Introduction

Hydrocarbon accumulation is commonly estimated from electrical logs. The result is considered as one of the most important parameters in reservoir development plans. The hydrocarbon saturation dependency on resistivity in a clean rock is commonly determined using the Archie equation (Archie 1942), which is expressed as follows:

$$S_{{\text{w}}} = \left[ {\frac{a}{{\phi^{m} }}\frac{{R_{{\text{w}}} }}{{R_{{\text{t}}} }}} \right]^{{\tfrac{1}{n}}}$$
(1)

where RO and Rt are the resistivity of rock fully and partially saturated with formation water having a resistivity of RW. SW is water saturation and a, m and n are Archie coefficients.

The Archie equation, in fact, is a combination of resistivity index (RI) and formation resistivity factor (FRF), which are defined as follows, respectively:

$${\text{RI}} = \frac{{R_{{\text{t}}} }}{{R_{{\text{o}}} }} = \frac{1}{{S_{w}^{n} }}$$
(2)
$${\text{FRF}} = \frac{{R_{{\text{o}}} }}{{R_{{\text{w}}} }} = \frac{a}{{\varphi^{m} }}$$
(3)

The saturation exponent (n) is commonly determined by injecting a fluid (gas/oil) into a cleaned, water-saturated rock sample. The sample resistivity (Rt) is measured at pre-defined steps with decreasing water saturations (SW). A log–log plot of RI versus SW typically yields a straight line with the slope of n.

Carbonate reservoirs are of great importance, because they contain almost 60% of the world’s hydrocarbon reserves. Relative to siliciclastics, carbonates are simpler in terms of mineralogy but are more complex in terms of pore structure and texture (Ham and Pray 1962; Focke and Munn 1987; Laubach 1988; Akbar et al. 2000; Lucia 2007; Tariq et al. 2020; Naderi-Khujin et al. 2016, 2020). The extensive biological origin of the sediments combined with various diagenetic processes yield complex pore structure and texture, which may greatly vary between reservoirs (Bauer et al. 2012; Abdolmaleki and Tavakoli 2016; Sharifi-Yazdi et al. 2019, 2020; Tariq et al. 2020). As a result of the high heterogeneity, they usually have low oil recovery factors (Heide 2008).

Various studies investigating the relationship between saturation exponent and other petrophysical parameters have shown that the wettability has vital effects on the value of n (Unalmiser and Funk 1998). Experimental studies have demonstrated that while Archie’s law is widely utilized, it may not always hold true in all cases. Theoretical considerations of electrical conductivity in porous media mainly follow the Archie’s law (Sheng 1990; Sahimi 1993). The applicability of the Archie equation is limited to strongly water-wet and clean rocks, which is not observed in most carbonate formations (Shamsi et al. 2001). However, these models cannot adequately explain the conductivity behavior observed in oil-wet systems, where the presence of oil and its interactions with the rock surfaces introduce additional complexities.

Saturation history, wettability, content of clay minerals and salinity of the brine phase affect the resistivity index (Keller 1953; Mungan and Moore 1968; Anderson 1986; Wei and Lile 1991; Sondena et al. 1992; Dengen et al. 1996; Unalmiser and Funk 1998; Shamsi 2001; Man and Jing 2001, 2002; Hamada et al. 2002; Fleury 2002; Cerepi 2002; Chen and Kuang 2002; Kurniawan 2005; Han et al. 2007; Li 2010a; Bauer et al. 2012; Moss and Jing 2015; Melani et al. 2015; Mohamad and Hamada 2017; Khan and Tariq 2018; Tariq et al. 2020). The saturation exponent is assumed to be 2 in many reservoir studies. However, it is only true in water-wet homogeneous formations (Schlumberger 1987). Keller (1953) demonstrated that the saturation exponent can exhibit values significantly different from the conventional value of n = 2. He found that n could vary from 1.5 to 11.7 for the same rock if the wetting conditions change. His findings shed light on the complexities of fluid behavior in porous media, emphasizing the need to consider different n-values to accurately characterize fluid saturation in various reservoir settings. Several studies have revealed the significant variability of the saturation exponent across heterogeneous reservoirs, ranging from approximately 20 in highly oil-wet environments to a more common value of 2 in highly water-wet settings (Donaldson and Siddiqui 1989; Melani et al. 2015).

