NMR and Multifractal Characterization of Pore Heterogeneity in Transitional-Marine Shales: A Case Study from the Longtan Formation, Sichuan Basin

Abstract

Transitional marine–continental shale reservoirs are typified by intricate pore architectures and pronounced heterogeneity; accurate characterization of their pore network and fluid mobility underpins reservoir appraisal and sweet-spot forecasting. Focusing on the Longtan Formation transitional shales in the Sichuan Basin, this study integrates NMR T₂ spectrometry, geochemical–mineralogical assays and multifractal analysis to elucidate multi-scale heterogeneity of the pore framework and its governing mechanisms. Results reveal that the investigated shales are characterized by low porosity (0.46–7.43%) and high bound fluid saturation (66.77–97.28%). Multifractal spectral width (Δα) and degree of multifractality (ΔD) serve as robust metrics of pore heterogeneity, correlating closely with rock composition (e.g., TOC and clay content). By combining multifractal indices, mineralogical assemblage and fluid movability, the samples are classified into three reservoir archetypes, with Type I (weakly heterogeneous—high quality) identified as the prospective developmental sweet spot. This work provides a theoretical and methodological backbone for quality assessment and play-ranking of transitional marine–continental shale reservoirs.

1. Introduction

Shales serve as both prime hydrocarbon kitchens and tight reservoirs for unconventional resources [1,2,3,4], whose extreme lithological heterogeneity and ultra-low permeability exert spatio-temporal controls on hydrocarbon storage and migration [5,6]. Efficient exploitation therefore hinges upon a quantitative appraisal of the intricate pore architecture and associated fluid mobility [7,8].

With the continuous refinement of pore-characterization technologies [9,10,11,12], multi-dimensional datasets of shale pore architectures are increasingly unveiled, offering critical insights into pore evolution and hydrocarbon storage mechanisms [13,14]. Numerous researchers have embedded fractal theory into shale pore investigations [15,16,17,18,19], leveraging nitrogen-adsorption, NMR and small-angle neutron-scattering data to quantify heterogeneity across various length scales. Conventional monofractal models, however, exhibit pronounced limitations in capturing the intense heterogeneity inherent to shales [20,21,22,23]. Consequently, multi-scale fractal theory—encompassing multifractal formalism—has been imported to portray pore-texture heterogeneity with higher fidelity [24,25,26], and has evolved into a pivotal tool for dissecting pore complexity in coals, tight sandstones and shales across disparate observational resolutions [27,28,29].

The Sichuan Basin, situated in central-western China, constitutes the country’s premier petroliferous province [30,31,32,33], hosting several tens of hydrocarbon-bearing stratigraphic units that command high scientific and commercial value [34,35,36,37,38]. From the Palaeozoic through the Mesozoic to the Cenozoic, marine, continental and transitional marine–continental successions are all well developed [39,40]. Multiple marine shale packages within this stack have already achieved commercial hydrocarbon production [41,42,43,44]. At present, the Upper Palaeozoic Permian Longtan Formation mudstone–shale sequence has emerged as a focal target for hydrocarbon researchers [31,45,46]; these transitional shales are endowed with substantial resource potential and are considered ripe for large-scale exploration and development, earmarked as a critical play for future shale-gas reserve growth and production increase in China [39,47,48]. Previous workers have conducted preliminary investigations into favourable lithofacies, organic-matter enrichment and pore development. Enze Wang et al. [48] examined the Permian Longtan Formation transitional marine shales in the Lintanchang area, the southeastern margin of the Sichuan Basin, through petrological and geochemical analyses, identifying four lithofacies: organic-lean calcareous shale, organic-lean mixed shale, organic-lean clayey shale and organic-rich clayey shale. They found that clay-mineral-associated pores constitute the dominant pore type, with clay-mineral content being the principal control on shale petrophysical properties, and TOC the foremost factor governing gas content; among these, organic-rich clayey shale deposited in tidal-flat–lagoon systems represents the most favourable lithofacies. Kai Yang et al. [39] performed geochemical analyses on the Permian Longtan Formation transitional marine shales in well F5 of the Sichuan Basin, finding a mean TOC of 3.04% and a mean thermal maturity of 2.46%, deposited under humid–hot climate, high terrigenous input, oxidising conditions and high sedimentation rates in open-water settings; terrigenous plant debris served as the primary organic-matter source, with terrigenous influx, redox conditions and sedimentation rate being the principal controls on organic-matter enrichment. Jizhen Zhang et al. [49] investigated coal-bearing transitional marine shales of the Longtan Formation in southern Sichuan, employing nitrogen adsorption–desorption experiments and conventional monofractal theory to reveal pore-structure characteristics and heterogeneity; they found that smaller pores developed larger specific surface areas, yielding high fractal dimensions (2.564–2.677). Nevertheless, criteria for sweet-spot selection within this succession remain sparse, and a pivotal challenge lies in evaluating the complexity of pore architecture and fluid mobility within these mudstones.

