**5. Discussion**

The results shown in the preceding section manifest that several multifractal parameters can be utilized to characterize the pore heterogeneity. In this study, only two multifractal parameters ( *D*10 and *D*−10) are selected to describe multifractal characteristics and evaluate pore heterogeneity in di fferent probability measure areas within tight sandstones. The multifractal parameter *D*10 are used to account for multifractal behaviors of pore network in higher probability density areas, while the parameter *D*−<sup>10</sup> represents the multifractal characteristics of pore network in lower probability density areas.

#### *5.1. Relationship between Petrophysical Parameters of Tight Sandstone and Multifractal Parameters*

Petrophysical property is the most direct performance of the pore structure of tight sandstone which can significantly a ffect fractal characteristics of pores. Figure 10 illustrates that multifractal parameters ( *D*10 and *D*−10) show a negative correlation with permeability, RQI and FZI, but they have no obvious relationship with porosity. Theoretically, porosity is mainly a ffected by the content of pore in the rock, especially large pore, which is independent of the complexity of pore distribution. Nevertheless, multifractal parameters mainly reflect the irregularity and complexity of pore geometry and pore network, with a significant influence on permeability, RQI and FZI. Meanwhile, not all pores

are suitable for fractal analysis, and fractal analysis from a NMR T2 spectrum only focus on pores with the pore size larger than dozens of nanometers or hundreds of nanometers [7], because the relaxation mechanism of fluid is quite complex in the smaller pore. In this case, the influence of bulk relaxation and di ffusion relaxation should be considered, and T2 cannot be directly simplified to surface relaxation T2s. Therefore, the porosity of samples as a quantity cannot e ffectively constrain the heterogeneity of the pore structure [29].

Besides, *D*−<sup>10</sup> shows a more sensitive response to permeability, RQI and FZI than *D*10, indicating that pore structure in lower probability density areas has a significant impact on petrophysical properties of tight sandstone reservoirs. From the T2 distributions and petrographic observations of sandstone samples, small-scale clay-dominated micropores associated with short T2 components (T2 < 10 ms) constitute the majority of the pore system, and only a few proportions are composed of the large-scale dissolution pores associated with long T2 components (T2 > 10 ms). Hence, pore system in lower probability density areas mainly consists of dissolution pores with low content; micropores dominate pore system in higher probability density areas. The formation of dissolution pores with larger pore scale, however, greatly improve reservoir properties. Therefore, the complexity of pore system composed of dissolution pores play a more important role in the petrophysical properties of sandstones.

**Figure 10.** The correlations between petrophysical parameters with multifractal parameters. (**a**) the correlations between *D*−10, *D*10 with porosity; (**b**) the correlations between *D*−10, *D*10 with permeability; (**c**) the correlations between *D*−10, *D*10 with RQI; (**d**) the correlations between *D*−10, *D*10 with FZI.

Furthermore, the correlation coe fficients of multifractal parameters with permeability, RQI and FZI show an increasing trend (Figure 10). This can be explained by the fact that RQI and FZI integrate porosity and permeability, which is the better petrophysical parameters for characterizing the pore structure of tight sandstone. Meanwhile, FZI is the combination of several microscopic pore structure properties, e.g., pore specific surface area, morphology and tortuosity, and therefore the highest correlation coe fficient [33]. Hence, FZI is a superior indicator of the pore structure heterogeneity of tight sandstone, especially in the lower probability density area.

#### *5.2. Relationship between Pore Structure of Tight Sandstone and Multifractal Parameters*

The relationships between multifractal parameters and NMR pore structure parameters of tight sandstone, including movable-fluid porosity (ϕ**m**), bound-fluid porosity (ϕ**b**), T2cutoff, T2gm, T35 and T50, are also analyzed, and the correlation coefficients are summarized in Table 5. *D*10 is highly associated with movable-fluid porosity, bound-fluid porosity, T2cutoff and T2gm, but it has no apparent correlation with T35 and T50 (Figure 11). Additionally, *D*−<sup>10</sup> shows few or no relationships with pore structure parameters (Table 5). This may be because small-scale micropores in higher probability density areas dominate the entire pore system of tight sandstone reservoirs, and thus, multifractal parameters of higher probability density regions are more useful for pore structure characterization of tight sandstone.

**Figure 11.** The correlations between multifractal parameter D10 and NMR pore structure parameters. (**a**) the correlation between *D*10 with movable-fluid porosity; (**b**) the correlation between *D*10 with bound-fluid porosity; (**c**) the correlation between *D*10 with T2cutoff; (**d**) the correlation between *D*10 with T2gm.

The T2cutoff is the efficient boundary which divides the total pore volume into movable-fluid pores and bound-fluid pores, according to whether the fluid within them can flow or not. For the studied sandstone samples, immovable bound-fluid mainly exists in clay-dominated micropores with poor connectivity, while movable fluid tends to exist in those dissolution pores which are connected by effective pore throats. T2cutoff value severely affects the proportion of different type of pores in tight sandstones. The larger the T2cutoff value, the more bound-fluid pores, and the fewer movable-fluid pores (Figure 12), which increase the heterogeneity of pore network in higher probability density areas.

