Method for Predicting Bound Water Saturation in Tight Sandstone Reservoirs Using Morphology and Fractal Models

Abstract

The nuclear magnetic resonance T₂ spectrum was used to identify the T_2 cut-off value, which is the key to determining the irreducible water saturation of a reservoir. In this paper, the saturation and centrifugal T₂ spectra of sandstone and coal samples were used to explore the correlation between each parameter and the T_2 cut-off value, using a single fractal dimension, a multifractal dimension and a spectral morphology discrimination method. The conclusions are as follows: (1) The T₂ spectra of nine sandstone samples in this paper can be divided into four types. Type A is represented by sample 2, wherein the T₂ spectrum shows a bimodal state and the area of the right T₂ spectrum (2.5~100 ms) is larger than that of the left T₂ spectrum (T₂ < 2.5 ms), indicating that the sample has good pore connectivity and belongs to the macroporous development sample. The B-type T₂ spectrum is unimodal, and the pore connectivity is poor, indicating that it is a large-pore development sample. The T₂ spectrum of the C-type sample is unimodal, and the pore connectivity is very poor, indicating that it is a mesoporous development sample. The T₂ spectrum of the D-type sample shows a single peak state, and the main T₂ is distributed within 0.1~2.5 ms. The pore connectivity is very poor, which indicates that it belongs to the small pore development type sample. (2) The single fractal model shows that, compared with other single fractal parameters, D₂ increases with the increase in the T_2 cut-off value, but the correlation is weak. Therefore, it is not feasible to predict the T_2 cut-off value using the single fractal dimension parameter. (3) The multifractal model shows that D₋₁₀–D₁₀ increases linearly with the increase in D₋₁₀–D₀, but there is no obvious linear correlation between D₀–D₁₀ and D₋₁₀–D₁₀, indicating that the low pore volume area in this kind of sample controls the overall heterogeneity of pore distribution. (4) The related parameters affecting the T_2 cut-off value include D₋₁₀–D₁₀, D₋₁₀/D₁₀, D₋₁₀–D₀, T_M and D₂. Therefore, based on the above five parameters, a T_2 cut-off value prediction model is constructed. The T_2 cut-off value calculated by the model is highly consistent with the experimental value, which proves the reliability of the model.

1. Introduction

With the continuous intensification of energy exploration and development and energy consumption, conventional oil and gas resources in China are no longer sufficient to meet the needs of social development, and the development of unconventional oil and gas resources is imperative. Tight sandstone gas has become one of the most important unconventional energy sources in China, and the development of pore and fracture structures in tight sandstone reservoirs is a key factor restricting the development of tight sandstone gas. Low-field nuclear magnetic resonance (LF-NMR) technology has the advantages of being non-destructive, fast and highly precise, and is widely used in fluid identification, pore fracture structure characterization and other fields [1]. Nuclear magnetic resonance logging curves can be used to distinguish movable fluid and immovable fluid, and the key parameter used to distinguish these is the T_{2 cut-off} value. The T_{2 cut-off} value is a key parameter in nuclear magnetic resonance logging, which reflects the boundary between the movable fluid and the bound fluid in the reservoir. In nuclear magnetic resonance measurements, T₂ relaxation time (also known as transverse relaxation time) is closely related to the fluidity of fluids in pores. Longer T₂ values correspond to larger pores and more fluid flowing, while shorter T₂ values indicate smaller pores. Based on saturated and centrifugal T₂ spectra, a T_{2 cut-off} value can be determined, which distinguishes the movable fluid from the confined fluid. The T_{2 cut-off} value is selected as a point where the T₂ spectrum undergoes significant changes during the centrifugation process. The T₂ values before this point correspond to the movable fluid, and the T₂ values after this point correspond to the confined fluid.

