We have used the same experiment system with a sand resin box as presented by Loeffler and Bano [21]. The resin box has a diameter of 2 and is 0.98 m high (Fig. 1) and was filled with a fine calibrated sand having a diameter between 0.3 and 0.5 mm and containing three buried pipes at a depth of 48 cm (a water-filled PVC pipe, a steel pipe and an air-filled PVC pipe). The GPR measurements were carried out with the PulseEkko1000 system (Sensors & Software, Canada). CMP and CO profiles were obtained with a 900 and 1200 MHz shielded antenna, respectively. We performed 71 parallel monostatic profiles (Pxx on Fig. 1) separated by 2 cm, while the distance between measurements was also 2 cm.

The data were processed with in-house interactive GPR software written in Matlab on a PC-station. The quality of the data was improved by removing the low-frequency component (known as the DC component) with a running average filter in time. It is important to note that during the processing we used a linear gain control, and not automatic gain control (AGC). A linear gain control allows for the detection of small changes in signal amplitude which may be obliterated with the use of an AGC filter. The same parameters were used for all data in order to keep the relative amplitudes of the signals from one data set to the other.

After the water of the first experiment shown by Loeffler and Bano [21] was drained and the sandbox had rested for several months (from February 2003 to April 2004), we acquired (in April 2004) a 3D data set for this condition of the sand, considered at this moment as “dry” (see below for more details). In Fig. 2, we present the CO data of the “dry” sandbox acquired in November 2002 and in April 2004. The three diffraction hyperbolae noticeable in Fig. 2 represent reflections from three buried pipes (from left to right: WPVC, steel and APVC pipes; see also Fig. 1). The bottom of the sandbox is well imaged (dashed line in Fig. 2); however, the travel time of the bottom reflections presented in Fig. 2b is slightly larger than the travel time of the bottom reflections in Fig. 2a. This is due to the average velocity of the radar waves, which decreases in the case of Fig. 2b because of the retention of water into the pores after drainage (February 2003).

Fig. 3 shows two CMP (CMP16 and CMP56) acquired on the dry sand (in April 2004). The CMP, located in the middle of the profiles P16 and P56 (see positions P1 and P2 in Fig. 1), do not show any noticeable difference and the average propagation velocity v (estimated from the bottom hyperbolic reflection) is the same on both CMP and its value is 0.105 m/ns. The sand is considered as homogeneous in this condition.

We next injected water (240 L) up to a level of 26 cm from the bottom of the box (i.e., the water table was at a depth of 72 cm), and acquired (1 month later) another 3D data set. The quantity of injected water allowed us to estimate a porosity (φ) of 27%. This value is, however, smaller than that (39%) obtained in the first experiment [21]; nevertheless, this is an effective value and the discrepancy is probably due to the settling of the sand (about 2 cm) and the retention of water in the pores after drainage in February 2003. This value of porosity is consistent with the ones given by Banton and Bangoy [3] or Todd [28] for fine and calibrated sand.

Afterwards, we injected 100 L of diesel fuel from the top of the sandbox above the water table. The injection point of the fuel (IPF) is shown in Fig. 1b. This fuel corresponds to the diesel found at a gas station and has a density of 0.84 g/cm³ [4], and therefore it is a light non-aqueous phase liquid (LNAPL). No settling of the sand has been observed after LNAPL injection. We repeated the same GPR acquisition scheme two more times:

• i) 12 days (in May 2004) after fuel injection;
• ii) 6 weeks (in June 2004) after fuel injection.

The relative dielectric permittivity or dielectric constant of LNAPL (κ_LNAPL) is considered constant and we, as many authors, (e.g. [7,8,26]) used a value of κ_LNAPL = 2. Note that the dielectric constant (κ) is dimensionless.

Fig. 4 shows the same profile T0 obtained with a 1200 MHz antenna before (with a water table at 72 cm depth) and after fuel injection. This profile is perpendicular to the Pxx profiles (crossing them in the middle; see Fig. 1) and goes over the steel pipe (at 50 cm depth) and the steel balls P1 and P2 (situated at 68 cm depth). The steel pipe is well imaged, while no diffractions coming from the steel balls P1 and P2 are observed, in both cases. The GPR data of Fig. 4, for the water table at 72 cm depth, do not show any clear reflections from the top of the saturated zone, which is the top of the capillary rise (of nearly 100% saturation) situated above the water table (e.g. [12]). This is a consequence of the existence of a transition zone above the saturated zone (e.g. [2]). Recall that in the transition zone the saturation decreases gradually, upward, from the saturated zone to the surface. Since the pipe is horizontal, we recorded it at a constant travel time of 9 ns (except for the semi-hyperbolic diffractions from the corners of the pipe) for the profile without fuel (Fig. 4a). By contrast, after fuel injection, the recorded travel time from the pipe is not constant. The travel time of the left corner of the pipe (at 9 ns in Fig. 4b) is shorter than the travel time of the right corner. Consequently, the area near the injection point (on the left of the pipe) has a higher propagation velocity than the area far from the injection point (on the right of the pipe). The dashed line indicates the reflections coming from the bottom of the sandbox, which is flat (nearly at 23 ns) in the case of Fig. 4a and considerably disturbed after fuel injection in Fig. 4b. The consequences of the pollution will be discussed in more detail in the following section.