Anderson (1986) carefully examined the measurements made by Mungan and Moor (1968) on artificial Teflon cores and those made by Sweeney and Jennings (1960) on carbonate cores for the effect of wettability on electrical properties of porous media. Both studies showed that the saturation exponent for oil-wet systems increases dramatically when the brine saturation is below a certain value. Attia and others reported that the break in the slope of log (IR) versus log (Sr) corresponds to the capillary pressure-based irreducible water saturation point (Attia et al. 2008). In a water-wet system, the values of n obtained from Mungan and Moor (1968) and Sweeney and Jennings (1960) measurements is about 2 and 1.6, respectively. Sweeney and Jennings (1960) also measured the resistivity index on neutrally wet systems and determined the resistivity index to be approximately 1.9. It is important to point out that brine was the displaced phase in all their experiments. Figure 1a summarizes the results obtained by Sweeney and Jennings (1960) on carbonate samples.

Fig. 1
figure 1

Experimental trends for resistivity index versus brine saturation (Dengen et al. 1996). a For strongly water-wet and neutral wet systems, the resistivity index can be approximated by a straight line (Sweeney and Jennings 1960). b Effect of hysteresis and wettability effects on saturation exponent (Wei and Lile 1991)

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Wei and Lile (1991) studied the saturation hysteresis in both water-wet and oil-wet sandstone samples (Figure 1b). They found that the n yielded a value of approximately 2 for both imbibition and drainage in the water-wet system. However, for oil-wet system, they found that the Archie’s law is only valid for high brine saturations (> 58%). In addition, they observed a significant difference in saturation exponents between drainage and imbibition processes when the brine saturation was less than 58% in an oil-wet system (Dengen et al. 1996).

The effect of saturation hysteresis on resistivity index was studied by Goddard et al. (1962) on sandstone samples. They observed significant differences in resistivities for mercury injection and withdraw processes when the mercury saturations were low. Also, Man and Jing announced that contact angle hysteresis, which leads to deferent pore scale physics, reveals hysteresis observed in both electrical resistivity and capillary pressure curves (Man and Jing, 2001). In another study, they concluded that the distribution and fraction of mixed-wet pores can lead to a variety of electrical resistivities, capillary pressures and residual oil saturation behaviors (Man and Jing 2002).

Carbonate rocks typically have far wider range of grain shape and size than most siliciclastic rocks. Clearly, several types of porosity may coexist in a carbonate reservoir, ranging from microscopic to cave size, which makes porosity and permeability estimation and calculation of reserves extremely difficult (Akbar et al. 2000; Mohsin et al. 2023). The pore system controls electrical resistance through fluid distribution in the pore spaces (Kułynycz 2017; Tavakoli et al. 2022). The transport properties such as the electrical conductivity and permeability depend not only on porosity but also are strongly sensitive to the connectivity of pore spaces and geometry (Cerepi et al. 2002). Determining different groups with more emphasis on the pore system is one of the other efforts to analyze the electric current (Sen 1997; Fleury 2002; Han et al. 2007; Nazemi et al. 2019, 2021; Najafi et al. 2023). Han et al. (2007) measured the electrical responses of several sandstones and carbonates. The samples were classified in four groups according to pore network. Each group generated different electrical response. For the group I, the measured RI–Sw curve followed an Archie behavior in the saturation range of Sw > 20% with an exponent of approximately 2 and showed a negative deviation at low saturation range (Sw < 20%). They attributed this negative deviation to the water film conduction at low saturation scale. They confirmed the effects of this film conduction by a numerical simulation. For the carbonate samples, the groups II and IV often had small values of n and slight electrical hysteresis. The RI–Sw curve of the group III (bimodal porosity), depends strongly on the microporosity configuration. If the micropores are isolated, the resistivity can sometimes increase at low saturation values (Han et al. 2007).

