Three-Water Differential Parallel Conductivity Saturation Model of Low-Permeability Tight Oil and Gas Reservoirs

Abstract

Addressing the poor performance of existing logging saturation models in low-permeability tight sandstone reservoirs and the challenges in determining model parameters, this study investigates the pore structure and fluid occurrence state of such reservoirs through petrophysical experiments and digital rock visualization simulations. The aim is to uncover new insights into fluid occurrence state and electrical conduction properties and subsequently develop a low-permeability tight sandstone reservoir saturation model with easily determinable parameters. This model is suitable for practical oilfield exploration and development applications with high evaluation accuracy. The research findings reveal that such reservoirs comprise three types of formation water: strongly bound water, weakly bound water, and free water. These types are found in non-connected micropores, poorly connected mesopores where fluid flow occurs when the pressure differential exceeds the critical value, and well-connected macropores. Furthermore, the three types of formation water demonstrate variations in their electrical conduction contributions. By inversely solving rock electrical experiment data, it was determined that for a single sample, the overall cementation index is the highest, followed by the cementation index of pore throats containing strongly bound water, and the lowest for the pore throats with free water. Building on the aforementioned insights, this study develops a parallel electrical pore cementation index term, ϕ^m′, to account for the differences among the three types of water and introduces a parallel electrical saturation model suitable for logging evaluation of low-permeability tight oil and gas reservoirs. This model demonstrated positive application effects in the logging evaluation of low-permeability tight gas reservoirs in a specific basin in the Chinese offshore area, thereby confirming the advantages of its application.

1. Introduction

In the 1940s, Archie proposed a power function relationship between formation factors and porosity, as well as the resistivity increase factor and water saturation, based on resistivity experiments and data analysis of water-saturated pure sandstone [1]. This seminal work established the fundamental basis for evaluating water saturation in well logging.

With the increasing complexity of geological conditions in field operations, the Archie equation is no longer suitable for evaluating a range of unconventional reservoirs, including shale, low-permeability tight sandstone, conglomerate, carbonate, and natural gas hydrates. The saturation models commonly employed in logging evaluation today represent new considerations integrated into the Archie equation. They can be broadly classified into four types: empirical models considering mud conductivity, models incorporating the cation exchange capacity of clay minerals, saturation models based on effective medium theory, and models considering the complex pore structure and the impact of multiple pores. In recent years, it has been observed in the field practice of unconventional oil and gas reservoir logging evaluation that saturation models accounting for the impact of multiple pores have demonstrated significant practical utility and have emerged as a focal point of research.

Zeng and Liu conducted a systematic analysis of the formation mechanism of low-resistivity oil and gas reservoirs and the additional conductivity effect of clay and proposed a dual-water model within a dual-porosity system [2]. Li et al. summarized existing saturation models, the formation mechanism of low-resistivity oil and gas reservoirs, and the factors influencing electrical properties and put forward a dual-porosity saturation model [3]. Building upon the heterogeneity and anisotropy of the formation and network conductivity theory, Li proposed a general relationship between the increase in resistance and water saturation. Under different formation conditions, the applicable saturation model can be determined by experimentally determining the undetermined parameters in the relationship and fitting methods. Recent developments have demonstrated that this model has also yielded favorable results in the logging evaluation of natural gas hydrate reservoirs [4,5,6]. However, the aforementioned models have limitations in their effectiveness in complex porous reservoirs, as their complexity makes them challenging to apply in actual oilfield operations and in determining model parameters, leading to poor application effectiveness in practical logging interpretation and evaluation in oilfields. Hence, this paper uses the example of a low-permeability tight sandstone gas field in the South China Sea to propose a three-water differential parallel conduction model suitable for low-permeability tight sandstone oil and gas reservoirs. The objective is to achieve precise logging water-saturation evaluation in low-permeability tight oil and gas reservoirs and support exploration work in actual oilfield operations.

The complex geological processes in low-permeability tight sandstone reservoirs have resulted in a complex pore structure characterized by the disappearance of primary pores and the prevalence of dissolution pores. As a result of this complex pore structure, traditional saturation models are ineffective for evaluating logging saturation in low-permeability tight sandstone reservoirs [7,8,9,10,11]. The saturation model based on the rock physics volume model utilizes the theory of multiple connected pores for electrical conduction, essentially taking into account the electrical conductivity of water in the formation. Therefore, through the study of pore structure characteristics and fluid distribution in low-permeability tight gas reservoirs, the differences in the electrical conduction properties of formation fluids in different types of pores can be clarified, thereby revealing the electrical conduction properties of low-permeability tight oil and gas reservoirs and establishing a highly necessary high-precision logging saturation model.

