1 Introduction

Shaly sandstone reservoirs are widely distributed and rich in reserves, which have been of great importance in the exploration and development of oil and gas (Salem 2000; El-Sayed 2020). However, because of the major differences in shale distribution forms and shale content in shaly sandstone, the resistivity of shaly sandstone is complicated (Merritt et al. 2016; Li et al. 2017; Bai et al. 2019; Zhao et al. 2020), which makes the recognization of the reservoir and the accurate calculation of water saturation are extremely difficult (Soleymanzadeh et al. 2021).

Based on the shale distribution forms, Schlumberger Ltd (1989) divides shale in shaly sandstone into laminated, structural, and dispersed shale. Different forms of shale distribution result in a significant difference in the resistivity of sandstone, which influences the accurate calculation of water saturation of shaly sandstone. Many researchers have proposed universal water saturation equations or ones customized for specific shale distribution forms (Cai et al. 2017; Dong et al. 2019), and some equations have good results in some cases. Based on resistivity experiments of sandstone samples, Archie (1942) proposed the classical Archie equation. However, this equation only applies to clean sandstone rather than shaly sandstone (Oraby 2021; Han et al. 2021). Afterwards, based on the assumption that the conductive network of shaly sandstone is consist of pores and shale, DeWitte (1950) deduced the conduction model of dispersed shaly sandstone. Poupon et al. (1954) simplified shaly sandstone into the thin interbeds of sandstone and shale and proposed a water saturation equation for idealized laminated shaly sandstone. However, it is rare to see such idealized laminated shaly sandstone in oil or gas fields. Therefore, this equation has not been widely used around the world. Based on the experiments of artificial shaly sandstone, Simandoux (1963) proposed a water saturation equation for shaly sandstone, known as the Simandoux equation. This equation has been widely used in the world, but it did not consider the distribution form of shale in shaly sandstone. Based on the statistic results of Oil Fields in Indonesia, Poupon and Leveaux (1971) deduced a water saturation equation of laminated shaly sandstone, namely, the Indonesian equation, which has been proven accurate in some Oil Fields around the world. In practice, these water saturation equations might be effective for one type of shaly sandstone but not applicable to others. This is mainly because that shale in shaly sandstone is not in an ideally laminated, structural or dispersed form but in more complicated distribution forms. For example, laminated shale might have different connectivity or tortuosity; structural shale might coexist with dispersed shale, and etc. Complicated shale distribution forms suggest more complex conductive patterns of shaly sandstone, which makes these water saturation equations less universal.

In recent years, with the rapid development of digital rock and simulation of rock physics (Yue et al. 2004; Wu et al. 2018; Andhumoudine et al. 2021; Liu et al. 2021, 2022; Li et al. 2022; Wang et al. 2022), many scholars start to use numerical simulations in the research of micro-scale conductivity mechanism and water saturation calculation of rocks, which have made some desirable progress. Yan et al. (2018) simulated the resistivity of sandstone with digital rock technology. Results show that when there are no cracks, the low resistivity of sandstone is mainly influenced by the porosity, clay content, clay form, salinity of stratum water, temperature, and wettability. Li et al. (2021) simulated the resistivity of sandstone and analyzed the micro-scale conductive pattern of sandstone. Results show that the resistivity of sandstone is heavily influenced by the particle size, particle sorting, and complexity of pores and throats. Based on the characteristics of the lamellar throat and considering the pore radius, throat radius, throat tortuosity and coordination number of pores, Wu et al. (2020) simulated the resistivity and cementation index (m) of tight sandstone. The simulation results coincide with the experimental results well. These studies have laid a solid foundation for the conductive simulation-based water saturation calculation in rocks.

By taking low resistivity laminated shaly sandstone in Zhujiang Formation of Neogene in Pearl River Mouth Basin as an example, this research extracts the shale distribution form and pore structures from microimages, simulates the resistivity of rock with finite element method and deduces water saturation equation, to accurately calculate the water saturation of laminated shaly sandstone. The proposed method in this research perfectly integrates the resistivity simulation of rock and the derivation of the water saturation equation, which fully corresponds to the water saturation equation with the shale distribution types of shaly sandstone. This provides a new mindset to calculate the water saturation of shaly sandstone.

