Heterogeneity of Pore and Fracture Structure in Tight Sandstone Using Different Fractal Models and Its Influence on Porosity–Permeability Variation

Abstract

The study of pore structure in low-permeability sandstone uranium deposits has become a key factor in the profitability of uranium mining. In this paper, pore and fracture distribution in the target sandstone were determined by using mercury injection parameters. Single and multi-fractal models are used to calculate the heterogeneity of pore and fracture volume distribution. Moreover, the correlation between compressibility and the heterogeneity of pore distribution has been studied. The results are as follows. (1) All the samples can be divided into three types by using maximum mercury injection volume and mercury withdrawal efficiency. Type A is represented by a lower maximum mercury injection volume (less than 0.5 cm³·g⁻¹) and a higher mercury withdrawal efficiency (larger than 25%). The volume percentage of pores whose diameter is less than 100 nm and 100~1000 nm in type A samples is larger than that of type B and C samples since in this type of sample, micropores are developed. (2) The fractal dimension value assessed using the Menger model has a good linear relationship with the thermodynamic model, which indicates that the abovementioned models have good consistency in characterizing the pore distribution of tight sandstone. Multi-fractal results show that the lower pore volume in the selected samples controls the heterogeneity of pore distribution in the overall sample. (3) As the effective stress increases, the permeability damage rate gradually increases in a power exponential equation. The correlation between porosity and compressibility is weaker, indicating that only a portion of the pore volume in the sample provides compression space. As the pore volume of 100~1000 nm increases, the compressibility decreases linearly, indicating that pore volumes larger than 1000 nm provide compression space for all the selected samples.

1. Introduction

The sandstone-type uranium resources in China account for 14.2% of total resources. This is a low-cost type of uranium that can be mined. In situ leaching of uranium has become an important method for uranium mining and metallurgy in China [1,2]. However, the lower porosity and permeability of tight sandstone have limited the efficient development of sandy mudstone uranium deposits in China [3,4]. The related literature indicates that the heterogeneity of sandstone pore structure restricts the seepage process of in situ leaching mining. The lower permeability coefficient of sandstone uranium deposits leads to lower single well injection volume, production capacity, dense well network, and small single well control area during the in situ leaching development of uranium deposits. Therefore, the study of pore structure in low-permeability sandstone uranium deposits has become a key factor in the profitability of uranium mining [5,6].

The relevant literature indicates that a large number of experimental tests have been used to quantitatively characterize the pore and fracture structure of tight sandstone, including argon ion polishing scanning electron microscopy, high-pressure mercury injection (HPMI), low-temperature liquid nitrogen and carbon dioxide adsorption testing (LT N₂ GA and CO₂ GA), among others [3,7,8]. Among them, HPMI tests have the advantages of being fast, simple, and having a wide range of test pore diameter, and have become the most commonly used method for characterizing the pore and fracture structure of tight sandstone [9,10,11,12]. The relevant literature indicates that the pore structure of tight sandstone in sandstone uranium deposits exhibits bimodal characteristics, and the complexity of large pore structures is significantly greater than that of small pore structures [13]. Additionally, tight sandstone gas has been taken as the study target, and HPMI tests have been utilized to study the pore structure of tight sandstone in different study areas [14]. Tight sandstone reservoirs include wide pore throats ranging from nanometer to micrometer scales, and complex pore geometries and throat structures exist. The micropore throat structure is considered an important factor affecting the macroscopic reservoir quality and fluid flow of tight sandstone [15,16,17].

Although HPMI tests could characterize pore throat distribution and connectivity, quantitative characterization of pore heterogeneity in tight sandstone has not been achieved. The fractal theory has been introduced to study the pore structure of tight sandstone by studying pore throat volume and specific surface area at different scales [12,13,18]. The multifractal theory divides entire pores into certain pore intervals, achieving a separate description of pore heterogeneity for divided pore intervals [19,20]. Single-fractal theories have been used to study the fractal dimension of tight sandstone, including the Menger and Sierpinski models. The results show that the sponge model can be used to characterize the heterogeneity of sandstone pore volume distribution, while the Sierpinski model can characterize the heterogeneity of pore-specific surface area distribution. As the content of mineral components increases, the fractal dimension of macropores gradually increases, while the fractal dimension of micropores tends to stabilize [9,21,22,23]. Multi-fractal theories are used to characterize the pore structure heterogeneity. The related literature has shown that the pore size distribution of tight sandstone exhibits multifractal characteristics, and the singular intensity range (Δα) can be used as an indicator to characterize pore throat heterogeneity. In addition, the throat of the sample exhibits larger heterogeneity than the pores [19,24,25]. It is worth noting that there are differences in the variation patterns of single and multiple fractal dimension values for pores with the same aperture, which also affects the promotion of the fractal theory in characterizing pore fracture structures in tight sandstone reservoirs.

