Calculation Method of Bound Water Saturation in Unconventional Reservoirs Using Fractal Theory

Abstract

The irreducible water saturation of reservoirs seriously restricts the efficient drainage of unconventional energy sources. NMR logging can be used to determine parameters such as total porosity, effective porosity, irreducible water saturation, and permeability, which play an important role in oil and gas identification. T_{2 cut off} value identification using the NMR T₂ spectrum is the key to clarifying the irreducible water saturation of unconventional reservoirs. In this paper, saturation and centrifugal T₂ spectra of sandstone and coal samples are used to study and calculate the T_{2 cut off} value, with methods including single fractal dimension, multi-fractal dimension, and spectrum morphological discrimination; in addition, the applicability of these three methods in characterizing T_{2 cut off} is discussed. According to the morphological difference of the saturated T₂ spectrum, relationships between morphological parameters and the T_{2 cut off} of four types of sample are described. The parameters related to T_{2 cut off} can be divided into two types: (1) the first type includes morphological parameters main peak position (T_M) and smaller-pore volume percentage (SPVP); with an increase of T_{2 cut off}, T_M increases linearly and SPVP decreases exponentially, and the correlation between SPVP and T_{2 cut off} is stronger than that of T_M. (2) The other type includes fractal parameters D₂ (fractal dimension of larger pore), D₋₁₀ – D₁₀, and D₋₁₀/D₁₀; with the increase of T_{2 cut off}, single and multi-fractal dimensions all increase linearly, and the correlation between D₂ and T_{2 cut off} is stronger than that of the multi-fractal dimension. When calculating the T_{2 cut off} of samples with macro-pores developed, spectrum morphological methods should be used preferentially, while the fractal dimension discrimination methods need be used for the T_{2 cut off} of samples with developed micro-pores. Then, the T_{2 cut off} value prediction and evaluation system are described. The overall results of this work can provide a theoretical basis for the inversion of bound water content in the original formation.

1. Introduction

Unconventional reservoirs are characterized by low porosity and extra-low permeability, which leads to fluid (water and gas) micro-distribution complexity under in situ geological conditions. This restricts unconventional oil and gas production. As a fluid occurrence parameter, water saturation (irreducible and movable water) plays an important role in controlling the desorption, diffusion, and seepage of coalbed, shale, and sandstone gas [1,2,3,4,5].

The low field nuclear magnetic resonance (LF-NMR) technique has several advantages, as it is non-destructive, fast, and of high precision. It is widely used in the fields of fluid identification and pore-fracture structure characterization, among others [6,7,8,9]. NMR logging can distinguish movable fluid from immovable fluid, with the key distinguishing parameter being the T_{2 cut-off} value. This value is related to the measurement accuracy of irreducible water saturation and permeability [9,10]. At present, there are two methods for T_{2 cut-off} value determination. One the one hand, a comparison of the T₂ spectrum integral curve can be obtained after water saturation and centrifugation shows the T_{2 cut-off} value under laboratory conditions [11,12]; however, this method has the disadvantages of high cost, a long cycle, and only generating a limited amount of data. One the other hand, the T_{2 cut-off} value of sandstone and limestone has been determined empirically as 33 and 93 ms, respectively; nevertheless, the T_{2 cut-off} values of different study areas, target layers, and reservoir types are quite different and with a large error [13].

One of the current research methods to predict the T_{2 cut-off} value is to use the saturated T₂ spectrum obtained from the NMR logging curve. After studying the relationship between the T₂ spectrum and the T_{2 cut-off} value for a large number of water-saturated samples, a calculation method for the variable T_{2 cut-off} using the difference in the T₂ spectrum morphological value was proposed [7,8,9,10]. Based on this, a normal distribution function was used to fit the T₂ spectrum of centrifugal bound water; then, a variable T_{2 cut-off} value was obtained [10]. To quantitatively characterize pore-fracture complexity, fractal theory is used to characterize pore distribution heterogeneity. Results show that there is a linear relationship between multifractal parameters (D₀−D₁₀) and the T_{2 cut-off} value; thus, a T_{2 cut-off} prediction model using multi-fractal parameters was proposed [14,15,16].

