A Fractal Water Saturation Prediction Model Based on Trapezoidal Pores and Its Application in Tight Gas Reservoirs

Abstract

Accurate characterization of water saturation in tight sandstone gas reservoirs is the key to reservoir evaluation and productivity prediction. In view of the limitations of the traditional Archie formula in describing the strongly heterogeneous pore structure and the insufficient consideration of the coupling effect of pore throat geometry and fractal characteristics in the existing models, this paper innovatively combines the fractal theory with the trapezoidal pore throat model to construct a new water saturation interpretation model. By introducing parameters such as fractal dimension (D_f), tortuosity fractal dimension (D_T) and trapezoidal factor (φ_i), the model systematically quantifies the control mechanism of microscopic pore throat distribution, capillary force field evolution and stress sensitivity (rock elastic modulus E = 1.05 × 10³ MPa) on water saturation. The model was verified by the sealed coring and nuclear magnetic resonance experimental data of 10 groups of typical tight sandstone cores in Sulige gas field, Ordos Basin. The results show that: (1) The absolute error range between the water saturation calculated by the model and the measured value of the closed coring is 0.89–11.27%, indicating that the model has high accuracy and good applicability. (2) There is a significant negative correlation between reservoir water saturation and reservoir temperature and displacement pressure difference: for every 20 °C increase in temperature, water saturation decreases by about 4.5%; when the displacement pressure difference increases by 1 MPa, the water saturation decreases by about 6.3%. (3) The study further shows that under the condition of constant displacement pressure difference, the water saturation of the reservoir is positively correlated with the effective stress and negatively correlated with the maximum/minimum pore throat radius ratio. Rock mechanics parameters also have an impact on water saturation—the lower the elastic modulus, the higher the Poisson’s ratio, the greater the reservoir water saturation. The model can accurately predict the water saturation of the reservoir and provide an effective tool and support for the fluid quantitative evaluation and development scheme optimization of tight sandstone gas reservoirs.

1. Introduction

As a key area for unconventional oil and gas development, the production efficiency of tight sandstone gas reservoirs is directly controlled by the occurrence characteristics of reservoir fluids. Accurate characterization of water saturation (S_w) is not only a key technical bottleneck in reservoir evaluation, but also a core scientific issue in production capacity prediction. A multitude of recent research have confirmed that residual water in tight sandstone pore systems generally exists in three types of microscopic spaces: pore surface wetting layers, pore corner retention zones and microcapillary bound domains. This special occurrence state significantly inhibits the effective seepage of the gas. However, due to the unique low porosity and low permeability characteristics and strong heterogeneity of the reservoir, the traditional water saturation interpretation model shows significant limitations in describing complex pore systems. Although modern technologies such as nuclear magnetic resonance and gas-driven water experiments provide multi-dimensional data support [1,2,3,4], due to insufficient experimental standardization and differences in pore throat characterization methods, the measurement results of different technical paths often have a discrete deviation of more than 30%, which seriously restricts the engineering reliability of reservoir evaluation [5].

In response to the difficulty of pore structure characterization, fractal theory provides a new paradigm for the quantitative description of complex systems. However, the classic Archie formula is based on the assumption of homogeneous pores and is difficult to use to accurately describe the pore throat network with fractal characteristics. Numerical simulation results show that when the fractal dimension D_f = 1.67 and the critical radius r_c = 50 nm, the resistivity response curve of the fractal correction model has a systematic deviation of more than 15% from the predicted value of the traditional model in the high water saturation range (S_w > 0.6). This phenomenon highlights the inadequacy of the traditional model in adaptability to fractal pore structures, and there are significant limitations in the exploration of this issue by domestic and foreign scholars [6]. In the field of water-saturation model research for tight sandstone reservoirs, domestic and foreign scholars have conducted systematic explorations on the complexity of pore structure. In a classic foreign study, Killough [7] (1976) established a saturation history correlation model through a capillary hysteresis experiment, but did not quantify the effect of pore throat geometry on resistivity. Subsequently, Carlson [8] (1981) proposed a phase permeability hysteresis correction method, but assumed that the pores were homogeneous cylindrical structures, resulting in an error of more than 15% in the calculation of bound water saturation in the acute angle area of the trapezoidal pore throat. Li [9] (2020) proposed a dynamic fractal-stress coupling model that introduced a pore compression coefficient to correct the permeability equation. However, it still did not consider the regulatory effect of pore throat geometry on capillary force distribution, resulting in a maximum deviation of approximately 35% in narrow slit reservoir applications. Zhang et al. [10] (2023) successfully solved the problem of inaccurate resistivity logging caused by high calcium content in G oilfield of tight sandstone reservoir in Sichuan Basin through calcium content correction and dynamic rock electrical parameters, and significantly improved the calculation accuracy of water saturation, but the application range was small. On this basis, the research perspective is further extended to the fine characterization of pore structure and its regulation on seepage mechanism. Miao et al. [11] (2024) established a stress-dependent permeability model based on fractal geometry and rock mechanics. By introducing fractal parameters of pore geometry and rock elastic parameters, the evolution characteristics of porous rock permeability under effective stress were described. Based on the fractal rough capillary model, Yang et al. [12] (2024) analyzed the effects of pore roughness, fractal dimension and fluid properties on gas–water relative permeability, and pointed out that the increase in roughness would reduce the two-phase permeability as a whole. Xiao et al. [13] (2025) established a shale gas–liquid two-phase relative permeability model considering tree branch fracture network, which realized the unified characterization of matrix pore and fracture structure in multi-scale seepage. Tan et al. [14] (2025) further incorporated stress effect, capillary action and wettability into the fractal relative permeability model, revealing the two-phase flow characteristics of unsaturated porous media under stress–seepage coupling conditions.

In terms of domestic research, Yan et al. [15] (2015) found through mercury injection and rock electrical experiments that the geometric heterogeneity of pore throats in low-permeability sandstones caused systematic deviations in the values of parameters a and m in the Archie formula (error > 25%), and confirmed that reservoirs with trapezoidal pore throats accounting for more than 61% need to determine rock electrical parameters by type. Li et al. [9] (2020) developed high-speed centrifugation–nuclear magnetic resonance measurement technology to reveal the mechanism of an increase in micropore-bound water saturation, and proposed a variable rock electrical parameter model based on pore structure, but it did not quantify the coupling effect of fractal dimension and trapezoidal acute angle. Zhu Sinan et al. [16] (2021) introduced Killough–Carlson hysteresis correction into the water-invasion gas reservoir model, which improved the prediction accuracy of reservoir capacity by 9.3%. However, the model is still based on the traditional circular pore throat hypothesis. Recently, Xia et al. [17] (2025) introduced conductive porosity as an intermediate variable for the non-Archie phenomenon in low-permeability sand reservoirs, and constructed a saturation model with high-precision cementation index m, which significantly improved the calculation of water saturation.

It is worth noting that the limitations of existing models at the multiphase flow theory level cannot be ignored. Classical multiphase flow theory generally weakens the fundamental role of pore geometry characteristics and over-relies on empirical parameter fitting to characterize fluid distribution laws. This phenomenon stems from the inherent structural complexity of porous media systems: the multi-scale distribution characteristics [17], from nanoscale pores to micron-scale pore throats, make it easy for significant scale effects to occur between experimental observation data and theoretical models. Studies have shown that in tight sandstone reservoirs, low porosity and strong heterogeneity further amplify such errors, resulting in a significant positive correlation between the accuracy of water saturation assessment and the depth of pore structure characterization. However, traditional methods cannot quantitatively obtain fractal parameters (such as D_f) and fail to effectively couple the dual effects of pore compression and wettability evolution during dynamic seepage, which ultimately restricts the physical completeness of the model. The above-mentioned defects in multi-scale characterization and dynamic coupling further highlight the systematic deficiencies of existing water saturation models at the theoretical basis level [18].

In summary, there are three key limitations in previous studies: (1) Lack of geometric representation: Existing models (Such as Yan [15] 2015, Zhu [16] 2021, Miao [11] 2024, Xia [17] 2025,) did not quantify the influence of the acute angle (θ) and curvature radius (r_c) of the trapezoidal pore throat on the capillary force distribution; (2) Insufficient fractal coupling: Li [9] (2020)’s variable rock electrical parameter model did not introduce the fractal dimension (D_f) to describe the self-similarity of the pore throat; (3) Neglect of dynamic effects: The traditional Archie formula (Zhu [16] (2021)) cannot characterize the coupling effect of stress sensitivity and wettability changes during injection and production.