Numerous scholars have extensively researched the occurrence state of fluids in tight gas reservoirs. Zhu et al. proposed a model wherein movable water occurs in large pores, while residual water is retained in small pores, through gas-driven water-saturation measurements on saturated water cores in conjunction with nuclear magnetic resonance (NMR) [12,13]. Ye et al. calibrated the relationship between nuclear magnetic resonance (NMR) movable water after centrifugation at 300 psi and the dynamic production of tight gas wells and suggested that movable water saturation can effectively characterize the water mobility in tight sandstone [14]. Gao et al. proposed the microscopic occurrence state of four types of formation water in tight gas reservoirs and derived corresponding cutoff values for three pore throat radii based on nuclear magnetic resonance (NMR) experiments under different pressure gas-driven water conditions [15]. The aforementioned scholars have gained a wealth of new insights into the occurrence state of fluids in tight reservoirs through nuclear magnetic resonance (NMR) experiments on rocks under varying water-saturation states. Furthermore, through field practices in oilfields, they have gathered evidence demonstrating that nuclear magnetic resonance (NMR) experiments are an effective method for studying reservoir pore structure and fluid occurrence state. However, the previous qualitative understanding of fluid occurrence in tight reservoirs has not been visually or quantitatively confirmed at the micro-scale level, and the quantitative characterization of fluids in different occurrence states is still outstanding. Therefore, conducting multi-level centrifugal nuclear magnetic resonance experiments and integrating micro-visualization simulation techniques to further investigate the occurrence state of reservoir fluids and achieve the quantitative characterization of fluids in different occurrence states represent the challenges and pivotal aspects in the study of low-permeability tight sandstone reservoirs.

2. Investigation of Fluid Occurrence

This study investigates the occurrence state of fluids in low-permeability tight sandstone reservoirs in the South China Sea region by conducting multi-level centrifugal force nuclear magnetic resonance experiments on core samples.

The MesoMR23-060H-I nuclear magnetic resonance imaging analyzer, produced by NIUMAG analytical instrument corporation, is employed for the multi-level centrifugal force nuclear magnetic resonance experiment. This instrument is primarily utilized for core nuclear magnetic resonance experiments, encompassing one-dimensional nuclear magnetic T₁, T₂ spectrum measurement, two-dimensional T₁-T₂ spectrum measurement, and nuclear magnetic resonance imaging measurement. The specific experimental procedures consist of six steps:

(1) Core saturation involves washing the core with oil, rinsing with salt, and vacuum-pressure saturation with water;

(2) Calibration of standard sample adjustment parameters is conducted using standard samples for the nuclear magnetic instrument, with standardized measurement parameters, including T_W, T_E, NECH, and others;

(3) The nuclear magnetic measurement of saturated water cores utilizes standardized measurement parameters to conduct nuclear magnetic resonance measurement on the core, yielding a one-dimensional nuclear magnetic T₂ spectrum;

(4) Centrifugation entails using a centrifuge to spin the core, resulting in 8 different core states with bound water at 50/100/150/200/250/300/400/500 psi, respectively;

(5) Post-centrifugation core nuclear magnetic measurement uses standardized measurement parameters to perform nuclear magnetic resonance measurement on the post-centrifugation core, obtaining a one-dimensional nuclear magnetic T₂ spectrum;

(6) Calculation of bound water saturation involves integrating the nuclear magnetic T₂ spectrum of the saturated water core and the post-centrifugation core to derive the T₂ cutoff value. The ratio of the area less than the T₂ cutoff value to the entire saturated water T₂ spectrum area represents the core-bound water saturation.

Figure 1 depicts the findings of a multi-level centrifugal force nuclear magnetic resonance experiment carried out on 6 samples from the South China Sea region. As the physical properties of samples 1 to 6 gradually deteriorate, the nuclear magnetic T₂ spectrum signals of less than 3 ms under saturated water conditions gradually strengthen, while the signals greater than 3 ms gradually weaken. Additionally, with the increase in centrifugal force, all core samples exhibit varying degrees of attenuation in the nuclear magnetic T₂ spectrum signals, highlighting differences in the pore structure and fluid properties of cores under distinct physical conditions. These experimental procedures enable the determination of bound water saturation for the 6 samples under different centrifugal forces, as well as the summarization of the basic physical parameters and changes in bound water saturation under different centrifugal forces, as detailed in

Table 1. Physical property parameters and bound water saturation under different centrifugal forces for six samples.

Sample	Lithology	Well	Depth m	Porosity %	Permeability mD	Pore Structure Coefficient $\sqrt{\mathit{K}\mathbf{/}\mathit{\varphi }}$	Bound Water Saturation/%
							50 psi	100 psi	200 psi	300 psi	500 psi
1	Fine sandstone	A	3760.0	12.12	35.56	1.713	33.44	27.89	18.94	18.12	13.23
2		B	3443.2	8.0	0.8	0.316	46.30	38.12	34.50	27.89	25.87
3		A	3717.3	8.7	0.84	0.311	62.04	49.39	37.75	26.04	22.10
4		B	3459.38	7.87	0.491	0.250	48.88	34.06	26.91	25.78	23.96
5	Siltstone	A	3699.0	5.11	0.11	0.147	78.25	70.01	60.16	50.68	47.62
6		A	3738.7	5.32	0.004	0.027	89.72	86.50	82.93	71.24	65.06

below.

Based on the data presented in Figure 1 and Table 1, a significant correlation between bound water saturation and rock properties is evident. Utilizing the pore structure coefficient $\sqrt{K/\varphi }$ to depict the intricacy of the rock’s pore structure reveals that as the pore structure coefficient decreases for cores 1 to 6, the pore structure becomes more intricate. Subsequent to centrifugation, the nuclear magnetic bound water saturation exhibits an increasing trend, signifying distinctions in fluid properties for rocks with varying pore structures. Consequently, utilizing multi-stage centrifugal force nuclear magnetic resonance experiments, the fluid occurrence state of reservoirs with diverse physical properties and pore structures can be examined.