In Fig. 5, we compare two CMP acquired on both sides of the steel pipe (as in Fig. 3) after fuel injection. The CMP acquired in this situation are very different and the reflections from the bottom of the box on the CMP above P1 (cf. Fig. 5a) arrive earlier than the corresponding reflections on CMP above P2 (cf. Fig. 5b). There is a large variation of the average velocities of the GPR waves. The sandbox cannot be considered as homogeneous anymore in this condition.

All the CO GPR profiles acquired 12 days (in May 2004) after fuel injection with a 1200 MHz antenna at the top of the sandbox do not show any clear reflections from the plume pollution. Therefore, in order to follow the lateral extension of the plume, we picked, in all profiles, the travel time of the reflections from the bottom of the box. Fig. 6 compares the results of the travel time (or the slowness) plots before and after fuel injection. Notice that the recorded time before injection (with a water level at a 72 cm depth) is not constant (Fig. 6a); a small difference (less than 2 ns) is observed between the zone situated near the water injection tap (with a recording time around 24 ns) and the zone situated far away from the tap (with a recording time around 22 ns). This is due to the non-homogeneous repartition of the water in the sandbox; the saturated zone is thicker near the tap than the zone far from the tap (see the left corner on the top of Fig. 6a). Hydrostatic equilibrium might not be completely reached at this state. We also observe three anomalies (in the middle in Fig. 6a) which are a consequence of the perturbation of the sandbox bottom reflection by the three pipes.

Fig. 6b shows the result after fuel injection. The influence of fuel on the travel time is important and obvious. The area near the fuel injection point (in blue) is characterized by a higher velocity than the area situated far away from the injection point (in red). The travel times vary between 20 and 24.2 ns. We also note that the influence of the fuel is not the same over the entire sandbox; this might be due to an inhomogeneous repartition of the water before fuel injection. Such high velocities (near the fuel injection point) can be explained by the replacement (at least partially) of pore water by the fuel. It seems that the pore water has been pushed away and replaced by the fuel by a lateral flow. The water, replaced by fuel, moves and concentrates in the surrounding areas less (or not at all) polluted (see the top of Fig. 6b). Despite the efforts made to fill the tank with homogenized sand (without layering), we observe that the lateral migration of the fuel plume is non-uniform. This non-uniform spreading of fuel and/or trapping of zones of water along the free-product level is a likely explanation for the numerous GPR hyperbolae observed by Sauck [24], and Sauck et al. [25] in the plume during their field observations in Carson City Park, Michigan (Sauck, personal communication).

Relative dielectric permittivities were calculated from average velocities derived from direct ground wave and reflection hyperbolae on CMP profiles on the one hand, and from the diffraction hyperbolae (due to different objects buried inside the sand) observed on CO profiles on the other hand. The velocities estimated by using the direct ground waves represent, in fact, velocities from the surface of the sand. The mean precision of the velocities was between ±0.01 and ±0.02 m/ns. The average velocities were subsequently converted into dielectric permittivities. Details of this approach are presented by Loeffler and Bano [21].

The results for each data set (dry sand, with water at 72 cm depth, before and after fuel injection) are summarized in Table 1. The values of the permittivities calculated from the measurements performed in 2002 are also shown in brackets. From this Table 1, we note that the values of the permittivities estimated in 2002 and 2004 (see the two first columns) are very close for the sand down to a depth of 50 cm. By contrast, the values of κ_bottom (the average over the entire height of the box) are different. For instance, in the first column, κ_bottom has the values 6.7 (estimated in 2002) and 7.3 (estimated in 2004), respectively. This has been discussed previously and confirmed by Fig. 2. While the values of the permittivities after fuel injection are estimated on two different points P1 and P2, the value of κ_bottom estimated at P1 is very different from the one estimated at P2. Remember here that the distance between the points P1 and P2 is 80 cm.

We repeated the same GPR measurements 45 days (in June 2004) after fuel injection and observed an unexpected and very clear GPR reflection situated in the vadose zone above the water table (far away from the fuel injection point). Fig. 7 shows the results comparing the GPR data acquired at different periods. We compare the same profile T0 obtained with a 1200 MHz antenna; the first one is acquired in May 2004 (Fig. 7a) and the second in June 2004 (Fig. 7b). In Fig. 7b, we remark that a clear reflection appears (indicated by R) at the end of the profile. It is situated between 1 and 1.4 m laterally, and at 8 ns depth (in time).

