*4.1. Permeability*

The permeability of testing sandstones was between 0.351 × 10−3–13.910 × 10−<sup>3</sup> mD under the confining pressure of 1100–1500 psi. The permeability of the medium sandstone was the highest, followed by the fine sandstone, and the siltstone was the smallest under similar pore pressure and confining pressure.

For the medium sandstone (HJD1-01), its permeability was between 11.160 × 10−4– 13.910 × 10−<sup>3</sup> mD under the confining pressure of 1100–1500 psi, with an average of 10.118 × 10−<sup>3</sup> mD (Table 2).


**Table 2.** Permeability results.

For the fine sandstone (HJD1-06, HJD1-08), the permeability was between 0.594 × 10−3– 1.188 × 10−<sup>3</sup> mD. The permeability of HJD1-06 was between 1.188 × 10−3–0.689 × 10−<sup>3</sup> mD and the average permeability was 0.917 × 10−<sup>3</sup> mD under 1100–1500 psi confining pressure. The permeability of HJD1-08 between 1.060 × 10−3–0.594 × 10−<sup>3</sup> mD and the average value was 0.806 × 10−<sup>3</sup> mD under the condition of 1100–1500 psi confining pressure (Table 2).

For the siltstone (HJD1-10), its permeability was between 0.351 × 10−3–0.643 × 10−<sup>3</sup> mD, with an average of 0.493 × 10−<sup>3</sup> mD (Table 2).

The grain size of sandstone has an obvious effect on permeability. The permeability of the sampling sandstone increases with the increase of grain size. The permeability of the medium sandstone (HJD1-01) with an average of 10.118 × 10−<sup>3</sup> mD was higher than that of the other samples. However, the permeability of the fine sandstone (HJD1-06 and HJD1-08)

and the siltstone (HJD1-10) was relatively close. The permeability of the fine sandstone was bigger than that of the siltstone under similar confining pressure and pore pressure.

The permeability and confining pressure are negatively correlated. The permeability gradually decreases as the confining pressure increases for the medium sandstone, the fine sandstone, and the siltstone (Figure 4). The decrease in permeability for HJD1-01, HJD1-06, HJD1-08, and HJD1-10 while the confining pressure increased from 1100 to 1500 psi was 2.75 × 10−<sup>3</sup> mD, 0.499 × 10−<sup>3</sup> mD, 0.466 × 10−<sup>3</sup> mD, and 0.292 × 10−<sup>3</sup> mD, respectively.

**Figure 4.** Relationship between confining pressure and permeability.

#### *4.2. LF-NMR Pore Characteristics*

The pore size distribution, total porosity, effective porosity, and the T2C value could be obtained by the T2 spectrum [1,14,46]. The T2 spectra of the sampling sandstones under the water-saturated condition and the centrifugal condition is shown in Figure 5a–c. T2 spectra for three sandstones range from 0.01 to 1889.65 ms. T2 spectra of HJD1-01, HJD1-06, and HJD1-10 show a wide bimodal distribution with a main peak and a subpeak. There is a significant difference in T2 distribution between the three sandstones (Figure 5d). The main peak of HJD10 is separate from the subpeak; however, the main peak of HJD1-01 and HJD1-06 is interconnected with the subpeak. The main peaks of HJD1-01, HJD1-06, and HJD1-10 are within 0.13–11.09 ms, 0.13–9.66 ms, and 0.01–1.38 ms. The subpeaks of HJD1-01, HJD1-06, and HJD1-010 are within 11.09–1889.65 ms, 9.66–666.99 ms, and 12.75–622.25 ms.

**Figure 5.** T2 spectra showing the porosity of different sampling sandstones: (**a**) NMR measurements of HJD1-01 under the water-saturated condition and centrifugal condition; (**b**) NMR measurements of HJD1-06 under the water-saturated condition and centrifugal condition; (**c**) NMR measurements of HJD1-10 under the water-saturated condition and centrifugal condition; (**d**) contrast of T2 spectra under the water-saturated condition between HJD1-01, HJD1-06, and HJD1-10.