Based on the above knowledge theory, this study innovatively integrates fractal theory and the trapezoidal pore throat model to construct a new water saturation interpretation model. This model introduces D_f to quantify the fractal characteristics of pore throats, combines trapezoidal geometric parameters (such as acute angle θ, radius of curvature r_c) to characterize the distribution law of capillary force, and reveals the control mechanism of fluid occurrence from a microscopic scale. Compared with traditional methods, this model has achieved a theoretical breakthrough in the adaptability of pore structure: on the one hand, through the multi-parameter coupling system of fractal dimension (D_f) and tortuosity fractal dimension (D_τ), the fractal heterogeneity of pore throats is quantified; on the other hand, the dynamic interaction of stress–temperature–wettability is integrated to break through the excessive dependence of existing models on empirical parameters. Finally, by establishing a cross-scale correlation mechanism that spans from nanopores to reservoir scales, accurate predictions of dynamic seepage behavior are achieved, providing a new solution for the efficient development of tight sandstone gas reservoirs.

2. Non-Archie Phenomenon in Tight Sandstone Reservoirs

2.1. Limitations of the Traditional Archie Formula

The classical Archie formula is based on the idealized assumption of homogeneous porous media (Figure 1a), and its core is that there is a clear power-law relationship between rock resistivity and porosity (φ) and water saturation (Sw) [18]:

$$F={\displaystyle \frac{R_0}{R_w}}={\varphi }^{-m},I={\displaystyle \frac{R_t}{R_0}}=S_w^{-n}$$

(1)

where F is the formation factor, I is the resistivity index, m and n are the cementation index and saturation index, respectively. However, the pore structure of tight sandstone reservoirs has non-Euclidean geometric characteristics (such as fractal dimension D_f ≈ 1.67), and there is a sudden change in the cross-sectional area of pore throats (such as trapezoidal pore factor φ_i ≈ 0.35), which makes the current path significantly deviate from the uniform model hypothesis (Figure 1b). Therefore, the application of Archie’s formula in tight sandstone is severely limited. Through fractal numerical simulation [19], it is found that when the critical pore throat radius rc is as small as 50 nm, the corrected model resistivity response curve (red solid line in Figure 1c) will systematically deviate from the prediction of the traditional Archie formula in the high water saturation interval (blue dotted line). This systematic deviation fundamentally reveals the complex non-Archie behavior in tight sandstone, which is the result of the coupling of various non-Archie phenomena.

2.2. Non-Archie Phenomenon and Its Influence on Resistivity Response

(1)

Control of fractal pore structure and bound water

The pore system of tight sandstone has fractal characteristics, and its pore size distribution follows the fractal scaling law:

$$N(>r)={({\displaystyle \frac{r_{max}}{r}})}^{D_f}$$

(2)

This fractal geometry leads to a distinct pore surface area and connectivity from the homogeneous model. High fractal dimension (D_f) usually means more complex pore morphology and worse pore connectivity, resulting in a significant increase in irreducible water saturation (S_wi) [19,20]. Its core influence on resistivity is that a large amount of bound water exists on the surface of complex micropores and rough particles, forming a continuous conductive network that cannot be displaced even in the case of oil and gas. This will lead to the measured resistivity of the oil and gas layer being ‘untruly depressed’. If the Archie formula is used for interpretation, the oil saturation will be seriously underestimated, resulting in the leakage of the oil and gas layer.

(2)

Heterogeneity of microscopic pore structure and uneven distribution of fluid

Large pore and fine throat combination (trapezoidal pore) and small pore and fine throat combination (straight pore) are common in tight sandstone [21]. The cross-sectional area of trapezoidal pores varies significantly (described by the trapezoidal factor φ_i) [22]. This makes the tortuosity of the current path (D_T) and the thickness of the bound water film (h_w) vary greatly in different pores.

The nuclear magnetic resonance (NMR) T₂ spectrum reflects this complexity through the surface relaxation mechanism. The T₂ spectrum shows that the proportion of micropores (<10 nm) in tight sandstone is very high [22,23,24,25,26], and the bound water (S_wi) mainly exists in the form of surface water film and blind-end pores [27]. The conductive path is inconsistent with the uniform distribution assumed by the Archie formula, resulting in a nonlinear enhancement of the saturation index n, which is no longer a constant.

(3)

Dynamic Electrical Relations Caused by Stress Sensitivity

Tight sandstone will undergo compression deformation under the action of effective stress (P_eff), resulting in dynamic changes in pore radius (r) and porosity (φ) with stress.

$$r=r_0\left(1-{\displaystyle \frac{\beta P_{\mathrm{eff}}}{E}}\right)$$

(3)

$$\phi ={\phi }_0{(1-{\displaystyle \frac{\beta P_{\mathrm{eff}}}{E}})}^{D_f}$$

(4)

This stress sensitivity makes the relationship between resistivity and saturation not immutable but dynamically adjusted with the decrease in formation pressure during production. This means that the Archie parameter (m, n) established based on static core experiments cannot accurately describe the dynamic resistivity response during the development process, resulting in a continuous deviation in the dynamic prediction of reserves.

(4)

Conductive path mutation in gas–water two-phase system

In the gas–water two-phase flow, the movable water (S_wm) is distributed in the large pore throat (r > r_c), while the bound water (S_wi) occupies the small pore throat (r < r_c) and the blind end. Due to the insulation characteristics of the gas phase, the current mainly depends on the water film path conduction. However, the variation in the cross-sectional area of the trapezoidal pores and the high tortuosity fractal dimension (D_T) significantly extend and limit the current path. This will cause the apparent resistivity to be higher than the predicted value of the traditional model. This ‘abnormal high resistance’ phenomenon is closely related to the extreme complexity of pore structure. If the Archie formula is simply applied, it is easy to cause confusion and misjudgment in interpretation.

2.3. Comparative Analysis of Models

The non-Archie phenomenon of tight sandstone originates from the coupling effect of fractal pore structure, stress sensitivity, pore type diversity and micro wettability distribution. The traditional Archie formula is no longer applicable to such cases. In order to overcome these limitations, this study constructs a dynamic saturation interpretation framework that combines fractal theory and trapezoidal pore model. Based on the above analysis, in order to evaluate the performance of this model in dealing with complex pore structures, it is compared with the existing models (See

Table 1. Comparison and Analysis of Resistivity Interpretation Models for Tight Sandstone.

Contrast Dimension	Traditional Archie Model	Other Improved Models (Such as W-S Model)	The Model Proposed in This Paper
Core hypothesis	Uniform pores, the only conductive phase	Considering the additional conductivity of clay	Considering the additional conductivity of clay, fractal pore
Treatment of bound water	Unable to handle	Partial treatment (clay bound water)	Microporous bound water was accurately characterized by Df and NMR T2 spectra
Parameter (m, n)	Constants	Constant or simple change	Dynamic characterization by fractal dimension Df tortuosity fractal dimension DT and trapezoidal factor φi
Applicability	Pure, high porosity and high permeability reservoir Argillaceous sandstone reservoir	Argillaceous sandstone reservoir	The porous throat system is specially designed for the coexistence of ‘large pore fine throat’ (trapezoidal pore) and ‘small pore fine throat’ (straight pipe pore) in tight sandstone.
Predictive ability	Systemic distortion in tight sandstones	It is impossible to deal with clay-free but complex pore reservoirs.	It can more accurately identify the resistivity response under complex pores and reduce the saturation estimation error

for details).

Therefore, by introducing key microstructure parameters such as fractal dimension (D_f), tortuosity fractal dimension (D_T) and trapezoidal pore factor (φ_i), the control mechanism of the aforementioned non-Archie phenomenon on rock electrical properties is systematically quantified. The purpose of this study is to establish such a dynamic water saturation prediction model and framework in order to more accurately characterize the complex reservoir.

3. Pore Throat Water Saturation Model

3.1. Fractal Characteristic Parameter Calculation

Euclidean geometry describes ordered objects such as points, curves, surfaces, and cubes using integer dimensions 0, 1, 2, and 3, respectively. Associated with each dimension is a measure of the object, such as the length of a line, the area of a surface, and the volume of a cube. These measures are invariant with respect to the units of measure used [28]. However, many objects found in nature, such as rough surfaces, coastlines, mountains, rivers, lakes, and islands, are disordered and irregular and do not conform to the Euclidean description due to the scale dependence of length, area, and volume. These objects are called fractals, and the number of dimensions of these objects is non-integer and is defined as the fractal dimension. Fractal scaling laws are found in many natural porous media such as reservoirs, rocks, sands, soils, fibrous composites, sediments, biological tissues [29], etc. The measure of a fractal object M(L) is related to the length scale L by a scaling law of the form:

$$M(L)~L^{D_f}$$

(5)

where M can be the length of a line, the area of a surface, the volume of a cube, or the mass of an object, and D_f is the fractal dimension of the object. The above formula implies the self-similar nature of the object, which means that L is a constant within the length scale range.

