associated with high null-space energy values (here >0.9) were ignored during
clustering. These gap cells, which are mainly located close to the top boundary, were filled by nearest
neighbor interpolation from adjacent cells.
The conceptual map is used to guide pilot point positioning and setting initial values for hydraulic parame-
ter (Kand S
s
) calibration. This involves transforming the conceptual map into a vector graph in order to
obtain a digital image of the cluster interfaces. The latter are displayed as black lines in Figure 4g. Then a
mesh is assigned to the vector graph using a finite element mesh generator. The mesh nodes are translated
Figure 4. (a and b) Reconstructed diffusivity tomogram and the associated null-space energy map. (c and d) Reconstructed specific storage tomogram and the associated null-space
energy map. (e) Computed hydrauli c conductivity tomogram using D 5K/Ss [see Brauchler et al., 2013]. (f) Resulting cluster distribution based on hydraulic conductivity tomogram.
(g) Pilot point distribution, cluster IDs, and Tikhonov regularization connections (gray lines). (h) Resulting hydraulic conductivity from pilot points inversion using K and Ss as a parameter
(longitudinal slice of the 3-D domain, Figure 8b, for comparison purposes).
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ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5511
into pilot point positions. Each pilot point an initial
value equal to the value of the corresponding clus-
ter centroid is assigned.
The selected mesh generator uses Delaunay trian-
gulation. The utilization of the mesh generator
leads to a higher pilot point density along the clus-
ter boundaries. This distribution is favorable
because along these boundaries the largest con-
trasts in hydraulic properties are expected. A maxi-
mum element size of 0.7 m and a resolution curvature of 0.3 are selected. Both parameters control the
mesh (i.e., pilot point) density and how fast it declines away from the cluster boundaries. The maximum ele-
ment size determines how big each grid element can be and the resolution curvature limits the mesh size
along a curved boundary [COMSOL Inc\., 2012]. Accordingly, the lower the values for each parameter are,
the more pilot points are allocated. The maximum number of nodes is set to 1200. Table 3 lists the number
of pilot points assigned to each cluster. The maximum element number, which controls the total number of
pilot points, is a compromise between available computational resources, available observations and
desired resolution.
The regularization step for interpolation among the pilot points is also guided by the conceptual maps.
Based on their neighborhood relationships, pilot points are connected in pairs, yielding separate networks
that delineate the structures observed in the conceptual maps. In Figure 4g, these networks are illustrated
as gray lines. For the maximum distance between a pair of pilot points 3 m were chosen, which equals the
average length of the clusters in horizontal direction, and for the calculation of the space of influence an
angle of 308was set [e.g., Jim
enez et al., 2013].
For interpolating between the pilot points and lateral extrapolation in 3-D, ordinary kriging is applied. The
diffusivity tomogram displayed in Figure 4a is utilized to derive the underlying semivariograms depicted in
Figure 5. As expected for fluviatile sediments, we find a larger range in the horizontal (2.3 m) than in the
vertical direction (1 m). During the subsequent calibration, the Kand S
s
values at the pilot point cells are
tuned, and by kriging the values of the other cells are filled. In the numerical model, pilot points exist only
in the vertical tomogram slice between the source and receiver well. For lateral extrapolation, we assume
stationary, horizontally isotropic geostatistical properties and consistently use kriging with the same semi-
variograms. However, it is clear that due to the missing field data away from the well couple, the resulting
3-D field will lose reliability in lateral distance. Alternatively, several tomograms from different source and
receiver wells may be collected from the field and combined for 3-D inversion, such as shown in Berg and Ill-
man [2012].
3.3. Simulation of Pumping Tests
In the numerical groundwater flow model domain, the refined child grid (13 m 34m32 m) is embedded
in the coarser parent grid (60 m 360 m 32 m). The selected refinement ratio between parent and child
Table 3. Number of Pilot Points and Connections for Each
Cluster
Cluster ID
Number of
Pilot Points
Number of
Connections
1 75 358
2 96 301
3 125 369
4 67 238
Total 363 1266
0 0.4 0.8 1.2 1.6 2.0
distance (m)
0
4·10
8·10
12·10
semi-variance
-8
-8
-8
2.2 2.3
Figure 5. Semivariogram derived from the diffusivity tomogram with horizontal (red) and vertical search direction (black).
