Experimental investigations on laboratory samples regarding the connection of spectral induced polarization to heterogeneity of hydraulic conductivity

Abstract

Knowledge of the heterogeneity of aquifers caused by sedimentary processes is essential for preferential flow path and flow time estimations due to the spreading of contaminants and ground water protection. Spectral Induced Polarization (SIP) method is a useful geophysical method that can be used to predict hydraulic properties from surface measurements, as there is a close connection between electrical and hydraulic parameters of rocks. Thus, the influence of aquifer heterogeneities such as sedimentary structures on the relationship between SIP parameters and hydraulic conductivity has been subject of some studies and is still an area of interest. In this paper we present the results of laboratory measurements on defined heterogeneous sand samples with the aim to improve the understanding of that connection. More specifically we investigated the dependence of hydraulic conductivity ($K$), as well as several SIP parameters like quadrature conductivity (${\upsigma }^{{\prime}{\prime}}$), in-phase conductivity (${\upsigma }{^\prime}$), phase shift ($\varphi$) and chargeability spectra ($m(\tau )$), which were calculated by Debye decomposition, on the composition of samples. For the laboratory experiments a measurement cell was designed to carry out hydraulic flow experiments and 4-electrodes SIP measurements on samples in the decimeter scale. The samples consisted of two different sands with different grain size distributions. The two sands were combined in varying volume shares and geometries as well as in parallel and serial orientation relative to hydraulic and electrical current flow. The experimental results show that the SIP parameters and $K$ are clearly dependent on the volume share of the sand components. In terms of preferential flow paths, known correlations to hydraulic conductivity could be reproduced, however the SIP parameters showed no dependency on the orientation of hydraulic heterogeneities. The results indicate that in samples, where the porosity and thus also the electrical conductivity amplitude are approximately homogeneous and only the grain surface area and hydraulic conductivity vary, the mean electrical parameters determined from the SIP data do not provide any information for recognising preferential flow within the scope of the measurement accuracy.

Introduction

In the last decades intense research was performed in the description and prediction of groundwater flow and solute transport in heterogeneous porous aquifers. As a result, there is a general acceptance that flow and transport in such media is governed by the spatial structure and variability in the hydraulic conductivity distribution (e.g., Sudicky 1986; Zheng and Gorelick 2003). Recently, the ongoing research in field measurements has led to a point that small scale heterogeneities in hydrogeological parameters such as hydraulic conductivity or porosity can be sufficiently captured and characterized e.g., by direct push investigations or hydraulic tomography, partly combined with geophysical measurements (e.g., Schmelzbach et al. 2011; Brauchler et al. 2012; Römhild et al. 2022).

In this regard, direct push technologies provide high resolution vertical profiles of e.g., hydrogeological parameters (e.g., Butler et al. 2002; Lessoff et al. 2011; Zschornack et al. 2013) whereas hydraulic tomography is proven to be an efficient method to characterize subsurface heterogeneity in hydraulic conductivity fields in both the lateral and vertical direction (e.g., Bohling et al. 2007; Brauchler et al. 2003; Zhao and Illman 2018).

However, while these two investigations can provide high resolution in depth, their implementation is invasive. Geophysical measurements can be conducted non-invasively from the surface or within boreholes. They give in-situ values of comparatively large sample volumes.

It has been shown that the Spectral Induced Polarization (SIP) method is promising to derive hydraulic parameters as they show a connection in various studies (e.g., Weller et al. 2015; Revil et al. 2012; Slater and Lesmes 2002; Börner et al. 1996). Therefore, a lot of research has been done on using the SIP method as a proxy for pore space related parameters, like hydraulic conductivity, porosity and grain surface area.

SIP parameters and hydraulic parameters are both influenced by pore space geometry and are related to the microstructure of the pore space. The complex nature of electrical conductivity in rocks without metallic components or clay is attributed to zones of unequal ionic transport characteristics along the pore channels, caused by charged interfaces and constrictions. Complex electrical conductivity has been most extensively studied in rocks without conductive minerals by Vinegar and Waxman (1984), Olhoeft (1985) and others.

Consequently, the challenge lies in the development of a petrophysical approach that allows the estimation of effective hydraulic conductivity in case of a heterogeneity on a larger scale than the grain size, preferably without further calibration or additional information. Therefore, the general goal is a quantitative interpretation of the SIP data regarding hydraulic properties and the formulation of a petrophysical model transforming the geophysical data into hydraulic conductivity.

So far, some work has been done regarding the influence of anisotropy on SIP on sands with electrically conductive particles. E.g., Revil et al. (2015) proposed a model for dispersed metallic conductors and Gurin et al. (2020) studied larger metallic inclusions in the centimeter scale. Nevertheless, more research is needed for materials without metallic components. Zhdanov (2008) provided a theoretical foundation for heterogeneous sands with grains of anisotropic shape. In this work however, we focus on heterogeneities on a more macroscopic scale.

Investigations on the subject of the connection between hydraulic conductivity and SIP parameters are so far mainly done in smaller-scale measurement cells with homogeneous samples in the laboratory (e.g., Slater and Lesmes 2002; Koch et al. 2011) or in field studies where the exact composition of the underground remains unclear (e.g., Fiandaca et al. 2018; Benoit et al. 2019;) or a combination of both (e.g., Attwa and Günther 2013).

In the previous laboratory studies predicting $K$ from SIP measurements extensive databases have been compiled to investigate the influence of numerous changes of petrophysical or geometrical parameters of the material on the results. However, these studies were usually carried out in cylindrical measurement cells with a volume of few cubic centimeters with restricted possibilities of installing multiple geological layers in one sample.