To confirm the presence of this reflection we also show in Fig. 7 the profile P56 which is perpendicular to T0 and goes over P3 (the steel ball at 38 cm depth), P2 (the steel ball at 68 cm depth) and the clay cake A (see also Fig. 1). In Fig. 7d (acquired in June 2004), we observe the same reflection (marked by R) which is flat and continuous over the entire profile. This reflection is certainly due to a dielectric permittivity contrast between two layers within the vadose zone. Loeffler [20] compared the reflection coefficients of the steel ball P3 and the horizon R for the signals marked by a vertical dashed line in Fig. 7d and found that they have, approximately, the same negative value. Therefore, the origin of this reflection might be from the contact between two zones: a high-saturation zone underlying a low-saturation zone, which gives a negative reflection coefficient. A possible mechanism for this to happen is that the fuel coating the sand grains may have effectively displaced the water from the pores such that a dielectric permittivity contrast exists (e.g. [1]), far away from the fuel injection point, within the vadose zone. A new hydrostatic equilibrium is achieved for this condition of the sand. Other laboratory and field GPR studies (e.g. [10,11,17]) have also shown the displacement of the water by the fuel through a lateral flow. In the next Section, we try to model by FDTD the contact between these zones. We notice here that no such reflection is observed in the profile of Fig. 7c acquired in May 2004. The black arrows on the top of each figure indicate the intersection between the profiles T0 and P56.

The synthetic profiles were computed using the two-dimensional (2D) forward-modelling code developed by Girard [14]. The modelling is based on a finite difference time domain (FDTD) numerical scheme as proposed by Yee [29]. The code works on dispersive and inhomogeneous media with conductive losses. The soil dispersion is modelled by exponential functions in time of the Debye or Davidson-Cole types, which are incorporated into the FDTD scheme. The code uses the piecewise-linear recursive convolution (PLRC) technique, presented by Kelley and Luebbers [19], and is extended to the power-law functions or non-exponential functions in the time domain for the dielectric response. The perfectly matched layer is used as an absorbing boundary condition to simulate an open space.

To model the profile T0, we assumed a non-dispersive medium, which is justified by the fact that the fuel is a non-conductive material and has a low value of permittivity (κ_fuel = 2) and a high quality factor (e.g. [30]). The model used is shown in Fig. 8 and consists of sand (in orange) which is divided into three layers based on their fluid content; dry sand (κ_sand = 4.6 and h = 32 cm), transition zone (32 cm thick, in which the saturation decreases gradually upward) and saturated sand (35 cm thick, water table and capillary rise). The saturated sand with fuel (κ_sf = 3.8) is represented in yellow, the mixture of sand/air/fuel (κ_saf = 4) in black, the mixture of sand/water/fuel (κ_swf = 15) in blue. The steel pipe (κ_steel = 81) and the soil (κ_soil = 4), are presented in red and brown, respectively. This conceptual model corresponds to a schematic distribution of the fuel concentration (below a point-pollution source) over a water table as described by Fetter [13] and Sauck [24].

To calculate the permittivity of the mixtures, we incorporated into the FDTD code the complex refractive index method (CRIM) for three- or two-phase medium (e.g. [22]) as follows:

Fig. 9 shows the modelled profile and the real profile T0 12 days after fuel injection. We observe that the modelling reflections from the pipe (P) and bottom (B) fit well with the real data. Nevertheless, some other signals, which are not visible in the real data, are observed in the synthetic profile. This is due to the discontinuities (interfaces) between different “layers” (zones) in our model of Fig. 8. See, for example, the passage from the layer of permittivity κ₁ = 3.8 to the layer of permittivity κ₂ = 15. However, the passage between “layers”, during the lateral fuel infiltration, is progressive and not as abrupt as represented in our model.

In order to verify our hypothesis about the origin of the reflection Horizon R, observed 45 days after fuel injection, we ran one more time the modelling of profile T0. We used the same model as in Fig. 8, but with an interface (marked by a green line situated 14–16 cm above the initial saturated zone) separating two zones: a highly saturated sand (κ = 45) underlying a less saturated sand (see Fig. 10). The result of this modelling is shown in Fig. 11a from where we notice that the modelled reflection R fits well (in time and amplitude) with the same reflection observed in the real data of Fig. 11b. The same remarks discussed in the case of Fig. 9 stand for the case of Fig. 11.

A 4D GPR survey was carried out on a sandbox contaminated with diesel fuel in order to follow the extension and the evolution of the fuel plume in the sand. The GPR data do not show any clear reflections from the plume pollution; however, GPR velocities are extremely affected by the presence of the fuel and the main changes are on the travel time anomalies. The lateral extension of the plume pollution in the vadose zone is shown by comparing the plots of travel times (or slowness) of the reflection from the bottom of the sandbox, before and after fuel injection. The differences between the two plots are obvious and the influence of the fuel is not the same over the entire sandbox. The zone situated away from the injection point shows an increase in travel time, which implies a low velocity anomaly. The pore water has been replaced by the fuel through lateral flow by creating a highly saturated zone far from the fuel injection point. Some weeks later, we repeated the same GPR measurements and observed a nice reflection situated on the opposite side of the fuel injection point. Finally, the forward FDTD modelling method in combination with dielectric mixing models gave theoretical support to understand and determine the physical characteristics and the volume of the contaminated sand and explain the origin of the observed reflections from the contaminated vadose zone. A next logical step is to combine the GPR with suitable laboratory measurements of fluid content in the sandbox experiment. The quantitative parameters given by GPR measurements can be used to constrain a lateral flow model, which requires many unknown parameters. In order to follow the lateral flow of the plume, a joint GPR and lateral flow modelling is necessary.