This value is related to the measurement accuracy of irreducible water saturation and permeability [2,3]. At present, there are two methods to determine the T_{2 cut-off} value. On the one hand, the T_{2 cut-off} value under laboratory conditions can be obtained by calculating the T₂ spectral integral curve obtained after saturation and the centrifugation of water [4]. However, this method has the disadvantages of a high cost, a long cycle and only limited amounts of data. On the other hand, according to empirical inference, the T_{2 cut-off} values of sandstone and limestone are 33 and 93 ms, respectively; however, the T_{2 cut-off} values of different research areas, target layers and reservoir types vary greatly, resulting in large calculation errors [5].

At present, the more commonly used method to predict the T_{2 cut-off} value is to use the saturated T₂ spectrum obtained from the NMR logging curve for conversion. After studying the relationship between the T₂ spectrum and the T_{2 cut-off} value of a large number of saturated samples, scholars proposed a method to calculate the variable T_{2 cut-off} value by using the morphological differences in the T₂ spectrum [6,7]. On this basis, the normal distribution function is used to fit the T₂ spectrum of the centrifugal bound water, and then the variable T_{2 cut-off} value is obtained. Subsequently, to quantitatively characterize the complexity of pore fracture structure, scholars have used fractal theory to characterize the heterogeneity of pore distribution. The results show that there is a linear relationship between the multifractal parameters (D₀–D₁₀) and the T_{2 cut-off} value. Therefore, a T_{2 cut-off} prediction model based on multifractal parameters is proposed [8,9].

In summary, the relevant literature has established a T_{2 cut-off} prediction model to achieve the quantitative prediction of irreducible water saturation in specific reservoirs [10,11,12]. However, there are still the following problems in this field. Firstly, the correlation results show that there is a relationship between T₂ spectral morphology and the T_{2 cut-off} value, but there is a lack of research on representative morphological parameters. At the same time, the relationship between multifractal parameters and T_{2 cut-off} values has been studied, but there are few studies on the correlation between single fractal parameters and T_{2 cut-off} values. Therefore, for specific reservoir types, the applicability of different prediction methods for the T_{2 cut-off} value of the same reservoir needs to be further studied.

Therefore, this paper takes the sandstone samples collected in the Qinshui Basin as an example, and uses low-field nuclear magnetic resonance technology to obtain the saturation and centrifugal T₂ spectra of different samples, as well as the T_{2 cut-off} value of each sample. On this basis, from a morphological point of view, the parameters including peak value, peak position and peak area corresponding to the hole are counted, and the relationship between the morphological parameters and the T_{2 cut-off} value is discussed. At the same time, single and multifractal models are used to realize the quantitative characterization of the heterogeneity of the pore fracture distribution of the target sample, and the correlation between the multifractal parameters and the pore fracture parameters is discussed. On this basis, the key parameters are selected from the morphological and fractal parameters, and the T_{2 cut-off} value is calculated using the single fractal dimension, multifractal dimension and spectral morphology discrimination methods, to construct the T_{2 cut-off} value prediction model.

2. Experimental Test and Theoretical Analysis

2.1. Experimental Test

The target samples were collected from the Ordos Basin in the western part of the North China Plate. The region is a typical superimposed basin on the edge of the craton, with a total area of about 37 × 10⁴ km². The basin is composed of the western margin thrust belt, Tianhuan sag, Yishan slope, western Shanxi bending fold belt, Yimeng uplift and Weibei uplift tectonic belt (Hu et al., 2020) [8]. The Benxi Formation in the Qinshui Basin is mainly composed of alternating marine and continental mudstone, sandy mudstone, claystone, limestone and limonite layers interbedded with coal lines, with a thickness ranging from 10 to 50 m. In this formation, tight sandstone is an important component, but its scale and thickness are relatively small, and the sand bodies are relatively independent of each other. The tight sandstone of the Benxi Formation is mainly composed of fine-grained sandstone, with small and tightly arranged particles and relatively low porosity and permeability; hence, it is called tight sandstone. This sandstone typically has high mechanical strength and stability. Due to the lower porosity and permeability of tight sandstone, its reservoir properties are relatively poor. However, through reservoir modification techniques such as fracturing to increase production, the physical properties of tight sandstone reservoirs can be significantly improved, thereby increasing the efficiency of oil and gas resource extraction. To study the pore distribution of tight sandstone reservoirs at different depths, 11 tight sandstone samples of Benxi Formation were collected from different exploration wells, with each having a sampling depth of 3000~3900 m.

