Influence of the Test Specimen Diameter on the Measurement of Electrical Resistivity in Sands by Using Laboratory Devices

Electrical resistivity tests have long been used in geo-environmental site characterization because they provide qualitative data to identify the presence of contaminants in soil. The measurement of soil electrical resistivity is affected by many intervening factors, which are usually studied separately in the laboratory and are important to ensure the correct interpretation of in situ test data. This paper presents the evaluation of the influence of the diameter of test specimens on electrical resistivity measurements of a particulate medium in the laboratory, using two different techniques, copper plates and a resistivity probe. The data were discussed considering Archie’s Law, and it indicates that the geometry of test specimen has a greater effect than the measurement technique. The larger test specimen allows more current pathways for electron propagation because electric current travels along the path of least resistance to the passage of electrons. The findings from this research can be used to guide a laboratory test investigation and its data interpretation.


Introduction
Multidisciplinary geotechnical research involves combining different methods for geo-environmental site characterization like: surface and subsurface exploration; geophysical surveys to supplement drilling; sampling methods; and laboratory tests of soil, gas and liquids samples.
The resistivity piezocone test (RCPTU) stands out among in situ tests as a tool for defining subsurface stratigraphy and simultaneously measuring electrical resistivity which is routinely employed in Geotechnical Engineering and Geophysics to characterize subsoil and in soil analyses (Bryson & Bathe, 2009), as well as in other areas of science such as Agronomy (Corwin & Lesch, 2005;Molin & Rabello, 2011), Archaeology and Geology (Schoor, 2002).
Although the electrical resistivity is an intrinsic property of each material, there are several intervening factors affecting it and several experimental researches offered significant insight on this subject.Those studies include that of Abu-Hassanein & Benson (1994), who analyzed compacted clays and demonstrated the influence of moisture content, compaction energy and the initial saturation of the test specimen on electrical resistivity.Daniel (1997) studied in laboratory two compacted soils (calcine tailing and pure quartz rock flour) and analyzed the impact of the degree of soil saturation, porosity and type of interstitial fluid on electrical resistivity in piezocone penetration tests.Aquino (2010) studied compacted clay and correlated electrical resistivity with soil index properties.Long et al. (2012), who studied marine clay, demonstrated that its electrical resistivity was determined by the salt content in the interstitial fluid and the percentage amount of clay in the sample.
Two different techniques are commonly used in the laboratory to measure electrical resistivity: conductive plates installed at the top and bottom of a test specimen (Abu-Hassanein & Benson, 1994;Aquino, 2010), or a resistivity module (RCPTU) embedded in a test specimen (Daniel, 1997;Pacheco, 2004).These techniques provide different descriptions of the variation in electrical resistivity, since their geometric, electrical and physical characteristics differ.
In this context, this paper evaluates the effect of compaction mold geometry on the electrical resistivity of construction sand, using a resistivity probe developed at UNESP Bauru (Fig. 1) and copper plates to determine the distribution of the electric potential in the particulate medium.
The analyzes of electrical resistivity were done in two different test specimens with 155 mm and 450 mm diameter, but equal height in order to verify the influence of distortions along electrical current propagation pathways and the distribution of stresses within the soil during the tests, as well as their effect on the electrical resistivity measurement.rials with different resistivity.Natural soils have different electrical properties because of their different genesis, grain size, structures, moisture content and temperature.The characteristics of any material and soils are therefore strongly related to electrical conductivity and can be quantified by means of geoelectrical properties.They are relevant for the indirect investigation of soils (Kong et al., 2012) and can be used in various issues, such as soil physical properties (Archie, 1941), soil and groundwater contamination (Campanella & Weemees, 1990), soil improvement and stabilization (Liu et al., 2006 and2007), sand liquefaction (Arulmoli et al., 1985), and soil microstructure (Fukue et al., 1999).
The three-phase composition of soil is another important factor in the analysis of electrical resistivity.The distribution of electrical current occurs in the soil's three components, i.e., the liquid, solid and gaseous phases.However, most of the electrical current travels through the interstitial fluid since electrical charges are transported more easily in the liquid environment, generating electrolytic current (Lunne et al., 1997).