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ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5512
grid is 5:1, resulting in total around 1 million cells. Fixed hydraulic heads are implemented to enforce the
regional hydraulic gradient (0.004) measured in the field.
The child grid simulates the source (P0/M25) and receiver well (P5/M17.5) from the field test. Along the
source borehole, five vertical screens are implemented. This resembles those screens in the field at which
the pumping tests were performed. At the receiver an array of six observation points is defined in the
model to simulate the multichamber well configuration.
For the calibration, the repeated pumping tests at the source well are simulated by transient flow modeling.
Each test simulates a 140 s pumping at 0.3 L/s. With five tests at different depths and six receivers, 30 pressure
response curves are recorded in total. Each pressure signal is discretized by 300 points. Two choices are com-
pared for fitting these curves to those measured at the field site. In the first one, only Kis calibrated, and S
s
is set
a constant value of S
s,mean
55.5 310
24
m
21
equal to the mean of the tomogram (Figure 4c). The initial values
for Kare specified equal to the centroids of the clusters. In the second one, both parameters are independently
adjusted, which means that the number of decision variables is doubled. However, to save computational time,
here the initial values of Kare selected according to the results from previous calibration of Konly.
The selected optimization algorithm (Levenberg-Marquardt) is suited for parallelization, and a cluster with a
desktop (Intel i7 3.4 GHz, 16 GB RAM) and a workstation (Intel Xeon E5 3.1 GHz, 64 GB RAM) was used. The
number of parallel runs was 20, distributed on both machines. A single model run took approximately 5
min. A total of 1130 models runs were needed. The full pilot points inversion took approximately 5 h.
3.4. Calibrated Hydraulic Parameter Fields
The hydraulic parameters at the Stegem
uhle site show low variability in comparison with the conditions at
other test sites such as NCRS, where similar experiments were conducted [e.g., Berg and Illman, 2012]. This
is reflected in the pressure response curves (Figure 6) which all show a similar behavior. Despite that, in
order to be able to resolve heterogeneous structures, we suggest to make full use of the measured informa-
tion while exploiting the degrees of freedom in the hydraulic model. This means, the calibration procedure
is applied to fit all and the complete pressure response curves, and this is achieved by not only calibrating
the spatial distribution of Kbut also S
s
.
Figure 6 compares the 30 measured pressure response curves with those calibrated by Kadjustment only.
The fitting error (root-mean-squared error, RMSE) is minimized to 5 310
24
m, and it is shown that most
0
0.005
0.01
146.93145.75 146.03 146.33 146.62 147.22
0
0.005
0.01
0
0.005
0.01
0.005
0.01
0
0.005
0.01
145.71
146.01
146.31
146.61
146.91
0
0 40 80 120 0 40 80 120 0 40 80 120 0 40 80 120 0 40 80 120 0 40 80 120
Drawdown [m]
Time [s]
Depth of observed interval (masl)
Depth of source interval (masl)
Observed
Simulated
Figure 6. Observed and modeled pressure responses by adjusting only hydraulic conductivity (K) during 3-D pilot point-based inversion.
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ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5513
curves are properly reproduced. This fitting error seems to be the lowest possible with the current paramet-
rization. In some cases, the later stages are not ideally captured. This can be seen for most of the responses
from the fifth interval at the receiver. This is improved by also including S
s
as a decision variable. The result-
ing modeled pressure curves, as depicted in Figure 7, fit better to the measurements, and we reduce the
RMSE to 3 310
24
m.
The resulting K-fields are visualized in Figure 8. Kriging variance or estimation error of the hydraulic parame-
ters assigned to cells increases rapidly from the vertical slice between source and receiver well, which con-
tains the pilot points. Hence, to those cells surrounding the shown central region, mean values are assigned
(also for Figure 9). Including S
s
in the inversion yields a very similar field, and only a central layer with higher
Kfound in Figure 8a appears less accentuated in Figure 8b. Figure 9 depicts the calibrated distribution of S
s
.
