Active thermal tracer testing is a technique to get information about the flow and transport properties of an aquifer. In this paper we propose an innovative methodology using active thermal tracers in a tomographic setup to reconstruct cross-well hydraulic conductivity profiles. This is facilitated by assuming that the propagation of the injected thermal tracer is mainly controlled by advection. To reduce the effects of density and viscosity changes and thermal diffusion, early-time diagnostics are used and specific travel times of the tracer breakthrough curves are extracted. These travel times are inverted with an eikonal solver using the staggered grid method to reduce constraints from the pre-defined grid geometry and to improve the resolution. Finally, non-reliable pixels are removed from the derived hydraulic conductivity tomograms. The method is applied to successfully reconstruct cross-well profiles as well as a 3-D block of a high-resolution fluvio-aeolian aquifer analog data set. Sensitivity analysis reveals a negligible role of the injection temperature, but more attention has to be drawn to other technical parameters such as the injection rate. This is investigated in more detail through model-based testing using diverse hydraulic and thermal conditions in order to delineate the feasible range of applications for the new tomographic approach.

Tracers are commonly used to get insight into the hydraulic properties of the subsurface on the aquifer scale and to identify dominant transport routes. Among the many tracers used for aquifer characterization, heat is frequently injected as a thermal tracer in boreholes or wells (Anderson, 2005; Hermans et al., 2015; Rau et al., 2014; Saar, 2011). From measured breakthrough curves (BTCs), aquifer heterogeneity and preferential flow paths are inferred (Bakker et al., 2015; Colombani et al., 2015; Klepikova et al., 2014; Leaf et al., 2012; Macfarlane et al., 2002; Vandenbohede et al., 2008; Wagner et al., 2014; Wildemeersch et al., 2014).

Main attributes of ideal tracers are their good detectability, their lack of
influence on the flow regime, conservativeness and nontoxicity to the
environment. Heat is an ideal choice because it is easily detectable by
means of traditional temperature sensors, distributed temperature sensors (DTSs)
or geophysical techniques (Hermans et al., 2014), and it can be monitored continuously in situ. Typically,
background variations are insignificant, and natural heating–cooling cycles
have smaller frequencies than the investigated thermal signals. It is also
ideal because moderate changes in temperature do not harm the environment,
and thus commonly no regulative constraints are imposed. However, due to
possible viscosity and buoyancy effects, and their relationship with hydraulic
conductivity (

Our starting point is the fact that for detecting preferential flow paths full analysis of thermal transport behavior may not be necessary. If we focus on characteristic parameters such as travel times or moments of the BTCs, the signal-to-noise ratio may be acceptable for much broader temperature ranges. Travel times of traditional solute tracers are related to the hydraulic properties of aquifers, assuming that the main transport process is advection. This is the case given a sufficient ambient hydraulic or forced gradient during the experiment (Doro et al., 2015; Saar, 2011). One important difference of heat tracer transport over traditional tracers is that diffusion takes place not only in the pore fluid but in the rock matrix as well. So while the tracer front of a solute tracer tends to be sharp, the thermal tracer front appears smoothed. This may make interpretation of BTCs more difficult.

Because thermal diffusion takes place, heat transport is affected not only
by the hydraulic properties but by the thermal properties of the aquifer
material as well. However, contrasts in thermal parameters are relatively
small compared to contrasts in

In previous studies on thermal tracer testing, diverse set-ups have been
chosen that differ with respect to heating method; injection volumes, rates
and temperatures; test duration; and well configurations
(Wagner et al., 2014). Mostly hot water is
infiltrated in an injection well, and BTCs are recorded in one or more
downstream observation well (Ma et al., 2012; Macfarlane et al., 2002; Palmer et al., 1992; Read et al.,
2013; Wagner et al., 2014; Wildemeersch et al., 2014). Insight into aquifer
heterogeneity is not well constrained by analysis of thermal signals
introduced and measured over long screens. To obtain a better definition of
the heterogeneity, observations in several wells or at different depth
levels need to be compared. Ideally a tomographic setup is chosen, where
multiple point injection (sources) and observation points (receivers) are
used. By combined inversion of all signals, the spatial variations in