A good review of various experimental observations on saturation exponent can be found in investigations carried out by Sen (1997), who proposed theoretical explanations based on the mixture of various pore sizes (micro, macro and mesopores).

In general, the resistivity index curve RI (Sw), cannot be described by a power law (second Archie’s law RI = Sw−n) and n is a function of saturation itself. Petricola and Watfa (1995) suggested that the microporosity may act as a parallel path for the current yielding decreased n values, and therefore, a gradual insensitivity of the resistivity to saturation, as observed in shaly sands. On the other hand, Dixon and Marek (1990) suggested that the microporosity leads to low n values (e.g., n = 1.45), although n does not depend on the saturation itself in a considered range.

A sketch of the various shapes of the RI (Sw) curves is shown in Fig. 2 (Fleury 2002). The shapes are grouped as follows:

  • Type I: may be typical of carbonates from the Thamama Formation,

  • Type II: bending upward at intermediate saturation levels and then flattening out at low saturations,

  • Type III: single slope at low saturation, extrapolation to Sw = 1 above IR = 1,

  • Type IV: typical of oil-wet systems, large “n” values sometimes increase further at low saturation. This type is also valid in clastics.

Fig. 2
figure 2

Schematic of different types of RI curves versus Sw observed experimentally. I, II and III types are due to the pore structure, however, IV group is due to the wettability effects and is not specific to carbonates (Fleury 2002)

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Bouvier et al. (1991) offered a relation between RI (Sw) and capillary pressure curves, which is associated with pore-size distribution data. Li and Williams (2006) established a theoretical relationship between resistivity and capillary pressure. The model was developed based on the fractal modeling of porous media. Another study demonstrated that capillary pressure and relative permeability can be estimated from resistivity data using analytically derived mathematical models (Li 2010b). Li applied a power law model to the relationship. The correlation between capillary pressure and resistivity index data was notably stronger in low permeability rocks (Li 2010b). In the studies of Man and Jin, who investigated the relationship between electric current and capillary pressure on samples with different wettability, a significant similarity was observed between the curves of capillary pressure and resistivity index (Man and Jing 2001, 2002).

Figure 3 shows the resistivity index versus water saturation and mercury capillary pressure curves for a sandstone sample. A strong correlation exists between resistivity and pore-size distribution, indicating a significant relationship between these two parameters (Sbiga 2012). It is noteworthy that a significant change in the slope of the resistivity index occurs at the point where mercury starts penetrating into the micropore spaces. It may be due to the possible micro-pores/irregular surfaces inside the sample.

Fig. 3
figure 3

Comparison of the variations of capillary pressure (a) and resistivity index (b) with saturation in a sandstone sample. There is a good compatibility between the trends of the curves (Sbiga 2012)

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While the initial increase in oil saturation predominantly impacts resistivity by filling larger pore spaces, significant water volume remains trapped in micropore spaces, contributing to the observed high apparent saturation exponent. However, the saturation exponent (n) decreases as the oil penetrates the micropores and displaces the water by sufficiently increasing capillary pressure with little influence on resistivity. The experimental measurements made by Sbiga (2012) on the relationship between resistivity index and brine saturation also confirm this observation. Therefore, it seems that there is a relationship between resistivity and pore sizes (macro or micro).

The determination of Archie parameters presents a continuous challenge in reservoir studies due to its inherent complexity. Therefore, even a small deviation in the two exponents of real reservoirs from their default values will lead to significant errors in the estimated reserves (Glover 2017). Understanding and prediction of the effects of pore structure and wettability on the electrical properties are real scientific challenges, both theoretically and experimentally (Clavier et al. 1984; Sen 1997; Herrick et al. 2001; Fleury 2002; Dernaika et al. 2007). Electrical properties of carbonates are affected by microstructures and wetting characteristics and separation of these effects is a complicated task. In this paper, using rocks with different properties under different laboratory conditions, these effects are separately analyzed and the concept of saturation coefficient has been examined from different angles. The most effective parameters are extracted and used to determine a model for the saturation coefficient.