A related T_{2 cut-off} prediction model has been established in the literature. However, there are at least three problems in this research field. Firstly, the work only studies the T_{2 cut-off} of specific reservoir types, and the applicability of the prediction model to different reservoirs needs to be further studied. Secondly, relevant results show that there is a relationship between the T₂ spectrum morphology and the T_{2 cut-off} value, but there is a lack of research on representative morphological parameters. Meanwhile, while the relationship between multifractal parameters and T_{2 cut-off} has been studied, the correlation analysis between single-fractal parameters and T_{2 cut-off} needs to be further investigated. Thirdly, parameters obtained from the published literature vary greatly, and the applicability of fractal models for specific study areas and rock types needs to be analyzed.

In this paper, the saturation and centrifugal T₂ spectra of sandstone and coal samples collected from Qinshui and Qianxi basins are used to obtain the T_{2 cut off} value using the NMR technique to derive a relationship between the T₂ spectrum morphology and the T_{2 cut-off} value. Then, single fractal dimension, multi-fractal dimension, and spectrum morphological discrimination methods are adopted to calculate the T_{2 cut off} value. Moreover, the correlation of representative parameters for various methods is analyzed, and the applicability of these three methods in characterizing T_{2 cut off} is discussed.

2. Study Area and Experimental Methods

2.1. Sample Preparation and Experimental Analyses

A total of 18 samples were collected from the underground coal mines of Qinshui basin, Shanxi Province, China, which is a sweet spot for coal bed methane and tight gas exploitation. The sample size was about 25 × 25 × 25 cm. The maximum vitrinite reflectance (R_{o, max}) for 18 samples ranged from 0.39 to 2.76%. Representative sandstone samples from Benxi, Taiyuan, and Shanxi Formations of the Upper Paleozoic in eastern Ordos Basin, China, were selected. These samples were sampled from 9 coring wells with a burial depth of 2280~3267 m and were prepared into cylinders with a diameter of 25 mm and a length of 35 mm.

The coal samples obtained in this study were taken from Western Guizhou and eastern Yunnan, which are in the Yangtze Block and cover an area of approximately 2.58  ×  104 km². The CBM resource is approximately 2.20–2.75  ×  1012 m³. The main coal-bearing strata in this area are the late Permian Longtan Formation and Changxing Formation, which belong to a delta–tidal–lagoon sedimentary system. After the coal-bearing strata were deposited, the study areas underwent Indosinian–Yanshan–Himalayan tectonic movements. Faults and folds generated by structural movements of different stages control the distribution pattern of CBM resources in this area. The evolution degree of coals in this region fluctuates from bituminous to anthracite coal, and the coal reservoirs have different evolutions, including deep metamorphism during the early Yanshanian period and structural–thermal metamorphism during the Yanshan period (Figure 1).

Figure 1. Sample types and locations of study areas.

2.2. Calculation Methods

2.2.1. Experimental Process

For NMR analyses, 25 mm diameter cylinders 30 mm in length were prepared from all coal and sandstone samples. The samples were placed in the drying oven before the tests, with the temperature adjusted to 105 °C for approximately 12 h. The dry samples were thereafter weighed by a balance. The samples were saturated with distilled water in a vacuum (i.e., without confining pressure) for 48 h. After 24 h of saturation, continuous T₂ spectrum detection was performed on the same sample. If the difference in T₂ spectrum change after a half-hour interval was small, it was considered that the sample reached saturation. By comparison, it can be seen that this type of sample could reach saturation after 24 h of vacuum pressure saturation. It is worth noting that this article uses distilled water saturation, so the influence of factors such as salinity was not considered. After saturation, the wet samples were weighed, and the porosity (%) of samples was calculated. After measuring the T₂ spectrum, the samples were centrifuged at a certain speed to compare water content before and after centrifugation (Figure 2a). Compared with the relevant literature, the saturation process of the samples is consistent with previous methods, so it is believed that these samples can be used for comparison.

