Aguilera and Aguilera (2003) published rigorous equations for dual porosity systems that were shown to be valid for all combinations of matrix and fractures or matrix and nonconnected pores. The model used in this paper is a combination of matrix and fractures, the model is composed of a rock skeleton, matrix pores, and fractures. The pore structure of rock determines the conductive path of rock, suppose the additional shale conductivity is not considered (core experiments of reservoirs in multiple research blocks show that the cation exchange capacity is minimal and can be ignored). In that case, the conductivity of matrix pores and fractured water is the same, but the conductive paths are different. Therefore, the rock is conductive in parallel through the matrix pore network and the fracture network. The model of the dual porosity reservoirs is shown in Figure 4, the matrix pores and fractures constitute a very complex fracture-pore system. Matrix porosity is the ratio of matrix pore volume to total volume which is calcuiated using Equation 1. ϕ b = V b V (1) where ϕ _b is the matrix porosity (%); V _b is the volume of matrix pores (cm³); V is the total volume of the composite system (cm³).

Rock fracture porosity is the ratio of matrix pore volume to total volume which is calculated using Equation 2. ϕ f = V f V (2) where ϕ _f is the fracture porosity (%); V _f is the volume of fracture (cm³).

Total porosity of rock is calculated by Equation 3. ϕ = V f + V b V (3) where ϕ is the total porosity (%).

According to the theory of dual porosity medium, the pore space of a tight sandstone reservoir is divided into matrix pore space and fracture pore space. According to the definition of water saturation, the total water saturation can be expressed as Equation 4. S w = ϕ b S w b + ϕ f S w f ϕ b + ϕ f (4) where ϕ _b is the matrix porosity (%); S _wb is the matrix water saturation (%); ϕ _f is fracture porosity (%); S _wf is fracture water saturation (%).

Under the background of fractures and non-connected pores, the physical properties of the matrix of the reservoir are relatively uniform. Therefore, when calculating the matrix pore saturation, we use the classical Archie formula (Archie, 1942), and the Archie formula is shown in Equation 5. S w b = a b R w ϕ b m b R t n b (5) where R _w is the formation water resistivity ( Ω ⋅ m ); R _t is the formation resistivity ( Ω ⋅ m ); a, b, m _b , n _b are the rock electrical parameters of the matrix, which are determined by rock resistivity experiments. The rock samples contain virtually no fractures, so the m and n obtained from the rock resistivity experiment are regarded as the m _b and n _b of the rock matrix in this paper. Subsequently, this paper will enhance the calculation approach of rock electrical parameters in accordance with the actual circumstances of the reservoir, thereby enabling the saturation calculation to be more rational and precise.

According to the permeability characteristics of the dual porosity, Tang et al. (2018) and Wang (2020) calculated the water saturation of fracture using Equation 6. S w f = 1 / R t − 1 / R x o + 1 / R m f ϕ f m f 1 / R w ϕ f m f n f (6) where R _xo is the resistivity of the flushing zone ( Ω ⋅ m ); R _mf is the resistivity of mud filtrate ( Ω ⋅ m ); m _f and n _f are the electrical parameters of fractures.

Shale is dispersedly filled in the intergranular pore space of sandstone, which is conductive in parallel with the water of formation. When the reservoir contains shale, the resistivity of the reservoir will decrease, and the accuracy of the traditional method for calculating saturation will also decrease. There are many saturation models involving the influence of shale. The Indonesian formula is a famous formula in formation evaluation by well logs, as shown in Equation 7. 1 S w n = V s h c c R s h + ϕ a R w 2 R t (7) where V _sh is the shale content (%); R _sh is the resistivity of the shale ( Ω ⋅ m ); cc = 1-V _sh /2. This formula is suitable for low-formation water salinity and shaly sandstone with V _sh less than 50%. This formula better solves the saturation calculation of shaly sandstone formation.