Klepikova et al. (2014) presented a passive thermal tracer tomography application for characterizing preferential flow paths in fractured media. Their method focused on modeling the fracture network with a sequential method which involves first identifying the location of fault zones on the temperature–depth profiles under ambient flow and pumping conditions. Next, an inversion of the temperature profiles is conducted to obtain borehole flow profiles, and the last step is to estimate the hydraulic properties from these flow profiles. This method provides cross-well connectivities. The work by Doro et al. (2015) is dedicated to the experimental design of cross-well forced gradient thermal tracer tomography. In their approach, a special multi-level injection system is necessary to induce the tracer into a horizontal layer. They also recommend limiting the temperature range to avoid buoyancy effects. Their proposed methodology to interpret the results is to use an inversion scheme developed by Schwede et al. (2014) for this specific experimental setup. This inversion method utilizes the temporal moment of measured BTCs and hydraulic head data together in a joint geostatistical inversion procedure (Illman et al., 2010; Yeh and Zhu, 2007; Zhu et al., 2009). This procedure is computationally demanding, and it assumes a multi-Gaussian distribution of hydraulic properties, which represents a strong restriction in comparison to the true conditions in the field.

In our work, we suggest a travel-time-based inversion procedure, which does not require a priori structural or geostatistical assumptions and is computationally efficient. It is motivated by Vasco and Datta-Gupta (1999), who presented a numerical approach to reconstruct the hydraulic parameters of an aquifer using solute tracer injections in a tomographic setup. As a core element, the transport equation is transformed into an eikonal problem using an asymptotic approach for the tracer transport solution. Their approximation uses the similarity of tracer front propagation to seismic and electromagnetic waves, but with the restriction that the tracer front is abrupt. This approximation can be used for hydraulic signals as well (Vasco et al., 2000), and the travel time of the hydraulic signal can be related to the hydraulic diffusivity of the system. Brauchler et al. (2003) further developed a travel-time-based inversion for Dirac and Heaviside hydraulic sources, using the early-time diagnostics of the signals. To improve spatial resolution, they applied staggered grids (Vesnaver and Böhm, 2000) during inversion. This inversion methodology was applied to several hydraulic laboratory and field experiments (Brauchler et al., 2007, 2011, 2013b; Hu et al., 2011; Jiménez et al., 2013). Brauchler et al. (2013a) also utilized travel times in a tracer experiment on rock samples on the laboratory scale. Their work revealed that for those samples transport was dominated by the rock matrix, but hydraulic parameters were not estimated.

In this study, we present a new formulation for inversion of spatially
distributed hydraulic conductivity using early tracer travel times. It
follows the same principles as presented by Brauchler et al. (2003) for hydraulic
tomography. Our objective is to obtain a versatile and efficient technique
for thermal tracer tomography, which, by focusing on early times, minimizes
the role of buoyancy and viscosity effects. In the following section, the
new inversion procedure is introduced. It is then applied to a
three-dimensional (3-D) high-resolution aquifer analog of the Guarani
aquifer in Brazil. We inspect the capability of the new approach to
reconstruct 2-D and 3-D sections with heterogeneous

Under high-Péclet-number conditions, when it can be assumed that the
thermal transport is dominated by advection, the propagation of an injected
thermal plume can be used to gain information about the hydraulic properties
of the investigated aquifer. Our goal is to calculate the hydraulic
conductivity,

In this study, a step function injection temperature signal is used for the active thermal tracer test. In this case the traveling time of the thermal tracer is associated with the propagating thermal front. The tomographic concept requires multiple independent thermal tracer injections at different depths. Temperature BTCs are recorded at multiple observation points, for example at different levels in a downgradient observation well. As common practice for such setups, the number of sources and receivers is one of the important factors that defines the significance and resolution of the results.

