Application of Multifractal Theory for Determination of Fluid Movability of Coal-Measure Sedimentary Rocks Using Nuclear Magnetic Resonance (NMR)

Abstract

A precise evaluation of the fluid movability of coal sedimentary rock is crucial to the effective and secure utilization of coal measures gas reserves. Furthermore, its complex pore structure and diverse mineral components impact the flow properties of fluids in pore structures, causing accurate evaluation of fluid mobility to be extremely challenging. Nuclear magnetic resonance (NMR) technology is currently a prevalent technique to assess unconventional reservoirs due to its capacity to acquire abundant reservoir physical property data and determine fluid details. The free-fluid volume index (FFI) is a crucial factor in assessing fluid movability in the application of NMR technology, which can only be derived through intricate NMR saturation and centrifugation experiments This research utilized nuclear magnetic resonance (NMR) tests on 13 classic coal-measure sedimentary rock samples of three lithologies to reveal the FFI value. Moreover, the association between mineral components, pore structure parameters, and FFI was then extensively analyzed, and a prediction model for FFI was constructed. The results indicate that the T₂ spectra of sandstone and shale own a bimodal distribution, with the principal point between 0.1 and 10 ms and the secondary peak between 10 and 100 ms. The majority of the T₂ spectra of mudstone samples provide a unimodal distribution, with the main peak distribution range spanning between 0.1 and 10 ms, demonstrating that the most of the experimental samples are micropores and transition pores. The calculated results of the FFI range from 7.65% to 18.36%, and depict evident multifractal properties. Porosity, the content of kaolinite, multifractal dimension (D_q), and the FFI are linearly positively correlated. In contrast, the content of chlorite, illite, multifractal dimension subtraction (D_min − D_max), multifractal dimension proportion (D_min/D_max), and singularity strength (Δα) possess a negative linear correlation with the FFI, which can be further used for modeling. On the basis of the aforementioned influencing factors and the FFI experimental values of eight core samples, an FFI prediction model was constructed through multiple linear regression analysis. The accuracy of the prediction model was validated by utilizing this approach to five samples not included in the model development. It was revealed that the prediction model produced accurate predictions, and the research findings may serve as a guide for the classification and estimation of fluid types in coal reservoirs.

1. Introduction

With a rise in demand for natural gas in past decades, exploring unconventional reservoirs has become a frequent topic [1]. In contrast to conventional reservoirs, unconventional reservoirs have nanopores and complicated pore structures, demonstrating ultra-low porosity and permeability, making efficient extraction of unconventional energy sources more challenging [2]. Accordingly, research methodology on the micropore size distribution pattern and heterogeneity of unconventional reservoirs and evaluating their impact on the mobility of reservoir fluids has relevant explanatory value for the evolution of unconventional oil and gas [3].

The micropore structure of a reservoir alters not only its physical properties but also the flow efficiency of fluids in porous media to a moderate extent [4]. Most of the time, two methods are required to describe pore structures accurately. The first type is direct imaging, which includes field emission scanning electron microscopy (FE-SEM) [5,6,7], the focused ion beam-scanning electron microscope system (FIB-SEM) [8,9,10], and computed tomography (CT) technology [11,12,13,14], which can obtain the microscopic morphology of pores. The second type comprises fluid injection methods, such as CO₂ and N₂ adsorption experiments [4,15,16,17], high-pressure mercury intrusion (MIP) [18,19,20], and low-field nuclear magnetic resonance (NMR) [3,21], which can quantify the pore volume, specific surface area, and pore size distribution (PSD) of rock samples.

Among them, NMR testing can promptly and non-destructively detect porosity, PSD, and FFI [22,23,24]. It has evolved into a critical technique for analyzing the pore structure of unconventional reservoirs [11,25,26]. In the past two decades, numerous researchers have utilized various cutting-edge experimental techniques to explore the impact of pore structure on the FFI. Zhang et al. utilized NMR technology to examine the pore structure and fluid mobility features of coal-measure sedimentary rock in the Ningtiaota area. They identified that porosity, permeability, and clay mineral composition would impact FFI [1]. Gao et al. utilized multiple experimental techniques to evaluate the critical determinants of movable fluid saturation in tight sandstone reservoirs and summarized that differences in micropore structure characteristics are the most crucial component [22]. Hui et al. examined the impact of pore structure on FFI in Ordos tight sandstone using MIP and NMR methods. They revealed that throat radius and pore mercury saturation influence the FFI, among multiple influencing factors [27]. Wang et al. investigated factors that reflected the FFI in lacustrine sedimentary tight sandstone in the Ordos Basin and revealed that the FFI in diverse pore throat combinations is typically motivated by microscopic pore structure factors [28].

