Introduction

Water saturation was defined as the ration of pore volume that occupied by water with total rock pore volume (Picket 1966). It was an important input parameter in formation evaluation, pore fluid identification and reserve estimation (Ferraris and Carnegie 2005; Liu and Hu 2010). Hence, accurately calculating water saturation has always been a major goal pursued by petrophysicists (Xiao et al. 2013, 2015). In conventional reservoirs, saturation evaluation problem had been overcome since the Archie’s equation was first raised in 1942 (Archie 1942). The pore structure, which was defined as pore size and connectivity, were relatively simple in such type of formation, and relationships between porosity and formation factor, water saturation versus resistivity index can be well expressed by power function (Guo et al. 2023; Gupta and Kamal 2023). The Archie’s equation had been widely used for a long time. However, the limitations and applicability of Archie’s equation were increasingly being revealed after complicated reservoirs were recovered, e.g., low resistivity contrast hydrocarbon-bearing reservoirs, low permeability to tight reservoirs (Mahmood et al. 2023; Qian et al. 2023). To improve water saturation prediction accuracy, many models had been proposed since 1950s (Simandoux 1963; Waxman and Smits 1968; Poupon and Leveaux 1971; Waxman and Thomas 1974; Clavier et al. 1984; Rasmus 1985; Givens 1987; Givens and Schmidt 1988; Song et al. 1995; Mao et al. 1999; Mo et al. 2001; Liu et al. 2009; Zhang et al. 2010; Li et al. 2012, 2020; Sarihi and Vargas-Murillo 2015; Hou et al. 2023). In low-resistivity contrast hydrocarbon-bearing reservoirs, several popular models were proposed to calculate water saturation after the genetic mechanism that decreased formation resistivity was considered. The Simandoux’s equation and Indonesia formula were used to calculate water saturation in shaly-rich sandstone reservoirs (Simandoux 1963; Poupon and Leveaux 1971). The Waxman-Smits’ equation and dual-water model were used in formations with strong additional conduction that caused by cation exchange (Waxman and Smits 1968; Waxman and Thomas 2007; Clavier et al. 1984). The Givens’ model was available in low resistivity contrast formation that caused by metal mineral conduction, and these metal minerals were contained in rock matrix (Givens 1987; Givens and Schmidt 1988). These models had played an irreplaceable role in water saturation prediction, and were still in use in clastic reservoirs at this stage. However, they were only usable in low resistivity contrast hydrocarbon-bearing formations. In tight sandstone reservoirs, the main factor that led to complex conductivity was pore structure (Xie et al. 2023). The other factors, e.g., high shale content, cation additional conduction, and the presence of conductive matrix, can be ignored. In recent years, the digital core technique has been widely used to simulate rock electrical properties (Liu et al. 2009; Walsh and Vogler 2020; Ammar 2021; Hou et al. 2023). Based on the digital core, the resistivity index that corresponded to low water saturation can be simulated to restore the complete conductivity. Digital core technique was well used in rocks with complicated pore structure, especially in double porosity media, e.g., volcanics and carbonatite (Ahmed et al. 2018). This technique was only used in laboratory to simulate rock electrical properties, it cannot be directly used in field applications at present.

To precisely calculate water saturation in field applications in tight sandstone reservoirs, many scholars raised novel models, such as three-water model, parallel conduction model (Mo et al. 2001; Zhang et al. 2010; Umezawa et al. 2021). Although these models were usable in tight sandstones, too many input parameters were involved, and difficult to be acquired. For example, the volumes of clay water and capillary moisture cannot be estimated from well logging except for nuclear magnetic resonance (NMR) logging (Coates et al. 2000; Dunn et al. 2002). To calculate water saturation in formation with complicated pore structure, many optimized models were raised. Liu et al. (2012) and Xiao et al. (2013) pointed out that pore structure heavily affected the conductivity in tight sandstone reservoirs, and derived a theoretical model to calculate cementation exponent from porosity. Meanwhile, Xiao et al. (2013) also raised a statistical model to predict saturation exponent from NMR logging to optimize the Archie’s equation. These proposed models had been verified to be usable in certain reservoirs (Gao et al. 2023; Li et al. 2023). However, the wide applicability was questionable. The conductivity in tight sandstone was dominantly determined by pore structure; whereas, cementation exponent was only predicted from total porosity (Mao et al. 1997; Xiao et al. 2013). In fact, porosity cannot be simply used to fully reflect rock pore structure. Many rocks contained high porosity but low permeability due to the contribution of poor pore connectivity (Mao et al. 2013). This made the raised model by Xiao et al. (2013) unable to work well in tight sandstone reservoirs. To establish a reasonable model that can be well used to predict water saturation, the pore structure should be first considered.