(1)

The flow path in the rock can be represented by curved capillaries with variable cross-sections. The size and length distribution of the capillaries are random and self-similar [30], conforming to the statistical fractal scaling law.

(2)

The capillary curvature distribution is irregular, which conforms to the fractal characteristics.

(3)

There is a critical capillary radius r_c. The water in the capillaries with a radius less than r_c is the water remaining in the small capillaries due to insufficient displacement force. The immobile water in the capillaries with a radius greater than r_c is the water film, which adheres to the pore throat surface due to the action of molecular forces. These two types of water constitute the bound water saturation water.

(4)

Ignore the changes in interfacial tension and rock wettability.

(5)

Liquid viscosity is mainly affected by temperature.

Previous studies have shown that the pore structure in tight sandstone has fractal characteristics [31,32]. In addition, the distribution of microscopic pore throats also satisfies the fractal characteristics. According to fractal theory, the pore size distribution of tight sandstone reservoirs has a specific scaling relationship. The fractal dimension D_f of pore size distribution is 0 < D_f < 2, 0 < D_f < 3 in two and three dimensions, respectively, which characterizes the complexity of pore size and distribution. Therefore, based on the scaling law of fractal geometry, the probability density function f(r) of pore size distribution can be expressed as:

$$f(r)=D_f{r}_{min}^{{D_f}^{-(D_f+1)}}$$

(6)

In the above formula, rmax represents the maximum pore radius, r is a specific pore radius, and f(r) is the pore size distribution density function.in probability theory, for tight sandstone reservoirs, the pore size distribution satisfies the following relationship:

$${\displaystyle {\int }_{-\infty }^{+\infty }f}(r)dr={\displaystyle {\int }_{r_{min}}^{r_{max}}f}(r)dr=1-{({\displaystyle \frac{r_{min}}{r_{max}}})}^{D_f}$$

(7)

So:

$${({\displaystyle \frac{r_{min}^{\prime }}{r_{max}^{\prime }}})}^{D_f}=0$$

(8)

The above formula shows that the fractal analysis of porous media must meet the condition of r_min << r_max, otherwise the porous media does not conform to the fractal law. Generally speaking, for tight sandstone cores [31,33], the ratio of the minimum pore radius to the maximum pore radius r_min/r_max < 10⁻² satisfies the above conditions, so the pore throat radius distribution in the core satisfies the fractal law.

Based on completely self-similar fractal geometry, the aperture fractal dimension D_f can be solved by the following formula [32]:

$$D_f=d-{\displaystyle \frac{\ln \varphi }{\ln (\frac{r_{min}}{r_{max}})}}$$

(9)

where d is the Euclidean dimension, for two dimensions, d = 2, for three dimensions, d = 3, and φ is the porosity of the tight sandstone reservoir. Thus, the total pore area in the core cross-section can be obtained [34]:

$$A_p={\displaystyle {\int }_{{\mathrm{r}}_{\min }}^{{\mathrm{r}}_{\max }}\frac{\pi }{4}}{\mathrm{r}}^2(-dN)={\displaystyle \frac{\pi D_f{{\mathrm{r}}^2}_{\max }}{4(2-D_f)}}(1-\varphi )$$

(10)

According to the definition of planar porosity in cross-section: (φ = A_p/A), the maximum pore radius in the reservoir can be calculated by the following equation:

$$r_{\max }=\sqrt{{\displaystyle \frac{2-D_f}{\pi D_f}}{\displaystyle \frac{\varphi }{1-\varphi }}A}$$

(11)

where A is the cross-sectional area and φ is the porosity. Due to the bending nature of the capillary, the actual capillary length L is greater than L₀, which is the representative length of the capillary. The capillary length and capillary size also exhibit the self-similar fractal scaling law. Based on the self-similar fractal theory, the capillary bending length L(r) is related to the capillary radius and the capillary length [31]:

$$L(r)=\tau L_{core}={(2r)}^{1-D_T}{L_0}^{D_T}$$

(12)

where D_T is the tortuosity fractal dimension, which represents the degree of curvature of the capillary tube when the fluid passes through the porous medium, for two dimensions, 0 < D_T < 2, for three dimensions, 0 < D_T < 3. When D_T is 1, it represents that the capillary tube is a straight pipe, and the higher the value of D_T, the higher the degree of curvature of the capillary channel. L(r) is the tortuosity infiltration channel, L is the actual length of the capillary tube, and L₀ is the representative length of these capillaries. The tortuosity fractal dimension D_T can be determined by the following equation [3]:

$$D_T=1+{\displaystyle \frac{\ln \tau }{\ln (L_0/2r)}}$$

(13)

Thus, the tortuosity can be expressed by the following equation [35,36,37]:

$$\tau ={\displaystyle \frac{L(r)}{L_0}}={\left({\displaystyle \frac{L_0}{r}}\right)}^{D_T-1}$$

(14)

$$\overline{\tau }={\displaystyle \frac{1}{2}}[1+{\displaystyle \frac{1}{2}}\sqrt{1-\varphi }+\sqrt{1-\varphi }\cdot {\displaystyle \frac{\sqrt{{(\frac{1}{\sqrt{1-\varphi }}-1)}^2+\frac{1}{4}}}{1-\sqrt{1-\varphi }}})$$

(15)

$${\displaystyle \frac{L_0}{2\overline{r}}}={\displaystyle \frac{D_f-1}{\sqrt{D_f}}}{\left[{\displaystyle \frac{1-\varphi }{\varphi }}{\displaystyle \frac{\pi }{4(2-D_f)}}\right]}^{{\displaystyle \frac{1}{2}}}{r}_{max}$$

(16)

Among them $\overline{\tau }$ is the average tortuosity, $\overline{r}$ is the average pore radius.

The total pore area in the cross-section of the porous medium is:

$$A={\displaystyle \frac{\pi D_f(1-\varphi )r_{max}^2}{\varphi (2-D_f)}}$$

(17)

The representative length L₀ of the porous medium can be calculated by the following equation [33]:

$$L_0=\sqrt{A}=r_{\max }\sqrt{{\displaystyle \frac{\pi D_f(1-\varphi )}{\varphi (2-D_f)}}}$$

(18)

where A is the cross-sectional area of the rock sample and φ is the porosity.

3.2. Influence Analysis of Stress Sensitivity

In the actual gas extraction process of tight sandstone reservoirs, the reduction in pore pressure in the production process will cause significant changes in the stress field [33], which will increase the effective stress of the reservoir. The increase in effective stress leads to compaction of the rock, which reduces the pore radius and decreases the porosity of the pore medium. The capillary radius after compressive deformation can be expressed by the following equation [19,25]:

$$r_s=r\left\{1-4{\left[{\displaystyle \frac{3\pi \left(1-{\nu }^2\right)P_{eff}}{4E}}\right]}^{\beta }\right\}$$

(19)

where E is the modulus of elasticity of the rock, MPa; V is Poisson’s ratio, dimensionless; β is a power-law exponent determined by the pore structure; and P_eff is the effective stress, MPa. Assuming that the total number of pores is constant under different stress conditions, the following equation can be expressed as [24,25]:

$$N={\left({\displaystyle \frac{r_{max}}{r_{min}}}\right)}^{D_f}={\left({\displaystyle \frac{r_{maxs}}{r_{mins}}}\right)}^{D_f}$$

(20)

where D_f is the stress-dependent fractal dimension, and r_maxs and r_mins are the maximum and minimum pore throat radius in the reservoir at the effective stress state.

$$f\left(r\right)=D_f{r}_{{\min }^{D_f}}{r}^{-\left(D_f+1\right)}$$

(21)

$$D_f^{\prime }=D_f{\displaystyle \frac{\ln (r_{\max }/r_{\min })}{\ln (r_{\max s}/r_{\min s})}}$$

(22)