The operational mechanism of the QEMSCAN 650F rock analysis system (FEI Company, Hillsboro, OR, USA) is to acquire the distribution morphology and relative content of diverse components within rock samples via backscattered electron (BSE) imaging and mineral identification, employing color coding to depict pores and distinct minerals. Considering the ratios of distinct components, pores, quartz, potassium feldspar, sodium feldspar, kaolinite, illite, and chlorite are recognized as the primary constituents, with an image resolution of 2 µm. Within this research, mineral scanning was performed on samples 3 and 4 to acquire high-resolution two-dimensional mineral scanning images, and mathematical morphology techniques were employed to simulate the distribution of gas and water phases in the two samples at water-saturation levels of 100%, 50%, and 25%, as depicted in Figure 2. Within the image, the pink component symbolizes the quartz framework, the blue component signifies formation water, and the red component denotes natural gas, with an image resolution of 2 mm × 2 mm.

Figure 2a,d depict simulations of the component distribution in two fully water-saturated low-permeability tight sandstone rocks. The figures reveal that the primary pore types in the low-permeability tight sandstone rocks consist of intergranular pores formed by late-stage diagenetic dissolution, residual intergranular pores, and a small number of mold pores. The prevailing pore sizes are medium to small, with a notable occurrence of dissolved micropores and ultra-micropores, indicating overall poor physical properties and complex pore structures. In Figure 2b,e, simulations are conducted to illustrate the component distribution at a water saturation of 50%. The simulations demonstrate that gas predominantly occupies the well-connected macropores, while a certain quantity of formation water is retained in the poorly connected mesopores and the non-connected micropores. The simulations in Figure 2c,f illustrate the component distribution at a water saturation of 25%. It is evident that the previously poorly connected mesopores are mostly displaced by gas, with only a certain quantity of bound water retained in the non-connected micropores and ultra-micropores.

Building upon the visual simulation and integrating the results of multi-level centrifugal nuclear magnetic resonance experiments, additional research was undertaken to analyze the state of fluid occurrence in low-permeability tight sandstone rocks. Figure 3 illustrates a comparison between the nuclear magnetic resonance T₂ spectra of cores under varying centrifugal forces and the simulated distribution of gas–water two-phase at different saturations.

Figure 3 illustrates a notable attenuation in the nuclear magnetic resonance T₂ spectrum signal, transitioning from the state of fully saturated water to a centrifugal force of 100 psi. The diminished nuclear magnetic resonance T₂ spectrum signal corresponds to the green segment in the figure, predominantly indicating the fluid signal in macropores with T₂ values surpassing the cutoff. This suggests that, under a centrifugal force of 100 psi, the water component existing in well-connected macropores is selectively displaced, aligning with the formation water in well-connected macropores, which is given priority for displacement by gas in the visual simulation. As this portion of water can flow freely under a relatively minor displacement pressure differential, it is classified as free water.

With further increase in the displacement centrifugal force, the nuclear magnetic resonance T₂ spectrum signal with T₂ values surpassing the cutoff exhibits no notable attenuation. The nuclear magnetic spectrum signal within the T₂ values falling within the cutoff continues to diminish as the centrifugal force increases, aligning with the blue segment in Figure 3. This suggests that with the increase in centrifugal force, the mobility of formation water in poorly connected mesopores gradually enhances, aligning with the formation water present in poorly connected mesopores in the visual simulation. Taking into account the substantial influence of the production flow pressure differential on the mobility of this water component, when the displacement pressure differential is lower than the initiation pressure gradient, this portion of water remains in an immobile bound state. Consequently, this water component is classified as weakly bound water.

The overall shape of the nuclear magnetic resonance (NMR) T₂ spectrum ceases to change as the displacement centrifugal force reaches approximately 300 psi. At this stage, the weakly bound water existing in mesopores has been displaced, and the residual T₂ spectrum signals are predominantly distributed to the left of the T₂ cutoff value, corresponding to the red segment of the T₂ spectrum in Figure 3, representing fluid signals within the micropores and inter-crystalline ultra-micropores. Additionally, there are still a small number of T₂ spectrum signals exceeding the T₂ cutoff value. Owing to the frequently observed strong hydrophilic characteristics of low-permeability tight sandstone reservoirs, there is consistently a certain thickness of water film on the surface of the rock matrix particles. Consequently, it is believed that this portion of T₂ spectrum signals exceeding the T₂ cutoff value indicates the presence of the water film signal. Since the water in the micropores, inter-crystalline ultra-micropores, and rock matrix particle surfaces remains immobile, it is classified as strongly bound water.

Following multi-level centrifugal force nuclear magnetic resonance and visual simulation studies, it is proposed that low-permeability tight sandstone reservoirs comprise three types of formation water: strongly bound water, weakly bound water, and free water. The alterations in the NMR T₂ spectrum due to varying centrifugal forces during displacement mirror the variations in fluid distribution within the core. The centrifugal force associated with the initial decay of the free water signal in the NMR T₂ spectrum is denoted as the weak capillary pressure P₁, whereas the centrifugal force at which the NMR T₂ spectrum signal decays to the extent that only the strongly bound water signal persists is labeled as the strong capillary pressure P₂.