Figure 2. T_{2 cut off} calculation process and related parameter division. (a) Experimental device; (b) T_{2 cut off} obtained; (c) Single-fractal parameter; (d) Multi-fractal parameter, the four different colors represent different samples.

2.2.2. T_{2 cut off} Value

The calculation process of the T_{2 cut off} value is shown in Figure 2b. It is noted that there are differences in the physical significance of T_{2 cut off}. When T₂ is less than the T_{2 cut off} value, smaller pores (T₂ is smaller than T_{2 cut off}) are filled with immovable water. When T₂ is larger than the T_{2 cut off} value, larger pores (T₂ is larger than T_{2 cut off}) are filled with movable water (Figure 2b) [17,18,19].

2.2.3. T₂ Spectrum Morphological Parameter

Based on the saturated T₂ spectrum, the following three morphological parameters are proposed:

• The main peak position (T_M) represents the T₂ value corresponding to the highest spectral quantity. The T_FPP of the two samples is 8 and 7 ms, respectively (Figure 2c).
• The pore volume percentage of smaller pores (SPVP) and larger pores (LPVP) represents the spectral area percentage within a certain T₂. The relationship between the T₂ value and pore diameter is studied in the relevant literature; however, the conversion coefficient among the two parameters is not uniform [20]. Based on the results of [11], pore-fracture systems are classified into smaller pores (0–10² nm, T₂ < 2.5 ms) and large pores (10²–10⁴ nm, 2.5 ms < T₂) by using the T₂ spectrum and the response characteristics (relaxation time) of different pore diameters. The equation for SPVP is given as

$$SPVP={\displaystyle \frac{\mathrm{S}1}{\mathrm{S}}}$$

(1)

where S₁ is the corresponding spectral area when T₂ is less than 2.5 ms; S is the total spectrum area.

2.2.4. Single Fractal Parameter

Equation (2) is used to calculate the single fractal dimension of the saturated T₂ spectrum:

$$\lg (V_p)=(3-D_w)\lg (T_2)+(D_w-3)\lg T_{2max}$$

(2)

where V_p is the percentage of cumulative pore volume (amplitude) in a saturated water state; T_2max is the maximum transverse relaxation time, ms; D_w is the NMR fractal dimension value in a saturated water state. The detailed calculation process is shown in Figure 2d. The results show that the fractal curve can be divided into two sections by using the T_{2 cut off} value obtained from the saturated water. A slope of T₂ less than 2.5 ms corresponds to smaller pore size distribution heterogeneity, and a slope of T₂ greater than 2.5 ms corresponds to larger pore size distribution heterogeneity [21].

2.2.5. Multi-Fractal Parameter

For strongly inhomogeneous reservoirs, the pore size distribution curve usually fluctuates randomly and may exhibit different types of self-similarity at different pore size intervals. Thus, it is difficult to fully characterize the homogeneity of pore space by a single fractal dimension. the calculation process of multiple fractals and the physical meaning of parameters are detailed in the literature [22,23,24,25,26].

It should be noted that the multifractal calculation results include two expressions, i.e., the a~f(a) singular spectrum and q~D fractal dimension spectrum. The values of these two expressions have a clear linear correlation, and thus we suggest that either expression can be selected for the multifractal calculation using NMR data. By taking the singular spectrum parameters as an example, the characteristic parameters mainly include a_min, a_max, a₀, a₀ − a_max, a_min − a₀, and A. Particularly, the right branch width (a_min − a₀), the left branch (a₀ − a_max), and a_min − a_max characterize the distribution heterogeneity of the data. For the NMR T₂ spectra, larger a_min − a₀ indicates a more heterogeneous pore size distribution in the low pore volume development area, while higher a₀ − a_max indicates a more heterogeneous pore size distribution in the high pore volume development area.