Methodology

Sample selection

All the factors related to the rock and rock-fluid properties such as lithology, pore type, pore size distribution and wettability, which affect the saturation exponent, have been studied. The factors affecting the electric current were divided into two groups and each group was individually investigated in selected carbonate rocks.

The first group factors include lithology, pore type and petrophysical properties. A total of 16 samples with different lithologies (limestone, dolomitic limestone and dolostone) and petrophysical properties were selected to investigate their effects on the electrical current. The electrical current experiments were designed to minimize the effect of wettability and performed at water–gas system with no need to restore the reservoir conditions or using oil.

In the second group, the effect of wetting on the electric current of 25 samples was investigated. The influence of the pore-size distribution was also studied. The samples with similar lithologies were selected from limestone rocks to minimize the influence of lithology and texture on the electrical current. Before testing, the samples were restored to reservoir conditions using crude oil. The wettability and pore size were determined by the Amott and mercury injection methods, respectively. The electrical current and saturation exponent tests were also performed using the crude oil in reservoir conditions.

First group of tests

These tests specifically examine properties such as texture, pore type and hydraulic flow unit (HFU). These properties are highlighted by the distinct resistivity indices and saturation exponents observed in the samples. The Dunham classification of carbonate rocks used for textural determination (Dunham 1962) focuses on the depositional fabric of carbonate rocks. In addition, pore type classification was performed according to Choquette and Pray’s classification (1970) based on the fabric of carbonate pore types, rather than petrophysical properties. All carbonate pore spaces are divided into fabric selective and non-fabric selective types in this classification. Unlike the fabric selective pores, non-fabric selective pores can cut across grains.

Generally, there is a weak positive correlation between porosity and permeability, especially in carbonates; therefore, hydraulic flow unit classification developed by Amaefule et al. (1993) was used to find a better correlation between porosity and permeability as main petrophysical parameters.

Resistivity and capillary pressure measured by porous plate method, were used to calculate the resistivity index and saturation exponent n. A brine-saturated sample was placed on a porous ceramic plate and sealed inside a steel cell. The humidified air entering into the cell at various pressures displaces the brine. The electrical resistivity of partially saturated plug sample (Rt) was measured at each injection pressure by placing it between two electrodes. The saturation exponent (n) was then calculated from Archie formula.

Second group of tests

The second group of tests were designed to highlight the effects of wettability on the resistivity index and saturation exponent. In this part, the selected samples were completely limestone with similar textures and pores; therefore, there are no prominent features for rock typing. Unlike the first group, the second group samples were tested using crude oil. Moreover, the wettability index was calculated using the Amott method (Amott 1959) and the irreducible water saturation was obtained during the tests. Both wettability and saturation exponent tests were performed at the wettability restoring conditions at the reservoir pressure and temperature.

The wettability is a rock-fluid property that considerably affects reservoir dynamics such as fluid flow and electric current in porous media. The Amott method was used to determine the wettability index by the forced and spontaneous drainage and imbibition. The Amott wettability index (WI) is defined between 1 and − 1, which WI = − 1 and WI = 1 refer to strongly water-wet and oil-wet, respectively. However, it shows an intermediate wetting sample if it is between 0.3 and − 0.3 (Anderson 1986).

The influence of the pore-size distribution (PSD) was also studied. The capillary pressure data obtained from the mercury injection is based on the Young–Laplace equation (Young 1805):

$$P_{{\text{c}}} = \, 2\sigma \cdot {\text{Cos}} \;\theta /r$$
(4)

where r is the pore radius, σ is the surface tension between two fluids, θ is the contact angle and Pc is the capillary pressure. This equation is basically used for the pore-size distribution determination. Large range of pore sizes will be covered as the injection pressure increases.