Fractal theory can precisely identify the heterogeneity of porous media and is currently extensively employed in investigating rock pore structure [29,30,31,32]. Yao et al. assessed the impact of fractal dimension on coal adsorption capacity by utilizing the Frenkel–Halsey–Hill (FHH) fractal model according to N₂ adsorption experiment data [33]. Zhang et al. analyzed the fractal dimension’s relevant aspects by integrating the fractal dimension with reservoir evaluation and pore structure parameters [34]. Li et al. explored the evolution of the fractal dimension and the quantitative relationship between fractal dimension and porosity, relying on the digital 3D rock model [35]. Wang et al. evaluated the micropore structure characteristics of shale using gas adsorption tests and fractal theory. They discovered that the fractal dimension strongly correlates with mineral composition, specific surface area, and adsorption volume [36]. Nonetheless, the single fractal model cannot effectively comprehend pore structure heterogeneity and cannot distinguish samples with the same fractal dimension but completely distinct pore size distributions. Consequently, multifractal theory is essential to represent coal reservoirs with complex pore structures.

Multifractal theory illustrates the generalization of the single fractal. It divides complicated fractal structures into multiple regions with the same singularity strength and generalized fractal dimension. Then it analyzes these regions’ singularity strength and generalized fractal dimension to derive specific multifractal features. Zhao et al. performed a multifractal analysis of the PSD of tight sandstone and discovered a positive correlation between clay minerals and heterogeneity [37]. Zheng et al. presented a T₂ cutoff algorithm that utilizes multifractal variables [24]. Wang et al. defined the pore structure of tight sandstone employing NMR technology and multifractal theory. They discovered that the pore radius and multifractal dimension adequately represent the microscopic pore structure of tight sandstone, in which the pore type considerably affects the distribution of crude oil [38]. Zhao et al. explored the pore structure characteristics of high-rank coal utilizing MIP and FIB-SEM technology in association with multifractal theory and discovered that the multifractal dimension is an essential index for describing the pore structure characteristics of high-rank coal reservoirs [39]. Currently, most existing studies employ a single fractal to evaluate pore structure heterogeneity. Multifractal analysis has the potential to grow as a supplementary method for quantifying the heterogeneity of pore structures. Furthermore, there are no published reports on the impact of a comprehensive analysis of reservoir pore structure properties, multifractal features, and mineral content on the occurrence characteristics of reservoir fluids.

This work aims to establish an FFI prediction model based on multifractal theory, which comprehensively considers the influence of pore structure characteristics and mineral components. The pore structure and fluid movability features of typical rock samples were explored using multifractal theory and NMR experiments. The primary variables affecting the occurrence of mobile fluids in coal reservoirs were distinguished by investigating the connection between mineral components, pore structure variables, and the occurrence properties of mobile fluids. A multiple linear regression model for predicting the FFI in the study area was established. This study serves as a guide for recognizing the features and fluid fluidity of rock samples, in addition to a theoretical foundation for assessing tight reservoirs in the study region.

2. Materials and Methods

2.1. Rock Samples

Thirteen rock samples with three different lithologies of shale, mudstone, and sandstone were accumulated from Quaternary and Cretaceous coal-bearing strata in the Daqiang Coal Mine, Tiefa Basin, and Liaoning Province. Following collection, rock samples were transported to the ground in self-sealing bags, where they were instantly encased with wax before being processed into uniformly sized cylinder cores (25 mm × 35 mm) in the laboratory.

All the samples’ densities were calculated using their dry weights and bulk volumes. We determined the water porosity and gas permeability based on GB/T 29172-2012. The rock samples’ permeability (K) was evaluated by passing high-purity nitrogen through them in a gas permeability testing system. At the same time, their water porosities (φ) were quantified using the water saturation method. The X-ray diffraction (XRD) measurements were performed utilizing a German Bruker D8 Discover X-ray diffractometer by SY-T 5163-2010.

2.2. NMR Measurements

The NMR relaxation time comprises the longitudinal relaxation time (T₁) and transverse relaxation time (T₂). In environments with low magnetic fields, T₂ has a considerably faster measurement speed than T₁ and can yield the same aperture distribution information as T₁. Consequently, T₂ is traditionally utilized for analyzing aperture distribution patterns [40]. The expression for the T₂ time is as follows [21]:

$$\frac{1}{T_2}=\frac{1}{T_{2b}}+\frac{1}{T_{2s}}+\frac{1}{T_{2d}}$$

(1)

where T_2b, T_2s, and T_2d represent bulk relaxation (ms), surface relaxation (ms), and diffusion relaxation (ms), respectively. The equation can be further simplified as [11]:

$$\frac{1}{T_2}=\frac{1}{T_{2s}}=\rho \frac{s}{v}$$

(2)

where s is the pore surface area (cm²); V is the pore volume (cm³); and ρ is the lateral relaxation strength of the surface (μm/ms).