3. Results and Discussion

3.1. Pore Type Classification Using T₂ Spectrum

To study the NMR T₂ spectra response of different reservoir types, the T₂ spectra of tight sandstone and coal samples were studied separately by using the peak number, main peak position, and pore structure of the T₂ spectra measured for the water-saturated state. Compared with unconventional reservoirs such as coal and shale, micro-pores of tight sandstone reservoirs are not developed, and the T₂ spectrum is dominated by a single peak [12]. The T₂ spectrum of 21 sandstone samples can be divided into three types. That is, a single peak of larger pores (Type A; Figure 3a), a single peak of meso-pores (Type B and C; Figure 3b,c) and a single peak of smaller pores (Type D; Figure 3d). Moreover, the single peak of meso-pores can be divided into right- (Figure 3b) and left- (Figure 3c) shifting forms. Specific definitions are as follows: Type A is characterized by a main peak greater than 100 ms, and the T_{2 cut off} is smaller than the main peak position; Type B is characterized by a main peak at 10~100 ms, and the T_{2 cut off} is smaller than the main peak position; Type C is characterized by a main peak at 1~10 ms, and the T_{2 cut off} is equal to the main peak position; and Type D is characterized by a main peak at 0.1~1 ms, and the T_{2 cut off} is larger than the main peak position. The smaller-pore volume percentage of Type D is the largest (86~92%), followed by Types C and B, and Type A is the smallest (4~27%) (Table 1).

Figure 3. Typical T₂ spectrum distribution of 24 sandstone samples. (a) Type A; (b) Type B; (c) Type C; (d) Type D.

Table 1. Summary of morphological parameters of all sandstone samples.

Type	Sample No.	T2 cut off (ms)	Peak Position (ms)	Adsorption Pore Volume Percentage	Seepage Pore Volume Percentage
A	No. 3	57.2	100	0.04	0.68
	No. 4	49.09	85	0.08	0.76
	No. 6	53.23	50	0.15	0.94
	No. 7	64.85	131	0.09	0.59
	No. 13	34.34	28	0.06	1
B	No. 8	4.82	21	0.2	0.93
	No. 10	7.79	4.78	0.37	0.96
	No. 11	8.39	6	0.28	0.93
	No. 12	4.64	3	0.49	0.96
	No. 16	6.67	6	0.34	0.98
	No. 9	16.01	20	0.11	0.97
C	No. 60	4.08	5	0.51	0.99
	No. 61	3.38	4.26	0.6	0.99
	No. 62	3.38	2.7	0.52	0.99
	No. 63	2.45	1.4	0.63	0.99
	No. 53	3.1	2.41	0.49	0.96
D	No. 49	0.48	0.19	0.86	0.99
	No. 52	0.49	0.31	0.92	0.99
	No. 54	1.7	0.61	0.62	0.99
	No. 57	1.6	0.39	0.78	0.99
	No. 59	0.31	0.19	0.83	0.99
	No. 66	0.5	0.25	0.74	0.97

Figure 4 and the related literature have shown that the pore fracture of low-middle-high ranking coal samples exhibits obvious differences, which also leads to T₂ spectrum morphology differences in all coal samples; the T₂ spectrum is dominated by single, double, and multi-peaks. The T₂ spectrum of 18 coal samples are divided into three types, that is, a single peak of larger pores (Type A; Figure 4a), three peaks (Types B and C; Figure 4b,c), and a single peak of smaller pores (Type D; Figure 4d). Moreover, three peaks of meso-pores can be divided into the right (Figure 4b) and left (Figure 4c) shift form. Specific definitions are as follows: Type A is characterized by a main peak greater than 100 ms, and the T_{2 cut off} is smaller than the main peak position; Type B is characterized by a main peak at 10~100 ms, and the T_{2 cut off} is smaller than the main peak position; Type C is characterized by a main peak at 1~10 ms, and the T_{2 cut off} is equal to the main peak position; and Type D is characterized by a main peak at 0.1~1 ms, and the T_{2 cut off} is larger than the main peak position. The smaller-pore volume percentage of Type D is the largest (32~92%), followed by Types C and B, and Type A is the smallest (3~24%) (Table 2). Overall, the main peak position, peak number, and spectral area ratio are three important parameters for identification of the type of T₂ spectrum.

Figure 4. Typical T₂ spectrum distribution of 18 coal samples. (a) Type A; (b) Type B; (c) Type C; (d) Type D.