2 Methods

2.1 Experiments

This research includes four types of experiments: micro-CT scan, X-ray diffraction (XRD) of mineral composition analysis, scanning electron microscope (SEM) observation, and casting thin section observation. Micro-CT scan is conducted by an Xradia MICROXCT-400 reconstruction scanner of Zeiss, and the results are used to analyze the shale distribution types in shaly sandstone. XRD of mineral composition analysis is conducted by a X’Pert-PRO-MPD diffractometer of PANalytical and the results are used to analyze the composition and content of minerals in shaly sandstone. The SEM observation is conducted by an FEI Quanta 650 FEG scanning electron microscope of Thermo Fisher Scientific. The casting thin section observation is conducted by a DM2500P polarizing microscope of Leica, and the results are used for image processing and the construction of conduction models.

2.2 Conduction model

Microimages of shaly sandstone can be obtained from the casting thin sections (Fig. 1a, d and h). The shale distribution form and pore structure of shaly sandstone can be obtained from image processing. Based on the distribution characteristics of shale in shaly sandstone, three types of micro-scale conduction models (conduction models of clean sandstone area, realistic shaly sandstone, and abstract shaly sandstone) were constructed (Fig. 1).

Fig. 1
figure 1

Construction process of three types of conduction models of shaly sandstone

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2.2.1 Conduction model of clean sandstone area

In order to construct conduction models of clean sandstone area, at first, 690 μm × 517 μm-sized microimages of clean sandstone area (Fig. 1d) in casting thin section images were selected. Then, by using image processing methods (such as the threshold method, Weszka and Rosenfeld 1978; Zhang 2019), the pore structures of the clean sandstone area were extracted, and the conduction models of fully water saturated clean sandstone area were constructed (Fig. 1e). Finally, by changing the ratio of oil and water in pores by morphological algorithm, the conduction models of clean sandstone area with different water saturations were constructed (Fig. 1f). The simulated resistivity results of conduction models of clean sandstone area will be input as the resistivity of clean sandstone area in the realistic conduction model and abstract shaly sandstone of shaly sandstone.

2.2.2 Realistic conduction model of shaly sandstone

In order to construct realistic models of shaly sandstone, at first, 9300 μm × 7000 μm-sized microimages which contain both clean sandstone area and shale area in casting thin section images were selected (Fig. 1a). Then, by using the image processing methods (such as the edge extraction method, Shiozaki 1986), the distribution structures of the sandstone area and the shale area were extracted, and the realistic conduction models of shaly sandstone were constructed (Fig. 1b and g).

2.2.3 Abstract conduction model of shaly sandstone

To construct abstract conduction models of shaly sandstone, based on analyzing the distribution features of shale area in 9300 μm × 7000 μm-sized microimages of shaly sandstone, the abstract conduction models were constructed (Fig. 1c). Then, by changing shale parameters including shale content, tortuosity of shale laminae, and homogeneity of shale laminae, a series of abstract conduction models were constructed (Fig. 1i, j and k). The definitions of tortuosity and homogeneity of shale laminae are shown in Eq. 1 and Eq. 2 (Fig. 1j and k).

$$\tau =\frac{l}{L}$$
(1)

where \(\tau\) is the tortuosity of shale laminae (dimensionless); l is the realistic length of shale laminae (µm); L is the linear distance of the two sides of shale laminae (µm).

$$s=\frac{h}{H}$$
(2)

where S is the homogeneity of shale laminae (dimensionless); h is the minimum thickness of shale laminae (µm); H is the maximum thickness of shale laminae (µm).