The purpose of this study was to establish models to optimize cementation exponent and saturation exponent after poor pore structure was considered in tight sandstone reservoirs. To establish these models, the Triassic Xujiahe Formation, which was located at the Northwestern Sichuan Basin, was chosen as an example. In these models, the change of cementation exponent and saturation exponent raised by pore structure were quantified, and models of predicting these two parameters were established to optimize Archie’s equation. Application of optimized Archie’s equation in our target tight gas sandstones illustrated that the calculated water saturation matched well with the core-derived results; whereas, the predicted water saturation was underestimated based on classic Archie’s equation once fixed cementation exponent and saturation exponent were directly used.

Effect of pore structure to electrical properties in tight sandstone reservoirs

Rock resistivity properties in our target tight sandstone reservoirs

Generally, the electrical properties of rocks can be expressed by using the Archie’s equation (Archie 1942). The Archie’s equation bridged the gap between porosity, resistivity, and water saturation, and can be expressed as follows:

$$ FF = \frac{{R_{0} }}{{R_{w} }} = \frac{a}{{\varphi^{m} }} ,$$
(1)
$$ Ir = \frac{{R_{t} }}{{R_{0} }} = \frac{1}{{S_{w}^{n} }} ,$$
(2)

where FF was the formation factor, and Ir was the resistivity index, they were all dimensionless. φ was the porosity in fraction, and Sw was the water saturation in fraction. R0 was the rock resistivity under fully water saturation, Rt was the rock resistivity under hydrocarbon saturated, and Rw was the formation water resistivity. The unit of them was Ω.m. a was a coefficient that related to lithology, m was the cementation exponent, and n was the saturation exponent. They were dimensionless. a, m and n were collectively referred to as rock resistivity parameters.

The Archie’s equation can be expressed by the crossplots of porosity and formation factor, water saturation versus resistivity index after sufficient core samples were drilled and applied for resistivity experiments in laboratory. Once rocks contained relative simply pore structure, crossplots of porosity and formation factor, water saturation versus resistivity index were regular, and fixed cementation exponent and saturation exponent can be acquired. However, in formations with complicated pore structure, simple power function cannot be used to express these relationships.

In our target Triassic Xujiahe Formation, 35 core samples were drilled for resistivity experiments in laboratory under simulated formation temperature and pressure environment, and crossplots of φ and FF, Sw versus Ir are acquired and displayed in Fig. 1. It notably illustrated that the relationships of φ and FF, Sw versus Ir cannot be simply expressed by power function, the trend lines were crooked. Although the correlation coefficient between Sw and Ir was high, the trendline was tortuous when Sw was higher than 75.0%. In addition, based on Fig. 1b, we could conclude that the resistivity index was equaled to 0.8969 when core samples were fully saturated with water (Sw was equaled to 1.0). This was not in line with theory and practice. Hence, no fixed cementation exponent and saturation exponent can be acquired from these core samples. If we compulsively use fixed parameters, unreliable water saturation would be calculated.

Fig. 1
figure 1

Crossplots of porosity versus formation factor (a), water saturation versus resistivity index (b) in our target tight sandstone reservoirs of Triassic Xujiahe Formation in northwestern Sichuan Basin

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Effect of pore structure to rock resistivity properties

To analyze the reason that caused the relationships between φ and FF, Sw versus Ir exhibited as non-Archie’s equation, we selected 18 out of 35 core samples to apply for nuclear magnetic resonance (NMR) and high-pressure mercury injection (HPMI) experiments in laboratory. Table 1 lists the experimental results of resistivity, NMR and HPMI for these 18 core samples. Porosity of these core samples only ranged from 4.48 to 12.29%, and permeability distributed from 0.023 to 0.298 mD. The selected 18 cores were representative, they can be used to reflect physical properties of our target Triassic Xujiahe Formation (Fig. 2c and d). Figure 2a and b indicates that our target tight sandstone reservoirs exhibited poor pore structure and weak rock physical properties, and thus weak relationship between core-derived porosity and permeability. This generated that the cementation exponent changed from 1.90 to 2.18, and the saturation exponent ranged from 1.78 to 2.14.