Compared to a conservative solute tracer, heat does not behave ideally. Diffusion is significant in aquifer matrix and pore fluid, while the viscosity and density of the groundwater are variable. Due to the highly diffusive behavior, the emerging thermal front cannot be considered as a sharp transition boundary. In order to obtain accurate results with the inversion, the complications from thermal diffusion need to be mitigated. Both diffusion and mechanical dispersion effects increase with travel time. Mitigation thus can be done by using an earlier characteristic time of the thermal front instead of the (peak of the first derivative) breakthrough time, thus using the fastest component of the heat transport–advection. The earlier characteristic time can then be corrected to the real breakthrough time using a conversion factor, as shown for hydraulic tomography by Brauchler et al. (2003) with a correction for the specific storage coefficient.

The propagation of a thermal front far from the source is described as a
one-dimensional (1-D) advection–diffusion problem considering thermal retardation:

Three steps of applying early-time diagnostics (ETD) on a
thermal breakthrough curve (BTC). (1) Identify the peak

Although Eq. (10) has three additional parameters – velocity (

The application of early-time diagnostics is illustrated in Fig. 1. We are
mainly interested in advective transport. However, thermal diffusion may also
be significant, smoothening and expanding recorded temperature BTCs, and thus
also affecting its derivative. The identification of the peak time through
the derivative

Major steps of inversion methodology:

To invert the tracer travel times, the SIRT algorithm (simultaneous iterative
reconstruction technique) is used to solve the eikonal problem, implemented
in GeoTOM3D (Jackson and Tweeton, 1996). The algorithm calculates the
transport trajectories between the sources and receivers and solves the line
integral of Eq. (2) along the trajectories – in a curve-based 1-D coordinate
system. To solve the line integral, the solution domain is discretized to a
grid. Initially a homogeneous velocity field is defined, and then the
velocity values of the cells are updated iteratively to minimize the
difference between the inverted and recorded travel times. The algorithm
results in mean tracer velocities, and they are transformed into

For discretization, instead of constructing a static regular grid, the staggered grid method (Vesnaver and Böhm, 2000) was used. Solving the problem on a regular grid would highly constrain the freedom of the solution to the geometry of the used grid and the source–receiver locations. By applying the staggered grid method, this constrain can be overcome, with the benefit that the nominal spatial resolution is increased. Otherwise, for a good spatial resolution using one fine grid, a large number of sources and receivers would be required or regularization terms would have to be applied. Staggered grids were successfully employed for hydraulic tomography by Brauchler et al. (2003) and for solute tracer tomography by Brauchler et al. (2013a). In this staggered variant, the problem is solved on different vertically and horizontally shifted versions of a low-resolution regular grid. The inverted results are different for the shifted grids, which are exploited by arithmetically averaging these results to arrive at a final tomogram. The inversion will be stable because of the coarse grids, while the resolution of the averaged tomogram will be as small as the displacements. Although this means that the travel time inversion step will be performed multiple times for one tomogram, it is still computationally affordable due to the marginal computation demand of a single coarse grid resolution.

Hydraulic conductivity,

Vertical cross section through the center of the 3-D Descalvado
analog data set showing the distribution of hydraulic conductivity (

To characterize the reliability of the results, the null-space energy map is
computed. This method has been applied for hydraulic tomography in several
studies (Brauchler et al., 2013a, b; Jiménez et al., 2013) and uses the
distribution of the inverted transport paths over the inversion grid. The
null-space energy map is calculated from the singular value
decomposition (SVD) of the tomographic matrix, which contains the length of
each inverted transport path in each grid cell. Values of the null-space
energy map are between 0 and 1; thus higher values mean higher uncertainties.
Based on the null-space energy map, non-reliable pixels can be deleted from
the tomogram. The resulting full inversion procedure, starting with the
tracer data and ending with the reliable part of the final

The presented methodology is developed and tested on the Descalvado aquifer
analog (Höyng et al., 2014) that is implemented in a finite-element heat
transport model (Fig. 3). This analog represents a 3-D high-resolution data
set obtained from mapping an outcrop of unconsolidated fluvio-aeolian
sediments in Brazil. These sediments host parts of the Guarani aquifer
system, one of the world's largest groundwater reservoirs. The analog is
based on five vertical outcrop sections that are recorded during ongoing
excavation and interpolated by multi-point geostatistics following the
procedure by Comunian et al. (2011). The spatial extent of the analog is
28 m

Parameterization of experimental setups, with base values and minimum–maximum ranges.