Through processing and calculating the overlapping nuclear magnetic T₂ spectra under varying centrifugal displacements, the calibration of strong and weak capillary pressures is accomplished. The nuclear magnetic T₂ spectra are partitioned into two segments based on the T₂ cutoff value, with separate calibration of the strong capillary pressure P₂ and weak capillary pressure P₁. By processing and calculating the nuclear magnetic data, the ratio of T₂ spectra signals between neighboring centrifugal forces is determined, reflecting the attenuation of nuclear magnetic T₂ spectra signals under neighboring centrifugal forces. If the signal attenuation is below 5%~10% of the saturated water-rock T₂ spectra signal, the strong and weak capillary pressures of the reservoir in the study area can be calibrated. Due to the probability of errors in calibrating the critical capillary pressure through a single rock core, through calculating and analyzing the multi-level centrifugal magnetic resonance experimental data of 205 low-permeability tight sandstone cores in a specific region of the South China Sea, the mean values of strong and weak capillary pressures for all rock samples are derived. Ultimately, the weak capillary pressure P₁ for this region is calibrated at 100 psi, while the strong capillary pressure P₂ is set at 300 psi.

The power function equation for fitting pore structure parameters ($\sqrt{K/\varphi }$) and the total bound water under the influence of centrifugal forces P₁ and P₂ is established, as depicted in Figure 4, yielding the definitive models for total bound water saturation (S_w1) and strongly bound water saturation (S_w2) for the South China Sea region, as illustrated in Equations (1) and (2).

$$S_{w1}=-0.1236Ln(\sqrt{K/\varphi })+0.4438$$

(1)

$$S_{w2}=0.281{\sqrt{K/\varphi }}^{-0.243}$$

(2)

In the equation, S_w1 represents the total bound water-saturation model in a certain area of the South China Sea; S_w2 represents the calculation model of strongly bound water saturation in a certain area of the South China Sea; K represents the rock sample permeability in mD; and ϕ represents the porosity in percentage.

By using the models for strongly bound water and total bound water saturation, the calculation of the three water-saturation components of a core with known porosity and permeability parameters can be achieved according to Equations (3)–(5), that is, the calculation of the saturation of strongly bound water, weakly bound water, and free water in a single saturated water rock sample.

$$S_{sb}=S_{w2}$$

(3)

$$S_{wb}=S_{w1}-S_{w2}$$

(4)

$$S_{fm}=S_w-S_{w1}$$

(5)

In the equation, S_w represents the water saturation; S_sb represents the saturation of strongly bound water; S_wb represents the saturation of weakly bound water; and S_fm represents the saturation of free water.

3. Investigation of Electrical Conduction Properties

The detailed subdivision of three water-saturation components in the core formed the basis for analyzing the relationship between the electrical conductivity of water-saturated rocks and the three water-saturation components in the comparative study, aiming to investigate the individual contributions of each component to the electrical conductivity of water-saturated rocks.

Electrical experiments on the rocks were conducted at room temperature (25 °C) using a sodium chloride solution with a salinity of 8000 ppm and an electrical resistivity of 0.696 Ω·m. Subsequently, electrical resistivity measurements were carried out after saturating 46 cores with a pressurized sodium chloride solution. The electrical conductivity of the 46 water-saturated cores was then compared with the three-water saturation of the same 46 cores. Figure 5 illustrates the relationships between the electrical conductivity of water-saturated rocks and strongly bound water saturation, weakly bound water saturation, and free water saturation, respectively.

Figure 5 reveals a negative correlation between the electrical conductivity of water-saturated rocks and the levels of strongly bound and weakly bound water saturation. Specifically, as the levels of strongly bound and weakly bound water saturation increase, the rock’s electrical conductivity gradually decreases. This phenomenon deviates from the traditional belief that higher reservoir bound water saturation corresponds to stronger rock electrical conductivity. Upon analysis, it is suggested that the poor physical properties and complex pore structures commonly found in low-permeability tight sandstone rock reservoirs lead to the approximation of the physical volume model of the rock, consisting of a large amount of rock matrix and a small number of pore components. The electrical conductivity of the reservoir is determined by the formation water present in the small number of pores. Even when the bound water saturation of the reservoir is high, the electrical conductivity is still limited by the pore volume, as the absolute content of conductive bound water is too low. Therefore, the traditional viewpoint that high bound water saturation corresponds to high electrical conductivity does not apply to low-permeability tight sandstone rock reservoirs.

An analysis of Figure 5 also reveals a positive correlation between the electrical conductivity of water-saturated rocks and free water saturation. With increasing free water saturation, which corresponds to improved physical properties and pore structures of the reservoir, the electrical conductivity of water-saturated rocks also increases.

An analysis of the intersection patterns of the three-water saturation and electrical conductivity of water-saturated rocks leads to the following conclusions regarding the electrical conduction characteristics of low-permeability tight sandstone rock reservoirs: higher levels of strongly bound and weakly bound water saturation correspond to lower electrical conductivity of water-saturated rocks, while the level of free water saturation determines the magnitude of the rock’s electrical conductivity. The electrical conductivity of low-permeability tight sandstone rock reservoirs depends not only on the level of bound water saturation but also on the absolute content of conductive formation water. Therefore, the traditional understanding that high bound water saturation corresponds to high electrical conductivity does not apply to low-permeability tight sandstone rock reservoirs.

4. Establishment of Saturation Model

Prior research indicates that formation conductivity is the result of various components, including pore water, wetting water film, clay, and conductive minerals [16,17,18,19,20,21,22]. Building on earlier studies of fluid occurrence in low-permeability tight sandstone rock reservoirs, it is suggested that these reservoirs contain three distinct types of pores: strongly bound water pores, weakly bound water pores, and free water pores. Examination of the contribution of electrical conductivity has shown that the three types of formation water have varying impacts on the rock’s electrical conductivity.