2.3 Methods of conductivity simulation

In this research, the AC/DC module in COMSOL Multiphysics software is used to simulate the resistivity of shaly sandstone with the finite element method. After importing the aforementioned conduction models into COMSOL Multiphysics software, assign values to the materials (assign resistivity to pore fluid and rock matrix), generate mesh, and insert constant voltage to both sides of the conduction models to form a return circuit. Then the electrical potential and the current density distribution of the conduction models were simulated in a constant electric field,. Based on the continuity principle of electric current, the integral of the current density was calculated and the total current was obtained. Finally, the resitivities of the conduction models were calculated using Ohm’s law.

2.3.1 Electric field simulation

For a constant electric field, Maxwell’s equations can be represented by Eq. 3. Meanwhile, based on the law of charge conservation, the current density of the constant electric field also satisfies Eq. 4 (Bird et al. 2014; Chen et al. 2016; Wu et al. 2022).

$$\left\{\begin{array}{c}\nabla \cdot E=\frac{\rho }{{\epsilon }_{0}}\\ \nabla \times E=0\end{array}\right.$$
(3)

where E is the electric field strength (V/m); ρ is the electric charge density (C/m); ε0 is the vacuum permittivity (F/m), which is around 8.8542 × 10− 12 F/m.

$$\nabla \cdot \varvec{J}=0$$
(4)

where J is the current density (A/m2).

2.3.2 Resistivity calculation

When the shape and size of the micro-scale conduction models of the rock are certain values, they exert electric potential in direction x, which is parallel to the direction of shale laminae. Based on the law of conservation of charge, in a constant electric field, currents that go through any yz section (vertical to shale laminae) are equal. Therefore, electric current (I) can be calculated by solving the integrals of current density on any yz section (Eq. 5). Based on Ohm’s law (Miller et al. 2015; He et al. 2018), substitute electric current (I) into Eq. 6, the resistivity of conduction models can be calculated.

$$I=\underset{{S}_{c}}{\overset{}{\iint }}{J}_{n}\cdot \text{d}{S}_{c}$$
(5)

where n is the direction x, y or z; I is the electric current (A); \({S}_{c}\)is the cross-sectional area of the conduction model (which is perpendicular to the direction of electric potential) (µm2)

$$R=r\frac{{S}_{c}}{L}=\frac{{{U}_{1}-U}_{0}}{I}\times \frac{{S}_{yz}}{{l}_{x}}$$
(6)

where R is the resistivity of the conduction model (Ω m); U1, U0 is the electric potential of input and output, respectively (V); r is the resistance (Ω); lx is the length of the conduction model (which is parallel to the exertion direction of the potential) (µm).

3 Geological setting and samples

3.1 Geological setting

Oil Field W13-2 is located in the middle of the Qionghai Uplift in the Pearl River Mouth Basin. This is a low-relief anticlinal structure. There are two reservoirs with distinct resistivities in Zhujiang Formation of Oil Field W13-2.

  1. (1)

    High resistivity sandstone reservoirs. The stratum below layer ZJ1-4 is categorized as tidal flats of coastal sedimentation. The reservoirs are mainly developed in the sand bar face of near shores. The rock type is fine sandstone with a porosity of 28.0–42.1% (36.7% on average) and a permeability of 100.0 mD to 4432.1 mD (896.4 mD on average). The main features of these reservoirs are low shale content and high resistivity.

  2. (2)

    Low resistivity sandstone reservoirs. The stratum between layer ZJ1-1 to layer ZJ1-4 is defined as a shallow sea deposit. The reservoirs are mainly developed in transitional belt facies. The rock type is shaly sandstone with a porosity of 24.0–35.5% (27.7% on average) and permeability of 0.01 mD to 100 mD (54.6 mD on average). The main characteristics of these reservoirs are high shale content and low resistivity.

3.2 Samples

This research mainly focuses on the low resistivity shaly sandstone reservoirs between layer ZJ1-1 to layer ZJ1-4. Rock samples were mainly drilled from 1050 to 1300 m underground. Among these rock samples, 2 are used for micro-CT scans, 18 are used for XRD of mineral compositon analysis, 4 are used for scanning electron microscope (SEM) observation, and 18 of them are used for casting thin section observation.

