# Theoretical Study on Geometries of Electrodes in Laboratory Electrical Resistivity Measurement
## 3 Experimental Tests and Results  
Experimental tests were conducted for verification of the derived equations in the previous section. A brine was selected as the material, and its electrical resistivity was measured during the experimental tests. Two brines were prepared by adding 2.16 g and 8.64 g of sodium chloride (NaCl) to distilled water (0.005 and 0.02 mol/L) and stirred over 5 min at 20 \({}^{\circ}\)C. The corresponding electrical resistivities of the brines were 18.88 and 4.78 \(\Omega\cdot\)m, respectively. The test setup is depicted in Figure 3. The container was made of acrylic and its size was \(0.275\times 0.215\times 0.13\) m; the brine is filled up to 0.125 m in height (the volume of the brine = 7.39 L). Cylindrical electrodes, with a radius (\(r\)) of 4 mm and height of 50 mm, were made of SUS303 to prevent corrosion and rubber rings were utilized to control and maintain the penetration depth of the electrodes. The electrical resistivity of the brine and the electrical resistance between the two electrodes were measured with a conductivity meter (Mettler Toledo S213, Mettler Toledo, Columbus, OH, USA) and LCR meter (Agilent HP 4263B, Agilent, Santa Clara, CA, USA) which can measure inductance, capacitance and resistance. A 1-V signal was applied, which was the maximum voltage of the measurement equipment, and a 1-kHz signal, which is the most commonly utilized frequency in petrophysics and other engineering fields, was applied to avoid the polarization effect [17; 18]. The electrical resistance was measured by varying the penetration depth (\(l\)) and the distance between the two electrodes (\(L\)). Experimental test cases are listed in Table 1.  
Figure 3: Test setup: measurement equipment, container and electrodes.  
Figure 4 shows the theoretical and experimental resistances, as a function of the distance between electrodes (\(L\)), in a brine with an electrical resistivity of 18.88 \(\Omega\)m. Lower electrical resistivity produced a higher current flowing through the brine (lower electrical resistance). Additionally, a deeper penetration depth (larger contact area of the electrode) resulted in a lower measured resistance. The predicted resistance from Equations (9) and (10) have a strong correlation until \(L=0.165\) m with the measured electrical resistance. However, the measured electrical resistance suddenly increased relative to the predicted value from \(L=0.205\) m. The deviation was attributed to the electrical boundary effect that results from distortion of the electric current near the boundaries. A similar phenomenon was reported in the previous research utilizing the non-conductive polycarbonate container for the experimental tests [18]. The authors stated that the container boundary affects the current flow during the electrical resistivity measurement. Note that the container in this study was made from a non-conductive material (acryl). The dashed line obtained from Equation (13), reflecting the container wall effect, gives an accurate prediction near the container wall. This means that the electrical resistivity of the acrylic is much higher than that of the brine, as assumed in the formulation process.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Electrical Resistivity, \(\rho\) (\(\Omega\)m)** & **Penetration Depth, \(l\) (mm)** &
\begin{tabular}{c} **Distance between Two** \\ **Electrodes, \(L\) (m)** \\ \end{tabular} \\ \hline \multirow{2}{*}{18.88} & 0 & 0.015 \\  & 0 & 0.045 \\  & 4 & 0.085 \\ \hline \multirow{4}{*}{4.78} & 8 & 0.125 \\  & 8 & 0.165 \\  & 16 & 0.205 \\ \hline \hline \end{tabular}
\end{table}
Table 1: List of experimental test cases.  
Figure 4: Evolution of electrical resistance with distance and depth (18.88 \(\Omega\)m): (**a**) \(|l/r=0\), (**b**) \(|l/r=1\), (**c**) \(|l/r=2\), and (**d**) \(|l/r=4\). Continuous lines are obtained from Equation (9) and dashed lines are obtained from Equation (13).