Table 2. Basic parameters of samples used in this study.

Sample No.	Φwater (%)	Φnitrogen (%)	Permeability (mD)	T2 cut off (ms)	Swi (%)	<2.5 (ms)	2.5~100 (ms)	First Peak Position (ms)
1	41.36	38.24	0.741	7	69.00	0.29	0.66	4.64
2	5.67	6.14	0.292	11	65.92	0.28	0.69	3.87
3	5.80	7.76	0.0064	2.7	77.90	0.78	0.16	1.55
4	4.95	5.18	0.055	33	66.18	0.21	0.74	42
5	4.55	5.36	0.091	60	59.29	0.10	0.69	60
6	5.79	6.53	0.054	30	63.98	0.19	0.79	41.6
7	10.69	10.76	0.222	35	67.79	0.15	0.75	18
8	5.85	6.82	0.0032	120	68.30	0.20	0.48	179
9	4.59	6.43	0.0060	62	69.28	0.23	0.64	86
10	5.28	7.03	0.725	38	45.56	0.04	0.73	71
11	3.67	4.21	0.098	11	62.42	0.29	0.67	6.69
12	3.44	4.22	0.383	40	53.93	0.08	0.72	49
13	2.91	3.56	0.359	46	28.67	0.09	0.30	258
14	3.28	3.88	0.064	21	63.69	0.32	0.52	2.68
15	3.68	4.49	0.044	6	52.31	0.40	0.60	1.86
16	3.27	4.10	0.047	16	53.54	0.28	0.57	2.23
17	1.54	1.85	0.017	2.7	92.85	0.92	0.08	1.55

3.2. Single Fractal Dimension Identification Method

Based on Figure 4, single fractal dimension of D₁ (smaller pores), D₂ (larger pores), and D_T (total pore-fracture system) is obtained by using the saturated T₂ spectrum. Figure 5a shows that D₁ is not 2~3, which serves as a practical boundary indicator for the validity of the chosen fractal model rather than a direct descriptor of the pore geometry itself.

Figure 5. Calculated single fractal dimension and relationship between fractal parameters and T_{2 cut off} of all coal samples (a) fractal dimension curve; (b,c) fractal dimension versus PV percentage; (d) T_{2 cut off} versus fractal dimension value.

The main reasons for this phenomenon are as follows. Sensitivity to Data Noise and Processing: The low-amplitude tail of the T₂ spectrum corresponding to small pores is particularly susceptible to noise and baseline drift. The calculation of D₁, which often relies on a linear fit in the log-log plot of the NMR-derived capillary pressure curve, can be significantly skewed by inaccuracies in this noisy region. Anomalous D₁ values may thus partly reflect the sensitivity of the model to data quality and signal-to-noise ratio at the measurement limits. Metric Choice and Normalization: The specific formulation of the NMR fractal model, including the choice of metrics for cumulative volume and pore radius (T₂ time), and the normalization procedures applied during data transformation can influence the slope of the log-log plot. Inconsistent or inappropriate handling in these steps, especially for the small-pore data segment, could artificially produce slopes corresponding to D₁ < 2. Inherent Model Simplification: The model employed assumes a single, contiguous fractal regime for the pore surface. Real coal pore systems are highly heterogeneous and may comprise multiple fractal sets or non-fractal structures. A calculated D₁ value outside the 2–3 range may highlight the inadequacy of a single-fractal approximation to capture the true complexity, suggesting the need for multi-fractal analysis or different geometric descriptors for that sub-interval of pores.

The D₂ and D_T of all samples are 2.5~2.95 and 1.98~2.51, indicating that larger pores have fractal significance. Figure 5b shows that there is a positive linear relationship between D₂ and SPVP, while the correlation between D_T and pore volume is weak. However, there is not an obvious relationship between LPVP and single fractal parameters (Figure 5c), which indicates that smaller pores are the key factor in restricting pore size distribution heterogeneity. Figure 5d shows that D₂ decreases linearly with an increasing T_{2 cut off} value, with an R² of 0.72.