4 Experiment results

4.1 Mineral composition

The mineral composition of the low resistivity shaly sandstone reservoir mainly contains quartz with a content of 47.0–75.0% (65.5% on average), then clay minerals with a content of 15.0–35.2% (24.0% on average). The content of calcite, dolomite, and other minerals is low. The low resistivity reservoirs have higher content of clay than the high resistivity reservoirs (Fig. 2a). The clay minerals of low resistivity reservoir are mainly illite/smectite mixed layers with a relative content of 53.2%, followed by kaolinite (17.6%) and illite (16.3%). The chlorite content is relatively low, which is 12.9% (Fig. 2b). The SEM microimages shows that the flake-shaped clay minerals (flake-shaped illite/smectite mixed layers, flake-shaped illite, and flake-shaped chlorite) cover the surface of grains and tend to form a large number of micropores (Fig. 2c, d and e). The clay minerals can adsorb stratum water and form a complete conductive network in rock.

Fig. 2
figure 2

Mineral compositon and SEM images of clay minerals

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4.2 Distribution form of shale

Research shows that there are three forms of shale distribution in shaly sandstone: laminated, structural, and dispersed shale (Schlumberger Ltd, 1989; El-Sayed 2020). The shale distribution form in Oil Field W13-2 is similar to the laminated shale, and there are significant differences between them, especially in the shape and scale of the shale laminae. In the scale of thin section images, shaly sandstone in Oil Field W13-2 can be divided into clean sandstone area and shale area (Fig. 1a, d and h). The clean sandstone area has a good pore structure and the capacity to reserve oil and gas (Fig. 1d). Meanwhile, the shale area has a very high content of shale and no reservable pores. The distribution direction of the shale area is roughly consistent with the formation boundary (Fig. 1h). The shale area in the shaly sandstone of Oil Field W13-2 is mainly distributed as irregular laminae (Fig. 3). This irregular laminated shale is significantly different from the ideal laminated shale, which mainly reflects in the tortuosity and homogeneity of shale.

Fig. 3
figure 3

Micro-CT images showing distribution form of clean sandstone area and shale area in shaly sandstone

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5 Simulation results

5.1 Current density of conduction model of clean sandstone area

This research selected 4 images of casting thin sections with the size of 690 μm × 517 μm, which were used to extract the pore structures of the clean sandstone area. Sixteen conduction models of clean sandstone area under 4 different water saturations were constructed. In these models, pore fluids include stratum water and oil. The stratum water analysis data in Oil Field W13-2 shows that the stratum water type is CaCl2, with a salinity of 33,000 mg/l. At a temperature of 74 °C, the resistivity of stratum water is 0.088 Ω m and the conductivity is 11.364 S/m. Quartz and oil have extremely high resistivity and can be regarded as insulators. The electric current density results of the conduction models of the clean sandstone area show that the current density in the stratum water occupied pore space is high (color from yellow to red) (Fig. 4). The more concentrated lines of current density show a better conductivity. Meanwhile, the current density in the oil occupied pore space and quartz is low (color blue). This phenomenon shows that the conductivity of the model of a clean sandstone area mainly relies on the stratum water in pores. After obtaining the current density distribution of conduction models of sandstone area through simulation, the resistivity of the conduction models of clean sandstone area under different water saturations (R0) (Table 1) are calculated through Eq. 5 and Eq. 6. Assign R0 as the resistivity of the clean sandstone area under different water saturations (Rsd) in the conduction models of realistic shaly sandstone and of abstract shaly sandstone.