Formation factor
The formation resistivity factor of soil (F) was first studied by Archie (1941), based on the laboratory determination of the electrical resistivity of a large number of sand samples saturated with solutions of water and sodium chloride, 20,000 mg to 100,000 mg of NaCl per liter, and with porosity varying from 10% to 40%.Archie (1941) determined that the formation resistivity factor is the ratio between the resistivity of sand saturated with water and sodium chloride solution (r 0 ) and the resistivity of a water and sodium chloride solution (r f ), Eq. 1: (1) By this equation, the formation resistivity factor is related with grain shape and it is considered that the electrical conduction varies solely as a function of the geometry of the pores formed between the grains, and hence, the mineralogy.According to Weemees (1990), the absence of electrical conduction through the particles is considered fundamental.
In the case of soil containing significant amounts of clay minerals, i.e., that can conduct electrical charges along the surface of its particles, resistivity measurements do not distinguish the electrical conduction phenomena, so the measured resistivity takes into account an apparent formation factor (Jackson et al., 1978).Therefore, the formulation to obtain the formation factor depends on the type of soil under analysis, which may involve more complex equations for clayey materials or simpler ones for sandy materials.
Based on empirical observations, Archie (1941) related the formation factor with sand porosity, Eq. 2, and this first mathematical relationship is known as Archie's law: where m is a constant determined experimentally and n is the soil porosity.Archie (1941) characterized the constant m as being dependent on the degree of cementation of individual particles; later, Guyod (1944) described m as being dependent on the type of soil parent rock (Jackson et al., 1978).
Based on his findings of marine sediments, Taylor-Smith (1971) proposed two simple formulations to describe the results of cohesive (sediment-rich clays) and non-cohesive materials, as follows: F = n -1.5 with n < 0.6 for sands, and F = n -2.0 with n < 0.6 for clays (Jackson et al., 1978).
Archie's formulation (1941), Eq. 3, underwent changes through the inclusion of the fitting constant a proposed by Winsauer et al. (1952), as reported Jackson et al. (1978), and also based on the study of unsaturated conditions, which resulted in Eq. 3: where a and s are constants that can be determined experimentally, r b is the apparent soil resistivity, r f is the interstitial fluid resistivity, and S r is the degree of saturation.Thus, the relationship presented in Eq. 3 is an important empirical relationship in studies of sand, which can be regarded as an indirect relationship of Archie's Law, which has been widely used by other authors such as Daniel et al. (2003), Pacheco (2004) and Oliveira (2004).
Parameter a is a constant that depends on soil porosity, whose value for unconsolidated soil varies around 1.0 (Oliveira, 2004).Jackson et al. (1978) studied the influence of the shapes of natural and synthetic particles on the formation of factor, as well as their particle size distribution curve, and demonstrated that exponent m is entirely dependent on particle shape.In their study, the values of m for marine sand varied from 1.2 to 1.9, and these values were not affected by the particle size distribution in sand, although the magnitude of m tends to increase with the increase in percentage deviation from particle sphericity.
According to Daniel et al. (2003), the values of constants m and s ranged from 1.3 to 2.2 and from 1.0 to 2.5, respectively.In contrast, in his studies of consolidated sandstone and washed unconsolidated sand stored in the laboratory, Archie (1941) found that constant m varied from 1.3 to 2.0 and constant s showed values close to 2.0 in the saturated condition.Daniel et al. (2003) also showed that factor S r -s varies much more than factor n -m , although exponents m and s are of the same magnitude (Fig. 2), because the degree of saturation (S r ) may vary from 0 to 1 in any soil, while soil porosity (n) typically varies from 0.25 to 0.50.Peixoto et al. (2014) studied the formation factor of a saturated sedimentary sand by means of permeability tests, using different concentrations of percolation fluids (Table 1).In that study, they applied two formulations to determine the formation factors: that of Archie (1941), represented by factor F, and that of Waxman and Smits (1968), which is presented as F* and was calculated by Eqs.According to Waxman and Smits (1968), C e = 0 for clean sands, and F* reduces to F, the usual formation resistivity factor, Eq. 5: The Eq. 5 can be rewritten as Eq. 6 for a better understanding:

Materials
The measurement of electrical resistivity was carried out on samples with the same height (197 mm) and with two different diameters (155 mm and 450 mm) in order to assess possible distortions on the current propagation pathways.The large diameter was designed to minimize distortions and possible edge effects when measuring the resistivity, i.e., the 450 mm diameter of the large test specimen was designed to be approximately three times that of the small test specimen (155 mm) (Table 2 and Fig. 3).
A washed sand was used and the main characteristics are presented in Table 3.The sample was prepared with drying-air and sifting it through four sieves; one with a nominal aperture of 9.52 mm and the other three with nominal apertures of 4.76 mm (Fig. 4) in order to eliminate the bigger particles and for a better homogenization of the test samples.