This study aims to integrate nuclear magnetic resonance (NMR) core measurements, geochemical–mineralogical assays and multifractal formalism, to quantify pore-texture complexity, fluid movability and rock-matrix heterogeneity in Longtan Formation (LTF) shale cores, and to elucidate the governing mechanisms. The resultant database and interpretative framework will underpin the future effective development of this marine–continental transitional shale succession.

2. Materials and Methods

2.1. Sample Description

The Upper Palaeozoic Permian Longtan Formation mudstone–shale package is extensively developed across central and southern sectors of the Sichuan Basin [46,48], with stratigraphic thicknesses generally ranging from 80 to 120 m. The succession is subdivided, in ascending order, into Tan-1, Tan-2 and Tan-3 sub-members [50,51], each corresponding to a third-order eustatic cycle of the late Permian. For this investigation, eleven full-diameter shale cores were retrieved from the Tan-3 sub-member of well YJ-1 in the southern basin, spanning 3021.44–3057.15 m true vertical depth. The samples display elevated thermal maturity [52] and have experienced intense hydrocarbon generation. Well location is shown in Figure 1a; lithology and corresponding depths are listed in Figure 1. The sample photos can be seen in Figure 2a–k.

2.2. NMR Experiment Procedures

All experiments were performed on a Numag MR Cores-XX high-precision bench-top NMR unconventional core analyser coupled with a TG20.5 high-speed centrifuge, with the temperature maintained constant at 25 °C (ambient). NMR core analyses strictly followed China National Petroleum and Natural Gas Industry Standard SY/T 6490-2014 “Laboratory Specifications for NMR Parameter Measurements on Rock Samples”. The principal workflow comprised: (i) oven-drying samples at 110 °C for 24 h [54], weighing and acquiring an initial NMR signal to establish the baseline; (ii) placing the dried plugs in a vacuum-pressure saturation vessel, saturating with deionised water under vacuum for 24 h, and recording the fully saturated T₂ spectrum; (iii) determining bulk volume by the buoyancy method and recording the saturated (wet) mass on an analytical balance; (iv) centrifuging the saturated cores for 6 h at predetermined speeds, with the rpm–g conversion listed in

Table 1. Centrifugal force parameters of samples.

Number	Speed (r/min)	Length (cm)	Angular Velocity (rad/s)	Centrifugal Force (MPa)	PSI
JY1#-1	10,000	4.87	1047.2	6.16	893.21
JY1#-2	10,000	4.87	1047.2	6.15	892.06
JY1#-3	10,000	4.19	1047.2	5.46	791.40
JY1#-4	10,000	4.90	1047.2	6.19	897.11
JY1#-5	10,000	4.86	1047.2	6.14	890.75
JY1#-7	10,000	3.88	1047.2	5.11	741.36
JY1#-8	10,000	4.78	1047.2	6.06	879.26
JY1#-9	10,000	4.92	1047.2	6.21	900.13
JY1#-10	10,000	4.89	1047.2	6.18	895.81
JY1#-11	10,000	4.92	1047.2	6.21	900.13
JY1#-12	10,000	2.48	1047.2	3.46	502.15

Note: # Weight measurement method.

; (v) acquiring the post-centrifuge T₂ spectrum and recording the corresponding mass. A total of 11 plugs were successfully measured; sample JY1-6 was damaged during preparation and excluded from all analyses.

2.3. Geochemistry and Mineralogy Analysis

Whole-rock mineralogy was quantified by X-ray diffraction (XRD) analysis, with the diffractometer operated at 40 kV and 40 mA. Fresh shale chips were selected and pretreated prior to measurement; the entire scan was performed at ambient temperature. The procedure strictly followed China Petroleum Industry Standard SY/T 5163—2018 “XRD Analytical Method for Clay and Common Non-Clay Minerals in Sedimentary Rocks”.

Total organic carbon (TOC) was determined with a carbon analyser. Prior to analysis, samples were ultrasonically cleaned, pulverised, acid-washed and low-temperature dried. The measurement complied with Chinese National Standard GB/T 19145—2003 “Determination of Total Organic Carbon in Sedimentary Rocks”.