Above all, the quantitative characterization of pore fracture structures in tight sandstone has been studied in the relevant literature. However, there are still shortcomings in this field. Firstly, all the fractal models have differences in the fractal structure of the same sample, and the applicability of fractal models to assess the heterogeneity of pore structure needs to be further studied. Secondly, dynamic variation in porosity and permeability under the effect of pore and fracture structure in tight sandstone needs to be further studied.

A total of 20 tight sandstone samples were collected from the same uranium mine and were used for scanning electron microscopy and HPMI test experiments. Based on those experiments, pore and fracture distribution in the target sandstone were determined. Based on mercury injection parameters (mercury removal efficiency and maximum mercury injection volume), single and multi-fractal models have been used to calculate the heterogeneity of pore and fracture volume distribution. Based on this, the correlation between the fractal dimension values of different models has been studied. Additionally, the pore fracture compressibility of typical samples was calculated by using a pressure pore permeability experiment. The correlation between compressibility and the heterogeneity of pore distribution has been studied. This study aims to compare the applicability of different fractal models to the heterogeneity of pore and fracture structure distribution in tight sandstone and to determine the optimal model for characterizing the heterogeneity of pore and fracture distribution in tight sandstone in the study area. This research’s result can provide a theoretical basis for the efficient development of tight gas in the future.

2. Experimental Methods and Related Theories

2.1. Sample Preparation and Experimental Methods

The tight sandstone samples collected in this experiment were collected from well X, and the basic information of all the samples is shown in

Table 1. Basic information of all the samples by pore structure and mineral contents.

Sample Number	Pore Volume (cm3·g−1)	Porosity (%)	Permeability (mD)	Mercury Withdrawal Efficiency (%)	Mineral Content
					Quartz	Feldspar	Calcite	Clay Mineral
1	0.191	4.6	0.0315	43.30	8	91
2	0.469	4.1	0.0487	38.53	33	63		2
3	0.253	2.1	0.0705	37.78	13	79		1
4	0.323	2.6	0.274	32.27	3	90
5	0.391	3.1	0.0725	42.50	20	70		2
6	0.542	4.4	0.225	37.47	26	73
7	0.202	2.1	0.330	44.37	6	87
8	0.844	6.8	0.923	31.12	7	90		2
9	1.126	8.8	0.627	37.76	10	81
10	1.02	8.3	0.300	37.6350	5	94
11	0.899	7.3	0.155	39.0789	4	94
12	0.974	7.9	1.96	37.7153	11	68
13	0.943	7.6	0.786	37.6354	8	75		1
14	1.068	11.6	0.406	11.5145	1	98
15	0.785	9.0	1.81	8.19569	9	87	1
16	0.918	8.2	0.762	16.4799	31	61
17	1.057	8.6	1.88	22.6168	7	45		1
18	1.44	12	5.56	18.9480	10	83		3
19	0.952	7.8	0.387	21.1926	10	79
20	1.045	8.1	3.02	18.1180	42	31		24

. The experimental steps are as follows. A total of 20 coal samples were taken for observation, and X-ray diffraction, HPMI experiments, and overburden porosity and permeability testing were conducted.

