Introduction

The Ruczaj district area in Kraków (Fig. 1) is a strongly developing part of the city, where many new buildings are being built. Unfortunately, there is an unexpected problem connected with the specific geological and hydrogeological conditions of this region, which compose of the soil, clay and gypsum layers. The building foundations are usually set on the textured gypsum basement; therefore, the soil and impermeable clay layers with variable thickness are to be removed. These gypsums layers are a part of the evaporative works that have been created as a result of Tortonian sea regression. The Miocene aquifer is associated with the layer of skeletal gypsum (Kleczkowski 1989; Rutkowski 1993).

Fig. 1
figure 1

Map of the survey area on the map of Kraków. See text for details

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In the case of rainfall, the infiltration water and water with SO42− from the gypsum basement flow into the zone surrounding the building foundations and became a great threat. So, the task of the engineering geologists is to determine preciously the thicknesses of the soil, clay layer and the depth of the top of the gypsum basement. There is a big resistivity contrast between those layers. Their location and approximate dimensions were determined.

Conductivity and resistivity of selective soil, rocks and water

Generally, the conductivity of soil, clay and gypsum varies in a broad interval and depends on the water content and its mineralization (Table 1). In Kraków, the conductivity of soil and clay ranges from 20–50 mS/m and 50–170 mS/m, respectively, and gypsum from 1 to 10 mS/m (Pasierb 2012). The building area often amounts to a few hundred square meters, and with such small area, the soil, clay and gypsum layers can be regarded as horizontal formations. Based on the mentioned consideration and an economic cost, we decided to use the Ground Conductivity Meters (GCM) to determine the spatial distribution of the thicknesses of the soil, clay and gypsum and to check the results of GCM we used the DC resistivity and GCM soundings. There were no sources of high noise nearby measurement area (Neska et al. 2013).

Table 1 Resistivity (ρ) and conductivity (σ) of rocks and water combined from different papers
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We have considered theoretically typical applicability of those methods. We tried to detect shallowly buried, low conductive gypsum layer in a low resistive environment. The justification of the application of the DC resistivity and GCM soundings, in this case, is very simple. There is a big resistivity contrast between layers occurring there.

The ground on which the Ruczaj settlement is mainly the Tortonian clays. In the upper part of the complex, there are thin gypsum inserts, partly eroded in the late Pleistocene and subjected to karst processes (Pociask-Karteczka 1994). These gypsum layers were largely devastated during the industrialization of the area. A thin layer of Holocene alluvial deposits is lying on them, probably embedded by the nearby Wilga River (right inflow of the Vistula) and its inflows, and in some places, also the Pleistocene weathered older basement (Gradziński 1972).

Clays and loams have very low electrical resistivity or high conductivity (Table 1), but gypsum is characterized by much higher resistivity. The resistivities of rocks vary in the wide intervals. The mineral water, which saturates rocks, is characterized by relatively low resistivities and the more water in the rock, the higher rock (Antoniuk et al. 2003; Plewa and Plewa 1992).

Measurements

The conductivity method relays on a measurement of a second electromagnetic field related to the eddy current in the study medium formed by the primary electromagnetic field formed by the alternating current in the transmitting coil. The frequency of the alternating current is in a range of the audio frequency (tens kHz). The measurement parameter of Ground Conductivity Meters (GCM) is the apparent conductivity σa (mS/m) of the heterogeneous medium, which is within the footprint of a used GCM system (McNeill 1980a). The apparent conductivity at different levels of depth is measured using the horizontal (HD) and vertical (VD) magnetic dipole with a different spacing. The zone around the transmitting coil with a given frequency, where the conductivity is measured, is called as a near zone or induction zone (McNeill 1980b).

The CMD Mini-Explorer and CMD Explorer as the Ground Conductivity Meters produced by the GF Instruments, s.r.o., are designed for induction profiling. The equipment can be configured as the HD and VD system for apparent conductivity measurement, and it has six different spacing options (s = 0.32, 0.71, 1.18 m for Mini-Explorer and 1.48, 2.82, 4.49 m for Explorer) and operated on a frequency of 30 kHz and 10 kHz, respectively. These configurations enable to measure the apparent conductivity of the investigated medium with different depth levels (CMD Explorer; Short guide 2016).

The GCM measurements were taken at the points with 1 m distance on a profile with 100 m long in the district Ruczaj of Kraków (Poland). To recognize the geological medium, benchmark DC resistivity soundings, with a measuring step of 25 m along the same profile, were carried out.

To verify GCM interpretation results, the DC measurements were taken using the Schlumberger four-electrode system (Klityński et al. 2014). The data obtained from both DC resistivity measurements and GCM soundings were quantitatively interpreted using two different algorithms Occam and Levenberg–Marquardt (LMA); then, the results were combined for the determination of the studied medium. The result of the LMA interpretation is a model with layers separated by the sharp boundaries, while the result of the Occam interpretation is a model with a curve presenting the resistivity, which fluently changes from layer to the other (Constable et al. 1987).

The LMA algorithm involves the iterative adjustment of the model curve to the empirical data (measurement curves) and the parameters of the geoelectrical cross section calculated on their basis. One-dimensional inversion by this method is based on optimization of the error function, i.e., deviations between field (empirical) curves and those calculated from the model using the Levenberg–Marquardt algorithm (Levenberg 1944).

Inverting 1D according to the Occam’s algorithm is a method of calculating one-dimensional resistivity distribution in a geological center (Oryński et al. 2019). The basic assumptions of this method are striving to achieve the most smooth and the most simple model (Jóźwiak 2013). It is an iterative algorithm in which the starting model is a homogeneous half-space, and during the iteration process, the resistivities of individual layers are modified (Nowozynski and Slezak 2013). The error minimization procedure was constructed in such a way that the resistivity contrasts were as small as possible (Reynolds 2011).