Based on Figure 3, single fractal dimensions of D₁, D₂, and D_T are obtained by using the saturated T₂ spectrum of sandstone samples. Figure 6a,b shows that the D₁, D₂, and D_T of all samples are 2.12~2.54, 2.5~2.95, and 2.23~2.65, respectively, indicating that smaller and larger pores all have a fractal significance. Figure 6c shows that there is a negative linear relationship between D₂ and T_{2 cut off}, and the correlation between D_T and T_{2 cut off} is weak, indicating that the single fractal parameter D₂ is the key parameter to predict the T_{2 cut off}. It is noted that there is a phased relationship between D_T and T_{2 cut off}. When T₂ is less than 20 ms, there is a linear negative correlation among those two parameters. When T₂ is greater than 20 ms, T_{2 cut off} increases linearly as D_T increases (Figure 6d). It can be seen from Figure 3 and Figure 4 that most samples with a T_{2 cut off} of 20 ms belong to Types B and C, and the smaller-pore volume percentage is about 50%; that is, pores with different diameters of this sample type are relatively uniform. This may be related to the significant differences in the bound water state between samples with macropore development and those with micropore development. Hence, the single fractal parameter D₂ (larger pore distribution heterogeneity) of all coal and sandstone samples has a good linear relationship with the T_{2 cut off} value, which can be used to predict the T_{2 cut off} value.

Figure 6. Single fractal dimension calculation and relationship between fractal parameters and T_{2 cut off} of all the coal samples (a) fractal dimension curve; (b,c) fractal dimension versus PV percentage; (d) T_{2 cut off} versus fractal dimension value [21].

The observed reversal in the correlation trend between the fractal dimension D and the T_{2 cut off} value at approximately 20 ms is interpreted as a signature of a structural transition within the sandstone pore system. For a T_{2 cut off} > 20 ms (focusing on larger pores), the positive correlation suggests that the geometry of effective flow channels becomes increasingly complex and tortuous with increasing pore size, likely due to dominant control by secondary diagenetic processes like dissolution. Conversely, for a T_{2 cut off} < 20 ms (incorporating smaller pores), the negative correlation indicates that the abundant micro-/mesopores—often associated with primary depositional textures or early diagenetic clay minerals—exhibit relatively more homogeneous shapes and poorer connectivity. Their inclusion “dilutes” the overall morphological complexity calculated for the pore network. This phase change highlights the dual nature of the pore system: a flow-dominant, diagenetically modified macro-pore network with high complexity, and a storage-dominant, depositionally influenced small-pore system with lower apparent connectivity. The ~20 ms threshold may thus mark the approximate boundary between these two genetically and functionally distinct pore populations.

3.3. Multi-Fractal Dimension Identification Method

According to Section 2.2.5, the multi-fractal parameter of the 18 coal samples is obtained by using the saturated T₂ spectrum. There is a good linear positive correlation between fractal spectrum parameters and singular spectrum parameters, with correlation coefficients greater than 0.9 (Figure 7). It is indicated that only one expression parameter should be used to describe multifractal results.

Figure 7. Relationship between the generalized fractal parameters and singular fractal parameters [24]. (a) D₋₁₀ − D₀ versus a₋₁₀−a₀; (b) D₀ − D₁₀ versus a₀−a₁₀; (c) D₋₁₀ − D₁₀ versus a₋₁₀ − a₁₀; (d) D₁₀ versus a₁₀.

Figure 8a,b shows that there is a positive correlation between D₋₁₀ and T_{2 cut off} in [15,17], and the R² is 0.78 and 0.90. When T_{2 cut off} is larger than 20 ms, the linear correlation coefficient between T_{2 cut off} and D₋₁₀ is less than 0.5. Figure 8 and Figure 9 show that the pore volume distribution heterogeneity of the maximum lowest value area is of great significance to the prediction of T_{2 cut off}.

Figure 8. Relationship between T_{2 cut off} and D₋₁₀ based on the data in four papers. (a) cited by [14]; (b) cited by [16]; (c) cited by [24]; (d) data from this paper.