A rock porosity vacuum pressure saturation device was used for conventional rock property testing, rock permeability testing, and physical property measurement. The mercury intrusion test was conducted using an Auto Pore 9250 II mercury intrusion porosimeter. Prior to measurement, all the samples were washed with ethanol and placed in a 110 °C oven. The samples were vacuum dried under high-temperature conditions to a constant weight, and free and bound water from the sample was removed. Then, pressure was applied in different stages to inject mercury into the sample until the highest pressure that prevented mercury from entering the sample, and then the pressure was gradually reduced. The amount of mercury injected and removed at different pressures was measured. In the experiment, the surface tension was 0.48 N/m, and the contact angle was 140 °C [26]. There are two commonly used pore partitioning methods, IUPAC and Hodots. The IUPAC method divides pores into micropores (<2 nm), mesopores (2~50 nm), and macropores (>50 nm). This method is too general when dividing pores larger than 50 nm, making it difficult to achieve a refined description of the pore volume at that scale. Therefore, the Hodot classification method is adopted in this paper, which divides pore fracture into small pores (<100 nm), mesopores (100–1000 nm), and large pores and micro-fracture (>1000 nm).

X-ray diffraction whole-rock analysis allowed us to obtain the relative content of mineral components and clay minerals of all the samples. A D8 FOCUS type X-ray diffraction Instrument was used. An environment with a temperature of 20 °C and a relative humidity of 70% was created. The sample was ground into 200 mesh particles, saturated with ethylene glycol at 60 °C for 8 h, and heated at 500 °C for 2.5 h for mineral composition analysis [27].

To study the stress sensitivity of different typical samples and study the influencing factors and mechanisms of stress sensitivity, a tight sandstone stress sensitivity experiment was conducted. The detailed description is as follows. Firstly, a 25 ×50 mm columnar sample is prepared for experimental testing after drying for 24 h. The experimental test instrument is a YC-4 type overlay pressure pore permeability tester. Secondly, an experiment was conducted in accordance with the petroleum and natural gas industry standard SY/T6385-2016 [28] “Method for determination of rock porosity and permeability under overburden pressure”. This was used as the test gas, and pore pressure was kept constant at 1 MPa. The confining pressure values of all the samples were sequentially increased using a flow pump (2, 5, 8, 10, 15, and 50 MPa), and the equilibrium time at each pressure point was stabilized at more than 30 min. The porosity and permeability of all the coal samples were measured at different pressure points at 20 °C [29].

2.2. Related Theories

The fractal models used in high-pressure mercury intrusion testing include two types, single and multifractal models. Among them, the single fractal model includes the Menger, Sierpinski, and thermodynamics models. In general, the fractal dimension ranges from 2 to 3. The larger the fractal dimension, the more complex the reservoir space is, and the larger the reservoir heterogeneity is. The smaller the fractal dimension, the better the storage and sorting properties are, and the weaker the reservoir heterogeneity is. The equations are as follows [20].

The Menger (M) model is shown in Equation (1).

$$\mathrm{l}\mathrm{g}\left(\mathrm{d}v/\mathrm{d}p\right)\propto \left(D-4\right)\mathrm{l}\mathrm{g}\left(p\right)$$

(1)

where D_M is the fractal dimension, dimensionless; p is the injective pressure, in MPa; and v is the total pore volume, in cm³·g⁻¹.

The Sierpinski (S) model is shown in Equation (2).

$$\mathrm{l}\mathrm{n}\left(v\right)=\left(3-D\right)\mathrm{l}\mathrm{n}\left(p-p_t\right)+\mathrm{l}\mathrm{n}a$$

(2)

where v is the total pore volume when injective pressure is value x, in mL; p is the injective pressure, in MPa; p_t is the threshold pressure, in MPa; Ds is the fractal dimension, dimensionless; and a is constant.

The thermodynamics model is shown in Equation (3) [13,14].

$$\mathrm{l}\mathrm{n}\left(w/r^2\right)=\mathrm{D}\mathrm{l}\mathrm{n}\left(v^{1/3}/r\right)+\mathrm{C}$$

(3)

where v is the total pore volume when the injective pressure is x, in mL; r is the pore diameter, in nm; and D_T is the slope of fractal curves, which represents pore surface fractal dimension, dimensionless.

Multi-fractal theories are used to characterize the pore structure heterogeneity. The related literature has shown that the pore size distribution of tight sandstone exhibits multifractal characteristics, and the singular intensity range (Δα) can be used as an indicator to characterize pore throat heterogeneity. In addition, the throat of the sample exhibits larger heterogeneity than the pores [2].