T2 distribution is closely related to the size of the pores. Generally, the longer T2, the bigger the pore size. In contrast, the shorter T2, the smaller the pore size. The characteristics of the pores in sandstone could be analyzed by the distribution position and the area of the T2 spectrum [14]. The area of the peaks reflects the number of pores within a certain size; the larger the area, the greater the number of pores. The peak width suggests the distribution of a certain kind of pore, while the number of peaks reflects the continuity of pores at all levels. The two peaks of the T2 spectra reflect the two main types of pores. Based on the T2 spectra curve of HJD1-01, HJD1-06, and HJD1-10, micropores correspond to T2 < 11.09 ms, and macropores correspond to 11.09 ms < T2 < 1889.65 ms. The micropores peak of the T2 spectra is the largest, indicating that the micropores are the most developed. The bigger the value of T2, the bigger the pore size of the sandstone. The pore size of the micropores from big to small is HJD1-01, HJD1-06, and HJD1-10, The pore size of the macropores from big to small is HJD1-01, HJD1-06, and HJD1-10 as well.

The T2 spectrum morphology of the peak in the saturated state is very close to that after centrifugation (Figure 5c). This means the residual water trapped in the pores cannot be removed by centrifugation, which indicates that the connectivity of the pores is poor. Otherwise, the T2 spectrum morphology of the peak after centrifugation is very different from the peak in the saturated state; this indicates that the connectivity of pores is good, and the pores are very conducive to gas flow. According to Figure 5a–c, the macropore connectivity in the three testing samples is relatively good, which is significantly better than that of the micropores. Furthermore, there are obvious differences in the connectivity of micropores among the three sandstone samples; the micropores' connectivity is in the order from good to poor: HJD1-01, HJD1-06, HJD1-10.

The connection between the main peak and the subpeak could be used to identify the connectivity among pores [14]. For instance, there is a gap between the main peak and subpeak in the siltstone (HJD1-10) in Figure 5c, meaning the connection between micropores and macropores is poor. In contrast, the main peak is interconnected with the subpeak in HJD1-01 (Figure 5a) and HJD1-06 (Figure 5b), which suggests the pore connectivity of HJD1-01 and HJD1-06 is better than that of HJD1-10 between micropores and macropores. In general, the pores in the siltstone (HJD1-10) had poorer pore connectivity than that of the medium stone and the fine stone (HJD1-01, HJD1-06) [47].

The distribution position and area of the T2 spectrum of medium sandstone, fine sandstone, and siltstone are different [47], which suggests that sandstone grains can affect pore characteristics. For example, the distribution area of T2 spectra representing macropores in HJD1-1 (Figure 5a) is much larger than that of HJD1-6 and HJD1-10 (Figure 5b,c).

The porosity of sandstones can be analyzed based on LF-NMR T2 [14,48]. Total porosity in the saturated state and centrifugal state can be obtained by separate NMR tests (Figure 5a–c). The maximum value of the cumulative porosity in the saturated water state is known as the total porosity [14,49]. There is residual water in the sample after centrifugation. The maximum value of the cumulative porosity after centrifugation could be seen as the residual porosity. The difference between the total porosity and the residual porosity is the porosity occupied by free water, which can be regarded as the effective porosity. The total porosity of HJD1-01, HJD1-06, and HJD1-10 was 4.14%, 2.80%, and 2.97%. The effective porosity of HJD1-01, HJD1-06, and HJD1-10 was 1.56%, 0.60%, and 0.51%. The proportion of the effective porosity to the total porosity for HJD1-01, HJD1-06, and HJD1-10 was 37.68%, 21.42%, and 17.18% (Table 3).