In order to study the pore properties and their compatibility with reservoir properties, various parameters have been directly and indirectly extracted from the capillary pressure results by mercury injection method (Jennings 1987).

Mean radius (rM), displacement pressure (Pd), line fraction slope, total mercury injection porosity (ØM), pore throat radius at peaks, pore space percentage in different pore throat size, pore throat sorting (PTS) and pore throat radius at 20 and 35 present mercury saturation are the parameters determined for each sample from the mercury injection test. A brief description of the methodology to get parameters is given as follows.

Mean radius, displacement pressure and slope were determined from the linear sections of the cumulative pore-size distribution curve. Displacement pressure was obtained by extending slope of plateau to the mercury injection pressure axis in capillary pressure curve which is the start of mercury influx into the rock sample. The rate of heterogeneity of porous media was also obtained using capillary pressure curve with a correlation coefficient above 95% which is equal to its slope of linear part. Mercury porosity (ØM) was determined as the ratio of injected mercury volume to total sample volume. The saturation data acquired from the mercury injection test, encompassing various pore sizes, were categorized into two groups including microporosity (< 0.5 μm) and macroporosity (> 1.5 μm). Pore-throat size distribution (PTSD) graph constructed from fraction of pore volume injected versus pore throat radius. In the homogeneous rock samples, only one peak in throat radius distribution was observed. However, in the heterogeneous samples, multiple peaks were identified and are referred to as “r mode” in this article. “r mode 1” and “r mode 2” are large peak and small peak, respectively. Pore throat radius at 20 and 35 percent mercury saturation were determined from pore-size distribution curve.

The aforementioned slope is equivalent to pore-throat sorting (PTS) introduced by Jennings using two saturation points of 25 and 75% (Jennings 1987).

$${\text{PTS = }}\sqrt {\tfrac{{\text{Third-quartile\,pressure}}}{{\text{First-quartile\,pressure}}}}$$
(5)

Results and discussion

To examine the impact of rock properties on the saturation exponent, the relevant samples were classified based on texture and pore type (as shown in Table 1 and Fig. 4). This classification led to the identification of four distinct rock types:

  • rock type 1: packstone with moldic pore type

  • rock type 2: grainstone with low porosity

  • rock type 3: dolo-grainstone with moldic and intercrystal pore type

  • rock type 4: dolostone with intercrystal pore type

Table 1 Petrophysical and geological properties of the studied samples
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Fig. 4
figure 4

Typical images of thin section for four rock type groups. Images were taken in polarized light and the pores are clearly visible. Differences in texture, size and type of pores can be seen in thin section images

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The integration of textural classification according to Dunham (1962) and pore type classification based on Choquette and Pray (1970) yielded the rock-typing approach employed in this study. Rock types were identified, classified and named based on the connectivity of their microstructures. Typical images of thin sections for each group are shown in Fig. 4. The group 1 is composed of limestones whose main porosity is moldic. Although this group shows high porosity, its permeability is low due to the unconnected pores. The second group is characterized by its significant low porosity, leading to notably poor permeability. The third group consists of dolomitic limestones, which is mainly dolo-grainstone. This group has different types of texture and pores (connected and unconnected). The fourth group comprised dolostones with intercrystalline pores, exhibits higher permeability compared to the other groups, primarily attributed to its connected porosity. Table 1 shows that the reservoir properties (porosity, permeability, HFU) are improved with the increase in the number of rock types.

Besides, four groups of HFU were also identified for the selected samples, as it is given in Table1. Wide ranges of porosity (3.54–25.4%) and permeability (0.16 mD-279 mD) were observed for the samples. The saturation exponents of the samples obtained by the porous plate method ranged from 0.75 to 3.8. Their dependency on the lithology and hydraulic properties (HFU) are shown in Figs. 5 and 6.