When performing multifractal analyses, the pressure or relative pressure interval of mercury intrusion, N₂, and CO₂ adsorption experiments are taken as the total interval, which is divided into N intervals with a scale of ɛ by dichotomy. Quality probability function p_i (ɛ) of the i-th interval is defined as the exponential function of scale.

$$p_i\left(\mathsf{\epsilon }\right)~{\mathsf{\epsilon }}^{a_i}$$

(4)

where a_i is the singularity index, which can reflect the local singularity intensity. The higher its value, the higher the smoothness, regularity, or uniformity of the data. On the contrary, the lower its value, the greater the degree of data change or the larger the heterogeneity.

For the box with multifractal behavior, the number of boxes N(ɛ) increases exponentially with the increase of scale ɛ

$$N_a\left(\mathsf{\epsilon }\right)~{\mathsf{\epsilon }}^{-f\left(\alpha \right)}$$

(5)

The number of boxes with singular strength in the mass function between α and α + dα is defined as N_α(ɛ). The multifractal singular spectrum is expressed as f(α), which is the fractal dimension of a subset with the same singularity index. The equation proposed by CHH-ABRA and JENSEN is used to calculate α and f(α).

$$\alpha \propto \left[{\sum }_{i=1}^{N\left(\epsilon \right)}{\mu }_i(q,\epsilon )ln{p}_i\left(\epsilon \right)\right]/ln\epsilon$$

(6)

$$f\left(\alpha \right)\propto \left[{\sum }_{i=1}^{N\left(\epsilon \right)}{\mu }_i(q,\epsilon )ln{\mu }_i(q,\epsilon )\right]/ln\epsilon$$

(7)

where the family of probability measures is expressed as μ_i(q,ε).

$${\mu }_i(q,\epsilon )={\displaystyle \frac{p_i^q\left(\epsilon \right)}{{\sum }_{i=1}^{N\left(\epsilon \right)}{p}_i^q\left(\epsilon \right)}}$$

(8)

The partition function Ꭓ(q,ε) is defined as

$$\mathrm{Ꭓ}(q,\epsilon )={\sum }_{i=1}^{N\left(\epsilon \right)}{p}_i^q\left(\epsilon \right)$$

(9)

where q is the order of statistical moment (−∞ < q < +∞), and the value range of Q in this paper is [−10, 10]. When q >> 1, the information of the high probability area (dense area/high-value area) is amplified. When q << −1, the information of the small probability region (sparse region/low-value region) is amplified.

Among the multifractal parameters, D and D₋₁₀−D₀ are used for the quantitative characterization of pore structure heterogeneity and pore connectivity. The concentration degree of the pore volume distribution can be indicated by D₋₁₀−D₀. The larger the value, the larger the variability of the pore space, the greater the fluctuation of local distribution of pore volume, the narrower the distribution interval, the greater the difference of pore size distribution, and the more complex the pore structure. The larger the D, the stronger the correlation between different pore sizes, the higher the degree of uniformity, and the better the pore connectivity.

The permeability damage rate and compressibility were assessed using overburden porosity and permeability tests.

To better characterize the influence of confining pressure on the pore fracture system, the parameter S_i/S₀ is defined to characterize the stress sensitivity of the pore fracture.

Dimensionless parameter S_pfi is defined as

S_pfi = S_i/S₀

(10)

where S_i and S₀ are dimensionless T₂ peak areas at confining pressures of P_i and P₀. In this section, P₀ means that the confining pressure is 0 when i is equal to 1. At 5, the confining pressure increases from 5 Mpa to 10, 15, and 20. At 50, respectively.

According to Equation (10), the higher the SPFI value, the smaller the compression space of the corresponding hole and fissure, and the weaker the stress sensitivity.

Compressibility is an important parameter used to characterize the change in porosity and permeability of coal reservoirs. For coal reservoirs, it is the pore compression rate, that is, the pore fracture compressibility [30].

$$C_f={\displaystyle \frac{1}{{\varnothing }_f}}({\displaystyle \frac{ꝺ{\varnothing }_f}{ꝺP}})P$$

(11)

where C_f is the compressibility of pores and fissures, in MPa⁻¹. ∅_f is the fracture porosity, in %. P is the pore pressure, in MPa.