Fig. 4
figure 4

Current density distribution of conduction models of clean sandstone area

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Table 1 Simulated resistivity results of conduction models of clean sandstone area under different water saturations
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5.2 Current density of realistic conduction model of shaly sandstone

This research selected 4 microimages of casting thin section of different shale contents with the size of 9300 μm×7000 μm, which were used to extract the distribution of the clean sandstone area and shale area. Twelve realistic conduction models of shaly sandstone under three different water saturations were constructed. There are two conducting materials in these models, the clean sandstone area and the shale area. The resistivity of the clean sandstone area is assigned by the simulation results of the conduction models of the clean sandstone area (Table 1), and the resistivity of the shale area is obtained by counting the resistivities of the shale layer in Oil Field W13-2. The resistivity of shale (Rsh) is from 1.17 Ω m to 1.46 Ω m, which is 1.29 Ω m on average. Namely, the conductivity of shale (σsh) is 0.775 S/m. The electric current density of the realistic model of shaly sandstone area shows that (Fig. 5), with a higher water saturation (Sw ≥ 80%), the current density in the clean sandstone area is higher than that in the shale area, and the conductivity of shaly sandstone mainly relies on the stratum water in clean sandstone area (Fig. 5b, c, f, g, j and k). Meanwhile, with relatively lower water saturation (Sw ≤ 60%), the current density in the shale area is higher than that in the clean sandstone area, and the conductivity of shaly sandstone mainly relies on the irreducible water in the shale area (Fig. 5d, h, and l).

Fig. 5
figure 5

Current density distribution of realistic conduction models of shaly sandstone

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5.3 Current density of abstract conduction model of shaly sandstone

This research constructed 41 abstract conduction models of shaly sandstone with the size of 9300 μm × 7000 μm to quantitatively analyze the influence of various shale parameters on the resistivity of shaly sandstone. These models consider various shale contents (Vsh = 10%, 15%, 20%, 25%, and 30%), tortuosity of shale (τ = 1.00, 1.01, 1.05, 1.10, 1.17, 1.28, and 1.38), and homogeneity of shale (S = 0, 0.07, 0.14, 0.3, and 0.6). Like conduction models of the realistic shaly sandstone, there are two conducting materials (the clean sandstone area and the shale area) in these models. Moreover, the settings of resistivity of these two conducting materials in these models are consistent with the setting in the realistic conduction models of shaly sandstone. The electric current density of the abstract models shows that (Figs. 6 and 7), with higher water saturation (Sw ≥ 80%), the current density in the clean sandstone area is higher than that in the shale area, and the conductivity of shaly sandstone mainly relies on the stratum water in clean sandstone area (Fig. 6a, b, e, f, i and j). Meanwhile, with relatively lower water saturation (Sw ≤ 60%), the current density in the shale area is higher than that in the clean sandstone area, and the conductivity of shaly sandstone mainly relies on the irreducible water in the shale area (Fig. 6c, d, g, h, k and l).

Fig. 6
figure 6

Current density distribution of abstract conduction models of shaly sandstone (variable content and tortuosity of shale)

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Fig. 7
figure 7

Current density distribution of abstract conduction models of shaly sandstone (variable homogeneity of shale)

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With the changes in tortuosity of shale laminae (τ), there are no significant changes in electric current density in the abstract models (Fig. 6). However, with the changes of homogeneity of shale laminae (such as S = 0.33, S = 0.07, and S = 0), the electric current density of the abstract models changes in narrower areas of shale laminae (Fig. 7).

5.4 Comparison of simulated resistivity and well logging

The simulated resistivity results of the realistic model show that (Fig. 8a): (1) With a lower shale content (Vsh≤12%), there is a significant difference between the resistivity of the oil layer, oil-water layer, and water layer. The resistivities of oil layers are the highest (2 Ω m ≤ R < 5 Ω m), which is followed by the oil-water layers (1.2 Ω m ≤ R < 2 Ω m), while the resistivities of water layers are the lowest (R < 1.2 Ω m). These reservoirs belong to high resistivity reservoirs. (2) With a higher shale content (Vsh > 12%), the resistivities of the oil layers (1.2 Ω m ≤ R < 2 Ω m) and resistivities of the water layers (R < 1.2 Ω m) are very close. Moreover, shale content increases gradually from 12 to 36%, and the difference between the resistivities of oil layers and water layers decreases. These reservoirs are known as low resistivity reservoirs.