The NMR experiments were conducted using the Chinese RecCore-04 NMR instrument in China, whose settings are listed in

Table 1. Parameters of NMR T₂ test.

Attribute	Parameter φ (%)
Spectrometer frequency (SF)	12 MHz
The constant magnetic field strength of NMR	0.12 T
Pulse sequence	Carr–Purcell–Meiboom–Gill (CMPG)
90° pulse width (P1)	14.80 μs
180° pulse width (P2)	28.8 μs
Spectral width (SW)	250 KHz
Waiting time (TW)	3000 ms
Echo time (TE)	0.18 ms
Number of echoes (NECH)	14,000
Signal–noise ratio (S/N)	200

. The experiment consisted of two distinct sections. First, each sample was desiccated in a 105 °C oven for 6 h. The cores were then saturated with water for 48 h in an instrument with a 20 MPa Chinese vacuum pressure. After the experiment, NMR analysis was performed on the samples. After 12 h of centrifugation, a second NMR procedure was carried out. Pore structure factors besides nuclear magnetic porosity, T₂ cutoff value (T_2c), and the FFI were identified utilizing NMR experiments.

2.3. Multifractal Analysis

The multifractal analysis depicts the self-similarity of rocks using the continuous function (multifractal spectrum). This research uses the box-counting method to examine the multifractal properties of NMR T₂ spectrum data of rock samples. Firstly, divide the research area into N compartments of equal size, ε (N = 2 k, k = 1, 2, 3,...):

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

(3)

where L is the side length of the box.

The probability mass function of the ith box with size ε can thus be written as [41]:

$$P_{i\left(\epsilon \right)}=\frac{N_{i\left(\epsilon \right)}}{\sum _{i=1}^{N\left(\epsilon \right)}{N}_{i\left(\epsilon \right)}}$$

(4)

where $N_{i\left(\epsilon \right)}$ is the total volume of pores in the ith box, and $P_{i\left(\epsilon \right)}$ is the probability mass function associated with the size scale of each box satisfying the exponential power relationship [41]:

$$P_{i\left(\epsilon \right)}\propto {\epsilon }^{{-\alpha }_i}$$

(5)

where ${\alpha }_i$ is the Lipschitz–Hölder singularity exponent. It is related to the region and reflects the probability of the region being located. The number of boxes with the same α value is defined as $N_{\alpha }\left(\epsilon \right)$, and $N_{\alpha }\left(\epsilon \right)$ satisfies [41]:

$$N_{\alpha }\left(\epsilon \right)\propto {\epsilon }^{-f\left(\alpha \right)},\epsilon \to 0$$

(6)

where f(α) is the multifractal singularity spectrum, which is the fractal dimension with the same α value. The α and f(α) can be calculated using the Chhabra and Jensen methods, with the expression [42]:

$$\alpha (q)\propto \frac{\sum _{i=1}^{N\left(\epsilon \right)}\left[u_i(q,\epsilon )lg\epsilon \right]}{lg\epsilon }$$

(7)

$$f(\alpha )\propto \frac{\sum _{i=1}^{N\left(\epsilon \right)}\left[u_i(q,\epsilon )lg{u}_i(q,\epsilon )\right]}{lg\epsilon }$$

(8)

where

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

(9)

where q is an order of the matrix with the range from−∞ to +∞; the value of q in this study is an integer in the interval [−10, 10]. The partition function is expressed as follows [43]:

$$X\left(q,\epsilon \right)={\sum }_{i=1}^{N\left(\epsilon \right)}{p}_i^q\left(\epsilon \right)\propto {\epsilon }^{-(q-1)D_q}$$

(10)

where D_q is the generalized dimension associated with q that is specified as [42]:

$$D_q=\frac{1}{q-1}\underset{\epsilon \to 0}{\lim }\frac{log\sum _{i=1}^{N\left(\epsilon \right)}{p}_i^q\left(\epsilon \right)}{log\epsilon },(q\ne 1)$$

(11)

when q = 1, D₁ can be expressed as:

$$D_1=\underset{\epsilon \to 0}{\lim }\frac{\sum _{i=1}^{N\left(\epsilon \right)}{p}_i\left(\epsilon \right)lg{p}_i\left(\epsilon \right)}{log\epsilon },(q=1)$$

(12)

when q < 0, D_q depicts the low probability measurement portion; when q > 0, D_q depicts the high probability measurement portion [44].