It is revealed that highest values of around 7 310
25
m
21
are characteristic for the upper part of the aquifer,
whereas the S
s
in the lower ranges around 4 310
25
m
21
. Comparison of Figures 8 and 9 nicely shows
how the structures are related. With regularization and kriging, these clusters stimulate the calibration of
zones that can be interpreted as individual sedimentary hydrofacies [e.g., Bayer et al., 2011]. The latter are
characterized by specific and fairly constant hydraulic properties, and this is reproduced here by the spatial
correlation between the structures for Kand S
s
in Figures 8 and 9.
The vertical parameter distribution between source and receiver well in the 3-D aquifer model can be
compared with the tomograms reconstructed based on travel time diagnostics (Figure 4). It is shown that
the basic layer structure is mantained, with higher Kvalues on the lower-right section and low values in
the upper section. Same geometries can be recognized in both cluster map and pilot point-based field
(Figures 4f and 4h). In comparison with the travel time-based Ktomogram (Figure 4e), full signal inversion
yields a decrease in the range of K. This is also true for the specific storage ranges. In comparison with
the S
s
-tomogram (from 4 310
25
to 8 310
25
1/m) (Figure 4c), a lower variability is observed in Figure 9
(from 4 310
25
to 7 310
25
1/m).
3.5. Validation of the Reconstructed Aquifer With Tracer Test Data
In order to validate the reconstructed hydraulic parameter fields for predicting solute transport, the uranine
and sodium-naphthionate tracer tests are used. Applying tracer data to validate an inversion procedure
Depth of observed interval (masl)
0
0.005
0.01
0
0.005
0.01
0
0.005
0.01
0.005
0.01
0
0.005
0.01
0 40 80 120 0 40 80 120 0 40 80 120 0 40 80 120 0 40 80 120 0 40 80 120
Drawdown [m]
Time [s]
Depth of source interval (masl)
146.93145.75 146.03 146.33 146.62 147.22
145.71
146.01
146.31
146.61
146.91
Observed
Simulated
Figure 7. Observed and modeled pressure responses by adjusting hydraulic conductivity (K) and specific storage (Ss) during 3-D pilot point-based inversion.
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ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5514
Figure 8. Reconstructed hydraulic conductivity of aquifer using (a) hydraulic conductivity and (b) hydraulic conductivity and specific
storage.
Figure 9. Reconstructed specific storage field using hydraulic conductivity and specific storage tomograms.
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ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5515
based solely on hydraulic data poses evident challenges. Tracer transport is not fully predictable if only
hydraulic data are used. Moreover, the tracer test was performed under a different hydraulic regime, and
therefore comparison between tracer simulation and measurements will elucidate the robustness of the
hydraulic inversion.
As explained, the two tracers were applied between the source and receiver wells used for the hydraulic
inversion. The measured BTCs of the tracers are illustrated in Figure 10. The tracers were monitored for 2
months, but here only the time of breakthrough is shown. A first visual inspection reveals that the curves
follow a nearly ideal shape with early steep increase of concentration and, after a peak is reached, tailing
sets in. With a closer look, we also recognize nonuniformities in both curves. The sodium-naphthionate BTC
shows a small step in the later phase after the peak has passed by. In contrast, before uranine reaches the
maximum peak, an apparent local peak already appears which widens the period when the highest concen-
tration is detected.
Our main question is whether the reconstructed heterogeneity is accurate enough for predicting the trans-
port of the tracers. Aside from this, we also ask if this complexity is needed at all. Therefore, results for the
reconstructed aquifer are compared to the simplest reference, which is simulation with a homogeneous sys-
tem. The main steps are implementation in a flow and transport model, specification of unknown transport
parameters, and comparison of both model results.
For simulation of the groundwater flow velocity field, the same flow model setup as for the pilot points
based inversion was used. The models make use of a 3-D grid in order to accurately capture the heteroge-
neity of the aquifer and to account for potential transversal spreading of the tracer. Several authors recog-
nize the importance of 3-D models [e.g., Liu et al., 2007]. For example, Illman et al. [2008] state ‘‘the
knowledge of detailed 3-D distributions of Kis critical in prediction of contaminant transport.’’ Steady state
conditions are assumed according to the static hydraulic settings during the experiment. The fixed head
boundary conditions establish a constant regional flow field, and the pumping well P0/M25 (Figure 2) is
configured with an extraction rate of 0.3 L/s to simulate the forced gradient conditions generated in the
field. In the heterogeneous model, the reconstructed K-field was implemented, in the homogeneous one,
the arithmetic mean of 1.5 310
24
m/s of the heterogeneous variant was chosen. By using steady state
models, the inverted S
s
-fields are not utilized. However, the K-fields are derived by coupled inversion of K
and S
s
values under transient conditions. This way, the information content of the transient hydraulic experi-
ment is exploited for the steady state flow simulation during the tracer test.
The transport code MT3DMS [Zheng and Wang, 1999] is selected for solving solute transport. The transport
model domain covers 10 m 34m32 m with a spatial discretization of 160 332 340 cells, summing up
to 204,800 cells. Computational time on a 2015 desktop (i7, 16 GB RAM, 250 GB SSD) was approximately 5
min per model run using the method of characteristics.
The value ranges of two unknown parameters for transport modeling, dispersivity and effective porosity
needed to be estimated. For this, two MCMC chains were run, one for the homogeneous model, and other
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
1.2
1.4
C/Cmax
time (days)
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
1.2
1.4
time (days)
a) b)
Figure 10. Breakthrough curves (BTCs) measured at the pumping well during the tracer tests and MCMC realizations for the (a) uranine and (b) sodium-naphthionate.
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ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5516
for the heterogeneous model. Both BTCs are adjusted and a summed up RMSE is computed. Each chain has
a length of 3000 model runs, with a burn-in of 1500. BTCs measured at the pumping well during the tracer
tests are compared with those resulting from the simulation with the obtained MCMC ensemble. In Figure
10, only those for the heterogeneous model are depicted. The BTCs simulated with the homogeneous
model are comparable and not shown here. For both the homogeneous and the heterogeneous model, the
BTCs obtained with different dispersivity and effective porosity realizations spread around the observed
BTC, and no bias or other systematic error is observed.
The nearly ideal shape of the BTCs indicates that the aquifer exhibits only a moderate heterogeneity, and
thus minor differences among homogeneous and heterogeneous model results exist. The homogeneous
variant can appropriately capture the general form of the BTCs given properly tuned dispersivity and effec-
tive porosity values [see also Ni et al., 2009]. The heterogeneous variant is similarly suitable, which however
also means that special characteristics of the BTCs are not resolved. This can be attributed to the limited
resolution of the tomograms and the reconstructed fields in order to delineate local hydraulic heterogene-
ities relevant for solute transport. However, the limitation of an even very detailed reconstruction of aquifer
heterogeneity based on a single source-receiver plane certainly plays a major role, because the tracer trans-
port occurs in a 3-D domain. Therefore, the discrepancy between the BTCs and the heterogeneous model
simulation may also be caused by the applied lateral extrapolation from the vertical source-receiver plane.
For example, the untypical spreading of the uranine BTC around the peak may be due to unseen lateral
aquifer heterogeneity that strongly sidetracks the tracer. The irregular behavior of the measured sodium-
naphthionate curve after around 5 days may indicate that a portion of the tracer is temporary split apart
from the main plume and reaches the pumping well with a time lag. This is observed as local peak and
accentuates the tailing of the earlier main tracer mass fraction. As our 3-D reconstruction is only based on
extrapolation from one profile, more adjacent and ideally differently oriented profiles would be needed for
capturing such lateral heterogeneities.
Lateral tracer loss is also supported by the mass recovery rates of 86% (uranine) and 50% (sodium-naphthio-
nate). The relative low recovery observed for sodium-naphthionate in comparison to uranine during the
test cannot be definitively attributed to a certain reason. As the shape of the tracer BTC does not support
distinctive retardation of the tracer, sorption processes are unlikely the reason. Due to the relatively long
duration of the test and moderate groundwater temperatures (10–128C), a microbiological degradation of
sodium-naphthionate in the aquifer seems possible. Rapid microbiological sodium-naphthionate degrada-
tion was observed by Goldscheider et al. [2001] for water samples with a certain storage period, depending
1
Longitudial dispersivity (m)
RMSE
0.5 1.5 2 2.5 3 3.5 40
0.4
0.2
0.3
0.1
0
Figure 11. Ensemble of MCMC-based results of longitudinal dispersivities for the homogeneous (circle) and reconstructed heterogeneous
model (cross); the RMSE denote the discrepancy of measured and simulated BTCs. The spread of RMSE for a certain dispersivity value high-
lights that realizations with same dispersivity but different effective porosity values exist.
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JIM
ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5517
on storage temperature. Studies for those kinds of processes are scarce and more research effort in this
direction is needed. Nevertheless, as normalized BTCs are employed for this study, a lower tracer recovery
does not compromise the results.
Although, the reconstructed model is not superior to a simple homogeneous alternative in delineating the
tracer BTCs, it appropriately resolves structures relevant for the transport of the tracers. This is revealed by
comparison of the estimated value ranges for the MCMC ensembles. Figure 11 shows that the RMSE for both
model variants are similar, and the effective porosities range between 0.1 and 0.27. This is the same for both,
and also within the broad range of previously reported values of 0.10–0.25, [Schlie,1989;Hu,2011;Meyer,
2011; D. Meischner, Nat
urliches Einzugsgebiet und Trinkwasserschutzzonen f
ur das Wasserwerk Stegem
uhle
der Stadtwerke G
ottingen AG, Survey for Stadtwerke G
ottingen AG, unpublished data, 1985]. However, disper-
sivity values need to be substantially higher when a homogeneous model is used. Best results are obtained
for a longitudinal dispersivity of a
L
52.67 m for the homogeneous model. The optimal fit for the heterogene-
ous case is at a
L
51.64 m, which shows that macrodispersive effects are simulated explicitly and correctly
through the reconstructed macroscale hydraulic heterogeneity. The value of a
L
51.64 m is still significant and
this denotes that heterogeneities exist at a smaller scale than the resolution of the hydraulic tomography at
this site and with this experimental configuration, which strongly influence the tracer spreading. By individual
tracer BTC fitting, the estimated values of a
L
51.57 m for uranine and 1.72 m for sodium-naphthionate slightly
deviate from the result of combined fitting. These differences are not judged as significant enough to identify
clear differences in the tracer-specific transport or associated with the different injection levels.
4. Conclusions
The presented work shows that the sequential travel time and pilot point-based approach can be applied
to high-resolution reconstruction of hydraulic parameters at a field site. It is demonstrated, for the only
slightly heterogeneous field site that the presented procedure can identify sedimentary structures. How-
ever, comparison of model-based predictions with the tracer tests at the site reveals that the reliability of
the derived aquifer model also exhibits limitations.
The tracer test delivered two BTCs, which show minor irregularities and this indicates the only moderate
heterogeneity at the Stegem
uhle site. Therefore, even a homogeneous model can provide similarly good
predictions as a heterogeneous variant with the reconstructed K-field. A main point is that crucial transport
parameters, especially dispersivity, need to be set. We have not prespecified these parameters but analyzed
suitable value ranges by applying a MCMC based search. In other words, for minimizing any bias we exam-
ined model validity within these ranges. Within these degrees of freedom, the reconstructed model per-
forms similarly well as a homogeneous one. This reflects that even though macroscale heterogeneities are
reconstructed, their combined effect on tracer spreading averages. Therefore, the tracer breakthrough
curves can also be predicted by a higher integral dispersivity in a much simpler homogeneous model. How-
ever, as pointed out in the theoretical study by Ni et al. [2009], even if a homogenous model can provide an
appropriate fit, it will not capture the scale-dependent increase of dispersion with transport distance [see,
e.g., Molina-Giraldo et al., 2011; Gelhar et al., 1992]. In contrast, by reconstructing transport-relevant struc-
tures, their effect on macrodispersion is explicitly simulated, and thus the heterogeneous model is more
suited for predicting solute transport along shorter or longer distances.