Fig. 8
figure 8

Comparison of simulated resistivity of realistic conduction models and resistivity of well logging

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Well logging data of Oil Field W13-2 shows that (Fig. 8b and c): (1) Reservoir B is a high resistivity oil layer with a shale content of 9.7% and a resistivity of 3.20 Ω m (Fig. 8b). Reservoir C is a water layer with a shale content of 8.8% and a resistivity of 0.81 Ω m (Fig. 8b). In Fig. 8a, reservoir B and reservoir C are categorized as high resistivity oil and water layers, respectively. The resistivities from well logging data of reservoirs B and C correspond with the trendline of the simulated resistivity. (2) Reservoir A is a low resistivity oil layer with a shale content of 29.1% and a resistivity of 1.28 Ω m. Reservoir D is a water layer with a shale content of 19.5% and a resistivity of 0.75 Ω m (Fig. 8b and c). In Fig. 8a, reservoir A and reservoir D are categorized as low resistivity oil layer and water layer respectively. The resistivities from well logging data of reservoirs A and D correspond with the trendline of the simulated resistivity. A comparison of the simulated resistivity and resistivity from well logging data shows that the irregular laminae distribution and high shale content is the primary cause of the low resistivity oil layers in Oil Field W13-2.

5.5 Influence of shale parameters on resistivity

The cross plots of shale parameters (shale content, tortuosity of shale laminae, and homogeneity of shale laminae) versus simulated resistivity from the abstract models of shaly sandstone are drawn to analyze the influence of shale parameters on the resistivity of shaly sandstone (Figs. 9 and 10).

Fig. 9
figure 9

Cross plots of simulated resistivity (R*) versus shale content (Vsh)

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Fig. 10
figure 10

Cross plots of simulated resistivity (R*) versus tortuosity of shale laminae (τ), and homogeneity of shale laminae (S)

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  1. (1)

    Influence of shale content (Vsh): With a higher water saturation (Sw ≥ 80%, water layer), there is a minor influence of shale content on the resistivity of sandstone, and the resistivity remains below 1.1 Ω m. Meanwhile, with lower water saturation (Sw ≤ 60%,), there is a major influence of shale content on the resistivity of sandstone, and the resistivity of sandstone significantly decreases when shale content increases (Fig. 9a–f).

  2. (2)

    Influence of tortuosity of shale laminae (τ): With a lower shale content (Vsh ≤ 15%), the tortuosityτof shale laminae have minor influences on the resistivity of sandstone (Fig. 10a and b). Meanwhile, with a higher shale content (Vsh ≥ 20%), the resistivity of sandstone decreases to some extent as the tortuosity of shale laminae increases (Fig. 10c, d and e).

  3. (3)

    Influence of homogeneity of shale laminae (S): With a higher water saturation (Sw ≥ 80%, water layer), no significant changes are observed in the resistivity of sandstone when the homogeneity of shale laminae increases. Meanwhile, with lower water saturation (Sw ≤ 60%, oil layer), the resistivity of sandstone increases when the homogeneity of shale laminae increases (Fig. 10f).

6 Water saturation calculation

6.1 Simulation based water saturation equation

Due to the high shale content and low resistivity of shaly sandstone reservoir in Oil Field W-12, the commonly used Archie equation cannot accurately represent the conductive chracteristics or calculate the water saturation of the reservoir. Previous numerical simulation results show that the realistic conduction model (Fig. 11b, c and e) and the idealized paralleled conduction model (Fig. 11a and d) are similar but remains apparent differences. The differences mainly lie in the tortuosity and homogeneity of shale laminae of the conduction models. This research uses variation coefficient (ξ) to represent the ratio of resistivity of the realistic conduction model (Rt) with the resistivity of idealized paralleled conduction model (Rparal) (Eq. 7). In the research area, the tortuosity of shale laminae (τ) is around 1.1, and the homogeneity of shale laminae (S) is around 0.2. Therefore, the variation coefficient (ξ) can be calculated from the simulated resistivity under these conditions (Fig. 10f). Variation coefficient (ξ) is positively correlated to the water saturation (Sw) (Fig. 11f and Eq. 8).