Figure 9. Relationship between T_{2 cut off} and D₋₁₀−D₁₀ and D₋₁₀/D₁₀ [14] (a,b), T_{2 cut off} vs D₋₁₀ − D₁₀ and D₋₁₀/D₁₀; (c,d) [16], T_{2 cut off} vs D₋₁₀ − D₁₀ and D₋₁₀/D₁₀ [24].

As mentioned in the related literature, D₋₁₀ − D₁₀ and D₋₁₀/D₁₀ were used to calculate the T_{2 cut off} value. It is shown that the above two parameters have a good linear positive correlation with the T_{2 cut off} value, and the correlation coefficients are greater than 0.9 (Figure 9a,b). For sandstone samples with a T_{2 cut off} of 0~70 ms, there is a good linear positive correlation between the above two parameters and the T_{2 cut off} value when the T_{2 cut off} is smaller than 20 ms. For coal samples with a T_{2 cut off} of 0~60 ms, there is a weak linear positive correlation between the above two parameters and the T_{2 cut off} value (Figure 9c,d). This indicates that parameters D₋₁₀ − D₁₀ and D₋₁₀/D₁₀ can be used to predict the T_{2 cut off} value of small-pore samples (the T_{2 cut off} value of those samples is smaller than 20 ms).

3.4. T₂ Spectrum Morphological Discrimination Methods

As mentioned in Section 3.1, the main peak position, peak number, and spectral area ratio are three important parameters in identifying the type of T₂ spectra (Figure 10). For sandstone samples with a T_{2 cut off} of 0~70 ms, the T_{2 cut off} increases linearly with a higher T_M value (Figure 11a). Its linear correlation is much higher than multifractal parameters (Figure 10). Moreover, the smaller-pore volume percentage decreases polynomially with a higher T_{2 cut off} value, and the fitting coefficient between this parameter and T_{2 cut off} is the highest, namely 0.93 (Figure 11b,c). However, there is no obvious relationship between the larger pore volume percentage and T_{2 cut off}. It is worth noting that due to laboratory conditions, this article does not discuss the sensitivity of noise, the selection of spectral smoothing windows, or the impact of the quality of nuclear magnetic resonance signal inversion on signal quantity. To mitigate this impact, we selected the same CPMG sequence number for analysis, but the influence of these external factors can still interfere with the signal.

Figure 10. Relationship between T_{2 cut off} and peak position in different types of T₂ spectra.

Figure 11. Correlation between T_{2 cut off} and peak position and the adsorption pore volume percentage of sandstone samples. (a) T_{2 cut off} vs Peak position; (b) T_{2 cut off} vs SPVP; (c) T_{2 cut off} vs LPVP.

For coal samples with a T_{2 cut off} of 0~70 ms, the T_{2 cut off} increases linearly with a higher T_M value (Figure 12a), and its linear correlation is stronger than multifractal parameters. Moreover, the smaller-pore volume percentage also decreases polynomially with a higher T_{2 cut off} value, and the fitting coefficient between this parameter and the T_{2 cut off} is lower, namely 0.88 (Figure 12b,c). However, there is no clear relationship between larger pore volume percentage and T_{2 cut off} (Figure 13d). It is noted that the fitting coefficient between this parameter and T_{2 cut off} is higher (R² is 0.96) when T_{2 cut off} is smaller than 20 ms. In conclusion, the smaller-pore volume percentage is a valid parameter for the prediction of the T_{2 cut off} value.

Figure 12. Correlation between T_{2 cut off} and peak position and the smaller-pore volume percentage of coal. (a) T_{2 cut off} vs Main peak position; (b) T_{2 cut off} vs SPVP; (c) T_{2 cut off} vs LPVP; (d) T_{2 cut off} vs SPVP.

Figure 13. Correlation between T_M and fractal dimension and adsorption pore volume percentage. (a) our data; (b) data cited by [24]; (c) our data; (d) data cited by [12].