The relationship between singular strength, $\alpha \left(q\right),$ and multifractal spectrum, $f\left(\alpha \right),$ is expressed as follows through the Legendre variation [42]:

$$\alpha \left(q\right)=\frac{d\tau \left(q\right)}{dq}$$

(13)

$$f\left(\alpha \right)=q\alpha \left(q\right)-\tau \left(q\right)$$

(14)

Typically utilized as an indicator of the heterogeneity features of subjects, singularity strength range, Δα, is defined as [43]:

$$∆\alpha ={\alpha }_{max}-{\alpha }_{min}$$

(15)

where α_max and α_min are the maximum and minimum values of singularity strength, respectively.

Multifractal parameters, including D_q, α(q), f(α), and Δα, i.e., are derived from the T₂ spectrum distribution data to assess the multifractal properties of the micropore structure of coal-measure sedimentary rock reservoirs in the study area.

3. Results

3.1. Petrophysical Properties

The density, helium porosity, NMR porosity, and gas permeability of all the numbered materials are shown in

Table 2. Petrophysical properties of the investigated rock samples.

Sample No.	Density (cm3/g)	He Porosity (%)	NMR Porosity (%)	Permeability (mD)
SS-1	2.53	9.27	9.12	0.0070
SS-2	2.48	10.24	9.87	0.0030
SH-1	2.52	4.15	3.85	0.0050
SH-2	2.51	7.75	7.45	0.0030
SH-3	2.69	9.89	10.13	0.0010
SH-4	2.37	4.95	5.05	0.0020
SH-5	2.42	4.21	4.03	0.0010
MS-1	2.43	9.15	9.4	0.0020
MS-2	2.45	10.73	10.69	0.0030
MS-3	2.38	9.18	9.73	0.0010
MS-4	2.44	10.01	10.45	0.0017
MS-5	2.41	9.62	10.17	0.0012
MS-6	2.42	9.73	9.85	0.0026

. In all experimental rock samples, the helium porosity range is 4.15–10.73%, and the nuclear magnetic porosity range is 3.85–10.69%. The measurement results of helium porosity and NMR porosity are similar (Figure 1), confirming the accuracy of the NMR porosity. Regarding the helium porosity, the variations for sandstone, shale, and mudstone samples are 9.27–10.24%, 4.15–9.89%, and 9.15–10.73%, respectively. The sandstone and mudstone samples provide a marginally greater helium porosity than shale samples. The span of gas permeability is 0.0012–0.0070 mD, with an average value of 0.0026 mD, all belonging to low-permeability rock samples.

3.2. Mineralogical Compositions

The content of minerals and clay minerals in the experimental sample is depicted in Figure 2. As illustrated in Figure 2, the testing sample consists primarily of quartz, plagioclase, and clay minerals, with the content spanning from 35 to 56%, 7 to 21%, and 25 to 50%, respectively. In addition, the shale sample contains a greater proportion of clay minerals than the sandstone sample. The predominant clay minerals are kaolinite and chlorite, with a trace quantity of illite. The variations for kaolinite and chlorite are 34 to 77% and 7 to 26%, respectively; the content of kaolinite is considerably more prevalent than chlorite and illite.

3.3. T₂ Distributions

Figure 3 depicts the testing samples’ saturation and centrifugal T₂ spectra. The solid black line indicates the T₂ signal increment curve in the saturated state, revealing the full aperture distribution of the numbered samples. Following the results from prior research [21], the location of each peak in the NMR T₂ spectrum can reflect the properties of pores and cracks of variable thickness. Micropores and transition pores with T₂ relaxation times of 0.5–2.5 ms, 20–50 ms, and >100 ms correlate to mesopores, macropores, and microfractures, respectively. As seen in Figure 3, the T₂ spectra of sandstone and shale samples exhibit a bimodal distribution, with the major peak occurring around 0.1 and 10 ms and the secondary peak appearing between 10 and 100 ms. T₂ spectra of mudstone samples are predominantly unimodal, with a scope of 0.1 to 10 ms. The mudstone sample (MS-4) possesses low signal quantities at 30–100 ms, demonstrating that the testing samples are predominantly microporous, and that the sandstone and shale samples contain tiny transition pores. Furthermore, relying on the centrifugal T₂ spectrum of the experimental sample, it can be determined that the primary peak centrifugal T₂ spectrum of the observed sample is noticeably smaller than the saturated T₂ spectrum, denoting that the bound fluid content in the micropores is intense and complicated to disperse under centrifugal force. The secondary peak centrifugal T₂ spectrum variation demonstrates a noticeable decreasing trend, revealing that centrifugal force precipitates a substantial quantity of the movable fluid in the transition pore.