The Descalvado aquifer is built up mainly by highly conductive sand and
gravel with a layered structure. The average hydraulic conductivity value is
approximately

Flow and transport are simulated as coupled processes, using the software
FEFLOW (Diersch, 2014) and the SAMG algebraic multigrid solver (Thum and
Stüben, 2012). The analog is embedded into a larger domain with
extrapolated homogeneous layers, to minimize lateral boundary effects. The
model mesh is generated with the Triangle algorithm (Shewchuk, 1996) and
progressively refined towards the analog. Close to wells, the elements are
refined to millimeter scale. The total extent of the model is
118 m

The aquifer is assumed to be confined. In order to simulate initial
steady-state conditions with regional groundwater flow in the direction of
the long axis,

We present reconstructions of

Focus is set first on 2-D reconstruction. Three profiles in the central plane
of the aquifer are selected (Fig. 3). This central plane constitutes a mapped
outcrop section with relatively high facies variability. It contains
heterogeneous structures of different sizes and contrasts, and it is chosen
for being sufficiently far away from the analog boundaries. The location of
profile 1 is depicted in Fig. 3. Figure 4a shows the relative locations of an
upstream injection well and downstream observation well used for all three
2-D profiles. The distance between the wells is 5 m for an investigated area
of 5 m

To examine further the role of aquifer heterogeneity, two additional profiles from the central plane of the analog are investigated. In both cases, the source–receiver geometries are kept the same (Fig. 4a). Profile 2 shows a similar layered structure to profile 1, but with fewer small-scale heterogeneities. The central conductive zone is thicker, providing better connection between the two wells. In profile 3, the central conductive zone is discontinuous, creating a different hydrogeological situation, with weaker connection between the two wells.

In the simulated setup, 6 sources and 6 receivers are employed (Fig. 4a), resulting in a set of 36 source–receiver combinations. The sources are defined as point injections with constant injection rates during the entire simulation time. The used injection temperature signal delineates a Heaviside step function, where the instantaneous change in temperature is arbitrarily set at 0.1 days after the start of simulation, which marks the beginning of the experiment. In order to record BTCs in all observation points even at very small injection rates and temperatures, extremely long simulation times are used (50 days). However, most of the breakthroughs occur during the first five days of the simulation.

The crucial technical design parameters for the experiments are the injection
rate,

In practice, the source of the injected water can be the investigated
aquifer, but note that in this case heating has to be well controlled to keep
the injection temperature constant. During a field experiment, the recorded
data are always distorted by noise. With the commonly used temperature
sensors, this noise is considered very small (Wagner et al., 2014), but still
the sensitivity of the temperature sensors is limited. To take this into
account when simulating the receiver points, those where the temperature
changes are smaller than 0.1

For the 3-D reconstruction, an exemplary case is defined with one injection and three observation wells forming a triangular prism (Fig. 4b) located close to profile 1. The base face is an isosceles triangle, and the observation wells are located along the baseline. The axis of this triangle is at the line where the 2-D profiles are located. The distance between the injection well and the central observation well is 6.5 m, and the length of the triangle base is 3 m. The configuration of the individual wells is the same, resulting in 18 observation points and 108 source–receiver combinations in total. The experiment was simulated using the base values from Table 2, employing the same Heaviside injection signals as in the 2-D cases.

Hydraulic conductivity profiles (see Fig. 3):

The following results are structured into four major parts. The first part is the inspection of the inverted tomograms for the three 2-D and one 3-D analog profiles. The second part is the validation of the method using the result of the 3-D reconstruction. The third part is a sensitivity analysis of the inversion procedure with respect to experimental settings such as injection rate and temperature. The fourth part reveals the application window of travel-time-based thermal tomography through rigorous testing with different sections, changing hydraulic conductivity contrasts and varying experimental parameters.

The left column of Fig. 5 depicts the analog profiles, and these are
contrasted with the inverted ones on the right. For better comparability, the
original analogs are upscaled (using the arithmetic mean of the values within
a cell) to the same grid as used for the results with
0.125 m

BTCs from 34 source–receiver combinations were used in one tomographic
experiment. During staggering, the tomographic inversion is performed on
16 different spatially shifted coarse grids. The uniform cell size of these
low-resolution grids is 0.5 m ^{®} Core^{™}
i7-4770 CPU 3.40 GHz).