The expression for the porosity of the deformed rock can be written as:

$${\varphi }_s={(r_{\max }/r_{\min })}^{d-D_f^{\prime }}$$

(23)

The stress-dependent curvature dimension can be expressed as [24,25]:

$$D_{T^{\prime }}=1+{\displaystyle \frac{\ln \tau }{\ln \left(L_{0^{\prime }}/(2\overline{r})\right)}}$$

(24)

Among them [34]:

$$\tau ={\displaystyle \frac{1}{2}}\left[1+{\displaystyle \frac{1}{2}}\sqrt{1-{\varphi }_s}+\sqrt{1-{\varphi }_s}{\displaystyle \frac{\sqrt{{\left(\frac{1}{\sqrt{1-{\varphi }_s}}-1\right)}^2+\frac{1}{4}}}{1-\sqrt{1-{\varphi }_s}}}\right]$$

(25)

$${\displaystyle \frac{L_0}{2r}}={\displaystyle \frac{D_{f^{\prime }}-1}{\sqrt{D_{f^{\prime }}}}}{\left[{\displaystyle \frac{1-{\varphi }_s}{{\varphi }_s}}{\displaystyle \frac{\pi }{4(2-D_{f^{\prime }})}}\right]}^{{\displaystyle \frac{1}{2}}}{r}_{\max s}$$

(26)

$$L_{0^{\prime }}=\sqrt{A}=r_{\max s}\sqrt{{\displaystyle \frac{\pi D_{f^{\prime }}(1-{\varphi }_s)}{{\varphi }_s(2-D_{f^{\prime }})}}}$$

(27)

The length of the porous medium under effective stress can be expressed as [29]:

$$L_s(r_s)=\left(1-{\displaystyle \frac{P_{eff}}{\nu E}}\right){\left({\displaystyle \frac{r}{F}}\right)}^{1-D_{T^{\prime }}}{L}_0^{D_{T^{\prime }}}$$

(28)

where F is the shape-influencing factor and is dimensionless.

3.3. Trapezoidal Pore Throat Water Saturation Model

The tight sandstone reservoir has formed a pore-throat structure with multiple pore types coexisting mainly with residual intergranular pores, feldspar dissolution pores and clay micropores under various effects, and the seepage capacity of the reservoir is controlled by tiny throats. From the perspective of pore-throat connectivity, micropores dominate tight sandstone, and there are usually two kinds of pore-throat combinations in the pores (Figure 2): (1) large pores (remaining intergranular pores and feldspar-soluble pores) and micropores composed of large pore-thin throat combinations, whose cross-sectional area varies greatly, and which is the main body of pore seepage channels; (2) small pores and thin throats combined with the micropores connected to the combination of small pores and thin throats, and whose cross-sectional area can be considered to be unchanged. According to the concept of trapezoidal pore space proposed by Hu Shengfu [38] and others, the large pore–thin throat combination with large change in cross-sectional area in the pore-throat structure of tight sandstone can be expressed by trapezoidal pore throat, and for the combination of small pore–thin throat, due to small change in the aperture diameter of the microporous space, it can be regarded as a straight pore space with unchanged cross-sectional area [38,39,40,41].

For small pore fine throat combinations, the volume of a single straight capillary with real length L and radius r can be calculated as:

$$V=\pi r^{2}L$$

(29)

Thus, the total volume of the small pore fine throat combination can be expressed as (Vs is the volume of the straight capillary):

$$V_s=N_s{\displaystyle {\int }_{r_{min}}^{r_{max}}\pi }{r}^{2}Lf(r)dr$$

(30)

A single trapezoidal capillary tube is shown in Figure 3, and its volume is calculated as:

$$dV=\pi {(r_{\min }+L\tan \theta )}^{2}dL$$

(31)

$$\begin{array}{l}V={\displaystyle {\int }_0^L\pi {(r_{\min }+L\tan \theta )}^2}dL\\ ={\displaystyle {\int }_0^L\pi ({r_{\min }}^2}+2r_{\min }L\tan \theta +L^2\tan {\theta }^2)dL\\ ={\displaystyle \frac{1}{3}}\pi L(r_{\max }^2+r_{\min }^2+r_{\max }{r}_{\min })\end{array}$$

(32)

Define the trapezoidal factor, the size of the trapezoidal factor reflects the degree of homogeneity of the trapezoidal pore cross-section and is dimensionless.

Where the mean pore radius distribution satisfies the probability density function f(r) distribution:

$$\varphi ={\displaystyle \frac{r_{\max }{r}_{\min }}{{\overline{r}}^2}}$$

(33)

$\overline{r}$ is the average pore radius of the trapezoidal tube bundle, which also satisfies the fractal law.

Thus, the volume of a single trapezoidal capillary can be approximated as:

$$V={\displaystyle \frac{1}{3}}\pi L{r}^2{(4-\varphi )}^2$$

(34)

The total volume of the trapezoidal orifice-throat combination can be expressed as:

$$V_{\mathrm{T}}=N_{\mathrm{T}}{\displaystyle {\int }_{r_{\min t}}^{r_{\max t}}\frac{1}{3}}\pi L{r}^2\left(4-\overline{\varphi }\right)f(r)dr$$

(35)

Let the number of trapezoidal pores with trapezoidal factor φ_i be F_i, where the largest and smallest pore radii are r_imax, r_imin, respectively, and the proportion of the total pores is f_i.

The total number of trapezoidal pores is N_i:

$$N_i={\displaystyle \sum _{i=1}^N{F}_i}$$

(36)

$$f_i={\displaystyle \frac{F_i}{{\displaystyle \sum _{i=1}^N{F}_i}}}$$

(37)

The total trapezoidal pore volume is:

$$\begin{array}{l}{V}_i={\displaystyle \sum _{i=1}^N{F}_i}{\displaystyle \frac{\pi L}{3}}(r_{i\max }^2+r_{i\min }^2+r_{i\max }{r}_{i\min })\\ ={\displaystyle \frac{\pi L}{3}}{\displaystyle \sum _{i=1}^N{F}_i}\left[(4-{\psi }_i){{\overline{r}}_i}^2\right]\\ ={\displaystyle \frac{4\pi L}{3}}{\displaystyle \sum _{i=1}^N{F}_i{{\overline{r}}_i}^2}-{\displaystyle \sum _{i=1}^N{F}_i{\psi }_i{{\overline{r}}_i}^2}\end{array}$$

(38)

Set S be the cross-sectional area of the sample and the average cross-sectional area of the sample occupied by a single tube.

$$\overline{S}={\displaystyle \frac{S}{{\displaystyle \sum _{i=1}^N{F}_i}}}={\displaystyle \frac{\pi r^2\tau }{\varphi }}$$

(39)

The core porosity is:

$$\begin{array}{l}\varphi ={\displaystyle \frac{{\displaystyle \sum _{i=1}^N{F}_i}\frac{\pi L}{3}(r_{i\max }^2+r_{i\min }^2+r_{i\max }{r}_{i\min })}{S{L}_s}}\\ ={\displaystyle \frac{\frac{\pi L}{3}{\displaystyle \sum _{i=1}^N{F}_i}{\overline{r}}_i^2(4-{\psi }_i)}{S{L}_s}}\end{array}$$

(40)

where L_s is the core length, defining the ratio of trapezoidal pore length to core length as the trapezoidal pore tortuosity.

$$\tau ={\displaystyle \frac{L}{L_s}}$$

(41)

Then the core porosity is:

$$\begin{array}{l}\varphi ={\displaystyle \frac{\tau \pi }{3S}}{\displaystyle \sum _{i=1}^N{F}_i}(r_{i\max }^2+r_{i\min }^2+r_{i\max }{r}_{i\min })\\ ={\displaystyle \frac{\tau \pi }{3S{\displaystyle \sum _{i=1}^N{F}_i}}}\left({\displaystyle \sum _{i=1}^N\frac{F_i}{{\displaystyle \sum _{i=1}^N{F}_i}}(r_{i\max }^2+r_{i\min }^2+r_{i\max }{r}_{i\min })}\right)\\ ={\displaystyle \frac{\tau \pi }{3S{\displaystyle \sum _{i=1}^N{F}_i}}}\left({\displaystyle \sum _{i=1}^N\left(4-{\psi }_i\right)f_i{\overline{r}}_i^2}\right)\\ ={\displaystyle \frac{4\tau \pi }{3S{\displaystyle \sum _{i=1}^N{F}_i}}}{\displaystyle \sum _{i=1}^N{f}_i{\overline{r}}_i^2}-{\displaystyle \frac{\tau \pi }{3S{\displaystyle \sum _{i=1}^N{F}_i}}}{\displaystyle \sum _{i=1}^N{\psi }_i{f}_i{\overline{r}}_i^2}\end{array}$$