3.4. Multifractal Characteristics

The microheterogeneity of reservoir rocks is a crucial variable in fluid migration and structural stability [33]. Mandelbrot proposed fractal theory in 1975 to assess the heterogeneity of complex porous media, which has since been extensively applied to estimate the heterogeneity of reservoir rocks [45]. This section analyzes the NMR multifractal features of entirely water-saturated samples based on the NMR-measured T₂ spectrum distribution and the multifractal theory model. The default setting for the multifractal moment order, q, was a continuous interval between −10 and 10, assuming a statistical interval order of q = 1. D₋₁₀ and D₁₀ represent D_min and D_max, respectively. Table 2 displays the results of multifractal parameter estimates for each sample.

Among them, D₀ is a capacity dimension, depending on whether or not each compartment contains pores and, therefore, cannot reflect the distribution of reservoir pores [46]. D₁ is the information dimension that can quantify pore distribution heterogeneity. D₂ is the correlation dimension that further indicates the level of the aperture distribution’s self-similarity [47]. The value of D₀ > D₁ > D₂ reveals multifractal features of NMR T₂ distributions for all numbered samples. In addition, the Δα reflects the X-axis breadth of the multifractal spectrum, which is the degree of dispersion of the multifractal probability measure distribution within the research scope. The greater the value of Δα is, the more complicated is the pore distribution and degree of heterogeneity of reservoirs. D_min − D_max and D_min/D_max are referred to as symmetrical multifractal subtraction and symmetrical multifractal dimension proportions, respectively, which can depict the curvature extent of the generalized dimension curve and are employed to identify the degree of variation of pore features. Comparing the low probability measurement area to the high probability measurement area, the higher the value of D_min − D_max and D_min/D_max is, the more complicated is the pore structure in the low probability measurement area.

As depicted in Figure 4, the multifractal discrimination parameters lg X(q,ε) and lg ε of sandstone (SS-1), shale (SH-1), and mudstone (MS-5) samples demonstrate an excellent linear correlation, implying that the pore structure of the experimental rock samples acquired from the T₂ distribution data conforms to the multifractal characteristics, which is in accordance with prior findings.

The connection between the multifractal dimension (D_q) and the variable q is depicted in Figure 5. In addition, the calculation results of NMR multifractal dimension is depicted in

Table 3. Calculation results of NMR multifractal dimension.

Sample No.	Dmin	D−2	D−1	D0	D1	D2	Dmax	Dmin − Dmax	Dmin/Dmax	Δα
SS-1	2.267	1.677	1.343	0.977	0.860	0.741	0.671	1.596	3.378	1.845
SS-2	2.144	1.611	1.343	0.980	0.860	0.748	0.679	1.464	3.155	1.701
SH-1	1.494	1.186	1.073	0.923	0.830	0.745	0.669	0.825	2.234	0.995
SH-2	1.596	1.187	1.075	0.926	0.840	0.741	0.667	0.929	2.394	0.998
SH-3	2.332	1.704	1.314	0.865	0.790	0.720	0.667	1.666	3.498	1.922
SH-4	1.946	1.605	1.392	1.000	0.880	0.751	0.683	1.263	2.848	1.418
SH-5	1.954	1.602	1.299	0.848	0.770	0.699	0.649	1.305	3.011	1.717
MS-1	1.350	1.043	0.912	0.782	0.730	0.683	0.631	0.719	2.140	0.872
MS-2	2.655	1.952	1.516	0.873	0.770	0.677	0.624	2.032	4.258	2.317
MS-3	1.822	1.345	1.072	0.791	0.730	0.677	0.625	1.197	2.913	1.398
MS-4	2.384	1.750	1.352	0.852	0.770	0.691	0.638	1.747	3.740	2.006
MS-5	1.438	1.113	0.955	0.797	0.740	0.692	0.639	0.799	2.249	0.951
MS-6	1.820	1.348	1.075	0.804	0.740	0.680	0.627	1.192	2.901	1.394

. As the number of test samples, q, increases, the D_q exhibits two distinct tendencies: when q > 0, the multifractal parameters determine the pore structure of the high probability measurement area by decreasing substantially, and when q < 0, they determine the pore structure of the low probability measurement area by reducing slightly [44]. Micropore growth in experimental rock samples has a greater probability of being distributed. Hence, when q > 0, the multifractal factors quantify the heterogeneity of micropores; when q < 0, they evaluate the heterogeneity of macropores. Moreover, using q = 0 as the border, D_min − D₀ is the breadth of the left branch of the generalized dimension spectrum, which is used to evaluate the heterogeneity of macropores, and D₀ − D_max is the breadth of the right branch, which is used to describe the heterogeneity of micropores. Each sample’s D_min − D₀ values are greater than its D₀ − D_max values, suggesting that the heterogeneity of macropores is stronger than that of micropores.