Methods
Four groups of tests were defined: copper plate with 155 mm diameter test specimen (CP 155); copper plate with 450 mm diameter test specimen (CP 450); resistivity device with 155 mm diameter test specimen (RD 155); and resistivity device with 450 mm diameter test specimen (RD 450).Nine tests were carried out varying the moisture content from hygroscopic moisture up to the fully saturated condition for each test group (36 tests total).The void ratio of the sample was controlled, however some values were higher than it was expected.In these cases, the test data  4.
This approach allowed analyzing the variation of the electrical resistivity values as a function of the soil physical indices for both test specimens (small and large), using the two aforementioned techniques (Fig. 5).
The test specimens were prepared by using a modified pluviation method (Fig. 6), sifting the sand through only one sieve with nominal aperture of 19 mm.The use of just one sieve instead a set of sieves as proposed by Pacheco (2004) was necessary because it was not possible to pass the saturated sand through a set of sieves.It is important to point out that the size of the sieve (19 mm) was kept constant in all the tests, regardless of the moisture content of the sand.
Measuring electrical resistivity by the copper plate and resistivity device techniques differs in terms of the position of the measuring instruments and the generated electrical field.The values of frequency, root mean square voltage, V rms (which means the corresponding voltage of alternate current) and sine-wave were kept constant in both techniques.The measured voltage and current values were inserted in the operating equations of the measuring devices to determine the electrical resistivity.Each test specimen was subjected to the following measurements: soil temperature, sand weight, electrical current, voltage, soil moisture content, and water and soil conductivity.The electrical measurement of each specimen was correlated with its respective soil physical index.

Formulation to determine the constants of Archie's Law
The results of the evaluation of the edge effect of the compaction mold on the washed sand were analyzed using Archie's Law (Eq.3).The ratio between soil electrical resistivity and fluid electrical resistivity is called the formation factor (F), so Eq. 3 can be rewritten as Eq.6:    To apply this formulation, however, it was necessary to first determine the empirical constants m, s and a, which were calculated using the data from the tests performed with the circular copper plates and the resistivity probe.The method used to calculate these constants was the same as that used by Daniel (1997), which is detailed below.

Calculation of the porosity exponent (constant m)
Under conditions in which the degree of saturation is constant, the constant m can be determined by isolating n -m of Eq. 6 and applying the logarithmic operation on both sides, which leads to Eq. 7: In Eq. 8, it is attributed the value of a single constant C to factor (S r -s /a) of Eq. 7: Thus, it is possible to plot the formation factor (F) on the di-log scale as a function of porosity (n) and obtain exponent m by including a power trend line.

Calculation of the exponent of the degree of saturation (constant s)
Given that m is a known parameter, S r -s can be isolated in Eq. 6, similarly to the way constant m was calculated, and the logarithmic operation can be applied on both sides of the equality, Eq. 9: The parameter s can be obtained by plotting the results of F.n m vs. degree of saturation and fitting the points by means of a power trend, since a is a constant.

Calculation of coefficient a
After identifying the exponent m and the exponent s, the constant a can be determined by plotting the formation factor (F) vs. n -m .S r -s and calculating the angular coefficient of the fitted points on a straight line, given the linear relationship between the axes.