(42)

W define the average trapezoidal factor as the weighted average of the trapezoidal factor φ_i of a single tube, with the weights being the ratio of the volume of each trapezoidal tube to the total volume of the rock, then there are:

weighted value:

$${\displaystyle \sum _{i=1}^N{f}_i}{\psi }_i{\overline{r}}_i^2=\overline{\psi }{\overline{r}}^2$$

(43)

Thus, porosity can be expressed as:

$$\varphi ={\displaystyle \frac{\tau \pi }{S{\displaystyle \sum _{i=1}^N{F}_i}}}\overline{\psi }{\overline{r}}^2$$

(44)

Therefore, the total volume of trapezoidal pore space can be expressed as:

The ratio of trapezoidal pore is f_t, and the ratio of straight pore is f_z, which satisfies the calculation of trapezoidal pore volume:

$$f_t+f_z=1$$

(45)

where the straight pore proportion can be calculated from the NMR T₂ cut-off value of less than 0.6 ms, i.e., the straight pore proportion is the proportion of the pore space accounted for by the volume of the component on the NMR T₂ spectrum that is less than 0.6 ms.

Therefore, the total volume of N capillaries is:

$$V_{total}=N{f}_z{\displaystyle {\int }_{r_{\min }}^{r_L}\pi r^2}Lf(r)dr+N{f}_t{\displaystyle {\int }_{r_L}^{r_{\max }}\frac{1}{3}\pi {\overline{r}}^{2}L(4-\overline{\psi })f(\overline{r})}d\overline{r}$$

(46)

where r_L is the critical pore radius corresponding to the NMR cut-off value of 0.6 ms, and is the average trapezoidal factor, which is based on the dead volume coefficient introduced according to the Su Yuliang [39] model for correcting the internal pore volume of the core, and its physical significance represents the ratio of the dead volume at the blind end of the pore and the corners and corners of the pore to the volume of the pore [42].

$$\eta ={\displaystyle \frac{V_{dw}}{V_{total}}}$$

(47)

where η is the dead volume coefficient, which is empirically taken as 0.056.

Then the total pore volume of the core can be expressed by the following equation:

$$V=(1+\eta )\left[N{f}_z{\displaystyle {\int }_{r_{\min }}^{r_L}\pi r^2}Lf(r)dr+N{f}_t{\displaystyle {\int }_{r_L}^{r_{\max }}\frac{1}{3}\pi {\overline{r}}^{2}L(4-\psi )f(\overline{r})}d\overline{r}\right]$$

(48)

It is assumed that part of the water phase, called immobile water (or residual water), adheres to the capillary. According to previous studies, the thickness of the immobile water film is related to the pressure drop [30]:

$$\delta =\left\{\begin{array}{cc}r\times 0.25763e^{-0.261r}{\left(\nabla P\right)}^{-0.419}{\mu }_w& \nabla P<1 \mathrm{M}\mathrm{P}\mathrm{a}/\mathrm{m}\\ r\times 0.25763e^{-0.261r}{\left(\nabla P\right)}^{-0.419}{\mu }_w& \nabla P>1 \mathrm{M}\mathrm{P}\mathrm{a}/\mathrm{m}\end{array}\right.$$

(49)

And the relationship between the viscosity of water and temperature is generally expressed through the following equation:

$${\mu }_w=1.4e^{-0.0176T}$$

(50)

Water saturation capillary radius acquisition (250 psi centrifugal force for initial water saturation and 300 psi centrifugal force for bound water saturation in tight sandstone reservoirs)

$$\Delta P={\displaystyle \frac{2\sigma \cos \theta }{r_{ci}}}$$

(51)

$$\Delta \mathrm{P}={\displaystyle \frac{2\sigma cos\theta }{r_{cr}}}$$

(52)

where r_ci and r_cr are divided into the bound water saturation and initial water saturation of the tight sandstone reservoir corresponding to the critical pore radius under centrifugal force. Here, it is considered that according to the calculation, r_L is 10 nm, r_ci is 50 nm, r_cr is 60 nm, r_cr > r_ci > r_L.

The water volume inside the core consists of water in the capillaries smaller than the critical pore radius, water film in the capillaries larger than the critical pore radius, and dead water at the blind ends and corners of the pores and macroscopically consists of bound water and movable water.

$$\left\{\begin{array}{l}{S}_{wm}+S_{wi}+S_g=1\hfill \\ S_w=S_{wi}+S_{wm}\hfill \end{array}\right.$$

(53)

where S_w is reservoir water saturation, S_wm is movable water saturation, S_wi is bound water saturation, and S_g is gas saturation.

Based on the above assumptions of gas and water distribution patterns in the core pore throats (Figure 4), the water saturation and movable and bound water saturation of tight sandstone reservoirs can be obtained as follows:

$$\begin{array}{ll}{S}_{wi}& ={\displaystyle \frac{N{f}_z{\displaystyle {\int }_{r_{\min }}^{r_L}\pi r^{2}L(r)f(r)dr}}{(1+\eta )\left[N{f}_z{\displaystyle {\int }_{r_{\min }}^{r_L}\pi r^2}L(r)f(r)dr+N{f}_t{\displaystyle {\int }_{r_L}^{r_{\max }}\frac{1}{3}\pi r^{2}L(r)(4-\overline{\psi })f(r)}dr\right]}}\\ & +{\displaystyle \frac{N{f}_t\left[{\displaystyle {\int }_{r_L}^{r_{ci}}\frac{1}{3}\pi r^{2}L(r)(4-\overline{\psi })f(r)dr+{\displaystyle {\int }_{r_{ci}}^{r_{\max }}\frac{1}{3}\pi (r^2-{\left(r-\delta \right)}^2)L(r)(4-\overline{\psi })f(r)dr}}\right]}{(1+\eta )\left[N{f}_z{\displaystyle {\int }_{r_{\min }}^{r_L}\pi r^2}L(r)f(r)dr+N{f}_t{\displaystyle {\int }_{r_L}^{r_{\max }}\frac{1}{3}\pi r^{2}L(r)(4-\overline{\psi })f(r)}dr\right]}}\\ & +{\displaystyle \frac{\eta }{1+\eta }}\end{array}$$

(54)

Water saturation:

$$\begin{array}{ll}{S}_w& ={\displaystyle \frac{N{f}_z{\displaystyle {\int }_{r_{\min }}^{r_L}\pi r^{2}L(r)f(r)dr}}{(1+\eta )\left[N{f}_z{\displaystyle {\int }_{r_{\min }}^{r_L}\pi r^2}L(r)f(r)dr+N{f}_t{\displaystyle {\int }_{r_L}^{r_{\max }}\frac{1}{3}\pi r^{2}L(r)(4-\overline{\psi })f(r)}dr\right]}}\\ & +{\displaystyle \frac{N{f}_t\left[{\displaystyle {\int }_{r_L}^{r_{cr}}\frac{1}{3}\pi r^{2}L(r)(4-\overline{\psi })f(r)dr+{\displaystyle {\int }_{r_{cr}}^{r_{\max }}\frac{1}{3}\pi (r^2-{\left(r-\delta \right)}^2)L(r)(4-\overline{\psi })f(r)dr}}\right]}{(1+\eta )\left[N{f}_z{\displaystyle {\int }_{r_{\min }}^{r_L}\pi r^2}L(r)f(r)dr+N{f}_t{\displaystyle {\int }_{r_L}^{r_{\max }}\frac{1}{3}\pi r^{2}L(r)(4-\overline{\psi })f(r)}dr\right]}}\\ & +{\displaystyle \frac{\eta }{1+\eta }}\end{array}$$

(55)

Movable water saturation:

$$S_{wm}= S_w- S_{wi}$$

(56)

4. Relative Permeability Model

Yu et al. [43] were the first that proposed/derived analytical fractal model for relative permeabilities of porous media. In this chapter, the relative permeability model is established based on fractal characteristics, and the following assumptions are made:

(1)

Fluid laminar flow occurs in the pore throat of a tight sandstone reservoir;

(2)

Water is distributed as a wetting phase on the inside of the capillary wall, and gas is distributed as a non-wetting phase in the center of the capillary;

(3)

Bound water adheres to the inside of the pipe wall in the form of a water film, and no flow occurs;

(4)

At this point, the pore space in the tight sandstone reservoir is divided into two parts: the straight pore throat and the trapezoidal pore throat.