Figure 6 depicts the multifractal spectra of all numbered samples. The NMR T₂ distributions of samples with water saturation can reveal details regarding heterogeneity through the shape and symmetry of the f(α) spectra. The multifractal spectrum has a distinctive concave-down parabola pattern. When α(q) rises, the left side of the crest experiences a rise in f(α), while the right side suffers a decline in f(α). Another variable for multifractal features is the singularity strengths range, which is denoted by Δα (α_max−α_min). NMR T₂ distributions of all samples are diverse, with values of Δα ranging from 1.786 to 2.573, with an average of 2.194. The connection between τ(q) and q is depicted in Figure 7, in which τ(q) significantly rises as q rises when q < 0, while, when q > 0, it slightly rises as q rises.

3.5. Free-Fluid Volume Index (FFI)

Fluids in porous media can be classified as movable fluids and bound fluids, depending on their occurrence properties. Movable fluids are unaffected by capillary forces and approaching under external pressure, while bound fluids are inhibited. Complex micropores containing abundant bound fluids characterize the pore structure of low-permeability reservoirs. With low-permeability reservoirs, permeability delivers a poor description of the fluid occurrence and flow characteristics.

Hence, the fluid occurrence properties of low-permeability reservoirs are typically described using the concept of the FFI. The FFI can be determined via NMR saturation and centrifugal tests, with the methodology being the most prevalent technique for deciding the FFI [1]. Figure 8 depicts the schematic diagram of the area technique and physical model for determining the FFI, where the bound fluid corresponds to the area of the centrifugal T₂ spectrum, and the movable fluid relates to the difference in the area of the centrifugal and saturation T₂ spectra. The FFI is the proportion of the volume of a movable fluid to the total volume of the fluid. As seen in

Table 4. Calculation results of FFI.

Sample No.	T2 (ms)	FFI (%)	BVI (%)	FFI/BVI
SS-1	2.23	13.42	86.58	0.155
SS-2	2.77	13.02	86.98	0.150
SH-1	1.86	10.16	89.84	0.113
SH-2	1.55	14.46	85.54	0.169
SH-3	2.83	8.81	91.19	0.097
SH-4	4.64	11.59	88.41	0.131
SH-5	1.55	12.07	87.93	0.137
MS-1	2.48	7.93	92.07	0.086
MS-2	2.97	10.56	89.44	0.118
MS-3	2.95	8.34	91.66	0.091
MS-4	2.87	8.65	91.35	0.095
MS-5	2.51	12.44	87.56	0.142
MS-6	2.71	11.59	88.41	0.131

, the estimated range of the FFI is 7.65–18.36%, with ranges of 11.77–18.36%, 9.38–13.74%, and 7.65–14.45% for sandstone, shale, and mudstone samples, respectively. SS-1 has the highest FFI (18.36%), whereas MS-1 has the lowest FFI (7.65%), indicating that SS-1 has the maximum fluid mobility and MS-1 has the bare minimum.

4. Discussion

4.1. Correlation between Physical Properties and FFI

Research has shown a clear association between the free fluid content of core samples and their pore structure and mineral composition [27]. This section reveals a prediction model for the FFI by investigating the connection between the pore structure factors (porosity, permeability, and multifractal parameters), mineral components, and the FFI. During the formation of the model, 13 samples were randomly split into two groups. One group consists of eight samples for developing the model, whereas the other group contains five samples (SS-1, SH-1, SH-3, MS-2, and MS-3) to validate the model’s accuracy.

Physical properties parameters are a comprehensive reflection of the rock pore system, with porosity and permeability being critical pore structure parameters determining the reservoir’s storage space and seepage capacity. In contrast, the FFI reveals the reservoir’s fluid fluidity. This section discusses the impact of porosity and permeability on the FFI. Figure 9 depicts that helium porosity, nuclear magnetic porosity, and FFI are positively correlated, with correlation coefficients of 0.801 and 0.894, respectively. In addition, no association between permeability and FFI is observed. The FFI indicates the number of free fluids in reservoir rocks, while porosity reveals the amount of reservoir space. The broader the porosity is, the larger is the space for storage fluids and the wider is the number of pores that can participate in fluid flow, which promotes fluid flow and enhances the FFI. The absence of a positive correlation between permeability and FFI can be attributable to the low porosity and permeability of the experimental samples utilized in this study. Previous research has demonstrated that more than permeability might be needed to define the permeability and fluid velocity of tight reservoirs accurately [28]. Unlike conventional reservoirs, tight reservoirs with a permeability below 1 mD display nonlinear flow. The FFI represents the flow pattern of fluids in porous media and extensively describes the microscopic features of reservoirs. It is more appropriate than permeability for characterizing the permeability of tight reservoirs and assessing reservoir quality.