3.5. Correlation Degree of T_{2 cut off}-Related Parameters

Parameters affecting T_{2 cut off} are discussed based on single and multi-fractal methods as well as morphological discrimination methods. A related parameter includes single fractal dimension D₂, multi-fractal dimension D₋₁₀−D₁₀, D₋₁₀/D₁₀, morphological parameter T_M, and smaller-pore volume percentage. Figure 14a,b shows that there is an obvious polynomial relationship between T_M and smaller-pore volume percentage; this is because a larger T_M of samples represents the development of larger pore volume, resulting in smaller pores being undeveloped. Affected by echo duration, echo interval, and related parameters, the T₂ range of 18 coal samples is greater than 1000 ms (Figure 4). The T₂ value range of this kind of samples is much larger than that of other types in the three papers; thus, the correlation between the parameters of this type of samples and T_{2 cut off} is weak.

Figure 14. Correlation between PV percentage and fractal dimension. (a) our data; (b) cited by [24]; (c) cited by this paper; (d) cited by [12].

Figure 14a,b shows that there is no obvious correlation between pore structure and multifractal parameters. Figure 14b shows that the relationship between D₋₁₀−D₁₀, D₋₁₀/D₁₀, and smaller pore volume can be divided into two sections, with a smaller-pore volume percentage of about 40%. Specifically, when the smaller-pore volume percentage is less than 40%, two parameters increase with higher pore volume. When the smaller-pore volume percentage is greater than 40%, D₋₁₀ − D₁₀ gradually decreases. This result is consistent with the plots in Figure 14. In addition, Figure 14c,d shows that there is a positive linear relationship between pore structure and single fractal parameter D₂.

3.6. Applicability Analysis of Different Calculation Methods

Above all, multifractal parameters D₋₁₀ − D₁₀ and D₋₁₀/D₁₀ are used to fit the linear regression equation, which can realize the accurate measurement of the T_{2 cut off} value [14]. For samples with larger pores developed, the T_{2 cut off} is 20~70 ms, and there is a lower precision in the prediction of the T_{2 cut off} value by using multifractal parameters. This is because D₋₁₀ − D₁₀ and D₋₁₀/D₁₀ are 2~3 and 6~8, and value range of the two parameters is small. This also leads to large errors in predicting a large T_{2 cut off} value.

The results also show that single-fractal parameters D₂, T_M, and SPPV have a good linear relationship with the T_{2 cut off} value of samples with larger pores developed. The comparison shows that the single fractal and curve type parameter has good applicability in predicting the T_{2 cut off} value of samples with larger pores developed (SPPV is larger than 30%). This is because the T_{2 cut off} value is mainly restricted by pore structure. The more developed the larger pores are, the larger the T_{2 cut off} value is. Morphological parameters T_M and SPPV can directly reflect the degree of pore development; thus, the above three parameters have good applicability in predicting the T_{2 cut off} value. More importantly, the above three parameters are relatively easy to obtain and have the advantages of fast calculation and high precision.

4. Conclusions

• The T₂ spectra of all coal and sandstone samples are divided into three types, that is, a single peak of larger pores (Type A), a single peak of meso-pores (Types B and C), and single peak of smaller pores (Type D). Type A is characterized by a main peak greater than 100 ms and a T_{2 cut off} smaller than the main peak position. Type B is characterized by a main peak of 10~100 ms and a T_{2 cut off} smaller than the main peak position. The main peak position, peak number, and spectral area ratio are three important parameters in identifying the type of T₂ spectrum.
• Related parameters include single fractal dimension D2, multi-fractal dimension D₋₁₀ − D₁₀, D₋₁₀/D₁₀, morphological parameter TM, and smaller-pore volume percentage. The single fractal parameter D₂ (larger pore distribution heterogeneity) of all coal and sandstone samples has a good linear relationship with the T_{2 cut off} value.
• When calculating the T_{2 cut off} of samples with macro-pores developed, spectrum morphological methods should be used preferentially. Fractal dimension discrimination methods should be used when the T_{2 cut off} of samples with micro-pores developed is calculated. Multi-parameters D₋₁₀ − D₁₀ and D₋₁₀/D₁₀ can be used to predict the T2 cut-off value of small-pore samples (the T_{2 cut off} value of those samples is smaller than 20 ms).