Low-field nuclear magnetic resonance (LF-NMR) was used [13]. Cylinders with a diameter of 25 mm and a length of 30 mm were prepared from sandstone samples. Before the test, the sample was placed in a drying oven and the temperature was adjusted to 105 °C for about 12 h. Then, the dried samples were weighed with a balance. The samples were saturated with distilled water in vacuum (i.e., no confining pressure) for 48 h. After saturation, the wet sample was weighed and the porosity (%) of the sample was calculated. After measuring the T₂ spectrum, the sample was centrifuged at a certain speed to obtain the centrifugal T₂ spectrum [14,15].

2.2. Model Calculation and Parameter Acquisition

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 area, in ms.

The percentage of pore volume of smaller pores (SPVP) and larger pores (LPVP) represents the percentage of spectral area in a certain T₂ spectrum range, in %; relevant literature has studied the relationship between the T₂ value and pore size. However, the conversion coefficients between the two parameters are not consistent. Based on the research results of Yao et al. (2010), the pore fracture system can be divided into small pores (0–102 nm, T₂ < 2.5 ms) and large pores (102–104 nm, 2.5 ms < T₂) using the T₂ spectrum and the response characteristics of pores with different pore sizes [4,16,17,18].

Based on the saturated T₂ spectrum, two kinds of fractal parameters are proposed, as follows:

The first type is the single fractal parameter, which is calculated using the single fractal model. The calculation formula is as follows:

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

(1)

In the formula, V_p is the pore volume percentage of the T₂ spectrum under a saturated water condition, in %; T₂ max is the maximum transverse relaxation time, in ms; and D_S is the fractal dimension in a saturated water state, which is dimensionless.

The relevant literature shows that the fractal curve can be divided into two sections using the T_{2 cut-off} value of saturated water [19,20]. The slope of the curve with a T₂ value of less than 2.5 ms corresponds to the heterogeneity of pore distribution with a smaller pore size (D₁), and the slope of the curve with a T₂ value greater than 2.5 ms corresponds to the heterogeneity of pore distribution with a larger pore size (D₂).

The second type is the multifractal parameter, which is calculated using the multifractal model. The specific calculation formula is shown in Zhang et al. (2020) [1]. The multifractal characteristics of NMR data under different test methods (reflecting the inhomogeneity of pore size distribution) are studied using a box counting method [21,22,23]. When analyzing the volume probability of the T₂ spectrum in the interval [A, B], it is necessary to determine the scale and measure. The expression of proportion and measure is as follows:

$$\epsilon =2^{-k}L$$

(2)

$$\overline{\Delta n_i}={\displaystyle \frac{\Delta n_i-\Delta n_{\min }}{{\displaystyle \sum _{i=1}^N(\Delta n_i-\Delta n_{\min })}}}$$

(3)

The T₂ value is divided into several equal-length intervals for NMR testing, and the interval scale is represented by this. The analysis range of NMR data is 0~1000 ms, as detailed in Zhang et al. (2020) [1]. The multifractal dimension can be divided into two languages, namely the singular fractal spectrum and the generalized fractal spectrum. The characteristic parameters of f(a) mainly include a_min, a_max, a₀, a₀–a_max, a_min–a₀ and a. This is a set of main languages describing multifractal local features, called a multifractal spectrum [24,25]. From the perspective of information theory, another set, q~D(q), is introduced, which is called the generalized fractal dimension [26].