The research area belongs to fractured tight sandstone reservoirs. Fractures developed in rocks enhance the productivity of oil and gas reservoirs but also complicate the evaluation of oil and gas saturation. When calculating the oil and gas saturation of fractured tight sandstone reservoirs, factors such as physical properties, fractures, and lithology should be considered. For such reservoirs, this paper proposes a saturation evaluation method considering pore types and shale content (Figure 5). Specifically, when the shale content exceeds 20%, the Indonesian formula is used to calculate saturation. If the shale content is less than 20%, the dual porosity model is used to calculate saturation.

Usually, intersection processing of multiple well logging methods is used to obtain accurate total porosity of the reservoirs. Compared with other well-logging methods, density and compensated neutron log responses are less affected by the heterogeneity of fractured reservoirs (Yong and Zhang, 1986). Therefore, we use the intersection of density and neutron logs to obtain the total porosity of the reservoir. The specific method is shown in Equations 8–10. ϕ = p o r d s c 2 + p o r n s c 2 2 (8) p o r d s c = ρ b − ρ m a ρ f − ρ m a − V s h ρ s h − ρ m a ρ f − ρ m a (9) p o r n s c = ϕ N − ϕ m a ϕ N f − ϕ m a − V s h ϕ N s h − ϕ m a ϕ N f − ϕ m a (10) where ρ s h , ρ m a , ρ f are the density logging values of mud, skeleton, and fluid, respectively (g/cm³); ϕ s h , ϕ m a , ϕ f are the neutron logging values of shale, rock matrix, and pore fluid, respectively (%).

Acoustic logging data basically do not reflect the fracture porosity but mainly reflect the matrix porosity of the rocks. The matrix porosity of the reservoirs is calculated by the acoustic volume model. The calculation formula is shown in Equation 11. ϕ b = Δ t − Δ t m a Δ t f − Δ t m a − V s h Δ t s h − Δ t m a Δ t f − Δ t m a (11) where Δ t s h , Δ t m a , Δ t f are the acoustic time difference logging values of shale, rock matrix, and pore fluid, respectively (μs/ft).

In reservoir evaluation, imaging logging, and dual lateral logging data are usually used to calculate fracture porosity (Zhao et al., 2012). In this paper, the fractures are picked manually based on the intuitive interpretation of FMI (Formation Micro Scanner Image), and the electrical imaging data are calibrated using the optical scanning imaging data of the cores. Finally, the fracture parameters, such as fracture porosity, were calculated (Chen and Tan, 2003; Lai et al., 2015).

Accurate model parameters m _b and n _b are the key to the quantitative evaluation of saturation. The porosity and the microscopic pore structure of the reservoirs affect the parameters of models (Ding et al., 2017). Capillary pressure measurement and nuclear magnetic resonance (NMR) measurement are the main methods to describe the microscopic pore structure of the reservoirs. In this paper, the standard NMR T₂ spectrum is selected to characterize the microscopic pore structure. Rock electrical and nuclear magnetic resonance experiments were carried out on 10 tight sandstone samples from Ahe Formation. The m _b value and n _b value of each core are obtained by rock electricity experiment, and the NMR T₂ spectrum of each core is obtained by the nuclear magnetic resonance experiment. Figure 6 shows the standard T₂ spectra of six tight sandstone samples. Six rock samples are saturated with NaCl solution, and the resistivity of NaCl solution is 0.138 Ω ⋅ m . The position and shape of the T₂ spectrum of the three rock samples are basically the same (Figure 6A), which indicates that the microscopic pore distributions of the three rock samples are similar. The amplitude and envelope area of the T₂ spectra of the three rock samples are different, which indicates that the porosity of the three rock samples is different. Under the condition that the formation water resistivity and micro-pore structure are basically the same, the parameters m _b and n _b are different when the porosity of the three rock samples is different. The results show that the change of porosity will lead to the change of m _b and n _b values. The position and shape of the T₂ spectra of the three rock samples are different, but the envelope area is the same (Figure 6B), indicating that the porosity of the three rock samples is the same, but the microscopic pore structure is different. Under the condition that formation water resistivity and porosity are basically the same, the parameters m _b and n _b are different if the microscopic pore structure of the three rock samples is different. The results show that the microscopic pore structure affects the values of m _b and n _b .