The analytical workflow is summarised in Figure 3.

2.4. Multifractal Theory

Multifractal theory [18,32,55,56,57]—an extension of fractal geometry—employs the continuous generalised-dimension spectrum (Dq) and the singularity spectrum (α–f(α)) to capture local irregularities and global heterogeneity within complex systems. Compared with a monofractal dimension, its principal merit lies in resolving scale-dependent probability distributions, thereby quantifying both the intensity and frequency of local singularities. In shale-gas research, this formalism has been deployed to characterise pore-framework heterogeneity and pore-size distribution. Here, NMR T₂ spectra are used to compute D₀ (capacity dimension), D₁ (information dimension), D₂ (correlation dimension) and spectral width Δα, thereby revealing the intricacy of pore development. The workflow proceeds as follows.

2.4.1. Data Pre-Processing

The raw NMR T₂ distribution must first be transformed by converting the relaxation-time axis to a logarithmic scale across the measurement window, yielding 100 dimensionless bins of equal logarithmic width △I (Equation (1)) [28,29,55].

$${\mathsf{\beta }}_{\mathrm{k}}=\log ({\displaystyle \frac{{\gamma }_k}{{\gamma }_1}}),\mathrm{k}=1, 2, 3, \dots \dots , 99, 100$$

(1)

Here, γ_k denotes the measured value, γ₁ the minimum detection limit, and β_k the dimensionless value after converting the PSD detection range.

2.4.2. Sub-Interval Partitioning and Probability Density Calculation

Within △I, the PSD interval is subdivided into K(η) = 2^t sub-intervals of length η. An interpolation scheme is applied to guarantee that every sub-interval contains at least one measurement. The probability density (percentage) of measurements within each sub-interval, K_k(η), is then computed, and a family of partition functions N_k(η) is constructed (Equation (2)) [58,59]. The q-value ranges: from −10 to +10 with step 1 (21 points in total, or 41 points if including half-steps; we used step 1 for clarity). Number of sub-intervals: K(η) = 2¹⁰ = 1024. Smoothing: No additional smoothing filter was applied to the raw NMR T₂ spectra; instead, the probability density was interpolated using a cubic spline to ensure each sub-interval contains at least one measurement point.

$${\mathrm{v}}_{\mathrm{k}}(\mathrm{q},\mathsf{\eta })={\displaystyle \frac{{\mathrm{K}}_{\mathrm{k}}(\mathsf{\eta })}{{\sum }_{\mathrm{k}-1}^{\mathrm{N}(\mathsf{\eta })}{\mathrm{K}}_{\mathrm{k}}(\mathsf{\eta })}}$$

(2)

Here, v_k(q,η) denotes the q-order probability within the k-th sub-interval of the PSD data, and ∑_k-1^N(η)K_k(η) represents the sum of q-order probabilities across all sub-intervals.

2.4.3. Calculation of the Multifractal Generalised-Dimension Spectrum Dq

Equation (3) gives the multifractal generalised-dimension spectrum Dq [24,60,61,62].

$$\mathrm{D}\mathrm{q}=\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathrm{q}-1}}{\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathsf{\eta }\to 0}{\displaystyle \frac{\mathrm{l}\mathrm{o}\mathrm{g}{\mathrm{K}}_{\mathrm{q}}(\mathsf{\eta })}{\mathrm{l}\mathrm{o}\mathrm{g}(\mathsf{\eta })}},q\ne 1\\ {\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathsf{\eta }\to 0}{\displaystyle \frac{{\sum }_{\mathrm{k}-1}^{\mathrm{K}(\mathsf{\eta })}{\{\mathrm{P}}_{\mathrm{k}}(\mathsf{\eta })\mathrm{l}\mathrm{n}{[\mathrm{P}}_{\mathrm{k}}(\mathsf{\eta })\left]\right\}}{\mathrm{l}\mathrm{i}\mathrm{n}(\mathsf{\eta })}},q=1\end{array}\right.$$

(3)

For q = 0, 1 and 2, the resulting D₀, D₁ and D₂ correspond to the capacity, information-entropy and correlation dimensions, respectively. For a partition of size η, the comparison between the mass probability within a given cell and the sum over all cells is quantified by ${\mathrm{v}}_{\mathrm{k}}(\mathrm{q},\mathsf{\eta })={\displaystyle \frac{{{\mathrm{P}}_{\mathrm{k}}(\mathsf{\eta })}^{\mathrm{q}}}{{\mathrm{K}}_{\mathrm{q}}(\mathsf{\eta })}}$.