This study proposes that the differences in the electrical properties of the three types of formation water stem from variances in the pore structures of the corresponding pore spaces. Furthermore, the electrical properties of the three types of pore spaces all follow Archie’s law. Therefore, a rock physical volume model is suggested, considering the parallel differences of the three types of formation water, as shown in Figure 6. The electrical conductivity of low-permeability tight sandstone rock reservoirs (C_t-tr) is derived from the parallel combination of the resistivity of the strongly bound water, weakly bound water, and free water, as depicted in Equation (6).

By utilizing Archie’s equation (Equations (7) and (8)) to express the electrical conductivity of the strongly bound water, weakly bound water, and free water in Equation (6) and assigning the corresponding cementation and saturation indices to the three types of pores, we derive the comprehensive enhancement of the parallel electrical conductivity of low-permeability tight sandstone rock reservoirs with varying formation water, as illustrated in Equation (9). Further exploration of the formation conductivity equation under complete saturation conditions, where the saturation of the three types of pores in Equation (9) is 100%, results in the precise expression of the electrical conductivity of low-permeability tight sandstone rock reservoirs under full saturation, as depicted in Equation (10).

$$C_{\mathit{\text{t-tr}}}=C_{fm}+C_{wb}+C_{sb}$$

(6)

$$F=C_w/C_o=a/{\varphi }^m$$

(7)

$$I=C_o/C_t=b/{S_w}^n$$

(8)

$$C_{\mathit{\text{t-tr}}}=({{\varphi }_{fm}}^{m_{fm}}{S_{wfm}}^{n_{fm}}+{{\varphi }_{wb}}^{m_{wb}}{S_{wwb}}^{n_{wb}}+{{\varphi }_{sb}}^{m_{sb}}{S_{wsb}}^{n_{sb}})C_w/ab$$

(9)

$$C_{\mathit{\text{o-tr}}}=({{\varphi }_{fm}}^{m_{fm}}+{{\varphi }_{wb}}^{m_{wb}}+{{\varphi }_{sb}}^{m_{sb}})C_w/ab$$

(10)

In the equation, C_t-tr represents the conductivity of low-permeability tight sandstone reservoirs in S/m; C_sb represents the conductivity of strongly bound water in S/m; C_wb represents the conductivity of weakly bound water in S/m; C_fm represents the conductivity of free water in S/m; F represents the factor of pure sandstone formation; C_w represents the formation water conductivity in S/m; C_o-tr represents the conductivity when the rock is fully saturated with water in S/m; C_o represents the conductivity of fully saturated pure sandstone in S/m; a/b is the lithology coefficient; ϕ is the rock porosity; S_w is the water saturation; m is the cementation index; n is the saturation exponent; ϕ_fm is the porosity of free water; m_fm is the cementation index of free water porosity; n_fm is the saturation exponent of free water porosity; S_wfm is the water saturation of free water porosity; ϕ_wb is the porosity of weakly bound water; m_wb is the cementation index of weakly bound water porosity; n_wb is the saturation exponent of weakly bound water porosity; and S_wwb is the water saturation of weakly bound water porosity.

In accordance with Archie’s first equation (Equation (7)), in the case of pure, clay-free, and 100% water-saturated sandstone, its resistivity exhibits a direct proportionality to the resistivity of the pore water, with the proportionality coefficient termed as the formation factor F. The formation factor F exhibits a power function relationship with the porosity of pure sandstone. Nevertheless, the intricate pore structure and lithology in low-permeability tight sandstone rock reservoirs render the traditional Archie’s equation inapplicable. Consequently, the formation factor F′ for low-permeability tight sandstone rock reservoirs is defined as the ratio of the pore water conductivity C_w to the conductivity of the fully saturated low-permeability tight sandstone rock reservoir C_o-tr. This derivation leads to the establishment of the cementation index term ϕ^m′, accounting for the variances in the electrical properties of the three types of pores, as depicted in Equation (11). Subsequently, ϕ^m′ is defined as the cementation index term, taking into consideration the parallel electrical conductivity of the three types of formation water.

$${\varphi }^{m\prime }=a\frac{C_{\mathit{\text{o-tr}}}}{C_w}=\frac{{{\varphi }_{fm}}^{m_{fm}}+{{\varphi }_{wb}}^{m_{wb}}+{{\varphi }_{sb}}^{m_{sb}}}{b}=\frac{{\varphi }^{m_{fm}}{(1-S_{w1})}^{m_{fm}}+{\varphi }^{m_{wb}}{(S_{w1}-S_{w2})}^{m_{wb}}+{\varphi }^{m_{sb}}{S_{w2}}^{m_{sb}}}{b}$$

(11)

In the equation, ϕ^m′ represents the porosity cementation index term considering the parallel conduction of three water differences; S_w1 is the total bound water saturation; and S_w2 is the strong bound water saturation.