3. Results and Discussions

3.1. Pore Types and Sample Types Assessed Using SEM and HPMI Tests

The identification of pore types in tight sandstone reservoirs through casting thin sections and SEM (Figure 1) resulted in the development of primary intergranular pores. There are seven types of pores, including pores, residual intergranular pores, intergranular dissolved pores, intragranular dissolved pores, heterogeneous pores, clay mineral intergranular pores, and micro-fractures. Among them, residual intergranular pores, intergranular dissolved pores, and intragranular dissolved pores are the most frequently developed. When the clay content increases, clay mineral intergranular pores are more frequently developed (Figure 1a–c). Intergranular dissolved pores are formed by unstable minerals, such as rock debris and feldspar, under the action of organic acids, with a pore size generally ranging from 0.01 to 0.20 mm. They mainly exist in the form of quartz secondary enlarged edges, with a small number of clay minerals, mainly illite aggregates, filling the intergranular spaces (Figure 1d–f). The size of intergranular pores between clay mineral particles is larger, while interlayer gaps are usually narrow, usually several tens of nanometers. Micro-fractures in the study area are also relatively frequently developed. As infiltration channels, fractures have a significant improvement effect on the physical properties of tight sandstone reservoirs (Figure 1g–i). However, it is difficult to determine whether they are primitive developments or artificially generated samples.

Conventional porosity and permeability tests indicate that the porosity and permeability of each sample are 2.1~12% and 0.03~3.01 mD. The reason why some samples have higher porosity and permeability is that the volume of micro-fractures in these samples is relatively larger (Figure 1g–i). The mineral content of all the samples is quartz and feldspar, with an average of 13.2 and 76.6%, respectively. In general, porosity, permeability, and mineral contents are different in all the samples. Classification of sample types is the basis for the refinement of pore fracture structures in tight sandstone reservoirs.

Based on pore structure parameters assessed using HPMI tests, the sample type has been determined by using maximum mercury injection volume (MMIV) and mercury withdrawal efficiency (MWE). The related literature has shown that maximum mercury injection volume represents total pore volume, and mercury withdrawal efficiency represents pore connectivity. The abovementioned parameter could comprehensively characterize pore structure (pore volume and pore connectivity). Figure 2 shows that all the samples can be divided into three types. Type A is represented by lower MMIV (less than 0.5 cm³·g⁻¹) and higher MWE (larger than 25%) values, indicating that the pore volume of this type of sample is relatively lower, but the pore connectivity is good. Type B is represented by larger MMIV (larger than 0.5 cm³·g⁻¹) and higher MWE (larger than 25%) values, indicating that the pore volume of this type of sample is relatively higher and the pore connectivity is good. Differing from type B, type C is represented by larger MMIV (larger than 0.5 cm³·g⁻¹) and lower MWE (less than 25%) values, indicating that the pore volume of this type of sample is relatively larger, and the pore connectivity is poor.

3.2. Pore Structure Heterogeneity Assessed Using Different Fractal Models

The mercury injection–removal curves of all the samples are shown in Figure 3. Figure 3a shows that there is a significant hysteresis loop between the mercury injection and mercury removal curves o type A samples, and the mercury removal curve during the higher-pressure stage is close to horizontal, indicating that pore connectivity is good. In this type of sample, a pore whose diameter is 300~1000 nm is the advantageous diameter, the pore diameter distribution curve is relatively uniform, and pores of all diameters have developed (Figure 3b). Figure 3c,d shows that there is a significant hysteresis loop between the mercury injection and mercury removal curves of type B samples, and the uneven distribution of pore fracture in this type of sample is higher than that in type A samples. Comparing type A and B, mercury injection and removal curves are almost identical when the injective pressure is larger than 1 MPa, pores whose diameter is 100 nm have the advantageous diameter, and the pore diameter distribution curve is relatively heterogeneous. The above results indicate that there are significant differences in the pore diameter distribution of different types of samples.

The results in Figure 4 show that the volume percentage of pores whose diameter is less than 100 nm and 100~1000 nm in type A samples is larger than that of type B and C samples, since in this type of samples, micropores are developed. However, the volume percentage of pores whose diameter is larger than 1000 nm in type B and C samples is larger than that of type A samples, since in those types of samples, macropores are developed, and the porosity/initial permeability of the two types of samples is larger (Table 1).