After calculation of null-space energy maps, a threshold of 85 % is found
suitable to constrain the

The reconstructed profiles in the right column of Fig. 5 shed a first light
on the capabilities of thermal tomography. First, we observe that for all
profiles the upper zone (in blue) cannot be reconstructed by the inversion.
Typically a considerable fraction of it is masked in gray due to the limited
contribution to heat transport, which is not surprising due to the low
hydraulic conductivity of this zone. In contrast, the tomographic approach
identifies the location of the highly conductive upper-central zone (in red)
rather well. This zone delineates the fastest travel route between the wells
for the heat tracer. Between the upper (blue) and central (red) zones is the
strongest contrast in the profiles. This strong contrast shadows the top of
the tomograms, because the transport is short-circuited through the high-

A striking feature is that the tomographic approach resolves the continuity
of the highly conductive upper-central zone in profiles 1 and 2, and it
detects the discontinuity in profile 3. Furthermore, the inverted value of
hydraulic conductivity of this zone
(

3-D distribution of hydraulic conductivity (

The promising findings as depicted in Fig. 5 support the applicability of travel-time-based tracer inversion for thermal tomography, even though thermal diffusion tends to blur advective travel times, which hinders a reliable inversion. However, by taking early arrival times of the recorded BTCs, this effect is minimized. Likewise, when preferential pathways exist, these will be detected by the first thermal breakthrough, which is least influenced by diffusion. As a result, travel-time-based thermal tomography appears especially suited for locating and characterizing high-conductivity zones.

With the 36 source–receiver combinations, exact profile reconstruction is not possible, since the tomograms appear to be smoothed. Fine-scale differences in the form of the high-conductivity zone are not reproduced in the tomograms. This is the same for the small facies mosaics that originally occur in the mainly orange lower-central zone. This zone seems mixed with the lower yellow zone, and the hydraulic conductivities of both zones are slightly underestimated. Despite the minor hydraulic contrast between both layers, however, the tomograms indicate locally a facies transition (especially in Fig. 5f). This is not identified in the tomogram of profile 1 (Fig. 5b). Here most small-scale structures exist in the lower-central part above. These cannot be resolved, but they detract from the transport routes of the thermal tracer and thus induce noise in the reconstructions of the lower-central and bottom layer.

Figure 6 shows the reconstruction of the selected 3-D section. The result is
presented the same way as the 2-D profiles, using an upscaled version of the
original analog for comparison. Three-dimensional staggering is employed,
resulting in 64 coarse grids in total. This requires 64 individual inversions
and thus a computational time that is drastically longer than in the 2-D
cases. With 20 iterations per inversion, the total computational time on the
same PC (Intel^{®}
Core^{™} i7-4770 CPU 3.40 GHz) was around 1 h
for 3-D inversion. The spatial resolution of the coarse grid is
0.5 m

To assess the reliability of the inverted result, the null-space energy map is calculated. For the 3-D application a limit of 95 % of reliability is used to accept reconstructed voxels. Lower values would substantially reduce the reconstructed volume, since non-reliable voxels are not presented. Generally, the reliability and thus overall result quality of the 3-D analysis is worse than for the 2-D cases. This is due to the fact that the inverted transport paths cover less of the domain of interest.

Figure 6a depicts the upscaled analog model, sliced in half at the central
plane where the injection well is located. The same method of presentation is
used for the reconstruction in Fig. 6b. To highlight the differences to the
2-D results, the inverted high-

In Fig. 6b, the central conductive zone of the aquifer is localized mainly
at the lateral boundaries close to the wells. Centrally,

For validation, the reconstructed 3-D

Considering the good reconstruction of the high-

The experimental setup may be crucial for the quality of the inversion results. For example, it is well known from related tomographic inversion studies that the feasible resolution depends on arrangement and the numbers of sources and receivers (Cardiff et al., 2013; Paradis et al., 2015). Here we focus on two technical design parameters, which are particularly crucial for thermal tomography when using heated water: the injection temperature and the injection rate. In the following sensitivity analysis, we question whether these need to be carefully tuned or not. Profile 1 is chosen for investigation, depicted again in Fig. 8a and 9a. Note that for forward simulation of travel times the full 3-D analog model is always used.