Still, in our application case, neither the homogeneous nor heterogeneous model variant perfectly predicts
the recorded tracer concentrations. When measurement errors can be neglected, we interpret inconsistencies
caused by unresolved lateral heterogeneity. The proposed sequential approach employs 3-D hydraulic simula-
tion and inversion, but structures are constrained only by the vertical 2-D travel time tomograms. For
improved structural reconstruction, additional tomograms between different source and receiver wells would
be needed. With these, more reliable results from hydraulic parameter interpolation rather than the presented
extrapolation can be expected. Aside from this, as Illman et al. [2012] demonstrate in a sandbox experiment,
the resolution by HT could be improved with a higher density of sources and receivers. As a result, so far unre-
solved microstructures could be detected and the value of the dispersivity would be further decreased.
The presented coupled inversion procedure shows to further refine K-tomograms and S
s
-tomograms
between the investigated source and receiver wells in comparison to travel time-based inversion only. A
main observation is that pilot point-based inversion reduces heterogeneity, although homogenization is
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ENEZ ET AL. PREDICTING SOLUTE TRANSPORT UTILIZING A HYDRAULIC TOMOGRAM 5518
not enforced through regularization. This may be due to the fact that the 2-D travel time tomograms are
based on a diagnostic of early arrival times, which are accentuated by the existence of high Kzones or pref-
erential flow paths. In contrast, the pilot point approach calibrates the full pressure response curves and cal-
ibrates a 3-D model, and by this a higher volume of the aquifer is referred to. Further insight could be
obtained, for instance, by employing different parts of the response curves for pilot point-based inversion.
As an innovative step, it is shown that including S
s
in addition to Kin the pilot point-based inversion is ben-
eficial for minimizing model misfit to field data. This has rarely been included in related work [e.g., Castagna
and Bellin, 2009]. However, this means also doubling the number of decision variables for the optimization
problem. This is potentially not desirable, as this eventually can overparameterize the problem and enhance
its ill posedness. In fact, the generated heterogeneous aquifer model can be considered as one solution of
many, and further alternative realizations fitting the data could be explored. As future work, we therefore
plan to envisage the diversity of several equally probable realizations, based on Kwith or without S
s
as free
parameters.
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Acknowledgments
The investigations were conducted
with the financial support of the Swiss
National Science Foundation to the
project ‘‘A field assessment of high-
resolution aquifer characterization: An
integrated approach combining
hydraulic tomography and tracer
tomography’’ under grant
200021_140450/1 and the CCES
funded project ‘‘RECORD Catchment.’
The helpful comments of the Associate
Editor and three reviewers are greatly
appreciated. Further thanks go to Gabi
Moser for language corrections. All
data are available by e-mailing the
corresponding author
(santos.jimenez@erdw.ethz.ch).
Water Resources Research 10.1002/2014WR016402
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We present a novel hydraulic tomography procedure, which is based on a travel-time based inversion and application of pilot points. The travel-time based inversion allows for the reconstruction of a diffusivity distribution, which displays the dominant structural elements of a heterogeneous aquifer. This information is used to guide pilot point based inversion of hydraulic conductivity distribution. We utilize the diffusivity distribution to optimize the spatial positions of the pilot points and the relationships among them. For the implementation of the relationships, graph theory is applied. The associated high-computational effort was encountered with subspace regularization and parallel computing. The developed inverse methodology was successfully applied to a three-dimensional sedimentary aquifer analogue. The validation shows that the sequential procedure strongly improves the misfit between predicted and true pressure response in comparison to the results, only when travel-time inversion is employed.