When evaluating saturation in low-permeability tight sandstone rock reservoirs, the accurate determination of water saturation is significantly influenced by complex lithology and intricate pore structures. Previous research indicates that the predominant lithology in low-permeability tight sandstone rock reservoirs is fine sandstone, often containing numerous siltstone reservoirs with generally high clay content. The additional conductive contribution of clay has a substantial impact on the accuracy of saturation evaluation from well logging. Hence, the clay component in actual low-permeability tight sandstone rock reservoirs cannot be disregarded. Simandous considered the conductive ability of clay to be approximately equal to that of siltstone and, based on the field exploration experience of mudstone, set the saturation exponent (n) for siltstone and the clay component to 1, thereby constructing a saturation model applicable for considering the conductive nature of clay in sand–mudstone [23], as depicted in Equation (12).

$$S_w={(\frac{a{R}_w{R}_{sh}}{a{R}_w{V}_{sh}{R}_t+R_{sh}{\varphi }^m{R}_t})}^{1/n}$$

(12)

In the equation, R_w represents the formation water resistivity, Ω·m; R_sh represents the shale resistivity, Ω·m; R_t represents the formation resistivity, Ω·m; and V_sh represents the shale volume.

In the course of exploration and development activities in the South China Sea Basin, the Simandous equation has been effectively utilized to assess water saturation in conventional gas reservoirs and some low-permeability gas reservoirs, yielding positive outcomes. However, its application in low-permeability tight sandstone rock reservoirs with poor physical properties and complex pore structures has not been optimal. Therefore, considering the conductive contribution of clay in the original equation, the parallel cementation index term ϕ^m′, accounting for differences in the three types of water conductance, replaces the pure sandstone cementation index term ϕ^m in the Simandous equation. This adjustment eliminates the influence of vertical pore structure differences on the accuracy of well-logging water-saturation evaluation, resulting in a three-water differential parallel conductive saturation model suitable for low-permeability tight sandstone rock oil and gas reservoirs, as depicted in Equation (13).

$$S_w={(\frac{a{R}_w{R}_{sh}}{a{R}_w{V}_{sh}{R}_t+\left[{\varphi }^{m_{fm}}{(1-S_{w1})}^{m_{fm}}+{\varphi }^{m_{wb}}{(S_{w1}-S_{w2})}^{m_{wb}}+{\varphi }^{m_{sb}}{S_{w2}}^{m_{sb}}\right]R_{sh}{R}_t/b})}^{1/n}$$

(13)

5. Model Parameter Calculation and Research

When assessing reservoir saturation using conventional models, the model parameters are frequently derived through regression fitting based on the outcomes of rock electrical experiments conducted on multiple core samples. However, employing a fixed set of rock electrical parameters during well-logging processing is inadequate for low-permeability tight sandstone rock reservoirs characterized by pronounced vertical heterogeneity. Traditional conventional models do not account for the vertical heterogeneity of reservoir pore structures and electrical properties. Furthermore, the widespread utilization of a single set of rock electrical parameters during evaluation not only falls short of meeting accuracy requirements but also disregards the wealth of electrical information obtained from rock electrical experiments.

The model parameters are derived through inverse modeling based on rock electrical experiments using the construction of the three-water differential parallel conductive saturation model. The parameters to be determined comprise the strongly bound water pore cementation index (m_sb), weakly bound water pore cementation index (m_wb), free water pore cementation index (m_fm), saturation exponent (n), and rock electrical coefficients (a, b). Owing to the variations in lithology, physical properties, and pore structure characteristics of low-permeability tight sandstone rock reservoirs in different regions, the model parameters for each region must be determined based on the specific rock electrical experimental data. The subsequent section provides an example using a specific area in the South China Sea to elucidate the method of obtaining model parameters through inverse rock electrical modeling.

Section 3 presents the derived general equation (Equation (9)) for calculating the three-water differential parallel conductivity C_t-tr in low-permeability tight oil and gas reservoirs. Utilizing this equation in conjunction with the three-water differential parallel conductive rock physical volume model, the fluid composition of the reservoir under varying oil and gas saturation conditions is extensively examined. The study yields two distinct fluid compositions under weak and strong hydrocarbon accumulation conditions: high-water-saturated oil and gas reservoirs with free water under weak hydrocarbon accumulation conditions and high-hydrocarbon-saturated oil and gas reservoirs with weakly bound water under strong hydrocarbon accumulation conditions, as illustrated in Figure 7.

The derivation of the electrical conductivity of low-permeability tight sandstone rock reservoirs under these two hydrocarbon accumulation conditions results in the oil and gas reservoir electrical conductivity in the presence of free water (C_t-tr1), as demonstrated in Equation (14), and the oil and gas reservoir electrical conductivity in the presence of weakly bound water (C_t-tr2), as illustrated in Equation (15).

$$C_{\mathit{\text{t-tr}}1}=\{{[\varphi (1-S_{w1})]}^{m_{fm}}{(\frac{S_w-S_{w1}}{1-S_{w1}})}^{n_{fm}}+{[\varphi (S_{w1}-S_{w2})]}^{m_{wb}}+{(\varphi S_{w2})}^{m_{sb}}\}{C}_w/ab$$

(14)

$$C_{\mathit{\text{t-tr}}2}=\{{[\varphi (S_{w1}-S_{w2})]}^{m_{wb}}{[(S_w-S_{w2})/(S_{w1}-S_{w2})]}^{n_{wb}}+{(\varphi S_{w2})}^{m_{sb}}\}{C}_w/ab$$

(15)

The electrical resistivity of a single core at various water-saturation states was measured in the rock electrical experiment. Based on the porosity and permeability of the core determined through physical property experiments, the rock electrical coefficient “a” was set to 1. By employing the Archie equation, the formation factor “F” and cementation exponent “m” of the single core were computed. Furthermore, the resistivity index “RI” and water-saturation “S_w” data points were fitted using a power function in a double-logarithmic coordinate system to determine the rock electrical coefficient “b” and saturation exponent “n”, as depicted in Figure 8.

