where \(\gamma\), g, h, and \(\theta\)  are surface tension (N/m), density of grain (kg/m3), gravitational acceleration (m/s2), height of the meniscus created, and contact angle of the liquid on the solid (degree) respectively. From this equation, the surface tension of each sample can be obtained, as shown in Table 5. Sample A has a low surface tension, so there is no force to hold the brine around the injected area, thus, the brine can flow easily to the bottom of part of the sample. On the contrary, for Sample C, which has a high surface tension, brine tends to distributed only around the area on the injection. As for sample B, the surface tension is not strong enough to hold a certain amount of brine, so that at 10 ml injection, some of the brine is retained at the injection site and some of the brine flowed to the bottom of the sample.  
Based on these observations, we can explain the data in Figure 4b, the sample that injected with brine from the middle position for three different grain sizes. The resistivity of Sample A (the sample with the largest grain size) is always higher than Sample B and Sample C in the area before the critical brine saturation. This result happens because the brine in Sample A tends to fill the pore in the lower position so that the majority of the brine is not in a significant area (between two potential electrodes). In contrast to Sample B and Sample C, the majority of brines are in significant areas. Likewise, the data in Figure 4a, the resistivity of the samples that injected with brine from the bottom position for three different grain sizes. Sample A always fills the lower pore compared to Sample B and Sample C, which sometimes fills the upper pore from the brine injection position. After the brine had passed the critical brine saturation, the brine saturation did not significantly affect the electrical resistivity, both for samples that are injected from the middle position or the bottom position.

4.3 Interpretation resistivity data of the samples that injected with brine from the top position               

Measurements have also been made on the samples that was injected with the brine from the top position (Az, Bz, and Cz). The results of the measurements are shown in Figure 11. In Figure 11a, the resistivity of sample Az is always lower than the resistivity of the sample Ax and Ay. Based on the previous analysis, the characteristic of sample A is that it is easy to let the brine pass through the pore space. Thus, in this case, the brine flows to the bottom of the sample while also wets some parts of the sample. The data in Figure 11a can be interpreted that for sample Az, the area between the two potential electrodes (P1 &P2) has higher fraction of brine that wets the sample compared to the sample Ay and Ax.
In Figure 11b, the resistivity of the sample Bz at low saturation, is higher than that of the sample By. In this situation, the brine is kept above the upper potential electrode (P2), due to the small pore size and low permeability. However, after saturation is increased, the brine eventually seeps down to the bottom part of the sample. Thus, in the sample Bz, the brine is more dominant in the area between the two potential electrodes compared to sample By. Consequently, the resistivity of the Bz sample is lower than that in sample By in this area. Subsequently, after the brine passes the upper potential electrode (P2), the resistivity of all samples tends to be the same (critical brine saturation).
In Figure 11c, the resistivity of sample Cz is always higher than the resistivity of the sample Cy in the area before the critical brine saturation. Sample B has low permeability and small pore size, thus, brine tends to be kept at the injection site. As the consequence, that the area between the two potential electrodes has a higher fraction of fluid that wets the grains in sample Ay compared to sample Ax and Az.

4.4 Calculation of electrical resistivity  

The calculation of resistivity has been done by using the Archie's equation and finite element method (FEM). The FEM method is applied to solve the Laplace equation (Garboczi 1998), where the input for the FEM is the digital images which were obtained from sample scanning using micro-CT. The brine saturation model is a simple fluid filling model developed by Fauzi, Mustofa, and Latief (2019). Fluid fills the pore space with a simple mechanism where the pore is gradually replaced with fluid up to a certain level of saturation. Electrical resistivity calculation results are shown in Figure 12.
Based on the results, the Archie equation and FEM only agrees with the experimental data on the samples that were injected with brine from the top position for all sample and samples, as well as the ones that were injected with brine from the middle for Sample B and Sample C. The m cementation exponents obtained were 1.9, and the saturation exponents obtained were 1.9.  For the sample where the brine was injected from the bottom position, there are no similar trend which can be explained by the Archie equation and FEM method at all. These phenomena show that the spatial distribution of brine, which is the most important part in conducting electricity in the sandpack, has a very significant effect, i.e., areas that do not have a sufficiently good spatial distribution of brine, cannot be explained or approached by the Archie equation and FEM method.
 

5. CONCLUSION                                

The result of electrical resistivity measurement of sandpacks using a four-electrode technique have shown that the spatial distribution of brine in the pore space, significantly affects the measurement of electrical resistivity. If the brine has sufficiently good spatial distribution in the area between the two potential electrodes (P1 and P2), the resistivity will be decreased significantly the saturation level increases. Meanwhile, if the brine is distributed mostly in the outside of the area between the two potential electrodes, the resistivity is not considerably reduced. In addition, observations of partial saturation measurement suggest that the critical brine saturation occurs when the brine spatial distribution has passed the upper electrode potential (P1) at exact saturation around 0.7–0.8 for all samples.
Electrical resistivity in the samples that have larger grain sizes is always higher than that of with the smaller grain sizes in the region before the critical brine saturation. This result is related to the permeability of the samples and the surface tension between brine and grain. Moreover, the calculation of electrical resistivity through Archie's equation and finite element method only applies to samples that have a good brine distribution in the area between two potential electrodes.