Fractal curves were obtained using the Menger model (Figure 5). The results indicate that fractal curves of all samples can be fitted as a line, and the average fitting degree is higher than 85%, indicating that pore distribution curves of all samples exhibit fractal characteristics. Note that fractal curves of type A samples can be divided into two parts when the injective pressure is 1 MPa (lop P is 0), indicating that the pore volume distribution of pores smaller than 100 nm and pores larger than 100 nm in type A sample exhibits different fractal characteristics. To study the fractal characteristics of different types of samples, only one fractal dimension value is calculated for type A samples. Figure 5d shows that the fractal dimensions of type A, B, and C samples are 2.1~3.0, 2.4~2.5, and 1.6~2.1, respectively. Therefore, the fractal dimension of type A samples is larger than that of types B and C, indicating that the pore volume distribution heterogeneity of this type of sample is larger than that of the other samples. The reason is that the volume of pores whose diameters are larger than 10,000 nm in type A samples is smaller, and only pores smaller than 10,000 nm are developed. The shorter pore size range results in increased heterogeneity of pore volume (Figure 5d).

Fractal curves were obtained using the Sierpinski model (Figure 6). The results indicate that the fractal curves of all samples can be fitted as a line, and the average fitting degree is lower than 70%, indicating that fractal characteristics of pore volume distribution are not obvious. Note that fractal curves of all the samples can be divided into two parts when ln P is 2, indicating that the pore volume distribution of pores smaller than 100 nm and pores larger than 100 nm of all the samples exhibits different fractal characteristics. Figure 6d shows that the fractal dimensions of type A, B, and C samples are 2.4~2.6, 2.5~2.6, and 2.5~2.7, respectively. Therefore, the fractal dimension of type A samples is smaller than that of types B and C, indicating that the pore volume distribution heterogeneity of this type of sample is smaller than that of the other samples. The results are different from the results in Figure 5. The reason is that the fitting degree of the fractal curve calculated by this model as a line segment is lower, resulting in low calculation accuracy. When the pore volume distribution is complex, segmented fitting should be performed for all the pores.

Fractal curves were obtained using the thermodynamics model (Figure 7). The results indicate that fractal curves of all samples can be fitted as a line, and the average fitting degree is higher than 95%, indicating that pore distribution curves of all samples exhibit fractal characteristics. Differing from the Menger and Sierpinski models, the fractal curve obtained using this model has the highest fitting degree. Figure 7d shows that the fractal dimensions of type A, B, and C samples are 2.4~2.7, 2.5~2.6, and 2.4~2.6, respectively. Therefore, the fractal dimension of type A samples is larger than that of types B and C, indicating that the pore volume distribution heterogeneity of this type of sample is larger than that of the other samples. The reason is that the volume of pores with diameters larger than 10,000 nm in type A samples is smaller, and only pores smaller than 10,000 nm are developed (Figure 7d).

The fitting curve of the pore partition function lg(q, ε) and scale ε shows that there is a strong linear relationship (Figure 8a), satisfying the requirement of scale invariance, which means that the macropores, mesopores, and micropores of all the samples exhibit multifractal characteristics. When q is less than 0, the fitting curve of the partition function has a negative slope. When q is larger than 0, the fitting curve of the partition function shows a clear positive slope of linear behavior and tends to be dense, indicating that the pores are concentrated in the small-scale pore size range. Compared to larger pores, the double logarithmic curve of micropores is sparser, indicating there are relatively stronger differences in pore size distribution within the corresponding scale range (Figure 8a).

There are significant differences among the Dq-q curves of all samples, with wide branches (q is less than 0) representing sparse areas (lower pore volume areas), and narrow branches (q is larger than 0) representing dense areas (higher pore volume areas). The left wide and right narrow shape of the Dq−q curve indicates that the pore size distribution of the samples is uneven and multiscale. The drastic changes in the left branch indicate that the heterogeneity of pore size distribution is more significant in smaller pore size ranges, while the right curve shows a smaller amplitude of change and a linear trend, indicating that the pore size distribution of larger pores in larger pore size ranges is more uniform (Figure 8b–d). In a single sample, D₋₁₀−D₀ is larger than D₀−D₁₀, indicating that lower pore volume affects the overall heterogeneity of the pore diameter distribution.