**Table 3.** Results of NMR analysis.


The T2C value can be obtained by the following steps. First, draw a line parallel to the *X*-axis with the residual porosity (Figure 5a–c). This parallel line has an intersection point with the saturated cumulative porosity curve. The X value corresponding to this intersection point is T2C [14,50]. T2C can divide the T2 spectrum into two segments: one segmen<sup>t</sup> corresponds to pores with water that could not be drained, and the other segmen<sup>t</sup> corresponds to pores with water that could be drained. Pores with water that could not be drained are known as immovable pores, pores with water that could be drained are called movable pores. Therefore, the pore size at the T2C value is taken for the separation between the immovable pore volume and the movable pore volume [14]. The T2 spectrum which is less than T2C corresponds to micropores with poorer pore connectivity, which are the immovable pores. The T2 spectrum which is bigger than T2C corresponds to macropores with better pore connectivity which are the movable pores [15]. T2C was 5.54 ms, 2.77 ms, and 0.85 ms for HJD1-01, HJD1-06, and HJD1-10, respectively. Almost all of the micropores were immovable pores in the three samples. The proportion of immovable pores to total pores for HJD1-01, HJD1-06, and HJD1-10 was 61.50%, 79.07%, and 81.95%, respectively. The proportion of movable pores to total pores for HJD1-01, HJD1-06, and HJD1-10 was 38.50%, 20.93%, and 18.05% (Table 3).

## **5. Discussion**

*5.1. Permeability Variation with Effective Stress and Grains*

The permeability decreases with the increase of the effective stress; the relationship between the permeability and the effective stress is close to the linear relation when the effective stress is between 405 psi and 808 psi because the R<sup>2</sup> of all the samples is more than 0.9757 (Table 4, Figure 6).

**Figure 6.** Relationship between the effective stress and permeability.

y = −0.75 ×

HJD1-10


 + 0.94 ×

**Table 4.** Linear fitting results between permeability (y, mD) and effective stress (x, psi).

The grain size of sandstone has a significant impact on sandstone permeability [51]. The permeability decreases in the order of medium sandstone, fine sandstone, and siltstone under similar effective stress. For the medium sandstone (HJD1-01), the fine sandstone (HJD1-06), and the siltstone (HJD1-10), the slope in Figure 6 is −6.95 × <sup>10</sup>−6, −1.39 × <sup>10</sup>−6, and −0.75 × <sup>10</sup>−6, respectively, as the effective stress increases from nearly 400 psi to 800 psi.

The rock permeability damage by the stress sensitivity could be expressed by the permeability stress sensitivity coefficient. The greater the stress sensitivity coefficient, the better the stress sensitivity will be. The calculation formula is as follows [38]:

$$\mathbf{S}\_{\mathbf{S}} = \frac{\left[1 - \sqrt[3]{\frac{\mathbf{K}\_i}{\mathbf{K}\_0}}\right]}{\log\_{10}\frac{\delta\_i}{\delta\_0}}\tag{2}$$

where SS is the permeability stress sensitivity coefficient, dimensionless; Ki is the permeability of the sample under δi, mD; K0 is the permeability under δ0, mD; δi is the effective stress at any time, psi; and δ0 is the initial effective stress, psi.

Generally, Ss would increase for the same sample with the increase of the effective stress (Table 5). It seems that the sample with lower permeability and finer grains has a higher SS. The permeability of the medium sandstone (HJD1-01) is much higher than that of the other samples (including HJD1-06 and HJD1-08) with similar initial effective stress, and the Ss of HJD1–01 is much lower than the other samples with similar effective stress. For the fine sandstone samples HJD1-06 and HJD1-08, the higher the permeability, the lower the stress sensitivity. This agrees with the reports that sandstones with very low permeability are affected by stress to a greater degree than those with higher permeability [52,53]. SS could be affected by the sandstone grains. The Ss of the medium sandstone is a bit lower than that of the fine sandstone and the siltstone. However, the Ss of the fine sandstone and the siltstone is relatively close (Table 5).