2.4.4. Multifractal Singularity Exponent α(q) and Spectrum Function f[α(q)]

Equation (4) gives the multifractal singularity exponent α(q) [25,63,64].

$$\mathsf{\alpha }(\mathrm{q})={\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathsf{\eta }\to 0}{\displaystyle \frac{{{\sum }_{\mathrm{k}-1}^{\mathrm{N}(\mathsf{\eta })}v}_{\mathrm{k}}(\mathrm{q},\mathsf{\eta }){\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{P}}_{\mathrm{K}}(\mathsf{\eta })}{\mathrm{l}\mathrm{i}\mathrm{n}(\mathsf{\eta })}}$$

(4)

For the PSD particle-size distribution of α(q), the multifractal spectrum function f[α(q)] is expressed as Equation (5) [25,63,64].

$$\mathrm{f}[\mathsf{\alpha }(\mathrm{q})]={\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathsf{\eta }\to 0}{\displaystyle \frac{{{\sum }_{\mathrm{k}-1}^{\mathrm{N}(\mathsf{\eta })}v}_{\mathrm{k}}(\mathrm{q},\mathsf{\eta })}{\mathrm{l}\mathrm{i}\mathrm{n}(\mathsf{\eta })}}$$

(5)

From these computations, two complementary parameter sets, the generalised-dimension spectrum q–Dq and the singularity spectrum α–f(α), are derived, to characterise the fractal attributes of the shale pore network and to quantify both the degree of heterogeneity and the complexity of pore development within the shale.

3. Results and Discussion

3.1. Rock Composition Characteristics

XRD analyses reveal that the Tan-3 member of the Longtan Formation (LTF) is mineralogically complex, comprising clay minerals, quartz, feldspar, calcite, dolomite and minor pyrite. No phosphate or Fe-Ti oxide minerals were detected above the XRD quantification limit (~1–2 wt%), though trace amounts may be present. The assemblage is dominated by clay and carbonate phases. Carbonate minerals exhibit the widest variability, ranging from 15.2% to 51.9% (average 35.5%, n = 11), whereas clay content is more tightly clustered between 21.0% and 51.2% (average 40.9%, n = 11) and constitutes the largest volumetric fraction. Quartz-rich phases are comparatively subordinate, varying from 11.1% to 23.8% (average 16.6%, n = 11) (Figure 4). Relative to marine shales, the quartz content is markedly lower [36,65], while the carbonate fraction is elevated compared with intra-formational counterparts [6], presumably reflecting enhanced marine influence during the late Longtan stage [51].

Organic carbon constitutes a critical component of shale, and the Longtan succession in the southern Sichuan Basin is generally characterised by elevated TOC. In this dataset, TOC ranges from 0.82% to 2.75% (n = 11) with a mean of 2.01% and a large variance. Samples “YJ1#-1” and “YJ#1-3” exhibit depressed TOC (<1.0%), whereas all remaining plugs record values > 1.5% (Figure 5), placing them at a moderate level within the formation [46,66].

3.2. Pore Structure and Fluid Mobility from NMR

NMR porosity is quantified via the response of hydrogen nuclei [14,67]; hydrogen-bearing phases native to the core contribute a finite background signal. The T₂ spectrum acquired on the oven-dried plug (Figure 6) is therefore taken as the baseline, enabling subsequent signal stripping of both the water-saturated and post-centrifuge T₂ distributions through baseline annihilation.

Porosity is calculated by comparing the signal intensity of each rock plug with that of calibrated porosity standards [68]. A calibration sequence was run concurrently to verify spectrometer performance. As shown in Figure 7a, water volume and signal amplitude exhibit a strong positive linear correlation (R² = 0.99), confirming optimum system status. Sample masses in the dry, water-saturated and post-centrifuge states, together with the derived porosities, are listed in

Table 2. Parameters of shale samples under different conditions (dry, water saturated, post-centrifuge).

