where Vp is the volume of the pore sample,  \(V_s\) is the volume of the sample, and \(V_g\) is the total volume of grain.  is obtained from the ratio mass of grain and density of grain in which varied between 2.57-2.63 gr/cm3. The mesh range used in Samples A, B and C are 10/20, 20/40 and 40/60, respectively. Although the grain size is different, the porosity is almost the same for all samples. These samples have relatively high porosity, as reported by Tickel, Mechem and McCurdy (1932), Sparlin (1974), and Das (2008). 
Analysis of sample digital images was also carried out using the methodology described in Latief et al. (2017). Three samples, with the height of 2-cm and the diameter of 1-cm, were scanned to obtain a high-resolution image for purposes of microstructure analysis and assumed to represent the sample to be measured. The scanning process was perfomed using the following parameters: The Source-sample distance 54.60 mm, voltage 65 kV, current 80 µA, exposure 125 ms, rotation step 0.8-degree, 180° scan rotation, and used a 0.1 mm aluminium filter.

2.2 Experimental Set-up  

The laboratory resistivity measurements were carried out using a four-electrode technique, where two 2-mm thick copper plates were used for current electrodes and two stainless steel rings were used for potential electrodes. The technique was selected to minimize the effects of polarization (Campanella and Weemees 1990; Abu-Hassanein, Benson and Blotz 1996; Verwer, Eberli and Weger 2011). Connections between resistivity meter and sample container are shown in Figure 1.  
In this study, we use direct current method to measure the electrical resistivity of saturated samples. The calibration of the instrument was carried out by measuring the copper and brass cylinders and compared with a reference, to ensure that measuring instrument is appropriately calibrated (see Table 2). Electrical resistivity measurements were also conducted on a sandstone sample using the direct current (DC) and alternating current (AC) methods as a comparison, where the measurement results showed the same resistivity values for both DC and AC (see Table 2). This comparison was also conducted by Löfgren and Neretnieks (2003), to measure the formation factors using the AC and DC.

2.3 Experimental Procedure                

The measurements were initially conducted on the dry samples. Subsequent to that, the measurements were conducted on the saturated samples where the injected fluid is a green coloured  6% NaCl brine solution (conductivity of 1.196 mS/cm), and the fluid is injected at around 20%, 40%, 60%, 80% and 100% saturation level. The brine was injected at three different vertical positions, i.e., the bottom of the sample (between electrodes C1 and P1, will be further referred as sample Ax, Bx, and Cx); the middle of the sample (between two potential electrodes P1 and P2, will be further referred as sample Ay, By, and Cy); and the top of the sample (between electrodes P2 and C2, will be further referred as sample Az, Bz, and Cz).  See Figure 1 for the illustration of the injection position.
The preliminary research was conducted to examine the relationship between the electrical resistivity with respect to time. It was found that for a single value of saturation, the electrical resistivity is decreased over time as shown in Figure 2. The electrical resistivity decreases until it reaches a constant point (the resistivity is nearly unchanged). The observed changes in resistivity over time are predicted due to the changes in the spatial fluid distribution inside the pore structure so that the electrical resistivity reaches a constant value when there is no longer significant change in the spatial fluid distribution inside the pore.
In order to clearly observed the phenomena, multiple consecutive measurements were done every 1000 minutes. The average of the resistivity when the value was no longer change significantly, was used to plot the log resistivity vs brine saturation (denoted with a red dot in Figure 2).  In Figure 2, the electrical resistivity was observed to reach a constant value after approximately 6000 minutes after the injection. Table 3 shows the time required to obtain constant resistivity on each sample where sat 1, sat 2, ..., sat 5 is the order of imbibition in each sample. There is no specific pattern regarding the time to reach a constant resistivity. However, there is a tendency that for higher saturation, the time to reach the constant value is relatively shorter.  
 