In the process of modeling the model parameters, the rock electrical coefficient “b” and saturation exponent “n” in the model were initially ascertained. Subsequent to acquiring the rock electrical coefficient “b” and saturation exponent “n” for all cores using the aforementioned power function fitting method, the statistical analysis of the relationship between the rock electrical coefficient “b” and saturation exponent “n” and the core parameters were conducted, as illustrated in Figure 9.

An analysis of the intersection graph of the rock electrical coefficient “b” and saturation exponent “n” with the core parameters revealed that the rock electrical coefficient “b” did not exhibit a distinct pattern with the rock physical parameters. The range of the rock electrical coefficient “b” was concentrated between 0.94 and 1.02. Consequently, the rock electrical coefficient “b” was approximated by its average value when utilized in the model. A discernible pattern was observed between the saturation exponent “n” and the core properties, indicating an overall increase in the saturation exponent “n” as the core properties improved. The fitting process unveiled a strong relationship between the saturation exponent “n” and the free water porosity, leading to the development of a calculation model for the saturation exponent “n”, as represented by Equation (16).

$$n=1.086e^{0.0305\varphi }$$

(16)

Upon obtaining the calculation model for the rock electrical coefficient “b” and saturation exponent “n” and employing the two oil and gas layer conductivity equations under two types of hydrocarbon accumulation conditions in Equations (14) and (15), in conjunction with the rock electrical experiment data, the rock electrical inversion modeling of the strongly bound water porosity cementation exponent “m_sb”, weakly bound water porosity cementation exponent “m_wb”, and free water porosity cementation exponent “m_fm” was conducted. The specific steps for the inversion are as follows:

Initially, under the known conditions of rock porosity and permeability, the model for strongly bound water saturation and total bound water saturation (Equations (1) and (2)) in the study area can be adjusted to derive the strongly bound water saturation (S_w2) and total bound water saturation (S_w1) for a single rock, as depicted by the red dashed line and green dashed line in Figure 8. When the water saturation in the rock (S_w) exceeds the total bound water saturation (S_w1), it indicates the presence of free water in a high-water hydrocarbon reservoir. When the water saturation in the rock (S_w) lies between the total bound water saturation (S_w1) and the strongly bound water saturation (S_w2), it signifies the presence of weakly bound water in a high-hydrocarbon reservoir, with only a small amount of weakly bound water and predominantly strongly bound water in the rock.

Subsequently, discrete data points are chosen from the rock electrical experiment data, representing water saturation between the total bound water saturation (S_w1) and the strongly bound water saturation (S_w2), aligning with the data points located between the red and blue dashed lines in Figure 8. At this juncture, the rock contains both strongly bound water and weakly bound water concurrently. Two data points near the blue dashed line are selected, and the rock resistivity and water saturation associated with these data points are inserted into Equation (15), with the rock electrical coefficient “a” set to 1, and the rock electrical coefficient “b” and saturation exponent determined using the constructed calculation model. This results in two simultaneous equations that include the strongly bound water porosity cementation exponent (m_sb) and weakly bound water porosity cementation exponent (m_wb).

Ultimately, the resistivity data for water-saturated rocks from the rock electrical experiment is inserted into Equation (14), with the rock electrical coefficient “a” set to 1, and the rock electrical coefficient “b” and saturation exponent determined using the constructed calculation model. This yields a three-variable equation encompassing the strongly bound water porosity cementation exponent (m_sb), weakly bound water porosity cementation exponent (m_wb), and free water porosity cementation exponent (m_fm). Through amalgamating the two simultaneous equations and the three-variable equation into a system of three nonlinear equations, the numerical solution for the three porosity cementation exponents of a single core is derived by utilizing the vpasolve function in MATLAB R2023a.

Utilizing the previously described rock electrical inversion method, the calculation and statistical analysis of the three porosity cementation exponents for all cores were performed. The intersection fitting of the strongly bound water porosity cementation exponent (m_sb), weakly bound water porosity cementation exponent (m_wb), free water porosity cementation exponent (m_fm), and the overall core cementation exponent (m) with core parameters is depicted in Figure 10. A comparison reveals that with the improvement of the rock’s physical properties, the cementation exponents generally exhibit an increasing trend. In the case of a single core, the overall core cementation exponent (m) exceeds the three types of porosity cementation exponents obtained through inversion. This suggests that in low-permeability tight sandstone reservoirs, the overall core cementation exponent (m) is akin to the series connection of the three types of porosity cementation exponents, with each type contributing differently to the overall core cementation exponent. Among the three types of porosity cementation exponents for a single core, the strongly bound water porosity cementation exponent (m_sb) is the largest, whereas the free water porosity cementation exponent (m_fm) is the smallest. This implies that inferior micropore structures are associated with larger cementation exponents, whereas superior macropore structures are associated with smaller cementation exponents.