Test data and analysis
The test data are separated in five groups, each of them comprising four prepared specimens according to the moisture content.In other words, the measurements were taken from two test specimens for each container diameter (155 mm and 450 mm), using the copper plates and the resistivity device for each diameter (Table 4).
The average temperature in these tests was 24.9 °C, with a variation of 2.5 °C between the maximum and minimum.Because the test specimens were grouped in sets of tests and the maximum variation in temperature for a given set was 1.3 °C (Table 4), the resistivity device curves were used in lieu of each test temperature, precluding the need to adapt a standard temperature to correct the values of electrical resistivity.
The first index to be calculated was the porosity exponent m, using Eq.6 to Eq. 8 and assuming S r -s equal to 1. So, only the test data from the samples with degree of saturation higher than 94.0% and with no edge effects (set D -CP 155 and CP 450 and set E -CP 155, RD 155, CP 450, RD 450) were considered.The porosity values were related graphically to the formation factor, on a graphic di-log scale as shown in Fig. 7.
Because m is an empirical constant, R 2 is assumed to have lower values.Although the calculated value of R 2 was equal to 0.6116, the calculated value of exponent m, of -1.894, falls within the range reported in the literature, and is similar to the value of -1.96 reported by Daniel (1997) for pure fine sand.In this calculation, the void ratio was between 0.30 and 0.40, while the results reported by Daniel (1997) were between 0.30 and 0.50, indicating an equivalence between the test data.
The Eq. ( 9) was used to calculate the exponent of the degree of saturation, s, with m equal to 1.894.The F.n m term was calculated using all the five set data and they are presented in Table 4.After that, exponent s was graphically obtained.The calculation of the exponent s presented an R 2 value of 0.8145 (Fig. 8), which is higher than that calculated for exponent m and adequate for this magnitude.Therefore, the calculated value of -1.258 for constant s is a good fitting for this parameter and falls within the range reported in the literature, which varies from 1.0 to 2.5, according to Daniel et al. (2003).
As for exponent m and exponent s, it should be noted that although they are of the same order of magnitude, the variation of factor S r -s is greater than that of factor n -m because the degree of saturation varies more widely than soil porosity.Thus, the value of exponent m, which presented the lowest R 2 , is related with the factor that varies the least according to Archie's Law.The data measured with the copper plates showed two trend lines with high values of R 2 (Fig. 10), indicating that the distribution of voltages varied with the diameter of the test specimen.A comparison of the values of R 2 of the test specimens revealed that the one with a diameter of 155 mm the highest value.Figure 11 illustrates the results of the tests performed with the resistivity device.In this case, also, the two points with a degree of saturation of about 5.0%, i.e., set A in Table 4, was ignored.
The average temperature in these tests was 24.9 °C, with a variation of 2.5 °C between the maximum and minimum.Because the test specimens were grouped in sets of tests and the maximum variation in temperature for a given set was 1.3 °C (Table 4), the resistivity device curves were used in lieu of each test temperature, precluding the need to adapt a standard temperature to correct the values of electrical resistivity.
The fitting curves obtained in the tests with the resistivity probe also showed two different trends, depending on the diameter of the test specimen (Fig. 11).Again, good R 2 values were obtained on both curves, the higher one obtained in the test on the 155 mm diameter test specimen.
The similarity of the fitted curves obtained by the two electrical resistivity measurement techniques thus allowed for a comparison of the measurements of the test specimen with the two diameters taken with the copper plates and the resistivity probe (Fig. 12).
The electrical resistivity measured for the same test specimen by using the resistivity probe and the copper plates showed very similar fitting curves, all with high values of R 2 , particularly for the small test specimen.The val-    ues of electrical resistivity showed higher variations as a function of the test specimen diameter than of the measurement techniques.
It was therefore possible to consider a single adjustment of the electrical resistivity data for each test specimen on the same curve, considering the two different devices used in this study (Fig. 13).
An analysis of the R 2 values presented in Fig. 13 leads to the conclusion that, for each test specimen diameter, the fitting of the two resistivity measurement techniques to a single curve was perfectly adequate, since the resulting values of R 2 are similar to those presented in Fig. 12, in which the curves were calculated separately.
In Fig. 13, the two extreme data from the two test specimens, with electrical resistivity of about 3.000 W.m and degree of saturation of about 17%, were the ones that presented the highest absolute difference in resistivity between the test specimen diameters.Hence, the high absolute difference in resistivity between the test specimens stems from the difficulty of current propagation at high values of electrical resistivity, a condition in which there is greater distortion of current propagating through a particulate medium.