In this work we rely on a laboratory set up with sample sizes of 18 cubic decimeters which allows us to precisely construct heterogeneous samples with the ability to measure hydraulic parameters as well as IP-spectra on the same sample and, above all, in an identical hydraulic state. With the measuring system developed as part of this study, we are able to produce a wide range of samples composed of unconsolidated sands with an exactly reproducible heterogeneous structure and complete water saturation. The geometry of the cell and the size and shape of the composite samples were selected in such a way that hydraulic and electrical inaccuracies and distortions, e.g. due to edge effects or mixing of the components, are minimized.

Theoretical background

SIP

Complex electrical conductivity ${\upsigma }^{*}$ is a parameter that describes the electrical properties of rocks and their frequency dependence. It is the reciprocal of the complex resistivity ${\rho }^{*}$. Considering a sinusoidal excitation, according to Ohm's law ${\upsigma }^{*}$ can be considered as the complex transfer function between the electric field strength $E(\omega )$ and the current density $J\left(\omega \right)$.

$$J\left(\omega \right)={\sigma }^{*}\left(\omega \right)E(\omega )$$

(1)

${\upsigma }^{*}$ consists of the in-phase component and the quadrature component, each of which is frequency dependent and contains separate information about the rock properties.:

$${\upsigma }^{*}(\omega )=\frac{1}{{\rho }^{*}\left(\omega \right)}={\upsigma }{^\prime}(\omega )+i{\upsigma }^{{\prime}{\prime}}(\omega )$$

(2)

The in-phase component of conductivity ${\upsigma }{^\prime}$ captures the ohmic charge transport processes while the quadrature component ${\upsigma }^{{\prime}{\prime}}$ accounts for the capacitive electrical processes and is multiplied by the imaginary number $i$. Usually however, the resistivity amplitude |${\rho }^{*}$| or conductivity amplitude |${\upsigma }^{*}$| is obtained from SIP-measurements with

$$\left|{\upsigma }^{*}\right|=\frac{1}{\left|{\rho }^{*}\right|}= \sqrt{{{(\sigma }{^\prime})}^{2}+{{(\sigma }^{{\prime}{\prime}})}^{2}}$$

(3)

where the also measured phase angle $\varphi$ between in-phase component and quadrature component yields the information about the relation between both:

$$\varphi ={\text{tan}}^{-1}\left(\frac{{\sigma }^{{\prime}{\prime}}}{{\sigma }{^\prime}}\right)$$

(4)

The Cole–Cole model (Cole and Cole 1941) is typically used as a transfer function in Eq. (1) to fit the SIP-spectra and is given by

$${\frac1{\sigma^\ast\left(\omega\right)}=\rho}^\ast\left(\omega\right)=\rho_0\left\lceil1-m\left(1-\frac1{1+\left(i\omega\tau\right)^b}\right)\right\rceil$$

(5)

with the low-frequency resistivity constant ${\rho }_{0}$, the chargeability$m$, the angular frequency $\omega$, the relaxation time constant $\tau$ and the Cole–Cole exponent$b$. In contrast to the well-known Debye relaxation model, the Cole–Cole model assumes that there is a distribution of relaxation times in the material, where the exponent $b$ describes the steepness of the dispersion. Analysis of the measured SIP data can furthermore be carried out by applying Debye decomposition in an approach that is suggested by Nordsiek and Weller (2008). While in the Cole–Cole models the chargeability is essentially determined by a distribution of relaxation times $\tau$ and the exponent$b$, the Debye decomposition allows for a more flexible and broadband fit of the relaxation times. The chargeability spectra are calculated by solving an equation system that arises from

$${\upsigma }^{*}\left(\omega \right)=\frac{{\sigma }_{0}}{1-\sum _{k=1}^{n}{{m}_{k}\cdot \left(1-\frac{1}{1+i{\omega \tau }_{k}}\right)}^{2}}$$

(6)

where ${m}_{k}$ is a spectra of chargeabilities for n computed relaxation times ${\tau }_{k}$ and ${\sigma }_{0}$ is the low-frequency conductivity constant, which represents the electrical DC-conductivity (Nordsiek and Weller 2008).

Relations between rock properties

With the goal to better predict the hydraulic conductivity ($K$) in sands from geoelectrical measurements, many works have been conducted in the last decades to refine the governing models (e.g. Hördt et al. 2009; Attwa and Günther 2013). The well investigated model called “electrical Kozeny-Carman-equation” is based on the previously by Pape et al. (1987) formulated PARIS equation (Börner et al. 1996)

$${K}_{pred}=\frac{a}{F{({S}_{por})}^{c}}$$

(7)

where the dependence of the predicted hydraulic conductivity ${K}_{pred}$ (m/s) on the specific pore surface area-to-porosity-ratio of the rock ${S}_{por}$ (in 1/µm) and the formation factor $F$ is described. In contrast to the PARIS equation, not only the formation factor is determined electrically, but also the specific surface. In (7) ${S}_{por}$ depends linearly on the quadrature conductivity σ'' (Börner et al. 1996). The constant values $a$ and $c$ were found via experiments on consolidated and unconsolidated sediments. The parameter $c$ lies in the range of $2.8< c<4.6$ for consolidated and unconsolidated clastics. The values for the constant $a$ presented in the literature (e.g. Börner et al. 1996; Hördt et al. 2009) differ from each other. Deviations are caused by using different measuring methods for hydraulic conductivity.

The pore surface area-to-porosity-ratio ${S}_{por}$ (e.g. 1/µm) is related to the specific surface area ${S}_{m}$ (m²/g) as follows:

$${S}_{por}=\frac{{S}_{m}{d}_{m}(1-\Phi )}{\Phi }$$

(8)

Here, ${d}_{m}$ is the matrix density and $\Phi$ is the porosity of the material, which influences the ${S}_{por}$-parameter in a complex way (Börner et al. 2017).

In their study of unconsolidated sediments Slater and Lesmes (2002) find the correlation between the electrically determined $K$ and the grain diameter at 10 percent of the grain size distribution ${d}_{10}$ to be statistically more evident for the investigated samples than the correlation between $K$ and ${S}_{por}$. For their measurements they find the Hazen type formulation (Hazen 1911)

$${K}_{pred}=0.002{({d}_{10})}^{d}$$

(9)

where $d=1.3 \pm 0.2$, ${d}_{10}$ is given in mm and ${K}_{pred}$ is given in m/s. Furthermore, Slater and Lesmes (2002) find the influence of $F$ on the $K$ estimation to be statistically negligible for the investigated samples and thus propose a similar equation to (7) with

$${K}_{pred}=\frac{a}{{(\sigma {^\prime}{^\prime})}^{c}}$$

(10)

where $a=0.0002 \pm 0.00003$ and $c=1.1 \pm 0.2$ when ${\sigma }^{{\prime}{\prime}}$ is given in µS/m and ${K}_{pred}$ in m/s. The negligibility of $F$ may be for sample series with slightly varying porosity. In general, this is not observable due to the dominant dependence of hydraulic conductivity on porosity according e.g. the Kozeny-Carman-equation (Kozeny 1927; Carman 1937) and the PARIS-equation (Pape et al. 1987).

The true formation factor $F$ was calculated according to Börner et al. (1996) and Weller et al. (2013) from the real and imaginary part of conductivity. The parameter $l$ varies in the range of 0.01 to 0.1:

$$F=\frac{{\sigma }_{w}}{{\sigma }{^\prime}-\frac{{\sigma }^{{\prime}{\prime}}}{l}}$$

(11)

In the case of weak interfacial effects and the associated low polarization of sands, the true formation factor $F$, which excludes interfacial conductivity, deviated only slightly from an apparent formation factor, which includes interfacial conductivity.

Estimation of effective parameters

In order to review the applicability of IP measurements for the detection of hydraulic anisotropies and heterogeneities, the expected resulting petrophysical parameters of a composed sample are calculated by averaging the petrophysical parameters of the separate sands. For the hydraulic measurements the parallel and series configurations are equivalent to horizontal and vertical flow in an anisotropic aquifer respectively (Langguth and Voigt 2004) and the same calculations can be made. The resulting hydraulic conductivity ${K}_{par}$ of a parallel configuration is thereby the arithmetic mean

$${K}_{par}=\frac{1}{T}\sum _{k=1}^{w}{K}_{k}{T}_{k} where T= \sum _{k=1}^{w}{T}_{k}$$

(12)

of the separate hydraulic conductivities ${K}_{k}$, weighted by the thickness $T$ of the sand package. The number of parallel or in-series sand packages will be $w=2$ in this work. The resulting hydraulic conductivity ${K}_{ser}$ of a series configuration is the harmonic mean

$${K}_{ser}=L\frac{1}{\sum _{k=1}^{w}\frac{{L}_{k}}{{K}_{k}}} where L= \sum _{k=1}^{w}{L}_{k}$$

(13)

where the separate hydraulic conductivities are now weighted by the length $L$ of their respective sand package.

The calculation of the resulting complex valued conductivity ${\sigma }^{*}$ of a sample was expected to be analogous to the calculations for the $K$ values. Considering Ohm’s law, the resulting ${\sigma }_{par}^{*}$ of a parallel circuit is

$${\sigma }_{par}^{*} =\frac{1}{T}\sum _{k=1}^{w}{\sigma }_{k}^{*}{T}_{k}$$

(14)

where ${\sigma }_{i}^{*}$ is the complex electrical conductivity of the separate homogeneous sand packages. When applying Eq. (2) we arrive at

$${\sigma }_{par}{^\prime}+{i\sigma }_{par}^{{\prime}{\prime}} =\frac{1}{T}\sum _{k=1}^{w}\left({\sigma }_{k}{^\prime}+i{\sigma }_{k}^{{\prime}{\prime}}\right){T}_{k}$$

(15)

for the in-phase and quadrature components of conductivity. Equating the coefficients of the real and imaginary parts of Eq. (15) now yields

$${\sigma }_{par}{^\prime} =\frac{1}{T}\sum _{k=1}^{w}{\sigma }_{k}{^\prime}{T}_{k}$$

(16)

for the in-phase component of conductivity and

$${\sigma }_{par}^{{\prime}{\prime}} =\frac{1}{T}\sum _{k=1}^{w}{\sigma }_{k}^{{\prime}{\prime}}{T}_{k}$$

(17)

for the quadrature component. As with the hydraulic conductivities, the same procedure can be done for the series-configurated samples, which leads to

$${\sigma }_{ser}{^\prime}=L\frac{1}{\sum _{k=1}^{w}\frac{{l}_{k}}{{\sigma }_{k}{^\prime}}}$$

(18)

and

$${\sigma }_{ser}^{{\prime}{\prime}}=L\frac{1}{\sum _{k=1}^{w}\frac{{l}_{k}}{{\sigma }_{k}^{{\prime}{\prime}}}}$$

(19)

respectively. It must be noted here that Kenkel et al. (2012) found that if a sample is composed of 2 materials with the same amplitudes of conductivity, the total phase angle can be calculated as an arithmetic mean, disregarding if the sample has a serial or parallel configuration. In that case ${\sigma }_{ser}^{{\prime}{\prime}}$ would be calculated in the same way as ${\sigma }_{par}^{{\prime}{\prime}}$ in Eq. (17).

To further analyse the chargeability spectra obtained from Debye decomposition, for the merit of simplicity it can be useful to reduce them to a single value even at the cost of some of the information. Therefore, according to a definition from Weller et al. (2015) the total chargeability ${m}_{total}$ and the mean chargeability ${m}_{mean}$ are calculated by

$${m}_{total}={m}_{mean}\bullet n= \sum_{k=1}^{n}{m}_{k}$$

(20)

where ${m}_{k}$ is the polarization magnitude of a single Debye relaxation term as defined by Eq. (6) and $n$ is the number of computed relaxation times.

Materials and methods

Experimental setup

The size of the experimental setup, which is displayed in Fig. 1, was chosen according to the measuring technique and the properties of the considered unconsolidated rocks. The setup consisted of a Plexiglas tank (0.5 m length, 0.3 m width, 0.12 m depth) which was stabilized by a wooden basement. The tank is limited on all sides by Plexiglas plates, so that at least a low overpressure can be set inside the tank.

Fig. 1

Schematic of the complete experimental set up

Containers that are filled with water to accommodate the current electrodes are placed at both ends of the tank. There is a permeable wall between the tank and the electrode container to keep the sample inside the tank but allow water flow. The electrode containers are also used to adjust the pressure difference for the flow through the tank. In general, a pressure level is set that lies above the cover of the tank, so that pressurized groundwater conditions are present. Examples of alike measurements are found in the literature for solid rocks (Wendland and Himmelsbach 2002), for artificial unconsolidated rocks (Schmalz et al. 2002; Al Hagrey et al. 1999; Boerner 2009) and at lysimeters (Stoffregen et al. 2002; Schmalholz et al. 2004).

The sample cell of the set up can be opened and closed by a lid on top of it, to allow the preparation of a sample. Through small holes inside of the lid, the potential electrodes can access the sample for the electrical measurement. The potential electrodes consist of KCl crystals inside a glass tubing to allow for a non-polarizing contact with the sample. As current electrodes, Pb plates are used that are placed under water on the side of each water container as shown in Fig. 1. Those plates establish a homogeneous, linear electrical field in the sample during electrical measurements which means that even though the two current electrodes have contact to the sample at a single point respectively, they are representative for the whole plane that is perpendicular to the current flow. Therefore the electrical as well as the hydraulic measurements can be regarded as one-dimensional measurements. The advantage of our setup in comparison to column experiments in other works is the possibility to create a heterogenous sample out of multiple homogenous areas in the setup that are not limited to layer boundaries that are oriented perpendicular to the current flow, but that can have any direction.

In preparation of a sample the water that would saturate the sample was prepared first. Its electrical conductivity was aimed to be around 100 µS/cm when the sample was finished. Therefore, NaCl salt was poured into distilled water until 85 µS/cm were reached, anticipating solution processes taking place after mixing the water with the sand. The electrical fluid conductivity as well as the fluid temperature were measured with the equipment LF 196 from WTW and the compatible sensor Tetracon 96–1,5 (also WTW), a portable measuring instrument, that allows quick and reliable fluid conductivity and temperature measurements with and without temperature compensation. The measurements were conducted in a non-calibrated mode where conductivity measurements are not preprocessed to a reference temperature.

The sand was positioned in the cell that is shown in Fig. 2b under water by pouring it from a pipe. That way it was ensured, that every grain makes contact with the water isolated from other grains, so that no air bubbles could form and full saturation was achieved. The porosity was calculated by weighing the sand before filling it into the measurement cell. The overall volume of the tank equalled 18 dm3 after reductions due to measuring devices. As shown in Table 1, an average porosity between 38 and 44% can be calculated for all sands by subtracting the quotient of their volume trough the cell volume from 1. The error was estimated by ± 0.1%. The change of the fluid temperature stayed within 0.3 K during all measurements which were carried out at room temperature.

Fig. 2

Sample holder for hydraulic and IP measurements of the samples; a top: top view, bottom: side view; b picture of the measurement cell with plugged in electrodes

Table 1 Grain sizes, pore space parameters and K values of the sand samples

A small and height flexible container with 20 L capacity was used to regulate the pressure difference in the Plexiglas container. The measurement of the pressure head is measured by means of two capillaries. The ends of the capillaries were located below the positions of the two potential electrodes as shown in Fig. 1.

Beside the calibration measurements with water as tank filling, saturated sand was used as material for the measurements. To achieve the maximum of saturation possible, the build in of the sample was carried out in layers of approximately 3 cm thickness. First the pore water got prepared as a chemical solution where NaCl was mixed with distilled water to the desired conductivity of 85 µS/cm. Then, for each layer, first the water got filled into the measurement cell. After that, the sand was poured in carefully to make sure that every grain made contact with the water surface individually. The sand got filled in as long as the height of the sand stayed just under the water surface. To eliminate any built-in air bubbles, the layer was stirred with a wooden rod in a repeatable manner afterwards. The next layer was then built on top of the previous one in the same procedure.

Finally, to close remaining holes in the sand matrix, the samples were settled by carefully knocking on the side walls. The created space at the top of the sample was then filled up with sand. Still, the way of sample preparation meant that the samples were unconsolidated.

For the measurement series of the composed samples it was necessary to create separate domains in the measurement cell where the different sands could be poured into. To achieve that, one or more thin Plexiglas plates were put into the measurement cell and were fixated on the walls by a piece of tape. After finishing the built-in of the sample where every sand was only poured into some of the created compartments, all Plexiglas plates were taken out of the measurement cell. The space that was previously occupied by those plates was then filled up with the sands that had built up on each side of them and some extra sand that was poured onto the top of the sample.

In order to achieve pressurized groundwater conditions, the lid was closed on the sample on top of a sealing fabric. Afterwards the hydraulic head was lifted by about 5 mm over the top of the sample. Next, since some chemical solution processes took place, that should be avoided during electrical measurements, the sample was given at least twelve hours of time to ideally achieve an equilibrium of fluid conductivity, before any electrical data was gathered. Altogether, the built-in, measurement and evacuation of a sample took at least three days and up to five days if leakages occurred, that needed to be closed.

The electrical measurements were carried out before the hydraulic measurements because the latter seemed to trigger solution processes of solubles in the samples which would increase the fluid conductivity. For an electrical measurement the two lead current electrodes and the two KCl potential electrodes were connected via cables to the SIP QUAD II, a SIP measurement device by Radic Research that can measure the phase shift and conductivity amplitude over a frequency spectra from 1.2 Hz${\times 10}^{-4}$ to 2.3 ${\times 10}^{5}$ Hz. The geometric factor G = 0.18 m of the measurement setup could be calculated as the quotient of the effective cross-sectional area divided by the distance of the potential electrodes. The input voltage at the current electrodes was always set to 6 V. The corresponding input current depended on the sample and was 1.5 mA ± 0.5 mA.

The measurement process started at a frequency of 2.3 $\times {10}^{5}$ Hz from which the measurement frequency got divided by two for the next measuring frequency. This was repeated until the lowest used measuring frequency of 1.9 $\times {10}^{-3}$ Hz. To improve the measurement accuracy, every frequency was measured multiple times its period length. The measurements at the highest frequencies were stacked 50 times while measurements at lower frequencies were stacked successively less frequently, down to 3 stacks at the lowest frequencies. The resulting measurement parameters resistance and potential were then calculated as an average over the repetitions. From the stacks a measurement error is automatically computed. For all measurements we observed a measurement error that was at least a magnitude lower than the measured value.

After finishing the electrical measurements and removing the electrodes from the experimental set up, the hydraulical measurements were started. By lifting the water reservoir with the mariotte tube, shown in Fig. 1, the pressure head in the left container was lifted over the overflow in the right container. That allowed a steady fluid flow through the sample. The pressure head difference was measured at the capillary tubes which led to the ground vertically under the electrode spots. The flow rate through the sample was calculated by measuring the amount of fluid volume that left the measurement cell over the needed time. The hydraulic conductivity $K$ could then be calculated by applying Darcy’s Law with

$$K = \frac{\frac{V}{t}}{{A \cdot \frac{\Delta h}{{l_{c} }}}}$$

(21)

where $V$ is the total volume that flow through the overflow in the right container in the measurement time $t$, $A$ is the effective cross-sectional area of the measurement cell, $\Delta h$ is the hydraulic head difference between the two capillaries and ${l}_{c}$ is the distance between the two capillaries.

Sands

The examined materials were five commercially available, unconsolidated sands that differed in their grain size distributions. Two sands were neglected before the investigations in this work, thus the naming of the sands starts with S3. The sands were previously sieved to a certain degree and cleaned. For experiments only material without clay and silt content was chosen so that a membrane could be used for hydraulic measurements which would let water through but hold any grains back. As a membrane we used plastic woven mesh fabrics with pore sizes of 100 to 300 µm. Almost the whole spectrum of grain sizes of sands was covered to have a large variety of material tested that could later be chosen from.

An optical grain size analysis showed that 4 sands, S3, S4, S6 and S7 are well and moderately sorted, whereas the fifth sand, S5, is poorly sorted. The grain diameters ${d}_{10}$, ${d}_{30}$, ${d}_{50}$ and ${d}_{60}$ at the 10th, 30th, 50th and 60th percentile of the grain size distribution were obtained from the optical grain size analysis data and are shown in Table 1.

Furthermore, the parameters specific pore surface area ${S}_{m}$, a calculated mean porosity of the installed pure sand samples and the hydraulic conductivity $K$ of the samples are listed in Table 1. The parameter ${S}_{m}$ was analysed through the adsorption of nitrogen molecules using the BET-method after Brunauer et al. (1938). The pore surface area-to-porosity-ratio can be calculated by Eq. (8). The mean porosities of the pure sand samples were derived from a simple calculation. The sand got weighted before pouring it into the measurement cell. That weight was then divided by the volume of the measurement cell which is 18 dm³. The hydraulic conductivity $K$ was measured in the experimental set up using the hydraulic head method as described above.

Experimental results—homogeneous samples

A series of measurements was carried out on the five different sands to subsequently decide on which two of them to use for combined measurements. The measured $K$ values and the complex electrical parameters are shown in Table 2. As can be seen there, the fluid conductivity values are significantly higher than 100 µS/cm for S5 and S6, which is probably because higher amounts of solubles are present in those sands. Even though it was the goal to have approximately the same fluid conductivity for every sample, the results that were gained from the presented measurements on the sands S5 and S6 were sufficient to decide on whether to use them further.

Table 2 Experimental results of the homogeneous sand samples; hydraulic conductivity, fluid conductivity, in-phase and quadrature component of conductivity at 1 Hz frequency and true formation factor (see Eq. (11))

The results of the SIP-measurements on the pure sand samples are shown in Fig. 3 in terms of the in-phase conductivity and the quadrature conductivity over the measured frequency spectrum. They were obtained from the phase and amplitude using Eqs. (3) and (4). It must be noted, that before displaying the phase spectra in Figs. 3 and 4 one processing step was already done. Where in Cole–Cole fits usually an EM-Term is subtracted for high frequencies, in this work the chargeabilities of low relaxation times are adjusted in a reproducible manner for every sample. This is done to compensate coupling effects in the experimental setup and to make the phase spectra further analysable at high frequencies. The chargeabilities spectra are obtained from a Debye decomposition of the phase spectra. At relaxation times between 0.001 s and 0.1 s, which correspond to 10 to 1000 Hz, the chargeabilities were observed to be too high because of coupling effects. Therefore, a constant chargeability value, obtained from the chargeability at a relaxation time of 0.1 s, was applied to that range. By then performing the Debye decomposition backwards, the corrected phase and quadrature conductivity spectra were calculated. These processing steps have been done for all the data in the subsequent figures.

Fig. 3

Conductivity spectra of the pure sand samples S3 to S7; the in-phase conductivity and the quadrature conductivity versus the frequency are shown for every sample

Fig. 4

a Relationship between d10 and K of the six sands, compared to the empirically derived relation from Slater and Lesmes (2002). The dashed lines show the deviation from Eq. (9) that is controlled by the exponent d $\text{d}=1.3\pm 0.2$ b Measured quadrature conductivity in dependence of the specific pore surface area measured with the BET-method; The values for the imaginary part of conductivity were calculated as a mean value between 1 and 100 Hz; The proportionality of the regression line is according to Slater (2007). c Measured K multiplied by the formation factor in dependence of quadrature conductivity; the displayed exponents refer to Eq. (7). Deviations could have been caused by different properties of the grain surfaces of the sands used

The complex conductivity components at a frequency of 1 Hz are displayed in Table 2. Their conformity to known correlations such as $\sigma {^\prime}{^\prime}$ to $K\bullet F$ and $\sigma {^\prime}{^\prime}$ to ${S}_{por}$, as shown in Fig. 4b, c, was then checked before deciding on further usage.

The in-phase conductivity spectra in Fig. 3 show very little change with respect to the measurement frequency which can be attributed to the fact, that the chargeability is expected to be comparatively low for the examined material. However, between the sands some differences in terms of the in-phase conductivities are visible which can be explained by the differences in electrical conductivity of pore fluid and the formation factor. I.e., the elevated in-phase conductivity of sample S6 can be explained by its low formation factor.

As for the quadrature conductivity spectra in Fig. 3 it can be seen that the two sands S3 and S5 show significantly higher values than the remaining four.

The previously described correlation of ${d}_{10}$ to $K$ was compared to the empirical relation (9) found by Slater and Lesmes (2002). Figure 4a shows that the investigated sands plot nearby the fitted curve of the Slater and Lesmes (2002) data and especially the 3 sands S3, S4 and S7 show a good concordance.

The comparison of ${\sigma }^{{\prime}{\prime}}$ with ${S}_{por}$ in Fig. 4b shows only a weak correlation for the sands investigated. As this correlation has been proven for many rocks, we suspect that the surface properties are influenced by a possible industrial cleaning of sands or that the BET measurements are not fully representative.

To verify the parameters in Eq. (7) for rocks with similar formation factors, the product of formation factor and hydraulic conductivity is compared with the imaginary part of the electrical conductivity. The relationship for the sands investigated is shown in Fig. 4c.

Experimental results—composed samples

From the five analyzed sands, two suitable sands were selected for the subsequent experimental studies on the influence of heterogeneity on the spectra of complex conductivity. Criteria for the selection of the sands S3 and S7 for further measurements were:

- sufficiently different phase level of approximately a magnitude at 10 Hz and below,

- different spectra shapes—one spectra has a phase peak, the other one does not,

- stability of the sand packings during flow,

- and consistency with well investigated correlations, such as ${d}_{10}$ vs. $K$ and ${\sigma }^{{\prime}{\prime}}$ vs. $K\bullet F$.

Those two sands were placed in the experimental set up with varied volume shares as well as varied orientations of the S3–S7 interface with respect to the direction of flow. This meant that some samples would represent a series scheme, some a parallel scheme and some a mixed scheme of those two. A couple of measurements were carried out on samples with a 50–50 volumetric distribution with a parallel and series pattern as well as a 4- and 8-segments pattern with smaller but a higher number of homogeneous spaces in the sample. The remaining measurements were done on samples with 75–25 or 25–75 volumetric portions of the S3 and S7 sands, respectively. The aim hereby was to find out if the volume share would be visible in the measured data and to further investigate the behaviour of series and parallel samples with different geometry.

For the samples shown in the lowest two rows in Table 3, it was necessary to move the homogeneous area of the sand with 25% volume share closer to the center of the measurement cell. This is because the interface of the two sands had to lie in between the two potential electrodes, which as shown in Figs. 1 and 2, are only 200 mm apart. To maintain consistency, the homogeneous area was placed in such a way, that the volume share of the two sands between the two potential electrodes would be the same as in the entire measurement cell.

Table 3 Composition of the sand samples; the diagrams are top views of the experiment i.e., the direction of current flow and water flow goes from left to right in the diagrams, the blue rectangles represent the water tanks on each side and the red dots indicate the potential electrode positions; ${\text{T}}_{\text{k}}$ and ${\text{L}}_{\text{k}}$ from Eqs. (12) and (13) are shown exemplary in two diagrams and work analogous for the other samples

Further information on the measurement series is presented in Table 3. The parameters derived from the measurements of the two homogeneous sand samples would then serve to characterize the homogeneous parts of the combined samples.

Figure 5 shows the resulting complex conductivity spectra of the measurement series of the combined samples as well as the pure S3 and S7 samples, which are working as the two limiting cases for the interpretation of the quadrature conductivity spectra.

Fig. 5

Conductivity spectra of the composed sand samples of the S3 and S7 measurement series; the in-phase conductivity and the quadrature conductivity in dependence of the measurement frequency are shown for every sample

Discussion

The values of the in-phase conductivity in Fig. 5 lie much closer together than in Fig. 3, since the amount of solubles and therefore the water conductivity seems to be similar in both sands. In terms of the quadrature conductivity spectra it can be seen that the mixed samples show a combinatory behaviour of the two pure sands, since they lie in between those two. Furthermore, the quadrature conductivity peak of S3 at around 2 Hz${\times 10}^{-2}$ is also visible in the mixed samples but gets less prominent which could be influenced by the S7 sand. Regarding the absolute phase values between $1{\times 10}^{-3}$ Hz and $1\times {10}^{3}$ Hz, a systematic behaviour can already be seen as the samples with a higher share of S3 show higher absolute phase values and thus lie closer to the pure S3 spectrum, whereas a higher S7 content causes lower absolute phase values, closer to the S7 spectrum.

Figure 6 shows the measured $K$ values vs. the volume share of S7 in relation to the total sample. The added dashed line with the description “arithmetic mean” corresponds to Eq. (12), whereas the dotted line with the description “harmonic mean” corresponds to Eq. (13). Regarding the measured $K$ values, a clear separation trend can be seen, where the samples with a parallel configuration plot closer to the arithmetic mean than the samples of a series configuration. The data at 75% volume share of S7 plots more distant from the lines of the arithmetic and harmonic mean. This is likely because of the large measurement inaccuracy that is connected to hydraulic experiments and effects all the data points, including the points of the pure sand samples. For the hydraulic measurements in this work a measurement inaccuracy of $\pm 30$% has been estimated. A better correlation would probably be achieved by minimizing the inaccuracy with a larger number of measurements.

Fig. 6

Measured K values in dependence of the sample composition (dots) and expected K values, calculated from the pure S3 or S7 samples (lines); The expected trend can be seen that samples of a parallel circuit show a higher hydraulic conductivity than the samples of the series circuits of the same volume ratio. In addition, the samples of the series connection are closer to the weighted harmonic mean and most of the samples of the parallel connection are closer to the weighted arithmetic mean of the pure S3 and S7 sands. It should also be noted that the measurement inaccuracy is typically high for hydraulic tests

Still, the results show that a parallel configuration with higher $K$ values simulates preferential flow in the measurement direction while a series configuration with lower $K$ values at the same volume share simulates preferential flow perpendicular to measurement direction. In other words, preferential flow exists parallel to the interface of the two sands inside the sample.

Considering the quadrature conductivity values at 1 Hz of the composed samples measurement series in Fig. 7, a clear dependency on the volume shares of the components can be observed, as expected. No correlation in terms of the orientation of the components in the sample is visible, as the differences between samples of the same volume share are very little in comparison and show no clear trend. The lack of dependency of quadrature conductivity on the sample configuration, i.e., series or parallel, came as a surprising observation and is one of the main results of this work. However, this could be explained by the fact that the amplitudes of conductivity of the used materials were very similar. Therefore the resulting quadrature conductivity must be close to the arithmetic mean, as has been found by Kenkel et al. (2012).

Fig. 7

Measured ${\sigma }^{{\prime}{\prime}}$ values in dependence of the sample composition (dots) and expected ${\sigma }^{{\prime}{\prime}}$ values, calculated from the pure S3 or S7 samples (lines); the value of the quadrature conductivity shows correlation to the volume shares of the components in the sand samples but no correlation to the orientation of the components in the samples. Instead, the measured values are all close to the curve of the weighted arithmetic mean of the S3 and S7 samples, which is assumed for a series connection

The in-phase conductivity, shown in Fig. 8 depends mainly on the fluid salinity and sample porosity, according to Archie’s Law. During the measurement series it was attempted to reproduce the same values for both parameters for every sample as well as possible. Thus, no significant differences are observed here.

Fig. 8

Measured ${\sigma }{^\prime}$ values in dependence of the sample composition; As the values of the in-phase components mainly depend on parameters like fluid conductivity and porosity which were attempted to keep nearly the same for all samples, the resulting values are all very similar

All the main SIP parameters that were calculated in connection with the composed samples measurement series are shown in Table 4. In order to find a comparable parameter that describes the capacitive properties of the samples, the mean and total chargeability were calculated according to Eq. (20). Since the same relaxation time spectrum has been used for all the samples, the mean chargeability and total chargeability are proportional to each other. Therefore, only one of the two parameters is depicted in Fig. 9 with the mean chargeability. It can be seen, that the distribution of the chargeability mean values is again very similar to the distribution of the phase shifts and the quadrature conductivities.