Fig. 5
figure 5

Water saturation versus resistivity index for four rock type groups. The saturation exponent decreases with an increase in the rock group number. Unlike other groups, the electric current of group number 3 changes its nature during fluid injection and thus gives two different saturation exponents

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Fig. 6
figure 6

Saturation exponent versus sample’s HFU. Once the hydraulic flow units (which is accompanied by the improvement of petrophysical properties) increases, the range of variation and the average amount of saturation exponent decrease

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The plot of resistivity index data versus the water saturation is shown in Fig. 5. Four groups can be easily distinguished in this plot. Three of the groups (rock types 1, 2 and 4) have constant slopes; therefore, they have unique saturation exponents. The third group was further divided into two sub-groups (rock type 3–1 and 3–2), each exhibiting distinct and unique slopes. This division allows for a more accurate representation of the saturation exponent based on lithological and hydraulic characteristics. In fact, the broken point in the group 3 is due to different pore types and mixed textures of grainstone and dolostone rocks. Initially, the slopes follow the pattern observed in group 4. However, a sudden change occurs leading to the emergence of two distinct “n” values. Figure 5 shows that the numbers of rock types increase when the saturation exponents decrease associating with increased pores connectivity. In other words, the saturation exponent for the groups with non-connected pores (e.g., moldic pore type) is high and for the groups with connected porosities (e.g., intercrystalline pore type) is low.

There is only one constant in the second Archie’s formula called saturation exponent (n):

$${\text{RI}} = \frac{1}{{S_{w}^{n} }}$$
(6)

where RI and Sw are the resistivity index and water saturation, respectively. However, some researchers added the constant “b” to it as:

$${\text{RI}} = \frac{b}{{S_{w}^{n} }}$$
(7)

where b is called the saturation distribution factor which depends on the pore structure, water distribution, clay content and distribution and wettability (Abdassah et al. 1998). Shamsi et al. (2001) claim that for many carbonates, assuming a linear relation between the resistivity index and formation water saturation is also incorrect. It is shown in Table 2 that the obtained b values for the samples are nearly 1, which establish the second Archie’s formula.

Table 2 Empirical formula and coefficient results of resistivity index in various groups
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Figure 6 shows HFU numbers versus saturation exponent. There are different saturation exponent values in different hydraulic flow units.

For the study of the wettability effects on the saturation exponent, the resistivity and wettability results of the 25 selected samples were investigated (Table 3). The results showed that the wettability of the samples was mainly between oil-wet to neutral. Besides, the saturation exponent increases with decreasing wettability index which is associated with increase in oil wetting (Fig. 7). Similar results were obtained by correlating the saturation exponents with the irreducible water saturation data (Fig. 8). As shown in the figure, the saturation exponent increases with decreasing the irreducible water saturation. As a result, by decreasing the irreducible water saturation, the wettability index decreases as well, although some irregularities can be seen with increasing oil wetting (Fig. 9). The results of Fig. 7, 8 and 9 confirm each other and show that the saturation exponent increases with increasing oil wetting.

Table 3 Petrophysical and experimental results of the studied samples
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Fig. 7
figure 7

Saturation exponent versus wettability index by Amott method. With an increase in oil wetting, the value of n also increases

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Fig. 8
figure 8

Saturation exponent versus irreducible water saturation. As Swi increases, the value of n decreases

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Fig. 9
figure 9

Wettability index versus irreducible water saturation. Considering the majority of samples, the irreducible water saturation decreases with increasing oil wetting

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Due to the scattering of data in the plot of saturation exponent versus wettability index as well as the effect of irreducible water saturation on saturation exponent results, the effect of the throat radius on the saturation exponent was investigated. Pore-throat sizes exhibiting higher frequencies were employed for the categorization of samples. Based on these results, the samples were divided into two groups, including r > 1.5 µm and r < 0.5 µm (Table 4). These groups are distinguished in the cross plot of saturation exponent versus wettability, highlighting the influence of the pore throat radius on the variation of saturation exponent (Fig. 10). Also, Fig. 11 shows that the samples with large throat radius have better petrophysical properties than the samples with smaller throat radius. In samples characterized by small pore throats, which predominantly exhibit neutral wettability, the saturation exponent demonstrates a gradual change with respect to the wettability index, displaying a relatively gentle slope. Conversely, in samples featuring a significant throat radius, which mainly are oil-wet, the saturation exponent changes with the wettability index by a steep slope.