**Table 5.** Stress sensitivity coefficient (SS) under different effective stress between 405 and 801 psi.

#### *5.2. Effects of NMR Pore Characteristics on Permeability Variation*

Generally, the permeability of coal-measure sandstone is controlled by the number of movable pores, effective porosity, and total porosity. The total pore volume from big to small was HJD1-01, HJD1-06, and HJD1-10. The number of micropores and macropores was HJD1-01, HJD1-06, and HJD1-10 in descending order. Almost all of the micropores were the immovable pores in the three samples. The macropore connectivity affects the permeability of the tested samples. Nevertheless, the macropore connectivity in the three testing samples was relatively close. Therefore, the pore volume of the macropores or movable pores was the key factor to the permeability. Large pores are critical to the reservoir quality of the tight sandstone [50]. Movable pores, which are large scale pores with good pore connectivity, dominate the permeability of the reservoir [1,15]. The greater the number of movable pores and the larger the total porosity and the effective porosity, the bigger the permeability of the sandstone. The total porosity of HJD1-01, HJD1-06, and HJD1-10 was 4.14%, 2.80%, and 2.97%. The proportion of movable pores to total pores for HJD1-01, HJD1-06, and HJD1-10 was 38.50%, 20.93%, and 17.17%. The effective porosity of HJD1-01, HJD1-06, and HJD1-10 was 1.56%, 0.60%, and 0.51%. The permeability from big to small was HJD1-01, HJD1-06, and HJD1-10.

Grain size, compaction, and mineral composition could affect the pore structure characteristics [24,37,54]. Finer grain-size sandstones have experienced stronger compaction and cementation during diagenesis in comparison with sandstones with coarser grain size [55]. Stronger compaction and cementation tightly arranged the sandstone's particles and decrease the residual intergranular pores. Therefore, sandstone with a finer grain size develops smaller pore space and a narrower throat [55], which contributes to less total porosity and effective porosity. Mineral composition affects pore-throat structure

parameters such as porosity, pore throat, and effective porosity as well [56]. Sandstones with a coarser grain size have larger rigid grains, such as quartz and feldspar [37]. In the contrary, finer grain-size sandstones have more ductile minerals such as mica and clays [54]. Rigid grains could sustain more pressure during compaction, which is beneficial for preserving intergranular pores [57]. Meanwhile, clay minerals occupy primary pores and cut pore throats, resulting in a decrease in effective porosity. Therefore, sandstone with a coarser grain size is more likely to develop bigger total porosity and effective porosity compared with finer-grained sandstone [26]. In the testing sandstone samples, the finer the grain sizes, the less the pore space. The coarser the grain size, the larger the pore space [55]. The coarser the grains, the larger the effective porosity and the larger the permeability. This suggests that the effective porosity of sandstone has a positive correlation with the permeability value. Therefore, the effective porosity of sandstone is a sensitive indicator for evaluating the permeability of the tight sandstone reservoirs.

The effective porosity has a dominant effect on the Ss, and Ss is negatively correlated with the effective porosity. Ss of HJD1-01 was only about 0.27–0.42 times that of the other samples when the effective porosity of HJD1-01 was 2.6–3 times that of other sandstone samples. It can be seen from HJD1-10 and HJD1-06 that Ss is not only affected by the effective porosity, but also by the total porosity. The effective porosity of HJD1-06 and HJD1-10 was 0.60 and 0.51, which appears closer. Ss increases as the effective stress gradually increases. However, in a certain stress scope, the Ss of the fine sandstone (HJD-06) with higher effective porosity may be bigger than that of the siltstone (HJD-10) with lower effective porosity. It is speculated that the total porosity of sandstone also has a certain influence on Ss due to the total porosity of HJD-10 being a bit bigger than that of HJD-06.

Another reason for the increase of Ss of fine sandstone and siltstone compared to medium sandstone may be the content of plastic minerals. Clay minerals tend to increase with the decrease of sandstone grain size. The content and type of plastic minerals are one factor that determines the difference in Ss; that is, the higher the content of plastic minerals such as mica and clay, the stronger the Ss of the tight rock reservoir.