Table 1 Experimental results of resistivity, NMR and HPMI for 18 core samples that drilled from Triassic Xujiahe Formation in northwestern Sichuan Basin
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Fig. 2
figure 2

NMR T2 distributions (a), crossplot of core-derived porosity and permeability (b) for 18 core samples with resistivity experiments. Statistical histogram of porosity (c), and permeability (d) for all 528 core samples in Triassic Xujiahe Formation

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Relationship between pore structure and cementation exponent

Figure 3a displays the crossplot of porosity versus formation factor for two typical core samples. These two core samples contained similar porosity and permeability. However, their cementation exponent and saturation exponent were absolutely different. Rock resistivity parameters of core no. 1 were observably lower than that of core no. 6. Based on the macroscopic physical parameters, this difference cannot be clearly interpreted. However, the micro-pore structure can well reflect this variety of cementation exponent and saturation exponent. Although the pore structure of these two core samples was poor, the movable pore volume of no. 1 core sample was relatively higher.

Fig. 3
figure 3

Crossplot of porosity versus formation factor (a), NMR T2 distributions (b) and HPMI curves (c) for two typical core samples with different cementation exponent and saturation exponent

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Figure 3b and c displays the corresponding NMR T2 spectra and HPMI curves for these two core samples. The pore structure of no. 1 core sample was much superior. The pore space consisted of large pores that flow freely and small pores that cannot flow, and the pore volume of free fluid reached to 46.98%. However, the pore volume of no. 6 core sample dominated by irreducible water, and the irreducible water saturation was higher to 62.88%. Meanwhile, the median mercury injection pressure of no. 1 core sample was half that of no. 6 core sample.

To quantitatively reflect the effect of pore structure to cementation exponent, crossplots of pore structure parameters with cementation exponent for these 18 core samples are analyzed and displayed in Figure 4a–f, separately. Some regularity can be concluded as:

  1. a)

    Cementation exponents were various but not a fixed value in tight sandstone reservoirs, a uniform cementation exponent cannot be acquired to calculate water saturation by using Archie’s equation.

  2. b)

    Cementation exponent was relevant with rock pore structure. The better the pore structure, the higher the cementation exponent.

  3. c)

    Parameters that heavily associated with cementation exponent were porosity and irreducible water saturation (Swirr, reflecting the relative content of small pores). Although other parameters, e.g., logarithmic mean of transverse relaxation time (T2lm), T2 cutoff, the average pore throat radius (Rm), and median mercury injection pressure (P50), also positively related to cementation exponent, the correlation coefficients were low enough to be ignored. In addition, these parameters cannot be directly acquired. Pseudo-capillary pressure curves need to be first synthesized from NMR logging, some additional errors were introduced (Xiao et al. 2016).

Fig. 4
figure 4

Relationships between cementation exponent and pore structure parameters for 18 core samples

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Relationship between pore structure and saturation exponent

In Figure 5a, we displayed the crossplot of water saturation versus resistivity index for three typical core samples, and the saturation exponents of these core samples changed from 1.786 to 2.137. In Figure 5b and c, we displayed the corresponding NMR T2 distributions and HPMI curves. Comparisons of these three figures illustrated that saturation exponent was inversely proportional to pore structure. From no. 6 to no. 15 core sample, saturation exponent decreased, the main peak of NMR T2 spectra moved to the left, and the pore volume of free fluid also decreased. Meanwhile, the location of MPMI curves moved upward.