Through the investigation of intersection rules, the three types of porosity cementation exponents are individually combined with the optimal core parameters to establish models for calculating the cementation exponent. This leads to the derivation of the calculation model for the strongly bound water porosity cementation exponent (m_sb) for a specific block in the South China Sea, as depicted in Equation (17); the calculation model for the weakly bound water porosity cementation exponent (m_wb), as illustrated in Equation (18); and the calculation model for the free water porosity cementation exponent (m_fm), as delineated in Equation (19).

$$m_{fm}=0.246{(1-S_{w2})}^{0.4261}$$

(17)

$$m_{wb}=1.0822e^{0.005(1-S_{w2})}$$

(18)

$$m_{sb}=1.2774e^{0.0195\varphi }$$

(19)

By substituting all rock electrical parameter calculation models back into Equation (13), we obtain the three-water differential parallel conduction model suitable for low-permeability tight gas reservoirs in a specific area of the South China Sea.

6. Application Effectiveness

Figure 11 below provides a comprehensive log evaluation of a low-permeability gas reservoir in the South China Sea. Exploration and development have confirmed that the interval from 4040 to 4060 m constitutes a well-charged, high-quality gas reservoir. The upper section (4040–4050 m) typifies a high-resistivity gas reservoir, with an average resistivity while drilling of 50 Ω·m. Conversely, the lower section (4050–4060 m) displays a resistivity that is less than half of the upper high-resistivity gas reservoir, ranging between 10 and 20 Ω·m, signifying a relatively low resistivity.

A comparison of the “three porosity” curves of the upper high-resistivity gas reservoir with the relatively lower low-resistivity gas reservoir reveals that the compensation density of the upper high-resistivity gas reservoir is smaller, and the time difference and compensated neutron porosity are larger, indicating that the upper high-resistivity gas reservoir possesses superior petrophysical properties compared to the relatively lower low-resistivity gas reservoir. Additionally, the comparison of the nuclear magnetic resonance (NMR) T₂ spectra signals indicates that the upper high-resistivity gas reservoir displays more macropore fluid signals with T₂ values exceeding 300 ms, whereas the lower relatively low-resistivity gas reservoir contains more medium micropore fluid signals with T₂ values ranging between 30 ms and 300 ms. Furthermore, a comparison of the porosity and permeability curves clearly indicates that both the porosity and permeability of the relatively lower low-resistivity gas reservoir are lower than those of the upper high-resistivity gas reservoir, confirming the more complex pore structure of the relatively lower low-resistivity gas reservoir.

The saturation analysis is depicted in the eighth track of Figure 10. A comparison of the evaluation results between the Simandous model and the three-water differential parallel conduction model shows that the Simandous model, by not considering the variation in the longitudinal pore structure of the formation, uses a fixed set of rock electrical parameters, leading to an overall overestimation of water saturation and “spikes” in water saturation in locally complex pore structure sections. On the other hand, the three-water differential parallel conduction model proposed in this paper finely characterizes the differences in longitudinal pore structure by incorporating the differential parallel conduction of three water components and adapting to changes in rock electrical parameters, making it more suitable for complex pore structures in low-permeability tight reservoirs. Comparing the evaluation results between the new model and the Simandous model shows that the new model’s water saturation is more consistent with the strongly bound water saturation. Overall, it increases the gas saturation evaluation of the lower complex pore structure section by 8–10%, addressing the issue of significant differences in gas saturation evaluation between layers caused by the relative low-resistivity “illusion” and resulting in a more accurate gas saturation evaluation.

7. Conclusions

This study addresses the limitations of current saturation models and the challenge of determining model parameters by constructing a three-water differential parallel conduction saturation model. The new model was used to evaluate low-permeability tight sandstone reservoirs in the South China Sea region, producing positive results and confirming the model’s advantages in accurately assessing logging in low-permeability tight oil and gas reservoirs with complex pore structures.

(1) This study provides a new understanding of fluid occurrence states in low-permeability sandstone reservoirs based on comprehensive results from multi-level centrifugal force nuclear magnetic resonance experiments and visualization simulations. It proposes three types of formation water occurrence: strongly bound water, weakly bound water, and free water. Through the calibration of centrifugal nuclear magnetic experiments, the quantitative evaluation of the three water components in the South China Sea region was enabled;

(2) Using rock electrical experiment data, this study explores the relationship between water-saturated rock conductivity and three-water saturation, and it reveals that the three types of formation water demonstrate variations in their electrical conduction contributions. Specifically, it finds that rock conductivity decreases as the saturation of strongly bound and weakly bound water increases, while the saturation of free water determines the size of rock conductivity. Additionally, the conductivity of low-permeability tight sandstone reservoirs is influenced by both the bound water saturation and the absolute content of conductive formation water. The traditional belief that high-bound water saturation corresponds to high conductivity does not apply to low-permeability tight sandstone reservoirs;

(3) Expanding on the knowledge of fluid occurrence states and electrical conduction properties in low-permeability tight sandstone reservoirs, the cementation index term ϕ^m′ was introduced to account for the variations in the electrical properties of the three types of pores. This study ultimately suggests a three-water differential parallel conduction saturation model that is well-suited for a detailed quantitative assessment of logging saturation in low-permeability tight sandstone reservoirs;

(4) This study presents an innovative method for determining saturation model parameters through the inverse solving of rock electrical experiment data. It reveals that the overall cementation index (m) of the core resembles the joint contribution of the cementation indexes of the three types of pores, and the contribution weights of the three types of pore cementation indexes to the overall core cementation index vary. As rock properties improve, the four cementation indexes exhibit an overall increasing trend. Additionally, for a single core, the overall cementation index is the highest, followed by the cementation index of pore throats containing strongly bound water and the lowest for the pore throats with free water.