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In groundwater hydrology, geophysical imaging holds considerable promise for improving parameter estimation, due to the generally high resolution and spatial coverage of geophysical data. However, inversion of geophysical data alone cannot unveil the distribution of hydraulic conductivity. Jointly inverting geophysical and hydrological data allows us to benefit from the advantages of geophysical imaging and, at the same time, recover the hydrological parameters of interest. We have applied a coupling strategy between geophysical and hydrological models that is based on structural similarity constraints. Model combinations, for which the spatial gradients of the inferred parameter fields are not aligned in parallel, are penalized in the inversion. This structural coupling does not require introducing a potentially weak, unknown, and nonstationary petrophysical relation to link the models. The method was first tested on synthetic data sets and then applied to two combinations of geophysical/hydrological data sets from a saturated gravel aquifer in northern Switzerland. Crosshole ground-penetrating radar (GPR) traveltimes were jointly inverted with hydraulic tomography data, as well as with tracer mean arrival times, to retrieve the 2D distribution of GPR velocities and hydraulic conductivities. In the synthetic case, incorporating the GPR data through a joint inversion framework improved the resolution and localization properties of the estimated hydraulic conductivity field. For the field study, recovered hydraulic conductivities were in general agreement with flowmeter data.
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Hydraulic tomography is a minimally invasive methodology that has the potential to yield information for mapping the spatial variability of hydraulic parameters at a level of detail that was not possible before by using a single hydrological test. In term of resolution and volume of investigation this methodology is similar to crosshole geophysical tomography but with an important advantage: it does not require petrophysical relationships to relate the measurements to the unknown parameters. Data for inversion are in fact transient heads that are directly related to hydraulic conductivity and specific storage through the flow equation. However, inversion of hydraulic tomography data is still in its infancy and the potential of this methodology for mapping hydraulic properties at the local scale is not fully explored. Most of existing studies focus on the experimental setup, or use inversion methodologies based on linearization of the constitutive equation to obtain an image of subsurface heterogeneity that is then compared with the real one, typically in a synthetic case study. Although these studies are very promising and shed some light into the capabilities of hydraulic tomography for mapping hydraulic properties, several issues are still open. In this work we explore how much complexity is warranted in the inversion of hydraulic tomography. In our exercise we consider a synthetic example with the tomography data obtained by solving numerically the transient flow equation in a know conductivity field. The inversion is then performed within a Bayesian framework by using the pilot point concept to contain the number of unknown within the acceptable limit of a few tens. This procedure was proven to be efficient in the inversion of crosshole ground penetrating radar data. In addition to the values of the hydraulic conductivity at the pilot points, we assumed unknown the parameters of the spatial variability model. The optimization is obtained by using the particle swarm algorithm, a robust stochastic evolutionary computation technique developed for the first time in the context of a simplified social model. We considered several experimental setups varying the horizontal spacing between the two wells, the vertical spacing between the positions of the stressed periods at the pumping well and the positions of the head measurements at the monitoring well. A first result of our exercise is that the particle swarm algorithm proved to be very effective in avoiding trapping to local minima of the objective function, without loosing in efficiency, when a large number of pilot points is utilized. We observed that the likelihood of the map of hydraulic conductivity obtained from inversion depend critically on the distance between the wells with respect to the horizontal integral scale, and that the parameters of the hydrogeological model of spatial variability are inferred with an acceptable, although variable, level of accuracy.
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An asymptotic expression for transient head variations, valid at low-frequencies, forms the basis for an efficient scheme for estimating hydraulic conductivity. The critical computational step is equivalent to solving the governing equation for steady state head. Thus, model parameter sensitivities, relating changes in head to changes in hydraulic conductivity, of the fully transient problem can be computed with the equivalent of four steady state head computations. A comparison of model parameter sensitivities computed using the low-frequency asymptotic approach and sensitivities computed using a purely numerical approach indicates good agreement. An inversion of synthetic hydraulic tomography data indicates that it is possible to estimate overall permeability variations using the technique. In an actual application to truncated crosswell pressure tests from the Raymond field site, we image two high permeability fracture zones, in agreement with a conceptual model of the region. The location of the two fracture zones correlates with the position of transmissive fractures, as measured by borehole conductivity logs.