We first inspect the role of the temperature of the injected water. In all
of our models, the ambient groundwater temperature is considered uniform and
10

Hydraulic conductivity

Hydraulic conductivity

Figure 8 depicts the inverted

The sensitivity of the injection rate,

By raising the injection rate, the reconstructed continuity of the central conductive zone improves (Fig. 9c–e). For our particular case, this is attributed to the setup. Since the top two observation points are located in the upper low-conductivity zone, this influences the reconstruction of the central high-conductivity zone.

In contrast, at the highest simulated injection rate of

The insight gained from variable injection rates and temperatures revealed
that the presented tomographic inversion method is robust within a broad
range but has limitations. But what exactly are the limits? We tested a
broad range of different scenarios to delineate a general application
window, where the inversion method can be used to reconstruct the distribution
of

Each inverted

The quantification is based on an estimated connectivity time between the
sources and the receivers. The connectivity time is calculated by converting
the

To condense the results into a normalized parameter space and plot
them in a 2-D coordinate system, two dimensionless parameters are selected:
the thermal Péclet number (

The proposed application
window of the thermal tracer tomography – related to the injection
parameters of the thermal tracer test (effective injection power

After evaluating approximately 100 different experimental scenarios, resulting in over 350 data points, the application window of the method is identified. In Fig. 10 continuous lines mark strict boundaries between feasible and infeasible regions (where beyond the line no reconstruction is possible), and dashed lines denote an approximate boundary where the result quality of tomograms starts to decrease in the lateral direction (relative decrease in result quality).

If

At low

The result quality gradually declines towards high

Early arrival times of tracer BTCs are specifically suited for identifying highly conductive zones in heterogeneous aquifers. In our study we formulated a procedure for combined inversion of multiple early arrival times measured during cross-well tracer testing. A tomographic setup with multi-level tracer injection and observation was implemented in a model with a 3-D high-resolution aquifer analog, and we examined the capability of the inversion procedure to reconstruct the heterogeneous distribution of hydraulic conductivity. Heat was selected as a tracer, which offers several advantages in comparison to many solute tracers, but its applicability is traditionally considered limited due to the higher diffusion and coupled thermal–hydraulic processes.

It is demonstrated that the tomographic interpretation of heat tracer signals is well suited for characterization of aquifer heterogeneity. By picking early arrival times, the impact of thermal diffusion, buoyancy and viscosity variation is minimized and, in this way, inversion becomes quasi-insensitive to the temperature range. The presented application window of tested parameters of thermal tracer tomography is wide, and it covers three orders of magnitude for thermal Péclet numbers and five orders of magnitude for injection power. A key principle is that the transport in the aquifer is dominated by advection, and injection of hot water causes minor distortion. This can be controlled, for instance, by establishing a forced gradient between injection and observation point by operating an adjacent pumping well.

The travel-time-based inversion is a fast and computationally efficient procedure, which delivers a tomogram in a few minutes with six sources and receivers. It is revealed that not only structures of mainly highly conductive zones could be reconstructed, but also the values of hydraulic conductivity were closely matched. This is appealing, keeping in mind that the presented eikonal inversion is based on a rough approximation of groundwater flow and transport by a wave equation. Yet when close to strong contrast boundaries, the procedure is not able to reconstruct low-conductivity zones due to short-circuit–shadow effects. To reconstruct these hidden features, a further calibration step or additional information would be required.

In the following, we present the mathematical procedure to transform the transport equation of a thermal tracer into the eikonal equation based on Vasco and Datta-Gupta (1999). First the solution of the transport equation is written as a series of wave functions. After neglecting the low-frequency components, the transport equation is turned into the eikonal equation. Lastly, the travel time equation is presented as a solution to the eikonal problem.

The

The aquifer analog data used in this paper (Bayer et al.,
2015) are accessible from the Pangaea database using the following link: