Spectral Induced Polarization measurements on New Zealand sands - dependence on fluid conductivity

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1 Near Surface Geophysics, 015, 13, doi: / Spectral Induced Polarization measurements on New Zealand sands - dependence on fluid conductivity Sheen Joseph 1, Malcolm Ingham 1 and Gideon Gouws 1 School of Chemical & Physical Sciences, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand School of Engineering & Computer Science, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand Received February 014, revision accepted September 014 ABSTRACT Spectral induced polarization (SIP) measurements explore the variation of the complex conductivity (σ) of a material with frequency. Much of this variation results from polarization effects associated with the electric double layer on the aces of pore spaces. The consequent dependence of the SIP signature on pore structure thus has the potential to provide a link to the hydraulic properties of the material, which have a similar dependence. We report here on the variation of the SIP signature of unconsolidated sands, typical of those found in coastal aquifers in New Zealand, with the conductivity of the pore fluid. The SIP parameters of the measurements are modelled in terms of a Cole-Cole model and demonstrate the independence of relaxation time on fluid conductivity. The contribution of ace conductivity to the overall conductivity is calculated and the variation of the imaginary part of the ace conductivity with fluid conductivity is tested against two models for the origin of ace conductivity. The measured hydraulic conductivity is also compared with estimates provided by three proposed equations relating hydraulic properties to structural and electric properties. INTRODUCTION The measured bulk conductivity of an unconsolidated material is a complex quantity with both in-phase and quadrature-phase parts. At high salinities this is often expressed in terms of a parallel combination of ionic conductivity through the pore fluid and a complex ace conductivity (e.g. Keller and Frischknecht 1966; Slater 007; Weller and Slater 01; Zisser et al. 010 ) malcolm.ingham@vuw.ac.nz σ = σ ionic + σ. (1) In a saturated material, the ionic conductivity is related directly to the conductivity of the pore fluid (σ w ) through the formation factor (F) (Archie 194) σ σ w ionic = F = φ n where ϕ is the porosity of the material and n is referred to as the cementation exponent. Combining (1) and () leads to σ = 1 σ w + σ + iσ F in which σ and σ are the in-phase and quadrature-phase parts of the ace conductivity. σ represents conduction along the ace of grains and pore spaces which is in phase () (3) with an applied electric field. In contrast, σ is the result of diffusive polarization effects on grain and pore aces (e.g. Weller and Börner 1996). The ace conductivity is frequency dependent, and when the bulk conductivity of a saturated material is measured as a function of frequency it is therefore found that not only does the magnitude of the conductivity vary with frequency but there is also a phase dependence on frequency. The measurement of the phase of the bulk conductivity as a function of frequency forms the basis of the geophysical technique of spectral induced polarization (SIP). Typical phase values are of the order of not more than a few tens of milliradians, reflecting the fact that σ ionic + σ is much greater than σ. There are several mechanisms that have been considered as responsible for ace polarization at frequencies below about 10 khz. These have recently been discussed by Kemna et al. (01) and include effects associated with the formation of the electrical double layer at the interface between mineral grains and the pore fluid. The form of the phase dependence is believed to be characteristic of the grain size and pore structure of the unconsolidated material and SIP is thus believed to hold potential as a proxy for the hydraulic properties of a material (e.g. Börner et al. 1996). Considerable recent research has focused on both understanding the mechanisms that lead to SIP (e.g. de Lima and Sharma 199; Chelidze and Gueguen 1999; Titov et al. 00; Chen and Orr 006; Leroy et al. 008; Vaudelet et al. 011; Revil 01) and on the correlation of measured SIP parameters with hydraulic proper- 015 European Association of Geoscientists & Engineers 169

2 170 S. Joseph, M. Ingham and G. Gouws FIGURE 1 Measured grain size distribution for the natural sand used in Samples 1 and. ties (e.g. Slater and Lesmes 00; Binley et al. 005; Slater 007; Revil and Florsch 010; Koch et al. 01). If useful correlations between SIP and hydraulic parameters are to be derived, it is first necessary to fully understand the effect of other variables on the SIP response of materials. One such variable is the conductivity of the pore fluid. Although for most practical situations the variation of the ionic conductivity with σ w is well described by equation (), the manner in which the real and quadrature parts of the ace conductivity vary with pore fluid conductivity is less certain. As reported by Hördt and Bücker (013) previous measurements of the salinity dependence of the SIP responses of both consolidated and unconsolidated samples (Slater and Glaser 003; Kruschwitz et al. 010; Revil and Skold 011; Weller et al. 011; Weller and Slater 01) have produced somewhat contradictory results with some samples showing an increase in the imaginary part of the conductivity with salinity, and some a decrease. In this paper we present the results of a new study of the variation of the SIP response of unconsolidated sand with pore fluid salinity. In addition to attempting to resolve some of the ambiguities apparent from previous work, the SIP measurements are also the first on unconsolidated samples typical of many of the shallow unconfined coastal aquifers in New Zealand which are extensively used as a source of water for both agriculture and human consumption. Many of these aquifers are potentially under threat from saline intrusion resulting from a combination of changing rainfall patterns, rising sea levels, and possible over-extraction. Thus, just as a knowledge of the variation of SIP response with fluid salinity is useful in interpretation of salt tracer tests, which presently rely on electrical resistivity tomography (ERT) (e.g. Müller et al. 010), it also may have a potential role in the assessment of intrusion risk. MEASUREMENTS The two separate samples used were sourced from a coastal location on the west coast of the lower North Island of New Zealand. These were of naturally occurring fine sand typical of that found in FIGURE Schematic of the sample holder and measurement system used for SIP measurements. the shallow coastal aquifers in New Zealand. Particle size analysis using laser diffraction showed a grain size distribution with a mean grain diameter of 168 μm and a standard deviation of 40 μm as shown in Fig. 1. Sands from further north on the west coast of the North Island have a very high titanomagnetite concentration (Nicholson and Fyfe 1958) which typically consists of about 80% magnetite. Coastal processes mean that sands further south also show considerable magnetic content with up to 40% of grains carrying significant magnetic susceptibility (Hawke and McConchie 005, 006). Measurements of the mass susceptibility of the sand samples yielded a value of (1. ± 0.03) x 10-6 m 3 kg -1. In comparison, clean quartz sand typically has a mass susceptibility of less than 0.6 x 10-8 m 3 kg -1 (Hunt et al. 1995). Previous studies (e.g. Mansoor and Slater 007; Nordsiek and Weller 008; Florsch et al. 011) have shown that magnetic content in sands/soils leads to an enhanced SIP signal and, as discussed below, this has implications for the required precision of the measurement system. Two separate sample holders of differing internal diameters (49 mm for Sample 1 and 76 mm for Sample ) were used in the measurements. The design of the sample holders is shown in Fig.. The sample holders consisted of PVC pipes with current electrodes made of 15 μm stainless steel mesh. The potential electrodes were stainless steel rings fitted into grooves in the internal walls of the PVC pipe and positioned midway between the current electrodes with a uniform separation between adjacent electrodes (L = 10 cm for Sample 1, 8 cm for Sample ). Steel electrodes can lead to electrode polarization effects and, to avoid the introduction of error into the phase measurements, many previous studies have used non-polarizing electrodes 015 European Association of Geoscientists & Engineers, Near Surface Geophysics, 015, 13,

3 Spectral Induced Polarization measurements on New Zealand sands 171 (e.g. Zimmermann et al. 008). Such errors are most significant when the degree of polarization is low. The magnitude of the phase error introduced by the use of the steel ring potential elec- FIGURE 3 SIP phase measurements made on water of various conductivities (in µscm -1 ). trodes was assessed by making measurements on water samples of various conductivities. The results of these measurements are shown in Fig. 3. It is clear that for fluid conductivities in the range from about 10 0 μscm -1 to several 100 μscm -1 the phase error introduced by electrode polarization is negligible in the frequency range Hz. At frequencies below 0.01 Hz the phase of about 1 milliradian due to electrode polarization is of opposite sign to phases typically measured in SIP, while for a fluid conductivity of greater than 500 μscm -1 there is a positive phase error of similar magnitude at slightly higher frequency. The significantly increased phase at frequencies above 10 Hz is not only outside the frequency range generally of interest for relating SIP parameters to hydraulic properties, but also partly a reflection of instrumentation effects (Breede et al. 01). As noted above and shown in Fig. 4 the samples in this study have significant magnetic content, which means that the polarization is higher than is normally encountered for clean sand. As a result the small phase error in the frequency range of interest confirms that the use of stainless steel rings as potential electrodes rather than point, non-polarizing electrodes is satisfactory. FIGURE 4 SIP measurements on saturated unconsolidated sand samples for different fluid conductivities (in µscm -1 ). 015 European Association of Geoscientists & Engineers, Near Surface Geophysics, 015, 13,

4 17 S. Joseph, M. Ingham and G. Gouws The measurements were made with a custom built impedance meter originally designed to make in-situ measurements of the low frequency dielectric permittivity of sea ice at discrete frequencies between 10 Hz and 100 khz (Ingham et al. 01). Modification to the instrument, coupled with PC control of a Tektronix AFG 310 signal generator using LabView TM, allowed measurements to be made at discrete frequencies between 0.01 Hz and 1 khz. The measurement accuracy of the instrument was validated by testing on capacitor-resistor networks. Impedance measurements were made for a range of pore fluid conductivities (σ w ) between 1 and 900 μscm -1. This covers the range from fresh water to a level of salinity just below the guideline given by the World Health Organization above which water becomes non-potable (WHO 1996). The conductivity of the fluid was varied by adding KCl to distilled water. Sand saturated in distilled water was packed into the tube and, prior to each measurement, fluid of a given conductivity was passed through the sample until the conductivity of the fluid leaving the sample holder stabilized. Each set of measurements (i.e. using a particular sample) started with fluid of the lowest conductivity. The resulting SIP measurements for the two samples, both the magnitude of the bulk conductivity in μscm -1 and the phase difference between current and potential in milliradians, are shown in Fig. 4. The SIP measurements show features typical of those observed by previous authors. For a given fluid conductivity the magnitude of the conductivity shows a slight increase with increasing frequency which results from the frequency dependence of the ace conductivity. For measurements on both samples the phase response shows a characteristic shape. At high frequency (approximately > 10 Hz) the rapid rise in phase is the result of a combination of instrumentation effects (Breede et al. 01) and Maxwell-Wagner polarization (Chelidze and Guegen 1999). The rise in phase with increasing frequency is also dependent on the magnitude of the current applied to the sample (e.g. the cross-over of phase curves for Sample for fluid conductivities of 80 and 100 μscm -1 ). However, at lower frequencies the phase is found to be independent of current. Below 10 Hz, a maximum in the phase is observed. The magnitude of this is dependent on the pore fluid conductivity with the phase maximum decreasing as the salinity increases. This is largely a result of the increase in ionic conductivity associated with increasing salinity. Apart from for the lowest (1 μscm -1 for Sample ) and highest ( μscm -1 ) fluid conductivities the phase peak occurs at an approximately constant frequency of about 0.1 Hz. COLE-COLE MODELLING To further analyse the data it is necessary to model the measured spectra in terms of a polarization model. The three most common such models include the Debye model of a single relaxation time and the Cole-Cole (Cole and Cole 1941; Pelton et al. 1978) and generalized Cole-Cole (Pelton et al. 1983) models which allow for a distribution of relaxation times. A more recent approach representing a spectrum of Debye relaxations is the Debye decomposition method (Nordsiek and Weller 008; Breede et al. 01). Given the likelihood of a range of relaxation times, but to retain a relative simplicity of representation, we have used the Cole-Cole model for which the bulk complex conductivity of a sample as a function of frequency is given by FIGURE 5 Variation of Cole-Cole parameters with fluid conductivity. Sample 1 squares; Sample diamonds. (a) DC conductivity, σ o, (b) mean relaxation time, τ, (c) total chargeability, m, (d) exponent c. 015 European Association of Geoscientists & Engineers, Near Surface Geophysics, 015, 13,

5 Spectral Induced Polarization measurements on New Zealand sands 173 TABLE 1 Cole-Cole parameters σ o, τ PM, m and c derived from fitting the measured SIP spectra for samples with different fluid conductivity σ w with equation (4). τ CC is the relaxation time corresponding to equation (5) calculated from equation (6). Sample 1 Sample σ w (µscm -1 ) σ o (µscm -1 ) τ PM (s) m c τ CC (s) c ( iωτ PM ) σ = σ + o 1 m c. (4) 1+ (1 m)( iωτ PM ) Here σ o is the DC conductivity, τ PM is the mean relaxation time, the exponent c is a reflection of the spread of relaxation times, and m is the total chargeability which is related to σ o and the high frequency conductivity σ by m = 1 σ o /σ. Equation (4) is derived by taking the reciprocal of the Cole-Cole model for complex resistivity as introduced by Pelton et al. (1978). An alternative approach is to apply a Cole-Cole model directly to the complex conductivity in which case σ is represented by σ o σ σ = σ + c (5) 1+ (i ωτ ) CC which is the formulation used by, for example, Revil and Florsch (010). Although the relaxation times in each formulation are different (Tarasov and Titov 013), they are related by c CC =τ PM ( 1 m ) 1/ τ (6) (Florsch et al. 01). As the high frequency rise in the SIP phase is not related to the ace polarization effects of interest, and requires a more complex model to adequately represent it, equation (4) has been used to fit the measurements on both samples only in the frequency range from Hz. Modelling was carried out in Excel using the data analysis package Solver. The results of this fitting, in terms of the variations in parameters σ o, τ PM, m and c with fluid conductivity are shown in Fig. 5. The derived parameters and the corresponding calculated values of τ CC are listed in Table 1. The linearity of the plot of σ o as a function of the fluid conductivity σ w allows estimation of the formation factor F for each of the samples, and gives values of F = 3.54 ± 0.0 for Sample 1, and F =.84 ± 0.0 for Sample. Assuming the validity of () the measured value of porosity of the Sample of 47 ± 1 % allows the cementation exponent n to be calculated as 1.37 ± Given that the broadness of the phase peaks shown in Fig. 4 means that the derived values of τ PM have an (unquantified) uncertainty associated with them, the results shown in Fig. 5(b) suggest that the relaxation time is not only independent of σ w, as also found by Weller et al. (011), but is also essentially the same for both samples. Similarly, both sets of data show a similar monotonic decrease in chargeability with increasing fluid conductivity (Fig. 5(c)), and a constant value for the exponent c 015 European Association of Geoscientists & Engineers, Near Surface Geophysics, 015, 13,

6 174 S. Joseph, M. Ingham and G. Gouws (Fig. 5(d)). Several authors (Leroy et al. 008; Revil and Florsch 010; Breede et al. 01) have previously reported that the relaxation time is related to the square of the grain diameter. Thus, given that the data are all measured on the same sand, the independence of τ PM with fluid conductivity is unsurprising. Similarly, the constancy of c, related to the spread of relaxation times, is expected. As for the decrease in SIP phase with increasing fluid conductivity, the decrease in m can be explained by the increasing dominance of ionic conductivity over polarization effects as the fluid becomes increasingly conductive. As the Cole-Cole modelling provides a value for the DC conductivity, assuming that the ionic conductivity is much greater than the DC ace conductivity, equation (3) may be rewritten as σ = σ + σ + iσ o and thus, for any fluid conductivity, σ ) = Re( σ σ o (7) σ may be estimated as. (8) The imaginary part of σ gives σ. Plots of σ and σ as functions of σ w at a frequency of 0.1 Hz (approximately the phase peak), and the variation in σ o (σ ionic ) with σ w are shown in Fig. 6. For Sample 1, values of σ and σ show a gradual increase with fluid conductivity. At lower fluid conductivities the same trend is shown by values for Sample. However, for Sample the two highest fluid conductivity measurements suggest that a maximum in both the real and imaginary parts of σ exists. Although this is a similar behaviour to that noted by Slater and Lesmes (00) who reported a broad maximum in σ over a fluid conductivity range of μscm -1, and both Slater and Glaser (003) and Weller et al. (011) reported a decrease in σ at high fluid conductivities, further measurements at high conductivity are clearly desirable. It should also be noted that the same measurements also lead to anomalously low values of τ, and high values of c compared to the other measurements (Fig. 5). This may result from difficulty in fitting a distinct model to these measurements due to the very low phase values and extremely broad maximum in the phase spectrum (Fig. 4). MODELS FOR SURFACE CONDUCTIVITY AND POLARIZATION Various models have been proposed for the origin of ace conductivity and polarization. These have recently been summarized by Weller and Slater (01) and include the diffuse layer polarization (DLP) model originally proposed by Börner (1991, 199), and the Stern layer polarization (SLP) model of Leroy et al. (008) and Revil and Florsch (010). In the DLP model polarization is assumed to occur in the electrical double layer surrounding pores, whereas in the SLP model polarization is restricted to the Stern layer adjacent to mineral aces. Weller and Slater (01) detail the development showing that the two models predict different relationships for the dependence of σ on σ w. Whereas the DLP model predicts that σw σ = ad (9) b + σ the SLP model results in σ d σ w w = cs + as (10) bs + σ w in which an additive term, c s, is included, as suggested by Skold et al. (011), that takes into account polarization due to the hopping of protons on the mineral ace. The variation with fluid conductivity of σ at 0.1 Hz is shown in Fig. 7. The two anomalously low values at higher fluid conductivity are those noted above from Sample. If these two values are excluded and the two ace conductivity models are fit to the data the SLP model (rms misfit 0.031) gives a FIGURE 6 Variation with fluid conductivity (σ w ) of the in-phase ( σ - squares) and quadrature-phase ( σ - circles) parts of the ace conductivity. Diamonds show the variation of the ionic conductivity (σ ionic ). Open symbols are Sample 1, shaded symbols are Sample. FIGURE 7 Fit of the DLP (dashed line) and SLP (solid line) models for ace polarization to the observed σ data at a frequency of 0.1 Hz. (Note that, as discussed in the text, the two anomalously low values from Sample are excluded.) Sample 1 squares; Sample diamonds. 015 European Association of Geoscientists & Engineers, Near Surface Geophysics, 015, 13,