**Figure 12.** The correlations between movable-fluid porosity, bound-fluid porosity and T2cutoff. (**a**) the correlation between T2cutoff with movable-fluid porosity; (**b**) the correlation between T2cutoff with bound-fluid porosity.

Figure 11d illustrates that T2gm has the negative correlation with *D*10. The T2gm is a comprehensive performance for the whole pore size distribution of rock, and the large T2gm value means a good pore structure in the tight sandstone reservoirs [52]. In this study, a large amount of short T2 components related to micropores not only dominate the whole T2 distributions of sandstone samples, but also constitute the main peak. Therefore, T2gm mainly reflects the characteristics of main peak corresponding to pore system in higher probability density areas. Therefore, the heterogeneity of pore network of the high probability density areas tends to decrease as T2gm increases.

**Table 5.** The correlation coefficients between multifractal parameters and pore structure parameters.


#### *5.3. E*ff*ect of miNeral Compositions of Tight Sandstone on Multifractal Characteristics*

Several studies have testified that mineral compositions, mineral content and contact between minerals have a significant impact on the pore structure of rock, and the influence of each mineral on pores in distinct lithology may also be different [5,8,30,45,53].

As shown in Figure 13a, quartz content shows the negative correlation with *D*10 and the weak negative relationship with *D*−10. This result demonstrates that quartz plays different roles in the pore structure of different probability density areas. Sandstone petrographic observations show that the studied tight sandstone samples mainly consist of sedimentary quartz grains (Figure 3a,b), which act as the main skeleton minerals of the tight sandstone to resist compaction for the preservation of pores. In general, the higher content of quartz contributes to the higher textural maturity of sandstone and regular pore structure, which decrease the complexity of the pore systems in higher probability density areas. Nevertheless, the tight sandstone samples contain some authigenic quartz grains filling parts of large dissolution pores, which lead to the irregular pore geometry in lower probability density areas and thus larger values of *D*−10.

According to Figure 13b, feldspar content is well correlated to *D*−10, but has no correlation with *D*10. The above relationships indicate that tight sandstone with high feldspar content tends to have relatively complex and anisotropic pore structure in lower probability density areas. The content of feldspar in tight sandstone is associated with the diagenesis process. After deposition, the Taiyuan Formation experienced a long period of deep burial and diagenetic processes, and the chemical environment of fluid changed since a large amount of organic acids and CO2 were generated during the early hydrocarbon generation of the organic matter in the source rock, which causes the extensive dissolution of chemically unstable aluminosilicates, such as feldspar. In this study, feldspar contents of all samples are low (average 5.22%), and secondary dissolution porosities as a result of partial/complete

dissolution of feldspars are well observed in the sandstone samples (Figure 3a,b), indicating that feldspar experienced the strong alteration. These dissolution pores provide extra pore space for gas storage, while feldspar dissolution is always accompanied by by-products (e.g., authigenic quartz, kaolinite, illite) which precipitate in situ or in adjacent pores, resulting in a rather complex pore network in lower probability density areas.

Di fferent from quartz and feldspar, clay content shows the weak positive correlation with *D*10, whereas it has a better negative relationship with *D*−<sup>10</sup> (Figure 13c). This finding indicates that the effect of clay minerals on the pore structure in di fferent probability density areas is rather complex. A high content of clay can not only reduce complexity of pore in lower probability density areas, but also increases the pore heterogeneity of higher probability density areas. This may be interpreted that tight sandstone samples have the high clay content (average 24.32%), and micropores associated with clay minerals predominate the pore system of the high probability measure areas. The SEM image analysis results show that a large amount of clay minerals fill in between quartz grains with slablike, flaky, and hair-like forms (Figure 3g–i), blocking the pore throat system, causing some large pores to be closed and semi-closed. The micropores gradually replace the large pores, and reduce the amounts of large pores with increasing clay content, which decreases the complexity of the pore network of lower probability density areas. However, clay-dominated micropores have di fferent characteristics within various types of clay minerals, which are also characterized by multi-type pores with multi-scale pore size ranging from nanoscale to microscale, such as intracrystalline micropores (mainly < 2 nm), within aluminosilicate layers, inter-crystalline micropores (2–50 nm) between clay particles, and interparticle micropores (>0 nm) between aggregated clay particles [54]. These factors complicate the pore network in higher probability density areas, resulting in higher *D*10.

**Figure 13.** The correlations between mineral compositions and multifractal parameters. (**a**) the correlations between *D*−10, *D*10 with quartz content; (**b**) the correlations between *D*−10, *D*10 with feldspar content; (**c**) the correlations between *D*−10, *D*10 with clay content.