Fig. 5
figure 5

Crossplot of water saturation versus resistivity index (a), NMR T2 distributions (b) and HPMI curves (c) for three typical core samples

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Figure 6a–f displays the crossplots of saturation exponent versus pore structure parameters for all 18 core samples. These figures notably exhibited that the value of saturation exponent was inversely proportional to pore structure for our target tight sandstone reservoirs. Rocks with poor pore structure always contained relatively high saturation exponent, and vice versa. Hence, it was illogical to calculate water saturation by using fixed saturation exponent for all types of formations.

Fig. 6
figure 6

Relationships between saturation exponent and pore structure parameters for 18 core samples

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Models of predicting cementation exponent and saturation exponent from NMR logging

Establishment of cementation exponent and saturation exponent prediction models

Figures 4 and 6 clearly indicated that cementation exponent was associated with porosity and irreducible water saturation; whereas, saturation exponent was heavily affected by rock porosity and the ratio of large pore volume to small pore volume (defined by (100-Swirr)/Swirr)). Hence, to accurately predict water saturation, we chose porosity and irreducible water saturation as sensitive parameter to establish cementation exponent prediction model, and porosity (100-Swirr)/Swirr) was chosen to calculate saturation exponent.

Using resistivity experimental results of 18 core samples as the data set, various cementation exponent and saturation exponent prediction models were established and listed:

$$ m = 0.967 \times \log_{10} \left( \varphi \right) + 0.005 \times S_{wirr} ,\,R^{2} = 0.935 ,$$
(3)
$$ n = 0.003 \times \varphi^{2} - 0.027 \times \varphi - 0.155 \times \frac{{100 - S_{wirr} }}{{S_{wirr} }},\,R^{2} = 0.876, $$
(4)

where Swirr was the irreducible water saturation in %.

To use Eqs. 3 and 4 to calculate cementation exponent and saturation exponent, porosity and Swirr should be first acquired. Porosity can be easily accurately calculated by combining with NMR and conventional logging, e.g., density or acoustic logging (Freedman et al. 1998; Abu-Shanab et al. 2005; Jain et al. 2019). However, Swirr calculation was heavily depended on T2 cutoff. Xiao et al. (2018) pointed out that various T2 cutoff should be used to calculate Swirr in heterogeneous formation. However, the number of core samples was not enough, and T2 cutoff distribution for 18 core samples was relatively concentrated (Fig. 7). Hence, we used unified T2 cutoff (defined as the average value of T2 cutoff for all core samples) to calculate Swirr. In our study, the average T2 cutoff for 18 core samples was 10.38 ms. Hence, it was used to calculate Swirr from NMR logging by using Eq. 5:

$$ S_{wirr} = \frac{{\mathop \smallint \nolimits_{{T_{2, \min } }}^{10.38} S\left( T \right)dt}}{{\mathop \smallint \nolimits_{{T_{2, \min } }}^{{T_{2, \max } }} S\left( T \right)dt}}, $$
(5)

where T2,min and T2,max were the minimal and maximal transverse relaxation time, separately, and their unit was ms. S(T) was a function that associated with transverse relaxation time.

Fig. 7
figure 7

Statistical histogram of T2 cutoff for all 18 core samples. The average T2 cutoff was 10.38 ms, and it was defined as the optimal T2 cutoff to calculate Swirr from NMR logging

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Reliability verification

To verify the reliability of Eqs. 3 and 4, we extended them into field applications to process NMR logging data in wells A and B. Consecutive cementation exponent and saturation exponent curves were predicted. We extracted cementation exponent and saturation exponent that corresponded to the depths of core samples. In Fig. 8a and b, we compared predicted cementation exponent and saturation exponent from field NMR logging with core-derived results, separately. The red dotted lines in these two figures were the 450 diagonal lines, they were used to indicate the consistency between two types of data points. These two figures clearly indicated that the data points evenly distributed around diagonal lines, these meant that predicted cementation exponent and saturation exponent were all accurate. Further quantitative analysis showed that the absolute errors between predicted cementation exponent and core-derived results were lower than 0.05; whereas, the absolute errors between predicted saturation exponent and core-derived results can be controlled within ± 0.07. These absolute errors were low enough for water saturation estimation (Xiao et al. 2013).