Number	Sample Volume (cm3)	Drying Sample Quality (g)	Saturation Sample Mass (g)	Centrifugal Mass (g)	Saturated Water Mass (g)	Water Volume # cm3	Porosity # (%)	NMR Porosity (%)
JY1#-1	24.56	64.43	65.74	65.38	1.31	1.31	5.33	5.46
JY1#-2	24.25	65.47	66.07	65.91	0.60	0.60	2.47	2.64
JY1#-3	20.92	56.58	58.14	57.53	1.56	1.56	7.46	7.28
JY1#-4	24.34	69.14	69.20	69.19	0.06	0.06	0.25	0.46
JY1#-5	24.12	68.32	68.63	68.60	0.31	0.31	1.29	1.02
JY1#-7	19.37	50.63	52.07	51.69	3.88	1.44	7.44	7.43
JY1#-8	23.77	67.31	67.46	67.44	4.78	0.15	0.63	0.48
JY1#-9	24.47	68.51	69.13	69.01	4.92	0.62	2.53	2.81
JY1#-10	24.32	66.73	67.28	67.27	4.89	0.55	2.26	2.02
JY1#-11	24.58	66.61	67.09	67.03	4.92	0.48	1.95	1.94
JY1-12	12.14	32.8	33.02	33.00	2.48	0.22	2.1	1.81

Note: # Weight measurement method.

. NMR-derived porosities are essentially identical to those obtained gravimetrically (Figure 7b), validating the reliability of the NMR approach for porosity quantification.

Figure 8a–k present the T₂ spectra of the eleven shale plugs in the fully saturated and post-centrifuge states. All spectra exhibit a tripartite-peak architecture, evidencing the coexistence of micropores, mesopores and macropores [69,70] and attesting to a broad pore-throat size distribution. The left-most peak dominates the signal, spanning relaxation times of 0.1–10 ms and reflecting the overwhelming contribution of micropores. The intermediate peak—commonly a shoulder of the first—centres at 10–100 ms and corresponds to mesopore populations. The right-most, relatively isolated peak (100–1000 ms) signifies limited yet discernible macropore development within the Longtan shale.

Comparative analysis of multi-state T₂ spectra acquired by NMR enables quantitative characterisation of core pore structure and derivation of key petrophysical parameters, i.e., total porosity, flexible fluid saturation (FFN) and bound fluid saturation (BFN). Signal contrast between dry and fully saturated plugs yields LTF shale porosities of 0.46–7.43% (

Table 3. Sample BFN and FFN.

Number	T2cutoff (ms)	Bound Fluid Saturation (%)	Flexible Fluid Saturation (%)
JY1#-1	1.1	72.82	27.18
JY1#-2	1	73.81	26.19
JY1#-3	0.5	60.77	39.23
JY1#-4	27.1	83.45	16.55
JY1#-5	1.7	90.32	9.68
JY1#-7	0.6	73.29	26.71
JY1#-8	31.8	84.92	15.08
JY1#-9	0.8	80.1	19.9
JY1#-10	221	97.28	2.72
JY1#-11	1.6	87.45	12.55
JY1#-12	18.1	93.08	6.92

Note: # Weight measurement method.

), mean 3.06%. The T₂ cutoff is a critical NMR parameter [28,70]; an orthogonal optimisation applied to the spectral difference between saturated and centrifuged plugs gives T₂ cutoffs of 0.5–221 ms, average 27.6 ms (Table 3). The T₂ cutoff varies from 0.5 to 221 ms, consistent with the strong pore-structure heterogeneity documented by multifractal parameters. BFN is uniformly high (60.77–97.28%, average 81.57%), evidencing poor pore connectivity.

3.3. Multifractal Characteristics of the Pore Structure

Multifractal analysis is a mathematical framework tailored for quantifying the heterogeneity of complex systems and is especially suited to characterising shale pore architecture, fracture networks and mineral distributions.

Table 4. Generalised-dimension characteristic parameters of LTF shale samples.

Number	D0	D1	D2	D0-D1	Dmax	Dmin	△D
JY1#-1	0.9277	0.8575	0.8192	0.0703	1.5646	0.7432	0.8213
JY1#-2	0.8948	0.8505	0.8147	0.0443	1.4381	0.7392	0.6989
JY1#-3	0.8447	0.7792	0.7410	0.0655	1.5448	0.6667	0.8781
JY1#-4	0.8621	0.8037	0.7767	0.0585	1.3756	0.7137	0.6619
JY1#-5	0.8447	0.7812	0.7504	0.0635	1.4842	0.6848	0.7994
JY1#-7	0.8256	0.7754	0.7394	0.0502	1.4218	0.6670	0.7548
JY1#-8	0.8750	0.8430	0.8349	0.0320	1.3616	0.8031	0.5586
JY1#-9	0.8373	0.7706	0.7337	0.0666	1.4402	0.6612	0.7791
JY1#-10	0.8325	0.7720	0.7361	0.0605	1.4606	0.6643	0.7963
JY1#-11	0.8710	0.8285	0.7834	0.0425	1.4223	0.7025	0.7198
JY1#-12	0.8815	0.8450	0.8037	0.0365	1.4324	0.7201	0.7123