3. Results and Discussion

3.1. NMR T₂ Spectrum Distribution and Type Division

Unlike unconventional reservoirs such as coal and shale, tight sandstone reservoirs do not develop micropores, and the T₂ spectrum is dominated by single peaks [9]. The T₂ spectra of nine sandstone samples can be divided into the following four types. Type A is represented by sample 2, wherein the T₂ spectrum shows a bimodal state, and the area of the right T₂ spectrum (2.5~100 ms) is larger than that of the left T₂ spectrum (T₂ < 2.5 ms), indicating that the sample has good pore connectivity and belongs to the macroporous development sample (Figure 1a). Type B is represented by sample 8, wherein the T₂ spectrum is unimodal. The area of the right T₂ spectrum is significantly larger than that of the left T₂ spectrum, indicating that the pore connectivity of this type of sample is poor and that it belongs to the macroporous development sample (Figure 1b). The C-type sample is represented by sample 10. The T₂ spectrum shows a single peak state, and the main T₂ is distributed in 1~50 ms, indicating that the pore connectivity of this type of sample is very poor, and that it belongs to the mesoporous development type sample (Figure 1c). The D-type sample is represented by sample 52, and the T₂ spectrum is unimodal, and the main T₂ is distributed in 0.1~2.5 ms, indicating that the pore connectivity of this type of sample is very poor and that it belongs to the small pore development sample (Figure 1d). The results in

Table 1. Sample base information sheet.

Sample Number	Φwater (%)	Φnitrogen (%)	Permeability (mD)	T2 cut-off (ms)	Saturated Water Swi (%)	<2.5 (ms)	2.5~100 (ms)	Main Peak Position (ms)
2	3.75	6.55	1.0173	2.98	42.91	0.41	0.51	14.97
8	3.25	3.50	0.1169	4.82	27.96	0.18	0.74	21.09
10	3.24	3.20	0.1007	7.79	71.19	0.33	0.62	4.78
11	5.74	5.41	0.3871	8.39	64.36	0.25	0.68	6.00
12	2.62	2.80	0.1116	4.64	63.28	0.46	0.50	3.02
16	3.55	3.20	0.0352	6.67	63.15	0.31	0.66	6.00
45	10.24	9.27	3.0358	1.30	11.49	0.13	0.86	18.82
52	2.22	2.01	0.0146	0.49	66.38	0.92	0.07	14.97
58	9.77	9.03	0.5730	5.65	51.93	0.27	0.72	7.54

show that the porosity and permeability of type A and B samples are larger than those of type C and D samples. This is because the large pores in the pores and fractures of the former are developed and the pore connectivity is better.

3.2. Correlation between Single Fractal Parameters and T_{2 cut-off} Value

Based on Equation (1), the single fractal dimension values of D₁ (small pores), D₂ (large pores) and D_T (total pores) of the same sample can be calculated using the saturated T₂ spectrum of the sample. Figure 2a shows that D₁ is not in the range of 2~3, which indicates that the distribution characteristics of small pores have no fractal significance. The D₂ and D_T of the sandstone samples are 2.77~2.97 and 2.06~2.45, respectively, indicating that the macropores and total pores in the sandstone samples have fractal characteristics. Figure 2b shows that D₂ increases with the increase in pore volume percentage, but the correlation is weak, while the correlation between D_T and pore volume parameters is weak. Compared with small pores, there is no significant correlation between the volume percentage of large pores and the single fractal dimension value (Figure 2c), indicating that small pores are the key factors restricting the heterogeneity of reservoir pore distribution. Figure 2d shows that D₂ increases with the increase in T_{2 cut-off} value, but the correlation is weak. Therefore, it is not feasible to predict the T_{2 cut-off} value using the single fractal dimension parameter.

3.3. Correlation between Multifractal Parameters and T_{2 cut-off} Value

As described in Section 2.2, the multifractal dimension values of all sandstone samples can be calculated using the saturated T₂ spectrum curve (Figure 3). The results show that the q~D(q) spectra of all sandstone samples are obvious inverse S-typed, which is also another typical feature of the sample pore size distribution conforming to multifractal characteristics. The spectral line can effectively characterize the complexity of the pore size distribution characteristics at different stages, to reveal the local difference characteristics of the mosaic in the whole. The results show that the D₀ of all samples is one, indicating that each partition box contains pores, corresponding to the Euclidean dimension of one-dimensional distribution. D₁ is the information entropy dimension, which is related to the entropy value of the system and can be used to characterize the heterogeneity of the overall pore distribution. The higher the value, the higher the pore distribution range and the stronger the heterogeneity. Figure 4 shows that there is a linear positive correlation between the singular fractal spectrum parameters and the generalized fractal spectrum parameters. Except for the correlation coefficient between D₀–D₁₀ and a₀–a₁₀, which is 0.61, the other correlation coefficients are greater than 0.9. Therefore, only the generalized fractal parameters are used to characterize the multifractal characteristics.