4.2. Correlation between Clay Minerals and FFI

Clay minerals have an essential function in defining the fluid mobility of rocks. Clay minerals have hydrophilicity and enhanced hydration of exchangeable cations, and water molecules are more likely to penetrate the mineral lattice than crystal minerals such as plagioclase and quartz. Variations in the microstructure of clay minerals due to water absorption impact the fluid mobility of rock samples [37]. As mentioned above, the clay mineral composition of the numbered sample includes kaolinite, chlorite, and illite. This section addresses the impact of different clay minerals on fluid mobility. Figure 9 illustrates that the content of kaolinite is highly associated with FFI, with a correlation coefficient of 0.850 (Figure 10a), whereas the content of chlorite and illite is negatively associated with FFI, with correlation coefficients of 0.783 (Figure 10b) and 0.712 (Figure 10c), respectively. Because of variations between their unique properties, the impact of different clay mineral components on the fluid mobility of rocks differs considerably. A portion of kaolinite results from the dissolving of feldspar; the greater the proportion of kaolinite is in this portion, the more evident is the dissolving transition. The pores created by the dissolving transition are favorable for fluid passage. Moreover, intercrystalline pores are capable of arising between the book-like kaolinites, and intercrystalline pores will enhance the fluid storage capacity. Nevertheless, kaolinite decreases the principal intergranular and secondary pore spaces, restricting fluid passage. Subsequently, kaolinite has two distinct impacts on the FFI. This study discovered that the beneficial effect of kaolinite on the FFI outweighs its adverse influence on the numbered cores. On the other hand, the presence of chlorite and illite will result in inadequate pore connectivity, as their increased hydrophilicity enhances fluid cohesion around pores and impede fluid movement.

4.3. Correlation between Multifractal Characteristics and FFI

Figure 11a depicts the association between the multifractal dimension D_q (q = 10, 2, 1, 1, 2, 10), and the FFI of the eight modeling examples is depicted in Figure 11. The graph illustrates that, when q < 0, the FFI value is positively associated with D_q, and the correlation coefficient is more significant than 0.77 as q changes.

When q > 0, there is an imperfect correlation between FFI and D_q. Figure 11b depicts the relationship between the multifractal parameter D_min/D_max and FFI, which is explored to investigate further the association between the FFI and multifractal dimension. The FFI and D_min/D_max are negatively correlated, with a coefficient of 0.793. The broader the D_min/D_max value is, the more heterogeneous are the macropores. The complicated system of the macropores can inhibit fluid migration, resulting in a decrease in FFI.

Moreover, as demonstrated in Figure 11c, there is a significant negative relationship between D_min − D_max, D_min − D₀, and the FFI, with correlations of 0.719 and 0.745, respectively. However, no significant correlation between D₀ − D_max and the FFI is observed. As previously demonstrated, D_min − D_max corresponds to the relative complexity of macropore and micropore structures. The greater the value of D_min − D_max is, the more complicated are the macropore structures. Additionally, the more excellent value of D_min − D₀ partially reflects the intricate nature of macropore structures. Consequently, as D_min − D_max and D_min − D₀ increase, so does the FFI. D₀ − D_max represents the heterogeneity extent of the micropore structure. Although micropores predominate in the experimental sample, not all micropores lead to fluid migration, and a substantial number of ineffective pores do not contribute to fluid migration. Thus, the micropore complexity has no noticeable effect on the FFI. According to Figure 11d, the value of Δα decreases as the FFI value rises, with a correlation coefficient of 0.711, indicating that the Δα is indeed a factor affecting the FFI. The greater the value of Δα is, the broader is the non-uniformity representing the pore structure of the reservoir. A significant negative correlation exists between the FFI and the Δα. In conclusion, there is a significant correlation between the multifractal factors of pore structure and the FFI, with the multifractal parameters representing macropore structure exhibiting a massive impact on the FFI.