Fig. 11
figure 11

Comparision of the realistic conduction model and the idealized paralleled conduction model

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$$\xi ={R}_{t}/{R}_{paral}$$
(7)
$$\xi =0.3{S}_{w}+0.78$$
(8)

For the idealized paralleled conduction model, assume the side length of the cube represented by the rock as L, the cross-sectional area as A = L2 and the volume as V = L3; and assume the length of the shale laminae as L, the cross-sectional area as Ash, the volume as Bsh = L×Ash, the relative volume as Csh = Bsh/V = Ash/L2 and the resistivity as Rsh; assume the length of the sandstone area as L, cross-sectional area as Asd, volume as Bsd = L×Asd, relative volume as Csd = Bsd/V = Asd/L2, resistivity as Rsd and porosity as φsd. When the electric currents flow to the formation in the vertical direction of the cross section, Eq. 9 can be obtained based on the principle of paralleled conduction. Multiply L2 by both sides of equation Eq. 9, and then Eq. 10 can be deduced, which leads to the idealized paralleled conduction expression (Eq. 11).

$$\frac{A}{L\times {R}_{paral}}=\frac{{A}_{sh}}{{L\times R}_{sh}}+\frac{{A}_{sd}}{L\times {R}_{sd}}$$
(9)
$$\frac{1}{{R}_{paral}}=\frac{{C}_{sh}}{{R}_{sh}}+\frac{1-{C}_{sh}}{{R}_{sd}}$$
(10)
$${R}_{paral}=\frac{{R}_{sh}{R}_{sd}}{{R}_{sd}{C}_{sh}+{R}_{sh}\left(1-{C}_{sh}\right)}$$
(11)

where Rparal is the resistivity of the idealized paralleled conduction model (Ω m).

Combine Eq. 7, Eq. 8, and Eq. 11, then Eq. 12 is obtained.

$${R}_{t}=\left(0.3{S}_{w}+0.78\right)\frac{{R}_{sh}{R}_{sd}}{{R}_{sd}{C}_{sh}+{R}_{sh}\left(1-{C}_{sh}\right)}$$
(12)

where \({R}_{t}\) is the resistivity of the realistic conduction model of shaly sandstone (Ω m), Sw is the water saturation (fraction).

Based on the image processing results of microimages of thin sections, the conversion relationship between the relative volume of shale laminae (Csd) and the shale content (Vsh) is obtained (Eq. 13). In clean sandstone area of the realistic conduction model, there is no shale, grains are well sorted, and the pores are well connected. Therefore, the Archie equation can be applied (Eq. 14). Because the porosity of the shale area (\({\phi }_{sh}\)) is 0, the effective porosity of shaly sandstone (\(\phi\)) can be represented by Eq. 15. In the clean sandstone area of the realistic conduction model, assign m = n = 2, a = b = 1, and combine Eq. 14 and Eq. 15, then Eq. 16 can be deduced.

$${C}_{sh}=1.43{V}_{sh}-0.01$$
(13)

where \({V}_{sh}\) is the shale content (fraction).

$${S}_{w}=\sqrt[n]{\frac{ab{R}_{w}}{{{\phi }_{sd}^{m}R}_{sd}}}$$
(14)

where \({R}_{w}\) is the resistivity of stratum water (Ω.m); m is the cementation exponent, n is the saturation exponent, a and b are coefficients, \({\varphi }_{sd}\) is the porosity of the clean sandstone area (fraction).

$$\phi ={\phi }_{sd}(1-{V}_{sh})$$
(15)
$${R}_{sd}=\frac{{R}_{w}{\left(1-{V}_{sh}\right)}^{2}}{{S}_{w}^{2}{\phi }^{2}}$$
(16)

When Eq. 13 and Eq. 16 are substituted into Eq. 12, the water saturation equation of shaly sandstone is obtained (Eq. 17).