The results show that there are differences in the multifractal characteristics among the different types of samples. D₋₁₀−D₀ of type B samples is significantly larger than that of the other two types (Figure 9a). Lower pore volume heterogeneity in these samples is higher than that in the other two types. Different from D₋₁₀−D₀, the difference of D₀−D₁₀ among the three types of samples is smaller, and the distribution uniformity of higher pore volume among different samples is smaller (Figure 9b). The abovementioned results show that the correlation between D₋₁₀−D₀ and D₋₁₀−D₁₀ is obvious. This indicates that the pore distribution heterogeneity of all samples is affected by lower pore volume, which results in the non-uniformity of the pore distribution under multiple classifications of type B samples being higher than that of the other two types (Figure 9c).

3.3. Applicability of Different Fractal Models in Characterizing Pore Fracture Structures

Clarifying the applicability of different models in characterizing the heterogeneity of pore distribution in tight sandstone is the basis for studying the fractal characteristics of tight sandstone.

Figure 10a shows a significant negative correlation between the fractal dimension obtained using the S and that obtained using the M model. Similar to Figure 10a, there is a clear linear relationship between the fractal dimension obtained using the S model and that obtained with the thermodynamic model (Figure 10b). The reason for this result is that the fractal curve fitting based on the S model is poor, which leads to a negative correlation between the fractal characteristics of the different models. Figure 10c shows that the fractal dimension value based on the M model has a good linear relationship with the thermodynamic model, which indicates that the M model and the thermodynamic model have good consistency in characterizing the pore distribution of tight sandstone. Multiple different multifractal correlations (Figure 10d–f) showed that D₋₁₀-D₀ and D₀-D₁₀ had a certain correlation, but the correlation was weak. However, the fitting degree of other D₋₁₀-D₁₀ linear relationships is higher (R₂ is 0.98), which also indicates that the lower pore volume in the selected samples controls the heterogeneity of pore distribution in the overall sample.

On this basis, the relationship between the fractal dimension and pore volume based on the S and thermodynamic models is discussed. In general, with the gradual increase of pore volume, the fractal dimension values under different models tend to decrease, which also shows that the heterogeneity of pore volume distribution can be characterized by the fractal model. It is worth noting that when the pore size is less than 1000 nm, the correlation between the pore volume and the fractal value of pore size at different stages is weak (Figure 11b,c). The reason for this result is that the selected samples are developed with pores larger than 1000 nm and that a relatively small pore volume, smaller than 1000 nm (except in type A), also leads to the weak correlation among the two types of samples. Figure 11d shows that there is a certain negative correlation between the fractal dimension value of the two different models and the pore volumes greater than 1000 nm, which also shows that the pore distribution heterogeneity represented by the fractal model selected for this kind of pore is multi-needle for the pore volume greater than 10,000 nm.

The relationship between the multi-fractal dimension and pore volume is discussed (Figure 12). It is worth noting that the correlation between multiple classification parameters and the overall pore volume is weak (Figure 12a–d). However, the correlation of pore volume at 100~1000 nm is good, as with the increase of pore volume at 100~1000 nm, the multi-typing parameters tend to increase (Figure 12c). Therefore, with the increase of 100~1000 nm pore volume, lower pore volume is distributed, and the heterogeneity is gradually enhanced. D₋₁₀-D₀ is the main factor restricting the heterogeneity of overall pore size distribution, and therefore, its correlation is good.

Figure 13 shows the relationship between mercury removal efficiency and the single fractal dimension and the multifractal dimension. The results show that with the increase of mercury removal efficiency, the single fractal model tends to increase gradually, which also shows that with the increase of pore connectivity, the heterogeneity of pore distribution tends to be complex. Different from the single fractal dimension, the correlation between the multifractal dimension and mercury removal efficiency is weak. The reason for this result is that multifractal parameters are mainly controlled by pore volumes at different stages, and the influencing factors are relatively complex. In general, compared with the multifractal model, the single fractal model can be better used to characterize the quantitative characterization of pore distribution of selected tight sandstone. Among them, the sponge model and the thermodynamic model are more practical in characterizing the heterogeneity of pore distribution of tight sandstone samples.