Table 4 Petrophysical and experimental results of the studied samples
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Fig. 10
figure 10

Saturation exponent versus wettability index in different pore sizes. Samples with a large throat radius are separated from samples with a small throat radius, so that it shows the effect of the size of the throat radius on the saturation exponent

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Fig. 11
figure 11

Petrophysical properties in different pore sizes. Samples with larger throat radii have higher porosity and permeability

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The results demonstrate that injecting oil into larger pores is easy. Therefore, some more regular variations could be seen for the electric current and saturation exponent. In contrast, in the fine throat radius associated with a high amount of irreducible water saturation, the saturation exponent shows high variations when the wettability index changes slightly. This may be due to the interruption of the electrical current paths by saturation changing. In general, the pore-throat radius and wettability of the rock can significantly control the saturation exponent.

Through the integration of diverse datasets including petrophysical data, pore-size distribution, wettability and electrical data, an attempt was made to explore the interplay between these variables with saturation exponent from a different perspective. For this purpose, all the mentioned data were placed against the saturation exponent and the relationships were extracted. Among these relationships, the ones presented in Table 5 were finalized and compared with laboratory data in Fig. 12. As can be seen in the offered relationships, various parameters are involved in calculating the saturation exponent. As the input parameters increase, the correlation coefficient value for calculating the saturation exponent also increases. In the first relation, the gas permeability value (Kair) and the largest peak of radius of the pore-size distribution (rmode1) curve are mentioned as the most effective parameters. Despite the lack of direct relationship between these two parameters, an acceptable correlation (R2 = 51) has been obtained with the saturation exponent. With the addition of the wettability parameter (Amot-Harvey wettability index), the correlation coefficient has increased significantly (Table 5). In subsequent relationships with the addition of porosity values from mercury injection and microporosity, the correlation coefficient has increased to 87%. The analysis of positive and negative signs in these relationships reveals that as permeability increases, the saturation exponent also increases. Conversely, when considering changes in other parameters, an inverse relationship is mainly observed. Based on existing relationships and considering the sign of the wettability index, it is observed that oil-wetting conditions lead to an increase in the saturation exponent. Conversely, water-wetting conditions result in a decrease in the saturation exponent. As the amount of porosity as well as microporosity increase, the saturation exponent decreases. The complex results of saturation exponent versus hydraulic flow units (Table 1) can be attributed to the different relationship of saturation exponent with porosity and permeability.

Table 5 The relationships extracted from various data for the calculation of saturation exponent (n). As seen, the value of the correlation coefficient increases as the input parameters increase
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Fig. 12
figure 12

Comparison of the saturation exponent (n) derived from the relationships presented in Table 5 with the laboratory values. From figure a to d, which are related to formulas 1–4 (Table 5), respectively, the value of the correlation coefficient has increased

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In addition to the relations for interpreting the saturation exponent, the amount of this exponent can be predicted based on the available data. However, comprehensive data and extensive experimental studies are required to accurately predict and model these relationships. Table 6 presents a relationship derived exclusively from the outcomes of the mercury injection tests, demonstrating strong correlation coefficient. Figure 13 shows a comparison of the saturation exponent obtained from the mentioned relation with the laboratory values. The amount of data used to extract this relationship is not large, but the general positive trend was achieved. Subsequent studies could potentially validate and reinforce this relationship through the inclusion of extensive datasets. The observed direct or inverse relationship between the parameters and the saturation exponent has revealed a level of complexity that necessitates further investigation.