$$\begin{array}{c}{S}_{w}=\frac{0.15{R}_{w}{\left(1-{V}_{sh}\right)}^{2}}{\left({1.43 V}_{sh}-1.01\right){\phi }^{2}}+ \\ \frac{1-{V}_{sh}}{2(1.43{V}_{sh}-1.01)\phi }\sqrt{\frac{0.09{R}_{w}^{2}{\left(1-{V}_{sh}\right)}^{2}}{{\varphi }^{2}{R}_{t}^{2}}-4{R}_{w}\left(1.43{V}_{sh}-1.01\right)\left(\frac{0.78}{{R}_{t}}-\frac{1.43{V}_{sh}-0.01}{{R}_{sh}}\right)}\end{array}$$
(17)

6.2 Validation of water saturation equation

To validate the proposed water saturation equation of shaly sandstone, the results of the proposed equation were compared with three commonly used equations, namely Archie (Archie 1942), Simandoux (Simandoux 1963), and Indonesian equation (Poupon and Leveaux 1971), and were compared with the water saturation of the sealed core samples. In these equations, the resistivity of shale is obtained from the average value of the well logging resistivities of shale layers (Rsh = 1.3 Ω m), and the resistivity of the water saturation is obtained from the stratum water analysis data (Rw = 0.088 Ω m). Comparison of the calculation results of water saturation of well A1 in Oil Field W13-2 shows that (Fig. 12): (1) For high resistivity reservoirs between intervals 1223 to 1235 m, there are only minor differences among the calculation results of the four water saturation equations, and all the calculated water saturations fit the water saturation of sealed core samples well. (2) For low resistivity reservoirs between interval 1115 to 1145 m, differences are considerably large among the calculation results of the four water saturation equations. The water saturation of the Archie equation is evidently lower than that of the sealed core samples, followed by the Simandoux equation and Indonesian equation, and the water saturation of the proposed equation fits that of the sealed core samples best. The cross plots of the water saturation of the sealed core samples versus the water saturation calculated from the four equations also show that the water saturation of the proposed equation correlates to the water saturation of the sealed core samples best, illustrating that the proposed equation is more applicable to the low resistivity shaly sandstone reservoirs in Oil Field W13-2 (Fig. 13). The proposed resistivity simulation-based water saturation calculation method provides a new mindset to evaluate the water saturation of rocks.

Fig. 12
figure 12

Comparison of calculation results of water saturation of well A1 in Oil Field W13-2

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Fig. 13
figure 13

Cross plots of water saturation of sealed core samples versus water saturation calculated from four equations

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7 Conclusion

  1. (1)

    The shale content of low resistivity shaly sandstone reservoir in Oil Field W13-2 is high, from 15.0 to 38.2% with an average of 25.1%. In the shaly sandstone, shale is distributed in irregular laminae (with a scale of a millimeter), and the shale content of the shale laminae is high. However, in other neighboring areas (with a scale of a millimeter), the shale content is low and the porosity is high. These areas can reserve oil and gas.

  2. (2)

    At a high water saturation state (i.e. water layer), the electrical conduction of shaly sandstone mainly relies on the clean sandstone area. Various shale parameters have minor influences on the resistivity of shaly sandstone. At a low water saturation state (i.e. oil layer), the electrical conduction of shaly sandstone mainly relies on the shale laminae. The resistivity of the reservoir is low and only has an extremely small difference from the water layer. In other words, this reservoir is a low resistivity oil layer.

  3. (3)

    Shale content and homogeneity of shale laminae impact the resistivity of shaly sandstone the most, while the tortuosity of shale laminae has a minor influence. Therefore, the irregular laminae distribution and high shale content is the leading cause of the low resistivity oil layers in Oil Field W13-2.

  4. (4)

    In the low resistivity oil layers of Oil Field W13-2, the conductive form of the clean sandstone area and shale laminae tend to parallel but remain obvious differences from the total parallel. The simulation results deduced that the water saturation equation of shaly sandstone are more accurate than other equations, providing an innovative mindset to calculate the water saturation of shaly sandstone.