Fig. 8
figure 8

Comparisons of measured cementation exponent (a) and saturation exponent (b) in laboratory with predicted results based on our raised optimized models

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Case studies

By using our raised models, several wells, which located in northwestern Sichuan Basin, were processed to acquire cementation exponent and saturation exponent curves from NMR logging, and they were substituted into Archie’s equation to optimize saturation estimation model. We consecutively predicted water saturation curves by using optimized Archie’s equation. To verify the superiority of our optimized models, we also calculated water saturation by using classic Archie’s equation, and the used rock resistivity parameters are fixed and extracted from Fig. 1. Figure 9 displays the comparison of predicted water saturations based on two different types of methods with core-derived results in one well. In this figure, conventional well logging curves were displayed in the first three tracks, and they were used to indicate effective reservoir, calculate porosity, and identify pore fluids, separately. In the fourth track, we displayed NMR T2 spectra (T2_DIST), which were acquired from Schlumberger’s CMR-Plus tool. The T2 spectra exhibited as typical bimodal to multimodal distribution, indicating complicated pore structure in our target tight sandstone reservoir. In the fifth and sixth tracks, formation porosity (POR) and permeability (PERM) were displayed, separately. They clearly indicated that formation porosity was lower than 14.0%, and permeability was lower than 1.0 mD. Comparisons of predicted cementation exponent (m) and saturation exponent (n) with core-derived results (Core_m and Core_n) were displayed in the seventh and eighth tracks, respectively. Good consistency between these two types of resistivity parameters illustrated that our raised models were reliable. In the ninth track, we compared predicted water saturations with core-derived results. In this track, Sw_Archie represented the calculated water saturation based on classic Archie’s equation, and used resistivity parameters were fixed. Sw_new was the predicted water saturation by using optimized Archie’s equation based on our raised models in this study; whereas, Core_Sw was core-derived water saturation, it represented the true value in our target formation. Comparisons illustrated that predicted water saturation based on our raised optimized models matched well with core-derived water saturation. Whereas calculated water saturation by using fixed resistivity parameters was all 3.0–5.0% lower than true water saturation. Our raised saturation optimized models were valuable in improving tight gas sandstone reservoirs characterization and reserve prediction.

Fig. 9
figure 9

Comparisons of predicted water saturations by using two methods with core-derived results in a well

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To quantize the accuracy of predicted water saturations by using two methods, we extracted calculated water saturation that corresponded to the depth of core samples. Figure 10 exhibits the crossplot of calculated water saturations by using classic Archie’s equation, optimized models, separately and core-derived results. The black dashed line was a 45° diagonal, it was used to quantize the accuracy. The red dotted lines were the ± 5% margin of error. This figure notably indicated that the predicted water saturations by using optimized models were close to core-derived results, and the absolute errors between them were lower than 5%. However, if we directly calculated water saturations by using classic Archie’s equation, they would be overestimated.

Fig. 10
figure 10

Comparisons of predicted water saturations based on two methods and core-derived results in our target tight gas sandstone reservoirs

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Conclusions

Water saturation estimation in tight gas sandstone reservoir was difficult by directly using classic Archie’s equation due to the effect of complicated pore structure. The resistivity parameters were heavily affected, this made crossplots of porosity and formation factor, water saturation versus resistivity index were divergent, and inaccurate water saturation should be calculated unless the input parameters were optimized.

Experimental results of 18 typical core samples illustrated that pore structure was directly proportional to cementation exponent, and inversely proportional to saturation exponent. This was to say, formations with better pore structure always contained high cementation and lower saturation exponent. Two parameters, which were porosity and irreducible water saturation, were sensitive to cementation exponent, and porosity and ratio of large pore volume to small pore volume ((100-Swirr)/Swirr)) were associated with saturation exponent. Two optimized models, which were used to calculate various cementation exponent and saturation exponent from NMR logging, were raised.

Field applications illustrated that the optimized models were valuable in predicting cementation exponent and saturation exponent. The absolute errors of them were lower than 0.05 and 0.07, separately. After optimized resistivity parameters were substituted into classic Archie’s equation, reasonable water saturation was precisely calculated; whereas, the predicted water saturation by using fixed resistivity parameters were overestimated.