This section evaluates the stability of the proposed measurement procedure by considering the stability of resistances measured between the current electrodes, Rmeasured, the potential difference measured between the potential electrodes, VP1 − VP2, and the measured current, I. The measurement method is correct in principle. However, inaccuracy in the set-up might lead to incorrect measurements that are expected to be unstable. Therefore, we can assess the validity of the measured values by checking their stability.
We checked the stability of measurement through six sequences of repeated measurements at six levels of relative humidity and constant temperature. In addition, we checked the adhesion performance through observations of the contact surface, because strong adhesion between electrodes and the sample is important in our experimental set-up.
5.1. Configuration of electrodes
Figure 4 shows the granite sample’s cylindrical surface and the electrode arrangement with the measurement instruments. We observed the contact surface by micro-focus X-ray computed tomography (CT) to confirm the contact state of the electrodes. The CT results in Fig. 5 show that the electrodes were well attached, despite the roughness of the surface, thus demonstrating the strong adhesion achieved by the proposed method.
5.2. Stability evaluation procedures
5.2.1. Data sampling with humidity and temperature setting
Rmeasured, I, and VP1 − VP2, were measured for 600 s in each measurement, which was repeated multiple times. Sampling was conducted every 1 s. To eliminate the charge between the current electrode and granite surface, all terminals were shorted after each 600 s measurement. The discharge time for each repeat measurement was set to 2 h during resistance measurements in the GΩ range. Resistance measurements in the MΩ range used a longer discharge time of 6 h due to the greater injected current amount (Table 2).
Six sequences of repeated measurements gathered data for Rmeasured, I, and VP1 − VP2. One sequence is considered as a group of repeated measurement data performed at fixed humidity and temperature. The six sequences considered relative humidity at six set values (40%, 50%, 60%, 70%, 80%, and 90%) and constant temperature (30 °C).
5.2.2. Separation of sample and contact resistance from measured resistance
We used forward modelling to separate Rmeasured into Rsample and Rcontact. This approach uses the potential difference between the potential electrodes, VP1−VP2, as the fitting parameter and the injected current, I, as the input parameter. Sample resistivity, ρsample, was estimated by grid searching with forward modelling, and it was converted to Rsample.
The numerical calculation code developed by Suzuki et al. (2017) was used for forward modelling. This code is a transformation of the Dey and Morrison (1979) formulation into a cylindrical coordinate system. It calculates the static potential distribution in a cylindrical medium generated by a constant current. This study assumes the cylindrical calculation area to be a homogeneous structure with the same size as the granite target. Measured I is used as the source constant current in the calculation.
At first, we estimated ρsample as a fitting parameter by comparing the measured VP1 − VP2 with values calculated using multiple ρsample values for multiple potential distributions in cylindrical samples. Of the VP1 − VP2 values subsequently extracted from these calculated potential distributions, the one most consistent with the measured value was selected. The corresponding ρsample was selected as true.
Next, we determined Rsample using Ohm’s law. We extracted the potential difference between the current electrodes, VC1−VC2, from the potential distribution of the determined ρsample substituted into Ohm’s law. As the calculated potential distribution does not include the potential drop by Rcontact, the value obtained by dividing the extracted VC1−VC2 by the measured I is Rsample, not including Rcontact.
This estimation process of Rsample is shown Fig. 5. The validity of the process was confirmed by additional experiments (Additional file 1) using a cylindrical plastic measurement target (52 mm diameter, 100 mm length; MC501CD R2, Mitsubishi Chemical Advanced Materials, Tokyo, Japan) (Additional file 1: Fig. S1). Its resistivity was 100−102 Ω·m at 23 °C. The experiments confirmed that sample resistivity can be determined precisely by our estimation process regardless of the arrangement of the current and potential electrodes.
We also obtained Rcontact from Rmeasured to determine Rsample. Assuming that Rcontact of the positive electrode is the same as that of the negative electrode in the two-terminal measurements, Rcontact is given by
(1)
In this study, Rcontact is determined by Equation (1) using the obtained Rmeasured and the estimated Rsample.
5.3. Results and Discussion
5.3.1. Inspection and processing of time-series data
The stability of the measured resistance, current, and potential difference was assessed using time series of the data of the type depicted in Fig. 7 for 40% relative humidity and 30 °C. The time-series data show transient phenomena. The current recorded for about 1 min after the start of measurement was larger than that specified by the resistance meter (0.9 nA; Table 2). This large current meant that it took several tens of seconds for the measured resistance and potential difference to stabilise.