Results and discussion

The measured apparent conductivities as a function of the GCM system arrays are presented in Fig. 2a, c, and they are compared with results of DC resistivity method as shown in Fig. 2b, d. The interpretation of this curve indicates three layers: The first is the soil with the conductivity ~ 6 mS/m with a thickness of several dozen centimeters. Beneath them, there is the clay layer with conductivity close to 100 mS/m and thickness about 3 m.

Fig. 2
figure 2

Results of quantitative interpretation of GCM and DC resistivity soundings; fitting of the observed and calculated GCM data at the site 25 m for HD (a) and VD (c), configuration (c), results of 1D inversion of DC and GCM HD data (b), results of 1D inversion of DC and GCM VD data (d)

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Under the clay layer, there is the gypsum complex with conductivity about 8 mS/m with thickness increasing from 12 to over 20 m in SE–NW direction (Fig. 2b, d). The deeper layer is not observed by the GCM method, due to its limited footprint. It is characterized by high conductivity (about 300 mS/m), which is recognized by the DC resistivity sounding method. This layer belongs to Miocene clay formation (Kleczkowski 1989). A very similar situation, as described above, is visible for the DC resistivity sounding results, presented in Fig. 3. However, in this case, the depth range is almost twice as large and up to 20 m. A high conductive layer, interpreted as compact alluvial clay, is strongly visible, and also there is a strong boundary between it and high resistive gypsum layer. The Miocene basement (named Wieliczka formation clays as shown in Fig. 3e) is much better visible for deeper DC sounding than in the previous case.

Fig. 3
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Results of quantitative interpretation of DC resistivity sounding at the site 25 m; fitting of the observed and calculated DC-R data (a), Occam and LMA models (b), colored Occam model of resistivity (c), colored Occam model of conductivity (d) and geological model (e)

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Interpretation of the data obtained by both DC and GCM soundings are performed using the LMA, Occam algorithms and the Interpex IX1D software. Parallel with the mentioned software, the authors also used the linear filtering method to make the curve presenting the GCM sounding data corresponding to the assumed geoelectrical model (Ghosh 1971; Das and Verma 1981; Koefoed et al. 1972), while the GCM curve for the horizontal arrangement model was calculated using the Abramova’s filter (Loke 2003). For the precise interpretation of the GCM sounding, the iterative method was used, which is upon on founding the minimum of the error function (RMS). The used interaction algorithms were taken from DLIB library (King 2009).

Results of quantitative interpretation are presented as a conductivity cross sections of 1D Occam inversion of DC sounding data (Fig. 4) and 1D Occam inversion of GCM data (Fig. 5) The results of the LMA algorithm are presented as a well in the place of soundings, while the results of Occam inversion are interpolated between soundings. Moreover, results of DC-R inversion are presented in two variants: up to 20 m of depth (Fig. 4a) and up to 6 m of depth (Fig. 4b). The first one corresponds to the maximum range, the second to be better compared with the GCM.

Fig. 4
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Conductivity cross sections using 1D Occam inversion of DC sounding data and results of 1D LMA inversion of DC soundings

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Fig. 5
figure 5

Conductivity cross sections using 1D Occam inversion of GCM data for HD (a) and VD (b) configurations and collating results of 1D LMA inversion of DC soundings and GCM data for the corresponding point located along with the profile

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The horizontal resolution of the interpreted DC data (Fig. 4a, b) is worse than that of the GCM soundings (Fig. 5a, b), since the distance between two points of the sounding measurements for DC amounted to 25 m, while for GCM, only 3 m.

The following figure shows the conductivity cross sections of 1D Occam inversion of GCM data for HD (Fig. 5a) and VD (Fig. 5b) configurations. They are compared with the results of 1D LMA DC resistivity results. The color scale used in these two figures is the same.

Conclusions

Due to the conductivity of the soil, impermeable clay and gypsum layers vary from a few to above 100 mS/m and there is a significant resistivity contrast; the conductive method (GCM) was successfully used at the Ruczaj district in Krakow. The determined conductivity and thickness are equal to near 6 mS/m, 40 cm, near 100 mS/m, 3 m for soil and clay layers, respectively. The thicknesses of the mentioned layers increase in a S–N direction. These results were similar to that measured by the DC sounding, but the vertical and horizontal resolutions of GCM data are better than that of the DC method. The effect is closely related to the greater sensitivity and shorter distance between the measurement points of the GCM method. Our results correspond very well with those received earlier, near to the studied area, using the VLF-EM and GCM method (Foltyn et al. 2015; Oryński et al. 2016; Orynski and Klitynski 2018).

The layers occurring at the level deeper around 7 m are invisible by the GCM method due to its limited footprint, but the deep layer was determined by the DC sounding method. It is worth adding such deep layer which need not be determined for the building engineering.

Though the footprint of the HD arrangement in GCM method is shallower than that of the VD arrangement, the fit curve for the data obtained by the GCM is better than those of the data gained from the DC method.

From the economic and the resolution point of view, the GCM method can be regarded as an alternative method related to the DC method, especially for the shallow zone (Tomecka-Suchoń et al. 2017).

In addition, it is worth noting that, regarding the fact that in GCM method, the device does not apply directly to the ground, there are some difficulties with the correct designation of the first layer conductivity. Therefore, it is much lower for GCM than for DC resistivity results (Kowalczyk et al. 2017).

It is worth adding the authors who not only qualitatively interpreted but also quantitatively determined the conductivity and thickness of the geological beds upon on the GCM data. The quantitative interpretation relays on creating the cross-sectional model correspondent to the measured data.