3. EXPERIMENTAL RESULT

3.1 Influences of injection location

The result of measurement on sample A, sample B, and sample C are displayed in Figure 3 as plots of log electrical resistivity (\(\rho_r\)) versus brine saturation (Sw). Figure 3a shows data from sample Ax and Ay; Figure 3b from Sample Bx and By; Figure 3c from Sample Cx and Cy. In the three sample categories, log \(\rho_r\) dependency on Sw is quite similar. The resistivity of the samples that are injected with brine from the bottom position is always higher than the samples that are injected from the middle position until it reaches a certain point. We defined this point as critical brine saturation \(S_{w^{ }}^o\)  in which for sample A and sample C, critical brine saturation is approximately at  \(S_w^{o_{ }}\)= 0.70; for sample B, the critical brine saturation is about \(S_w^o\)  = 0.6. Beyond this critical brine saturation, the resistivity of the three sample categories is relatively the same up to the full saturation state. Based on these results, the three regions can be defined in the data as follows: region 1, at low brine saturation; region 2, at intermediate brine saturation; region 3, at the highest brine saturation. Regions 1 and 2 are prior to the critical brine saturation, while region 3 is beyond the critical brine saturation up to the fully saturated state.  
In region 1, the electrical resistivity decreases with increasing brine saturation for the three data sets. There are differences in the pattern of the decrease in the resistivity between each sample that are injected brine from the bottom position which will be further referred as samples X (Ax, Bx, and Cx) and the samples that are injected brine from the bottom position which will be further referred as samples Y (Ay, By, and Cy). In this region, the resistivity of samples X is slowly decreasing, on the other hand, the resistivity of samples Y decreases rapidly. In region 2, the opposite condition occurs where the resistivity of samples X decreases dramatically by approximately two orders of magnitude, while the resistivity of samples Y decreases monotonously. Region 3 is initiated with a critical brine saturation until the state of full saturation: sample A in the range Sw = 0.78-1.00; sample B in the range Sw = 0.59-1.00; sample C in the range Sw = 0.73-1.00. In this region, the change of resistivity of all samples is considered insignificant, compared to other regions.

3.2 Influence of grain size  

In this study, log electrical resistivity (\(\rho_r\) ) versus brine saturation (Sw) was also plotted as a function of grain size, as displayed in Figure 4. Figure 4a and 4b are the data obtained from the samples that are injected with brine from the bottom position (Samples X) and middle position (Samples Y) respectively. . The results show that Sample X and Y have different ρ vs Sw pattern. For Samples X (Figure 4a), we can define three regions in the data to explain the relationship between resistivity vs brine saturation. In the region 1 (at low saturations, Sw = 0-0.4), the electrical resistivity decreases gradually. In the region 2 (at intermediate saturation, Sw = 0.4-0.8), the resistivity decreases rapidly approximately in three orders of magnitude. In the region 3, at higher saturation, the resistivity tends not to change, even almost constant. For Samples Y (Figure 4b), the resistivity decreases significantly up to saturation of 0.8. Afterwards, the resistivity changes gradually until the state of full saturation for all samples. For both Samples X and Samples Y, the critical brine saturation occurs at approximately saturation of 0.8. While the log \(\rho_r\) vs Sw pattern of the two sample categories is unique, there are similarities between the two sample categories. In region before the critical brine saturation, the resistivity of samples with larger grain size is always higher than that of with the smaller grain size.

3.3 Physical properties of sample from digital images               

The results of the three samples scanning can be seen in Figure 5. Based on the digital images, it can be seen that Sample A has the largest grain size compared to Sample B and Sample C. The grain size of Sample B seems to be slightly larger than the grain size of Sample C, although no significant difference can be seen. Estimation of porosity through digital images and measurement showed good agreement, as shown in Table 4.
Beside calculating the bulk porosity, 2D porosity analysis was also performed on each vertical slice of the image. 2D porosity calculation results show that the samples are quite homogeneous (see Figure 6b). Thus, the injected brine is assumed to have the same behaviour in all pores of the sample. In addition, the pore size distribution of the samples was also analysed (see Fig. 6a). The pore size distribution which was calculated is both the connected and isolated pores. However, the isolated pores are most unlikely exist due to the fact that the sample is constructed from unconsolidated sand. In Sample A, the pore size is quite large, as the consequences of large grain sizes. As for Sample B and C, the pore size distribution is quite similar, although for Sample B it is slightly larger than Sample C. Pore size distribution shows a similar tendency, where the peak is higher for the smaller grain size.
The result from digital image analysis of the scanned samples is summarized in Table 4. The specific surface area S is defined as the ratio between the surface area of the grains to the total total volume of the sample. Specific surfaces area is used to calculate the permeability which obtained through the Kozeny-Carman equation, as shown in the equation (3) (Mavko, Murkeji and Dvorkin 2009)