4.4. FFI Prediction Model

The FFI of unconventional reservoir materials is affected by numerous factors, and the fundamental mechanism of action is intricate. As previously stated, a strong correlation exists between FFI and porosity, clay mineral composition, and multifractal parameters. This section evaluates the abovementioned parameters and constructs a multiple-regression prediction model for the FFI. Multiple linear regression is an effective technique for predicting dependent variables subject to various factors by identifying the most practical combination of multiple explanatory variables. Multiple linear regression analysis is founded on the belief that the association between each element and the dependent variable is linear, and that each factor is linearly independent. Before developing a multiple regression prediction model, it is essential to perform collinearity diagnostics for each element to eliminate the independent variables with linear correlation. In accordance with the outcomes of the model collinearity diagnosis, the multifractal parameters (D_min/D_max, D_min − D_max, and Δα) display an apparent collinearity phenomenon. Consequentially, the multifractal properties of the rock sample can be specified by selecting D_min/D_max, which has a significant relation with FFI among the multifractal parameters. In addition, no correlation between permeability and FFI is observed. This section asserts that porosity, φ, the content of kaolinite, chlorite, and illite, as well as D_min/D_max, are the explanatory variables of the FFI prediction model. SPSS is utilized to conduct multiple linear regression analysis on the FFI and construct a multiple linear regression prediction model for the FFI, illustrated by the following formula:

$$\mathit{F}\mathit{F}\mathit{I}=\mathrm{a}+\mathrm{b}\mathsf{\phi }+\mathrm{c}({\mathit{D}}_{\mathit{m}\mathit{i}\mathit{n}}/{\mathit{D}}_{\mathit{m}\mathit{a}\mathit{x}})+\mathrm{d}{\mathit{C}}_{\mathit{k}}+\mathrm{e}{\mathit{C}}_{\mathit{C}}+\mathrm{f}{C}_I$$

(16)

where a is a constant; b, c, d, e, and f are the weight coefficients of porosity, multifractal parameters, the content of kaolinite, chlorite, and illite, $\phi$, C_K, C_C, and C_I is porosity, the content of kaolinite, chlorite, and illite (%).

Before performing calculations, it is essential to standardize the parameters, eradicate the impact of dimensions and units, and convert them into scores suitable for regression analysis. The standardized parameters are then inputted into the SPSS software to undertake multivariate linear regression analysis on the FFI, estimate model parameters using the least-square method, and execute significance tests on the model and its parameters. Among the above five parameters, porosity, D_min/D_max, and the content of chlorite have a noticeable impact on the FFI. Therefore, the FFI multiple linear regression analysis model is established as follows:

$$FFI=22.676+1.653\phi -4.458({\mathit{D}}_{\mathit{m}\mathit{i}\mathit{n}}/{\mathit{D}}_{\mathit{m}\mathit{a}\mathit{x}})-1.432C_C$$

(17)

This model’s correlation coefficient is 0.85 (Figure 12), indicating excellent fit outcomes. The FFI of five additional samples was calculated and contrasted to the experimentally determined FFI values to validate the established model. Figure 11 depicts the intersection of experimental and predicted data, with the red line demonstrating that predicted and experimental results are equivalent. Based on Figure 11, the predicted and experimental results are well matched, suggesting that the model is suitable for predicting the FFI of rock samples with limited permeability. Moreover, the weight coefficients of this model may be employed to measure the influence exerted by influential factors. The broader the absolute value of the influence weight coefficient is, the larger is the factor’s influence on the FFI. A positive influence weight coefficient value indicates a favorable connection between the factor and the FFI, and vice versa. As demonstrated by Equation (17), porosity and the multifractal parameter, D_min/D_max, are positively correlated with FFI, and the content of chlorite is negatively associated with FFI, in accordance with prior investigations. The order of the influence weights of the above three factors on FFI is D_min/D_max > the content of chlorite > porosity, with D_min/D_max exhibiting the highest influence weight coefficient and chlorite content displaying the lowest influence weight coefficient, revealing that the heterogeneity of the pore structure is a crucial aspect that impacts the fluid fluidity of tight reservoirs.

5. Conclusions

• The pore structure and multifractal properties of three typical coal-measure sedimentary rocks from China’s Daqiang Coal Mine were explored utilizing NMR experiments, and a FFI prediction model was created using multifractal theory. The main conclusions are as follows.
• The experimental samples consist primarily of micropores and transition pores, and the range of FFI is 7.65% to 18.36%.
• Different physical parameters and mineral components have multiple effects on the FFI. Porosity, kaolinite content, and FFI exhibit a linear positive correlation, whereas chlorite and illite depict a linear negative correlation, with no apparent correlation between permeability and the FFI.
• The selected samples have prominent multifractal characteristics, in which the multifractal dimension (D_q) is linearly positively correlated with the FFI, and the multifractal subtraction (D_min − D_max), the multifractal dimension proportion (D_min/D_max), and the singularity strengths (Δα) are negatively correlated with the FFI.
• Utilizing NMR data from eight samples and multifractal theory, a prediction model for FFI was constructed and verified employing experimental data from five samples. The anticipated outcomes correspond closely to the experimental data.