On the basis of determining the factors affecting the electrical properties of tight sandstone, it is necessary to determine the response relationship between the parameters and pore structure so as to provide a basis for the accurate calculation of saturation. Through the above analysis, it can be concluded that porosity and pore structure have a significant influence on the model parameters of rocks. The logarithmic mean of the NMR T₂ spectrum is usually used to describe the variation in the reservoir’s microscopic pore structure (Zhang and Shi, 2005; Guo et al., 2022). The larger the logarithmic mean value of T₂ (T_2lm), the larger the pore size of the reservoirs. And the smaller the T_2lm, the smaller the pore size of the reservoirs. Figure 7A shows the relationship between mb value and porosity ϕ of core analysis. It can be seen from the Figure that the m _b value and the porosity present a power function relationship, and the m _b value increases with the increases of porosity ϕ. Figure 7B shows the relationship between the m _b value of the core analysis and the T_2lm values. The statistical results show that the m _b value first increases with the increase of the T_2lm value, and then the m _b value decreases with the increase of the T_2lm value.

Figure 8A shows the relationship between the n _b value of core analysis and porosity ϕ. The value of n _b increases with the increase of the porosity ϕ and presents a power function relationship. Figure 8B shows the relationship between the n _b value of the core analysis and the T _2lm value. The statistical results show that as the T _2lm value increases, the n _b value decreases firstly, and then increases.

The experimental results show that the correlation between model parameters and any single factor is not particularly high. To reduce the error of single actor regression calculation, m _b and n _b values are calculated using the multi-factor fitting regression method. Multifactor regression considers the complex interaction between multiple parameters and reflects the actual properties of rocks more accurately. According to the results of the rock resistivity and nuclear magnetic resonance (NMR) experiment, porosity and T₂ logarithmic mean value are selected as influencing factors, and the calculation model of m _b value and n _b value is established by using multiple regression methods. The model is shown in Equations 12, 13. m b = 1.401 ϕ b 0.1524 − 0.0004826 T 2 l m 2 + 0.004525 T 2 l m (12) n b = 4.447 ϕ b − 0.3701 + 0.0002246 T 2 l m 2 − 0.01689 T 2 l m (13) where m _b and n _b are the rock electrical parameters of the matrix; ϕ _b is the porosity of the matrix (%); T _2lm is the logarithmic mean of T₂ (ms).

Figure 9 shows the calculation results of the model. The m _b and n _b values calculated by the multiple linear regression method have a reasonable correlation with the m and n values measured by the experiment. The coefficient of determination for the m _b value is 0.8377, and for the n _b value is 0.7041, indicating that the calculation accuracy of this method is high, and porosity and pore structure are important factors affecting the model parameters of rocks.

In order to obtain electrical parameters for fractures, Tang et al. (2018) and Wang (2020) assume that the rock is a regular hexahedron with side length l, a diagonal fracture with angle β and width d _f passes through the rock, and a regular hexahedron hole with side length c develops inside the rock. According to the theory, the formula for calculating the parameters of fractured rock is shown in Equation 14. m f = − log l 1 c + d f + l − c ⁡ cos ⁡ β l ⁡ cos ⁡ β d f + c l − c d f log l 2 cos ⁡ β − c 2 d f + c 3 / l 3 (14) where d _f is the fracture width (mm); β is the fracture angle (dega). Substituted into Equations 5, 6, the matrix saturation and fracture saturation are calculated, and the total water saturation of the reservoir is calculated by Equation 4.

For the n _f , there is still no method to determine it. We assume it is equal to n _b in this paper.