Conclusions
The electrical resistivity data obtained by the two techniques were quite similar and allowed for the fitting of a single curve as a function of the test specimen diameter.Although the geometric and electrical characteristics of the techniques used in this study differ and provide different soil voltage distribution, the two techniques were equally efficient in measuring the electrical resistivity of sand prepared in the laboratory.
It was found that the test specimen geometry, i.e., its diameter, was more relevant in determining the soil electrical resistivity value than the testing technique (copper plate or resistivity probe).The difference in the distribution of voltage in the test specimens with different diameters is the reason for it.The 450 mm test specimen allows more current pathways for electron propagation, because electric current travels along the path of least resistance to the passage of electrons, so the electrical resistivity values are lower than those obtained in the 155 mm diameter test specimen in the same test condition (degree of saturation, porosity and void ratio).It is concluded that the resistivity data obtained for both test specimen diameters and by the two different techniques are valid and can be employed to study soil resistivity in laboratory.
The use of electrical resistivity in geo-environmental site characterization tests is qualitative, i.e., it varies within a range of values.Therefore, when a reference value is exceeded, it is a strong indicative of the presence of contaminants, requiring additional tests and soil, liquid and gas sampling for further site investigation.In these cases, the findings from this research can be used to guide the laboratory test investigation and its data interpretation.

4 and 5 .
The formation factor resulted in F* equal to 7.0, confirming Archie's Law (F) for sandy soils.
where: C 0 , C e and C w are the specific conductance (conductivity) of the core, clay exchange cations and equilibrating salt solution, respectively.
Figure 2 -Sensitivity of Archie's Law to variations in soil porosity and degree of saturation.Source: Daniel et al. (2003).
were not considered.So, the test data were divided into 5 sets (A, B, C, D and E), each of them containing one test data of each test groups (CP 155, CP 450, RD 155, RD450).Twenty tests with the moisture content varying from hygroscopic up to the fully saturated condition for each test group were considered.A total of 5 sets (A, B, C, D and E), one test data of each test groups (CP 155, CP 450, RD 155, RD450) is presented in Table

Figure 3 -
Figure 3 -Containers used in the electrical resistivity tests performed with copper plates and the resistivity probe.

Figure 4 -
Figure 4 -Pluviation device used for homogenization of the sand before the tests.
, São Paulo, 39(2): 157-165, May-August, 2016.161 Influence of the Test Specimen Diameter on the Measurement of Electrical Resistivity in Sands by Using Laboratory Devices

Figure 5 -
Figure 5 -Test specimens and measuring instruments: (a) 450 mm diameter tested with the resistivity device, (b) 155 mm diameter tested with copper plates, using a standard weight on top of the upper plate.Figure6-Pluviation device used for molding the test specimens.

Figure 6 -
Figure 5 -Test specimens and measuring instruments: (a) 450 mm diameter tested with the resistivity device, (b) 155 mm diameter tested with copper plates, using a standard weight on top of the upper plate.Figure6-Pluviation device used for molding the test specimens.

Figure 7 -
Figure7-Definition of the exponent m by using data points with a degree of saturation higher than 94%.
Figure 9 -Definition of the coefficient a for m = -1.894and s = -1.258.

Figure 10 -
Figure 10 -Electrical resistivity of the test specimens with diameters of 155 mm and 450 mm measured with the copper plates.

Figure 11 -
Figure 11 -Electrical resistivity of the test specimens with diameters of 155 mm and 450 mm measured with the resistivity device.

Figure 8 -
Figure 8 -Definition of the exponent s assuming a m value equal to -1.894.

Figure 12 -
Figure 12 -Electrical resistivity of the test specimens with the two diameters measured with the resistivity probe and with the copper plates.

Figure 13 -
Figure13-Adjustment of electrical resistivity as a function of test specimen geometry, using both resistivity measurement techniques.

Table 2 -
Dimensions of the containers used in the electrical resistivity evaluation tests performed with copper plates and with the resistivity probe.

Table 3 -
Washed sand main characteristics.

Table 4 -
List of test sets as a function of temperatures (T soil ), soil resistivity (r b ), fluid resistivity (r f ), formation factor (F), moisture content (w), void ratio (e), porosity (n) and degree of saturation (S r ).