According to the above, D₀–D₁₀ characterizes the heterogeneity of pore volume development area distribution, and D₋₁₀–D₀ characterizes the heterogeneity of pore volume underdevelopment area distribution. Taking Figure 1d as an example, when T₂ is less than 2.5 ms, the pore volume ratio of small pores is greater than 80%; therefore, this part of the region can be characterized by D₀–D₁₀. On the contrary, when T₂ is greater than 2.5 ms, the corresponding macropore volume ratio is less than 20%; therefore, this part of the region can be characterized by D₋₁₀–D₀. It can be seen from Figure 5a that the correlation between D₀–D₁₀ and D₋₁₀–D₀ is weak, indicating that the distribution heterogeneity of the low value area and the high value area of pore volume in this kind of sample is strong. At the same time, Figure 5b,c show that, with the increase in D₋₁₀–D₀, D₋₁₀–D₁₀ increases linearly, but there is no obvious linear correlation between D₀–D₁₀ and D₋₁₀–D₁₀, which indicates that the low value area of pore volume in this kind of sample controls the overall heterogeneity of pore distribution.

Figure 6a shows that there is no correlation between the cut-off values of D₁₀ and T₂. Figure 6b shows that there is a linear positive correlation between the cut-off values of D₋₁₀ and T₂, wherein the R₂ value is 0.75. Figure 6c,d show that there is a linear positive correlation between T₂ and D₋₁₀–D₁₀ and D₋₁₀/D₁₀ cut-off values, with R₂ being 0.68 and 0.76, respectively. Figure 6e shows that there is a linear positive correlation between D₋₁₀–D₀ and T_{2 cut-off} value, wherein the R₂ value is 0.75, which indicates that the heterogeneity parameter of the pore volume distribution of regional pore size can be used to predict the T_{2 cut-off} value.

3.4. Correlation between T₂ Spectral Morphological Parameters and T_{2 cut-off} Values

Peak position, peak value and T₂ spectral area ratio are three important parameters used to distinguish T₂ spectral types. Figure 7a shows that the correlation between T_M and the volume percentage of small pores and large pores is weak. Figure 7b shows that the correlation between T_M and the single fractal dimension D₁ is weak, but the linear fitting degree with D₂ is high, and the R₂ value can reach 0.45. Figure 7c shows that the correlation between T_M and the multifractal dimension is very poor. The above results show that the physical meaning of the morphological parameters is completely different from those of the fractal parameters.

In addition, Figure 8 shows that, except for the single fractal parameter D₂, the pore volume percentage has a weak correlation with other single and multifractal parameters. This also shows that the pore volume percentage is different from the fractal parameters and can be used as one of the parameters for T_{2 cut-off} value prediction.

When the T_{2 cut-off} value of the sample is 0~10 ms, the T_{2 cut-off} value decreases with the increase in the T_M value (Figure 8a), but its linear correlation is much weaker than with the multifractal parameters (Figure 6c,d). In addition, there is no significant correlation between the volume percentage of small and large pores and the T_{2 cut-off} value (Figure 9a–c). It is worth noting that this result is not consistent with the results of Zhang et al. (2023), which were mainly affected by the number of samples and the type of reservoir. As a result, the correlation between the morphological parameters and the T_{2 cut-off} value is weak [9].