3.4. Porosity–Permeability and Compressibility Under the Effect of Pore Fracture Structure Heterogeneity

Figure 14a shows that the effective stress and permeability exhibit a power exponential relationship: as the effective stress increases, permeability gradually decreases in a power exponential equation. The permeability damage rate can be obtained by Equation (11). The results show that, as the effective stress increases, the permeability damage rate gradually increases in a power exponential equation (Figure 14b). The compressibility of the abovementioned samples is 0.016 and 0.0023 MPa⁻¹, respectively, obtained using Equation (11). The results show that the compressibility of sample Shan338 is larger than that of the other sample, since in this sample, a micro-fracture is developed.

Figure 15a shows that the effective stress and permeability exhibit a power exponential relationship: as the effective stress increases, permeability gradually decreases in a power exponential equation. The permeability damage rate can be obtained by Equation (11). The results show that, as the effective stress increases, the permeability damage rate gradually increases in a power exponential equation (Figure 15b). The compressibility of the abovementioned samples are 0.0017 and 0.0027 MPa⁻¹, respectively, obtained using Equation (11). Based on the results of five samples, it can be concluded that the compressibility of samples B and C is generally lower.

The relationship between pore volume and compressibility was studied (Figure 16). Figure 16a shows that the correlation between porosity and compressibility is weaker, indicating that only a portion of the pore volume in the sample provides compression space. As the volume of pores with diameters 100~1000 nm increases, the compressibility decreases linearly, but as the volume of pores whose pore diameter is larger than 1000 nm increases, the compressibility increases linearly (Figure 16b,c). This indicates that pore volumes larger than 1000 nm provide compression space for all the selected samples. Note that compressibility decreases linearly with the increase of mercury removal efficiency. This may be related to the small number of samples tested in this experiment (Figure 16d).

The relationship between the fractal dimension and compressibility was studied (Figure 17). The results show that compressibility decreases linearly with the increase of the fractal dimension when assessed using the M and T models. The reason is that higher fractal dimensions represent pore structures of selected samples that are complex, with poor pore connectivity, resulting in a pore volume that is difficult to compress under effective stress.

The relationship between the multi-fractal dimension and compressibility was studied (Figure 18). The results show that compressibility decreases linearly with the increase of D₋₁₀-D₀ and D₋₁₀-D₁₀; this phenomenon is consistent with the results in Figure 16. The reason is that a higher fractal dimension represents a pore structure of selected samples that is complex, with poor pore connectivity, resulting in a pore volume that is difficult to compress under effective stress.

4. Conclusions

Based on our experiments, pore and fracture distribution in the target sandstone was determined. Based on mercury injection parameters, single and multi-fractal models were used to calculate the heterogeneity of pore and fracture volume distribution. Based on this, the correlation between fractal dimension values of different models was studied. Additionally, the pore fracture compressibility of typical samples was calculated using a pressure pore permeability experiment. The correlation between compressibility and the heterogeneity of the pore distribution has been studied. The conclusion are as follows.

(1)

All the samples can be divided into three types using maximum mercury injection volume and mercury withdrawal efficiency. Type A is represented by lower MMIV (less than 0.5 cm³·g⁻¹) and higher MWE (larger than 25%) values, indicating that the pore volume of this type of samples is relatively lower, but the pore connectivity is good. The volume percentage of pores whose diameter is less than 100 nm and 100~1000 nm in type A samples is larger than that of type B and C samples, since in this type of samples, micropores are developed.

(2)

Therefore, the fractal dimension of type A samples is larger than that of type B and C samples, indicating that the pore volume distribution heterogeneity of this type of sample is larger than that of the other samples. The reason is that the volume of pores with diameters larger than 10,000 nm in type A samples is smaller, and only pores smaller than 10,000 nm are developed.

(3)

The fractal dimension value obtained with the M model has a good linear relationship with the thermodynamic model, which indicates that the M model and the thermodynamic model have good consistency in characterizing the pore distribution of tight sandstone. Multi-fractal results show that the lower pore volume in the selected samples controls the heterogeneity of pore distribution in the overall sample.

(4)

As the effective stress increases, the permeability damage rate gradually increases in a power exponential equation. The correlation between porosity and compressibility is weaker, which indicates that only a portion of the pore volume in the sample provides compression space. As the pore volume of 100~1000 nm increases, the compressibility decreases linearly, which indicates that pore volumes larger than 1000 nm provide compression space for all the selected samples.