The large current at the start of measurement was interpreted as an inrush current. The increase of observed resistance after the current had settled probably corresponds to charging. Both of these effects shifted the measured resistance to higher values than their actual values. Therefore, it is reasonable to adopt the minimum value observed in each 600 s of resistance data, which likely includes the smallest effects of charging and inrush current.
On the other hand, after the inrush current, the current and potential difference became almost constant in the time-series data, making the data points equivalent. For standardised selection, the current and potential difference at the time of minimum resistance were chosen, as indicated by the dashed line in Fig. 7.
5.3.2. Stability of repeated measurements
Table 3 shows the stability of temperature and relative humidity in the six sequences. In each case, temperature varied by at most approximately 0.5 °C, and humidity varied by at most approximately 3%. The measured temperature and humidity were far more stable than those of the outside air.
Figure 8 and Table 4 show the results of repeated measurements in the six sequences and their statistical comparison, respectively. Figure 8(A) confirms that the specified current of 0.9 nA was injected correctly without leakage current. Because the current flowing through the ammeter is the sum of the injected current and the noise current, the observed current is larger than the specified current of 0.9 nA. Fluctuations of measured currents in each sequence were limited to about ±0.1 nA in the GΩ range and about ±1 nA in the MΩ range. We defined the fluctuation at each sequence as the standard deviation of all measurements in each sequence. The measured potential differences and resistances in each sequence fluctuated within about ± 20% of the mean, except at 80% humidity. The potential differences and resistances considerably decreased with increasing absolute humidity, with measured resistance being especially sensitive to absolute humidity even within each sequence.
Figure 8 indicates the high reproducibility of the current, potential difference, and resistance measurements. The performances of the two- and four-terminal measurements are respectively demonstrated by the stability of the resistance values and of the potential difference and current values. Dividing potential difference by current gives the sample resistance between potential electrodes P1 and P2 without considering the sample shape or electrode position. Kariya and Shankland (1983) found standard deviations of 0.73–0.98 for log electrical resistivity at temperatures of 500–1000 °C for dry rocks of granitic composition and texture. In comparison, our results in Table 4 have standard deviations of log resistance ranging from 0.01 to 0.25 in each sequence. The standard deviation of log resistance (i.e., the quotient of potential difference and current) with error propagation is 0.08–0.38 in each sequence, except at 80% humidity. These results indicate the high measurement accuracy achieved here compared with previous studies. Furthermore, the statistics were calculated for each sequence in this study, but the standard deviation would become smaller if the average and standard deviation were calculated not by sequence but by narrow bins of absolute humidity.
All the results in Table 4 show the greatest standard deviation for the 80% humidity sequence. This was attributed to the sample resistance being close to the boundary between the GΩ range and the MΩ range at 80% humidity. When the sample resistance is below the lower limit of the set range, the voltage generated by the injected current becomes very small so that the resultant small current reduces the measurement accuracy. These results are expected to be improved by frequent switching of the measurement range when the sample resistance falls outside suitable limits.
5.3.3. Estimation of resistance between current electrodes and contact resistance
Figure 8 shows the high sensitivity of Rsample, and possibly of Rcontact, to changes of absolute humidity. Figure 9 shows estimated values for Rsample and Rcontact. Not only Rsample, but also Rcontact, greatly decreased with increasing absolute humidity, further showing the necessity of controlling humidity in the laboratory for measurements of dry rock resistance.
Rcontact was much larger than Rsample, and accounted for most of each Rmeasured value in Fig. 8(C). This suggests that the area of the current path present on the electrode bonding surface is small with respect to the apparent electrode size: current appears to flow between the rock surface and the electrode only through part of the contact area observed by CT scanning.
Figure 8(C) confirms the dependence of Rmeasured on absolute humidity, which is attributed to moisture absorption by the sample. Alvarez (1973) and Okuyama (1972) reported that moisture greatly changes the resistance and resistivity of dry rock. Alvarez (1973) concluded that the adsorption of water molecules to minerals changed the resistance of rock samples, as also suggested by the present results.
Linear fitting to the estimated results (Fig. 9) is used to investigate whether the relationship between absolute humidity and Rsample can be expressed by a simple function. The estimated Rcontact appears mostly consistent with the fitting results, which implies the presence of an exponential relationship between absolute humidity and Rcontact. We interpret the changes in Rcontact to reflect atmospheric moisture penetrating the contact surface and filling minute gaps between the electrode and rock surface, thus increasing the contact points. It is reasonable to assume that moisture adsorption would occur even at the contact surface. The observed exponential relationship suggests that the number of contact points on the contact surface changes exponentially with changes in humidity. The fitting function assumes an exponential relationship for Rcontact = , where C and a are constants, and HA is absolute humidity.