3.5. T_{2 cut-off} Value Prediction Model Based on Multifractal Parameters

In the above content, the parameters affecting the T₂ cut-off value are found by using the single fractal and multifractal methods and the morphological discrimination method. The results show that the relevant parameters affecting the T₂ cut-off value include D₋₁₀–D₁₀, D₋₁₀/D₁₀, D₋₁₀–D₀, T_M and D₂. Therefore, taking D₋₁₀–D₁₀, D₋₁₀/D₁₀, D₋₁₀–D₀, T_M and D₂ as the prediction parameters for the T_{2 cut-off} value of sandstone samples, a multiple linear regression analysis is carried out using SPSS software v19.0 (SPSS Inc., Chicago, IL, USA), and the prediction model for the T_{2 cut-off} value is obtained. The calculation formula is as follows:

T_{2 cut-off} value =7.386 − 3.30 × (D₋₁₀−D₁₀) − 0.29 × D₋₁₀/D₁₀ + 5.40 × (D₋₁₀−D₀) − 0.24 × T_M − 1.14 × D₂

(4)

The results show that the positive correlation coefficient of the model can reach 0.88. The model is used to predict the T_{2 cut-off} value of the samples in the study area. The basic information of the samples to be tested is shown in

Table 2. Test model sample information sheet.

Sample	D−10–D10	D−10/D10	D−10–D0	TM	D2	T2C-NMR
1	2.54	5.07	2.17	1.36	2.95	1.60
2	1.74	3.30	1.50	0.39	2.99	1.60
3	5.03	12.61	4.46	4.26	2.98	2.68
4	4.04	9.45	3.52	3.80	2.99	2.03
5	4.43	8.78	4.00	2.41	2.99	3.10

and Figure 10. The comparison shows that the error range between the T_{2 cut-off} value calculated using Equation (4) and the T_{2 cut-off} value measured using centrifugal experiment NMR is less than 5% (Figure 10), indicating that the T_{2 cut-off} values calculated by the above two methods are highly consistent. Therefore, it can be considered that the model established in this paper has good applicability.

4. Conclusions

In this paper, the parameters including peak value, peak position and peak area corresponding to small holes are counted, and the relationship between the morphological parameters and the T_{2 cut-off} value is discussed. At the same time, the single and multifractal models are used to realize the quantitative characterization of the heterogeneity of the pore fracture distribution of the target sample, and the correlation between the multifractal parameters and the pore fracture parameters is discussed. On this basis, the key parameters are selected from the morphological and fractal parameters, and the T_{2 cut-off} value is calculated using the single fractal dimension, multifractal dimension and spectral morphology discrimination methods, in order to construct the T_{2 cut-off} value prediction model. The following conclusions are drawn.

(1)

The T₂ spectra of sandstone samples in the study area can be divided into four types. The A-type T₂ spectrum shows a bimodal state, and the area of the right T₂ spectrum is larger than that of the left T₂ spectrum, indicating that the samples have good pore connectivity and belong to the macroporous development type samples. The B-type T₂ spectrum is unimodal, and the pore connectivity of the samples is poor, indicating that they belong to the macroporous development sample. The T₂ spectrum of the C-type samples is unimodal, and the pore connectivity is very poor, indicating that they belong to the mesoporous development sample. The T₂ spectrum of the D-type samples is unimodal, and the main T₂ is distributed within 0.1~2.5 ms, and the pore connectivity is very poor, indicating that they belong to the small pore development sample.

(2)

The single model shows that small pore volume is the key factor restricting the non-uniformity of reservoir pore distribution. Compared with other single fractal parameters, D₂ increases with the increase in the T_{2 cut-off} value, but the correlation is weak. Therefore, it is not feasible to predict the T_{2 cut-off} value using the single fractal dimension parameter.

(3)

The multifractal model shows that D₋₁₀–D₁₀ increases linearly with the increase in D₋₁₀–D₀, but there is no obvious linear correlation between D₀–D₁₀ and D₋₁₀–D₁₀, indicating that the low pore volume area in this kind of sample controls the overall heterogeneity of pore distribution.

(4)

The relevant parameters affecting the cut-off value of T₂ include D₋₁₀–D₁₀, D₋₁₀/D₁₀, D₋₁₀–D₀, T_M and D₂. Therefore, the T_{2 cut-off} value prediction model is constructed based on the above five parameters. The T_{2 cut-off} value calculated by the model is highly consistent with the experimental value, which proves the reliability of the model.