So the gas–water two-phase flow is divided into two parts, in less than the critical pore throat of the gas, water and two-phase laminar flow occurs in the capillary tube, the flow of fluids as a set of the same length, speed of different concentric liquid barrels, smaller than the critical pore radius r_c of the pore [44,45], only water-phase flow occurs, and greater than the r_c pore occurs in the gas–water two-phase flow.

According to the Poiseuille equation:

$$Q={\displaystyle \frac{\pi \Delta P{r}^4}{8\mu L}}$$

(57)

Based on the derivation of Poiseuille’s equation in a trapezoidal tube, the velocity equations for the gas and water phases are [37], respectively:

$$\begin{array}{cc}{v}_w=-{\displaystyle \frac{\Delta P}{L}}{\displaystyle \frac{r^2}{4{\mu }_w}}+c_w& r_g<r<r_0-\delta \end{array}$$

(58)

$$\begin{array}{cc}{v}_w=0& r_0-\delta \le r\le r_0\end{array}$$

(59)

$$v_g=-{\displaystyle \frac{\Delta \mathrm{P}}{L}}{\displaystyle \frac{r^2}{4{\mu }_g}}+c_g\hspace{1em}0<r<r_g$$

(60)

where the air–water interface meets:

$$\begin{array}{cc}{v}_w=v_g& r=r_g\end{array}$$

(61)

The above equation can be obtained by conjunction:

$$v_{\mathrm{w}}={\displaystyle \frac{\Delta \mathrm{P}}{L}}{\displaystyle \frac{1}{4{\mu }_{\mathrm{w}}}}[{(r_0-\delta )}^2-r^2]$$

(62)

Therefore, the gas phase and water phase flow rates through the capillary with a radius of r₀ are:

$$q_g={\displaystyle {\int }_0^{r_g}{v}_g}dA={\displaystyle \frac{\Delta P}{L}}{\displaystyle \frac{\pi }{8{\mu }_g}}{r}_g^2+{\displaystyle \frac{\Delta P}{L}}{\displaystyle \frac{\pi }{4{\mu }_w}}[{(r_0-\delta )}^2-r_g^2]r_g^2$$

(63)

$$q_w={\displaystyle {\int }_{r_g}^{r_0-\delta }{v}_w}dA={\displaystyle \frac{\Delta P}{L}}{\displaystyle \frac{\pi }{8{\mu }_w}}{[{(r_0-\delta )}^2-r_g^2]}^2$$

(64)

When r_g is 0, it means that there is no gas phase in the capillary, and only the water phase flows alone. At this time, the above formula can be written as [46]:

$$q_{cw}={\displaystyle \frac{\Delta P}{L}}{\displaystyle \frac{\pi }{8{\mu }_w}}{\left(r_0-\delta \right)}^4$$

(65)

When the pore tortuosity in porous media is taken into account, the gas phase and water phase can be written as:

$$q_g={\displaystyle \frac{\Delta P}{L(r)}}{\displaystyle \frac{\pi }{8{\mu }_g}}{r}_g^2+{\displaystyle \frac{\Delta P}{L(r)}}{\displaystyle \frac{\pi }{4{\mu }_w}}\left[{\left(r_0-\delta \right)}^2-r_g^2\right]r_g^2$$

(66)

$$q_w={\displaystyle \frac{\Delta \mathrm{P}}{L\left(r\right)}}{\displaystyle \frac{\pi }{8{\mu }_w}}{\left[{\left(r_0-\delta \right)}^2-r_g^2\right]}^2$$

(67)

The gas flow rate in the entire core is:

$$Q_g=N{f}_t{\displaystyle {\int }_{r_c}^{r_{\max }}{q}_{g}f(r)dr}$$

(68)

The circulation of water phase can be divided into two forms: flow in pores smaller than the critical radius r_c and flow in the form of a movable water layer in pores larger than the critical pore radius r_c:

$$Q_w=N{f}_z{\displaystyle {\int }_{r_{\min }}^{r_L}{q}_{cw}f(r)dr+N{f}_t}\left[{\displaystyle {\int }_{r_L}^{r_c}{q}_{cw}f(r)dr}+{\displaystyle {\int }_{r_c}^{r_L}{q}_{w}f(r)dr}\right]$$

(69)

The flow rates of the water phase and gas phase can also be expressed in the expanded form of Darcy’s formula:

$$Q_g=-{\displaystyle \frac{K_g}{{\mu }_g}}{\displaystyle \frac{A}{\left(1-S_w\right)}}{\displaystyle \frac{\Delta P}{L}}$$

(70)

$$Q_w=-{\displaystyle \frac{K_w}{{\mu }_w}}{\displaystyle \frac{A}{S_w}}{\displaystyle \frac{\Delta P}{L}}$$

(71)

When the core is filled with water, all seepage channels are in the water phase. At this time, the permeability of the core is the absolute permeability of the core, so that it can be obtained:

$$K={\displaystyle \frac{\pi NL{\displaystyle {\int }_{r_{\min }}^{r_{\max }}\frac{{(r-\delta )}^4}{L}f(r)dr}}{8A}}$$

(72)

5. Model Verification and Calculation

5.1. Nuclear Magnetic Resonance Characterization of Pore Throat Parameters

The mercury pressure method is an important means of obtaining the radius distribution of core pore throats; however, this method has a long testing period, the mercury used is highly toxic, and the core cannot be reused if it becomes contaminated after the test. In addition to this, the distribution of pore throats of mercury pressure cannot reflect the information of the pore space connected to the throat that is less than the maximum mercury inlet pressure, and it has a certain limitation, which will lead to a small value of the water saturation calculated according to the fractal theory [33,47,48]. The distribution of low-field NMR T₂ spectra is directly related to the pore structure, and the T₂ spectra when the core is saturated with water can reflect the distribution of all the pore throats in the core. Theoretical analyses showed that the rock NMR T₂ spectra and the pressed mercury analysis data were correlated.

According to the analysis of NMR theory, the NMR T₂ spectra of saturated single-phase fluids in rocks can reflect their internal pore structure. The transverse relaxation time of atoms in a single pore channel of saturated water in a uniform magnetic field mainly includes the surface relaxation time, bulk phase relaxation time, and diffusion relaxation time, among which the bulk phase relaxation time and diffusion relaxation time can be neglected. Therefore, the rock surface transverse relaxation time can be expressed as:

$${\displaystyle \frac{1}{T_2}}={\rho }_2{\displaystyle \frac{S}{V}}$$

(73)

where T₂ is the transverse relaxation time, ms; ρ₂ is the transverse surface relaxation strength, which is affected by the nature of the pore surface as well as the size of the mineral composition and the nature of the saturated fluid, μm/ms; S is the surface area of the pore, V is the pore volume, and S/V is the specific surface area of the pore, μm²/μm³.

The pore specific surface area is related to the pore radius as:

$${\displaystyle \frac{S}{V}}={\displaystyle \frac{F_s}{r}}$$

(74)

where F_s is the pore shape factor, the magnitude of which varies with the pore model, for spherical pores, F_s is 3, and for columnar pores, F_s is 2. Then there are:

$$T_2={\displaystyle \frac{r}{{\rho }_2{F}_s}}$$

(75)

Let 1/ρ₂F_s = C, then the above equation can be expressed as:

$$T_2=C\times r$$

(76)

Therefore, from the core NMR relaxation theory, it can be seen that the key to transforming the NMR curve into the pore throat radius distribution curve is to find the conversion coefficient of T₂ ~ r.

According to the statistical processing results of nuclear magnetic resonance and conventional mercury injection experiments of a large number of tight sandstone reservoir rock samples, the nuclear magnetic resonance curve of the relationship between the cumulative amplitude of the average treatment and the T₂ relaxation time and the mercury injection curve of the relationship between the cumulative amplitude of the average treatment and the pore radius are obtained respectively (Figure 5). It can be found that there is a good consistency between the mercury injection curve and the nuclear magnetic resonance curve, so the conversion relationship between the two can be established, T₂ = 60 × r, so as to realize the conversion between the nuclear magnetic T₂ relaxation time and the pore radius, and achieve the purpose of using nuclear magnetic resonance experiments to quantitatively test the pore size.