Table 6 The relationship extracted from mercury injection data with acceptable correlation coefficient
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Fig. 13
figure 13

Comparison of the saturation exponent (n) obtained from various mercury injection data with its laboratory measured values

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The comprehensive examination of the first group revealed a significant impact of connected pore types on the saturation exponent. In these series of experiments conducted in a water–gas system, all the fine and large pore spaces were filled by the water. Therefore, the characteristics of the porous space (e.g., pore type and size) affect the water distribution ,and then, the electric current. However, in the second group of experiments performed by restoring the wetting conditions, both oil and water play pivotal role in fluid distribution. Despite the first group experiments, water as a non-wetting phase was present in the porous space and the wettability has changed with the presence of reservoir oil. Therefore, in addition to the characteristics of the porous space, the type of wettability affects the fluid distribution, and then, the electric current. The results also showed that the size of the porous space and the wettability are effective on the saturation exponent. Moreover, the radius of throat along with the type of pores and wettability were the main factors affecting the electric current and saturation exponent. Finally, the amount of porosity, permeability and hydraulic flow units showed different relationships with the saturation exponent, especially in the low hydraulic unit, which could be due to the mismatch of porosity and permeability in samples with poor petrophysical properties (low hydraulic flow units). The provided relationships can be used to predict the amount of saturation exponent using existing data, especially pore-size distribution.

In this study, two distinct rock groups were examined under different laboratory conditions to investigate the diverse factors influencing electric current. Both rock and fluid-related aspects were explored. The separate study of samples and test conditions showed the effects of different parameters on the saturation coefficient and improved the understanding of the concept of saturation exponent. As a result of this study, an empirical relation with a high correlation coefficient was established for calculating the saturation exponent. These findings provide assistance to engineers in accurately predicting the saturation exponent, enhancing their ability to make informed decisions and optimize reservoir management strategies. Nevertheless, as evident from the outcomes, the interplay between rock and rock-fluid parameters has been evident, posing a challenge in isolating their individual effects through separate analysis. Moreover, this overlap introduces complexity in the process of grouping and determining the exponent within these rock types.

Conclusions

  • Studies of carbonate samples with different rock types (various lithologies, textures, porosities and pore types) with similar wettability (water-wet) showed that there is a negative correlation between saturation exponent and pore connectivity. The saturation exponent is high in the samples with non-connected pores (e.g., moldic porosity) and low in the samples with connected porosity (e.g., intercrystalline porosity). However, exceptional irregularity showed that the heterogeneity in the pore types and texture cause the variations of the saturation exponent value as a function of the saturation, itself.

  • Saturation exponent and its variations reduced with increasing the hydraulic flow unit number. This relationship indicates complexity in hydraulic units with low porosity and permeability. This complexity is also seen in the relationship between porosity and permeability with saturation exponent.

  • Studies of samples with similar rock properties (limestone with similar textures and pores) with different wettability (oil-wet to neutral) showed that saturation exponent increases with decreasing wettability index and irreducible water saturation. However, the presence of large pores, which is a factor of heterogeneity in the rock, affects the value of the saturation exponent and causes the data to scatter in the curve of the saturation exponent wettability.

  • Similar to the above result, the experimental relationships showed that the permeability, the largest peak of radius of the pore-size distribution and wettability significantly affect electrical resistivity and saturation exponent. The relationship presented in Table 6 can be used to predict the amount of saturation exponent using mercury injection data, although increasing the volume of available data may lead to a more robust relationship.

  • In this study, both rock and fluid-related aspects were investigated to understand the diverse factors influencing electric current. The individual analysis of similar samples and specific test conditions showed the effects of different parameters on saturation exponent. Nevertheless, the interplay between rock and rock-fluid parameters is evident, presenting a significant challenge in isolating their individual effects. Moreover, this overlap introduces complexity in the process of rock grouping and determining the exponent within these rock types.

  • Considering the interplay between the porosity and permeability and their influence on wettability, the saturation exponent’s complexity is most pronounced in samples with low porosity and permeability. Moreover, as pore types significantly contribute to the heterogeneity observed in carbonate rocks, it becomes evident that pore space heterogeneity is the principal determinant of the saturation exponent.