Table 4 Overview of the measured and calculated IP parameters for the S3 S7 measurement series;${\upsigma }_{\text{w}}$: conductivity of the pore fluid, ${\upsigma }_{0}$: DC conductivity, ${\sigma }{^\prime}$(1 Hz): in-phase conductivity at 1 Hz, ${\sigma }^{{\prime}{\prime}}$(1 Hz): quadrature conductivity at 1 Hz, ${m}_{\text{mean}}$: mean chargeability over the relaxation time spectrum, ${m}_{\text{total}}$: total chargeability

Fig. 9

Mean m values from Debye decomposition in dependence of the sample composition (dots) and expected mean m values, calculated from the pure S3 or S7 samples (lines)

We were able to determine that mean values or values of a single frequency for imaginary part and chargeability have only small effects of heterogeneity. The analysis of the spectra by means of Debye decomposition now aimed to identify and quantify possible indications of heterogeneity in the shapes of the spectra. Figure 10 gives an example of the calculations carried out after performing Debye decompositions of the IP spectra. Shown are all the samples with a composition of 50% S3 and 50% S7. The aim was to find out if the frequency dependent chargeability of a composed sample shows a combination of the characteristics of the chargeability spectra of the separate pure sand samples and if these characteristics are influenced by the type of the sample i.e., series, parallel or mixed. To assess that, two theoretical chargeability spectra were calculated in the bottom diagram: one as an arithmetic mean of the separate S3 and S7 spectra (1) and a second one as a harmonic mean of the same two spectra (2). Those were then compared to the chargeability spectra that are obtained from Debye decompositions of the samples that are combined of S3 and S7 and are represented by the solid lines in the bottom diagram. It becomes clear that the latter lie closer to the calculated arithmetic mean spectrum.

Fig. 10

DD-spectra for all samples with a 50–50 volume ratio; bottom: the chargeability spectra is calculated normally for a parallel circuit-like behaviour of the sample (2) and a second time calculated where a series circuit-like behaviour is assumed (1). The actual measured samples show a higher similarity to (1). middle: phase spectra derived from the two chargeability spectra are displayed together with the measured phase spectrum and a better fit to the spectrum of the series-like behaviour can be seen, top: The in-phase conductivity of all measured spectra is shown

The middle diagram of Fig. 10 presents the phase spectra of the same samples. It includes two calculated hypothetical phase spectra, obtained from the two theoretical chargeability spectra (1) and (2). Here, similarly to the bottom diagram, it can be seen, that all the spectra of the composed samples follow the shape and magnitude of the calculated spectrum for the arithmetic mean. The same behaviour was also observed for samples of different volume shares. Therefore, we conclude that for the material used in this work, a Debye decomposition cannot help to differentiate between a series and parallel composition, even though it analyses the whole measured frequency spectrum.

The top diagram has no significance in that regard and is only shown for the purpose of verifying consistency of the theoretically calculated spectra.

Numerous publications are dedicated to the derivation of very detailed pore space properties from the SIP data. That is possible because the published spectra often show more or less pronounced relaxation effects. However, these studies are limited to homogeneous samples and very small samples of a few cm3 in size. However, such significant relaxation maxima are not observed in real-scale SIP measurements to assess the soil and groundwater zone. The experimental facility set up here and the experiments carried out with it are intended to contribute to a better understanding of the role of heterogeneity of that magnitude. Ultimately, these investigations show that for deriving hydraulic properties from geophysical, especially SIP measurements, the use of composite samples is more expedient compared to ideal homogeneous samples.

Conclusion

For this study we designed and constructed a cuboid experimental set up with room for samples of 18 dm3 volume to allow multiple homogeneous areas in order to examine both the hydraulic and electrical behaviour on the same composed sample.

The aim was to examine the influence of the electric measurement orientation relative to a heterogenic feature like preferential flow in a certain direction inside the sample by installing two sands with different hydraulic conductivity in a parallel or series arrangement in a sample and thus modelling either preferential flow parallel to measurement direction or perpendicular to it. For all these composed samples in the measurement series it can be concluded that the hydraulic conductivity behaved in the desired way since series-like samples showed lower integral $K$ values in the measurement direction than the corresponding parallel sample. Also, the volume share of the two sands had the expected impact, where a higher volume share of a sand with a higher $K$ value could be correlated to a higher total $K$ value of the sample.

In order to conclude an influence of the hydraulic heterogeneity on the parameters phase shift, quadrature conductivity or chargeability, it would be expected that the chargeability spectra show a characteristic trend, depending on the built-in orientation of the layer boundary for samples with the same volume ratios. However, no such correlation as e.g., known from Ohm’s law for parallel and series circuits, has been seen. Still, further experiments that satisfy an adjusted model representation might correlate in that regard. To investigate that further, measurement series are needed with material that has different hydraulic and electrical properties. Nevertheless, the IP-parameters phase shift in Fig. 8 and chargeability in Fig. 9 displayed a correlation to the volume share of the sample.

The results found in this study can be useful to understand e.g., field measurements on layered sands or even heterogeneous hydraulic settings of any kind. In any case, further experiments are necessary, especially with regard to further optimization of the components used for the composition of the heterogeneous samples with regard to greater differences in their hydraulic and pore space properties.