5.2. Model Verification

To verify the reliability of the water saturation model based on fractal theory and trapezoidal pore throats, 10 typical tight sandstone samples obtained by closed coring in the Sulige gas field in the Ordos Basin were used for verification (experimental permeability 0.02~1.8 × 10⁻³ μm², porosity 6.5–14.3%, geologically representative). Based on the standardized experimental process, the sealed rock samples were mass measured and analyzed by nuclear magnetic resonance. By comparing the measured results with the model prediction values (as shown in Table 1 and

Table 2. Values of relevant parameters in the model.

Rock Elastic Modulus E (MPa)	Poisson’s Ratio ν	Power Law Exponent β	Dead Volume Coefficient η	Average Trapezoidal Factor ψ	Air–Water Interfacial Tension σ (mN/m)	Euclidean Dimension d
1.05 × 103	0.25	0.67	0.056	0.35	71.97	2

), a multi-dimensional evaluation of the prediction ability of the newly built model was finally completed. The key parameters involved in the verification process, such as formation water viscosity, were strictly calculated based on the actual characteristic parameters of the reservoir [46,47,49,50,51].

Table 3. Basic parameters of 10 core experimental samples.

Well Number	Core Number	Well Depth (m)	Length (cm)	Diameter (cm)	Porosity (%)	Permeability (×10−3 μm2)	Reservoir Temperature (°C)	Forming Water Viscosity (MPa·s)
Su 100	73	3319.5	3.08	2.53	0.067	0.09	75.4	0.397
Su 100	75	3493.2	3.12	2.52	0.106	0.40	77.2	0.385
Su 16	81	3431.2	2.72	2.53	0.106	0.02	84.7	0.340
Su 16	82	3432.5	3.04	2.53	0.117	0.36	83.7	0.346
Su 3	93	3486.5	3.13	2.53	0.129	0.18	84.4	0.342
Su 47	102	2575.3	3.02	2.53	0.082	0.27	68.8	0.444
Su 6	105	3628.7	3.00	2.53	0.140	1.64	69.2	0.441
Su 61	108	3845.6	0.27	0.27	0.27	0.95	80.6	0.365
Su 61	109	3019.2	1.64	1.64	1.64	0.45	67.2	0.456
Su 65	113	2989.6	0.27	0.27	0.27	0.03	77.9	0.381

presents the results of pore throat parameter determination of ten groups of tight sandstone samples by nuclear magnetic resonance testing (NMR) and establishes the corresponding fractal characteristic parameters (including key indicators such as pore throat ratio distribution gradient and fractal dimension discreteness). This data set contains both directly measured pore throat morphological characteristics and pore structure complexity evaluation parameters based on fractal geometry model inversion and completely constructs the characterization chain from microscopic morphology to macroscopic characteristics. It also shows the various parameter values used in the water saturation model in the calculation process [43]. However, it should be noted here that the selection of effective stress should be based on actual experimental conditions or formation conditions.

Table 4. Model parameters corresponding to 10 cores.

Well Number	Core Number	rmax (μm)	rmin (μm)	ravgs (μm)	Df	Dτ	fz	ft
Su 100	73	12.93	0.0035	0.69	1.6709	1.2212	0.0125	0.9875
Su 100	75	8.97	0.0040	1.00	1.7091	1.1881	0.0079	0.9921
Su 16	81	3.60	0.0035	0.04	1.6768	1.2088	0.0500	0.9500
Su 16	82	7.47	0.0050	0.23	1.7065	1.1878	0.0056	0.9944
Su 3	93	7.47	0.0050	0.23	1.7198	4.2655	0.0150	0.9850
Su 47	102	4.32	0.0040	0.09	1.6419	6.4812	0.0205	0.9795
Su 6	105	22.37	0.0040	1.20	1.7722	3.9625	0.0090	0.9910
Su 61	108	5.18	0.0350	0.07	1.6217	3.7038	0.0260	0.9740
Su 61	109	8.97	0.0040	0.09	1.7479	3.8880	0.0200	0.9800
Su 65	113	5.20	0.0060	0.02	1.5959	8.0740	0.0060	0.9940

In the table, r_max/r_min—maximum/minimum pore throat radius; r_avgs—average pore throat radius; D_f-aperture fractal dimension; D_τ-tortuosity fractal dimension; f_z-straight pore ratio; f_t-trapezoidal pore ratio.

systematically compares the water-saturation data sources obtained by the three methods, including: (1) core sealing test benchmark value (closed coring method), (2) nuclear magnetic resonance spectrum interpretation value (NMR technology), and (3) mathematical modeling inversion value (new water-bearing model). To verify the prediction accuracy of the model, a two-layer error analysis system is constructed (as detailed in

Table 5. Comparison of three methods for determining water saturation.

Well Number	Core Number	Water Saturation Measured by Closed Coring (%)	NMR Analysis Results		Model Calculation Results		Water Saturation Calculation Error Analysis
			Sw (%)	Swi (%)	Sw (%)	SWi (%)	Error 1	Error 2
Su 100	73	43.20	42.50	25.06	43.76	40.7	1.30	2.96
Su 100	75	47.23	45.50	34.32	46.81	43.47	−0.89	2.88
Su 16	81	52.25	50.23	57.91	53.6	49.61	2.58	6.71
Su 16	82	43.26	45.50	35.30	48.07	44.63	11.12	5.65
Su 3	93	43.56	42.78	38.28	48.25	44.78	10.77	12.79
Su 47	102	49.02	48.52	33.61	51.73	47.94	5.53	6.62
Su 6	105	40.00	40.24	23.89	41.29	38.38	3.23	2.61
Su 61	108	48.77	47.10	29.35	49.48	45.98	1.46	5.05
Su 61	109	45.23	42.56	31.77	47.31	43.92	4.60	11.16
Su 65	113	46.88	48.10	32.97	49.71	46.15	6.04	3.35

): error 1 represents the absolute error value between the characterization model and the measured value of the sealed core (0.89–11.27%), and error 2 reflects the matching discreteness of the model and the nuclear magnetic analysis data (2.61–12.79%). According to the double cross-validation results, it is proved that the model has the ability to predict water. Saturation in the tight sandstone environment meets the engineering accuracy requirements and can provide a theoretical tool for the prediction of reservoir bound water distribution and the optimization of development plans.

In order to verify the applicability of the gas–water two-phase relative permeability model, this paper selects the phase permeability test data of Su 61-109 core under different displacement pressure differences (ΔP gradient: 1.25 → 3.67 MPa) for model verification. Comparing the calculation results of the gas–water two-phase relative permeability model with the measured data of the core (as shown in Figure 5), it can be seen that the gas–water phase permeability curve obtained by numerical simulation has highly consistent evolution characteristics with the experimental test results, and its morphological consistency is as high as more than 92%; and the difference rate of the inflection point position of the two-phase relative permeability is less than 8%, which proves that the model is reliable in the simulation of dynamic seepage process. It is worth noting that the increase in displacement pressure difference causes the gas–water phase permeability curve to show an overall leftward shift trend, which is specifically manifested in the coordinated changes in the decrease in gas phase relative permeability (0.71 → 0.62) and the increase in water phase relative permeability (0.78 → 0.9). At the same time, the gas–water isotonic point saturation is reduced from about 67% to about 62%, and the range of the two-phase co-permeability zone is significantly expanded.

Through the analysis of the model mechanism, it can be seen that under the constraint of constant reservoir water saturation, when the gas–water distribution in the pore throat reaches a dynamic equilibrium, the increase in the displacement pressure difference mainly enhances the water phase seepage capacity through two paths: (1) The high pressure difference causes the bound water film thickness to decrease, effectively releasing the restricted water flow space; (2) The enhanced hydrodynamic force (enables the water phase to overcome greater capillary resistance, thereby activating more retained water in the micropores to participate in the flow. This physical process fully explains the evolution law of the phase permeability characteristics observed in the experiment [52].

5.3. Model Calculation

The formation of water saturation in tight sandstone reservoirs is affected by the coupling of multiple physical field parameters [53,54]. The saturation-interpretation model constructed based on fractal theory reveals that reservoir porosity, pore-structure fractal dimension (D_f = 1.677), thermodynamic conditions (T = 25–85 °C) and reservoir-forming dynamics together constitute a saturation-control-factor system. This study uses this model as a theoretical framework and combines the measured data of 73 core (φ = 0.067%, K = 0.09 mD) to systematically quantify the interactive influence mechanism of reservoir-temperature field, displacement-pressure field (P = 8 MPa–15 MPa), and rock-mechanics parameters (Poisson’s ratio ν = 0.25, elastic modulus E = 1.05 × 10³ MPa) on water saturation (S_w).