7 Spectral Induced Polarization measurements on New Zealand sands 175 TABLE Measured and predicted values of permeability for Sample. significantly better fit than does the DLP model (rms misfit 0.064). Again, confirmation of this observation will require additional measurements with higher values of σ w. SIP SPECTRA AND HYDRAULIC CONDUCTIVITY The hydraulic conductivity of Sample was measured in-situ using the constant head method. Several authors have attempted to deduce relationships between SIP parameters and hydraulic properties (e.g. Slater and Lesmes 00; Binley et al. 005; Zisser et al. 010; Revil and Florsch, 010; Koch et al. 01). Perhaps the simplest such relationship is based on the Kozeny-Carman equation (Kozeny 197; Carman 1937) and can be written in the form (Freeze and Cherry 1979; Revil and Cathles 1999) 3 d φ k = (11) 180(1 φ) where k is the permeability and d is the diameter of uniformly spherical grains. This also leads to a relationship between k, the formation factor F, and the cementation exponent n (Revil and Cathles 1999) d k = (1) 3n F( F 1) and, by relating the mean relaxation time (τ CC ) to the mean grain size and the diffusion coefficient (D) of the ionic species in the fluid (Schwarz 196; Revil and Florsch 010; Koch et al. 01), also to k = D τ CC 4n F( F 1) k (m ) Measured (.3 ± 0.1) x Equation (11) (5.6 ± 0.5) x Equation (1) (4.9 ± 0.3) x Equation (13) (3. ± 0.5) x (13) Using the measured porosity, the mean grain size and the calculated values for n and τ CC, and taking the diffusion coefficient for the KCl solution to be x 10-9 m /s, the predicted values for permeability using equations (11), (1) and (13) compared to the measured value (calculated from the measured hydraulic conductivity) for Sample are listed in Table, It was previously suggested by Koch et al. (01) that the Kozeny-Carman equation (11) tends to overestimate the permeability, while equation (1) leads to an underestimate. While a single measurement does not allow for any significant conclusion to be drawn, for Sample both predicted values are over estimates. Using equation (13), in which the mean grain diameter is related to the mean relaxation time and the diffusion coeffi- cient, gives an estimate which is much closer to the measured value of permeability. It should be noted, however, that this value is very dependent on the chosen value (from a significant spread of suggested values) for the appropriate diffusion coefficient for aqueous solutions of KCl. CONCLUSIONS In making the first SIP measurements on New Zealand aquifer materials we have confirmed that the polarization effects are largely independent of pore fluid conductivity, with only the total chargeability appearing to have any significant variation with σ w. Future measurements of the SIP signature of these materials may therefore be interpreted in terms of the hydraulic properties of the material. We have also confirmed, at least for the Sample, that common proposed relationships between permeability and SIP spectra, such as the Kozeny-Carman equation, appear to overestimate the permeability. Decomposition of the measured complex conductivity into the separate components of σ ionic, σ and σ has also allowed proposed polarization models to be tested. The results of this suggest that the Stern layer polarization model for low fluid conductivities gives a better fit to the variation of σ with fluid conductivity shown by our data than does the fit from the diffuse layer polarization model. However, many more additional measurements at higher fluid conductivities are necessary before any such general conclusion can be drawn. Having demonstrated the independence of polarization parameters on fluid conductivity, future measurements will be based on samples with a much wider variation in grain size, to attempt to draw relationships between SIP spectra and hydraulic parameters that are applicable to New Zealand s shallow coastal aquifers. ACKNOWLEDGEMENTS We are grateful for constructive comments from Andre Revil and two anonymous referees which helped us to improve this manuscript. REFERENCES Archie G.E The electrical resistivity log as an aid in determining some reservoir characteristics. Transactions of the American Institute of Mining and Metallurgical Engineers 146, Binley A., Slater L., Fukes M. and Cassiani G The relationship between spectral induced polarization and hydraulic properties of saturated and unsaturated sandstone. Water Resources Research 41(1), W1417. doi: /015WR0040. 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