5.3.1. Temperature

Specifically, it can be seen from Figure 6 that: (1) Every 20 °C increase in temperature gradient causes S_w to decrease by about 4.5% (dominated by the thermal expansion effect of pore water); (2) An increase in displacement pressure of 1 MPa can reduce S_w by 6.3% (the energy required to break through the capillary threshold pressure is reduced).

At the same time, according to the quantitative analysis results of Figure 6, under the thermodynamic equilibrium condition of constant reservoir temperature (such as T = 85 °C), water saturation (S_w) and displacement pressure difference (ΔP) show a significant negative correlation; Based on the fractal theory, Liu et al. [55] (2017) calculated the irreducible water saturation under different pressure and temperature conditions, and also concluded that the higher the temperature, the greater the pressure difference, and the lower the irreducible water saturation of the core. That is, the increase in displacement pressure difference will lead to a decrease in water saturation, and its mechanism mainly includes two aspects: (1) The increase in pressure difference reduces the critical pore throat radius (so that more bound water in small-sized pore throats is activated and displaced), and (2) The increase in gas drive flow rate in larger pore throats causes shear thinning of the water film, effectively reducing the volume of capillary retained water. In addition, under the condition of constant displacement pressure difference, the increase in temperature weakens the stability of the water film by reducing the viscosity of water (considering the wettability characteristics of the hydrophilic rock model), and finally causes the water saturation to decrease; it is also worth noting that the increase in displacement pressure difference will amplify the effect of temperature on water saturation; that is, the regulation effect of temperature change on water saturation is more significant under high pressure difference conditions. These responses jointly determine the distribution characteristics of the water phase in the reservoir. The model proposed in this paper is derived based on the hydrophilic rock model, so by the same token, its water saturation is negatively correlated with temperature; that is, the water saturation decreases with increasing temperature [56].

5.3.2. Effective Stress

Combined with the experimental data analysis and Figure 7, it can be seen that the reservoir water saturation is significantly positively correlated with the effective stress [55], and its mechanism can be divided into two coupling processes: (1) pore–capillary dynamic response: the rise in the effective stress induces the compression of the pore radius of the rock, which leads to the contraction of the actual cross-sectional area of the capillary bundles, and the process makes the inner diameter of the capillary bundles lower than the critical threshold; (2) the effect of the weakening of the repulsion efficiency: under the constant differential pressure of the repulsion Weakening effect of replacement efficiency: under constant replacement pressure difference, the reduction in the cross-sectional area of capillary bundles directly reduces the proportion of capillary tubes reaching the critical radius, thus restricting the fluid mobility—the proportion of bound water dominated by capillary resistance expands, which ultimately manifests itself in the systematic increase in water saturation in the reservoir. Essentially, the phenomenon reflects the dynamic equilibrium between porous media deformation and capillary force competition in stress-sensitive formations. This understanding is mutually confirmed with the research conclusions of YAO et al. [57] They found that as the effective stress increases, the pore connectivity becomes worse, and some pores even become inaccessible pores, which will also lead to an increase in core water saturation.

5.3.3. Throat Radius

From the simulation results of the gradient of pore-throat ratio parameter in Figure 8 below, it can be seen that the water saturation of the reservoir is significantly negatively correlated with the ratio of the minimum/maximum pore-throat radius under a specific replacement pressure difference. CHEN et al. [58] constructed the irreducible water saturation model by introducing the fractal theory. Under the condition of a certain temperature, it is concluded that the larger the maximum displacement pressure (the smaller the critical pore-throat radius), the smaller the irreducible water saturation of the rock, and the ratio of the two shows a significant negative correlation. By fixing the maximum pore-throat radius and regulating the minimum pore-throat radius (as a pore microstructure parameter), the study reveals that the pore system exhibits two key evolutionary features when the ratio decreases: (1) microstructural anisotropy: decreasing the ratio will exacerbate the non-homogeneity of the pore structure, resulting in an exponential increase in pore-throat surface area, which will significantly strengthen the wettability and adsorption of the pore surface to the molecules of aqueous phases; (2) the critical sieving effect: the sub-critical pores will become more and more heterogeneous as the ratio decreases. Ratio shrinks, the proportion of subcritical pore throats (radius less than the critical value) in the reservoir expands, and the capillary resistance formed inside these pores exceeds the driving force conditions, forming a water-phase retention zone [55]. This law quantitatively portrays the competition mechanism between the capillary force and the replacement force in the low-permeability reservoir from the perspective of the pore structure coupled with the fluid dynamic equilibrium.

5.3.4. Rock Mechanics Parameters

Liu et al. [59] (2019) obtained that with the increase in water saturation, the Lame constant, bulk modulus, Poisson’s ratio, longitudinal wave impedance, longitudinal wave velocity and longitudinal/transverse wave velocity ratio of rock samples increase with the increase in water saturation, and the variation range is large. The lower the elastic modulus and the larger the Poisson’s ratio, the higher the water saturation of the reservoir. According to the results of numerical simulation (Figure 9, Figure 10 and Figure 11), the rock mechanical parameters and water saturation show a clear correlation: the lower the modulus of elasticity and the larger the Poisson’s ratio are, the higher the water saturation of the reservoir is. The intrinsic mechanism is the coupled stress-deformation control mechanism—the reduction in elastic modulus will weaken the deformation resistance of the rock skeleton. The pore structure will be more prone to plastic compression during stress loading, resulting in the contraction of the equivalent cross-sectional area of capillary bundles and the increase in the proportion of subcritical pore throats. A high Poisson’s ratio will strengthen the lateral strain effect and induce the geometrical elongation of capillary bundles. High Poisson’s ratio enhances the lateral strain effect, induces the geometrical elongation and specific surface area expansion of capillary bundles, and significantly enhances the wetting and retention of the aqueous phase on the pore wall. It is worth noting that, in the low effective stress domain (e.g., when the injection and extraction pressure difference is small) [60,61,62], the modulation amplitude of elastic modulus and Poisson’s ratio on water saturation is below the sensitivity analysis threshold (relative offset < 5%), and its effect can be included in the second-order modification term of the geological-engineering multifactorial coupled model, or can be ignored. Furthermore, regarding the selection of mechanical parameters, this study primarily focuses on deformation parameters (i.e., elastic modulus and Poisson’s ratio) because they directly govern the pore volume compressibility and fluid flow characteristics within the reservoir. While rock strength parameters (such as compressive strength and cohesion) are critical for determining reservoir stability and failure boundaries, their influence is less dominant during the production process considered in this model, which is primarily characterized by elastic deformation. Future work could further integrate these strength parameters to characterize the water saturation evolution under plastic deformation or rock-failure conditions.

6. Conclusions

Based on fractal theory and trapezoidal pore throat model, this study successfully constructed a dynamic interpretation model of water saturation suitable for tight sandstone gas reservoirs. By systematically introducing pore fractal dimension (D_f), tortuosity fractal dimension (D_T) and trapezoidal factor (φ_i), the model realizes the fine quantification of the heterogeneity of complex pore throat structure, and couples the dynamic effects of multi-physical fields such as formation temperature, effective stress and displacement pressure, which breaks through the theoretical limitation of the traditional Archie formula in describing the non-Archie phenomenon of tight reservoirs.

Model validation and mechanism analysis show that: (1) The model has high-precision prediction ability. Based on the verification results of 10 sets of sealed coring samples in Sulige gas field, the absolute error between the calculated water saturation and the measured value of the core is controlled within the range of 0.89–11.27%, which is also well matched with the results of nuclear magnetic resonance (NMR) analysis, and is significantly better than the error level of the traditional method. (2) The multi-factor coupling mechanism is clear. For every 1 MPa increase in displacement pressure difference, the water saturation can be reduced by about 6.3%. For every 20 °C increase in temperature, the water saturation can be reduced by about 4.5%. The increase in effective stress increases the water saturation by up to 8.2 percentage points by compressing the pore throat. (3) The dynamic seepage simulation is reliable, and the gas–water two-phase relative permeability curve is highly consistent with the experimental test results (morphological consistency > 92%), which accurately describes the dynamic law of isotonic point saturation with displacement pressure.

This model provides a reliable theoretical tool for the accurate evaluation of water saturation in tight sandstone gas reservoirs, and has important guiding significance for reservoir fluid identification, movable water prediction and development scheme optimization.