Note: # Weight measurement method.

shows that all shale and siltstone samples satisfy the hierarchy D₀ > D₁ > D₂, confirming that the LTF shale pore structure exhibits multifractal characteristics and rendering multifractal parameters more appropriate for capturing scale-dependent heterogeneous pore textures. Table 4 further lists D_min (approximated as D₋₁₀), D_max (approximated as D₁₀) and ΔD (approximated as D₋₁₀ − D₁₀), representing the generalised dimensions at extreme q-values and the overall heterogeneity degree across the measured pore-size interval. In this study, Dq decreases monotonically with increasing q for all samples (Figure 9), exhibiting a mirror-image “S”-shaped decay curve diagnostic of multifractal systems and signalling pronounced heterogeneity in pore or fracture distributions. The curve can be subdivided into seven segments: −10 ≤ q ≤ −4: corresponds to low-probability domains such as macropores and fractures; gentle Dq decline indicates relatively uniform distribution and weak heterogeneity, implying consistent sizes and spatial arrangement of large pores or fractures. −4 ≤ q ≤ −3: accelerated Dq drop caused by negative-order weighting reveals minor clustering of large pores or fractures, probably induced by local mineralogy or diagenetic overprint. −3 ≤ q ≤ −1: steepest Dq decline marks the transition from low- to high-probability regions with growing heterogeneity, representing the shift from mesopores to micropores where pore populations become increasingly non-uniform. −1 ≤ q ≤ 0: decelerating Dq descent as q approaches zero; D₀ (q = 0), the capacity dimension, gauges overall geometric complexity and indicates moderate connectivity and high self-similarity of the pore network at large scale (Despite a relatively high D₀ (≈0.9) indicating good spatial coverage of the pore network over the measurement scale, the pore-throat connectivity is poor, as evidenced by bound fluid saturations exceeding 70%. This apparent contrast reflects the fundamental difference between geometric space-filling (captured by D₀) and hydraulic connectivity (captured by fluid mobility parameters).). 0 ≤ q ≤ 2: continued deceleration of Dq drop reflects highly uneven, strongly heterogeneous regions where small-pore sizes and spatial arrangements are extremely variable, controlled by organic matter, clay minerals and micro-fractures. 2 ≤ q ≤ 4: extremely slow Dq decline indicates stabilising weights on high-probability regions; small-pore distributions remain heterogeneous, yet their rate of change diminishes, suggesting local clustering or incipient ordering. 4 ≤ q ≤ 10: Dq approaches convergence, yet significant heterogeneity persists in extreme high-probability (nanometre-scale) regions, corresponding to sparse or intrinsically complex structures such as intra-organic pores or inter-clay micropores.

The singularity exponent α quantifies the intensity of local probabilistic heterogeneity within the system: low α corresponds to high-probability, highly fluctuating domains (e.g., pore-dense or mineral-rich “hot spots”), whereas high α reflects low-probability, smoothly varying regions such as sparse macropores or fractures. f(α), the singularity spectrum, denotes the fractal dimension of the subset characterised by a given α and is interpreted as its abundance within the overall fractal architecture.

Table 5. Singularity-spectrum characteristic parameters of LTF shale samples.

Number	Δα	f(α)min	f(α)max	f(α)max − f(α)min
JY1#-1	1.0101	0.0217	0.9277	0.9060
JY1#-2	0.8653	0.1079	0.8948	0.7869
JY1#-3	1.0516	0.0718	0.8447	0.7729
JY1#-4	0.8103	0.1497	0.8621	0.7124
JY1#-5	0.9719	0.0374	0.8447	0.8072
JY1#-7	0.9182	0.0507	0.8256	0.7749
JY1#-8	0.7014	0.1537	0.8750	0.7213
JY1#-9	0.9402	0.0948	0.8373	0.7424
JY1#-10	0.9506	0.1805	0.8325	0.6520
JY1#-11	0.8833	0.0496	0.8710	0.8213
JY1#-12	0.8799	0.0367	0.8815	0.8448

Note: # Weight measurement method.

shows that the total spectral width Δα (αmax − αmin) is comparatively low, indicating moderate heterogeneity despite appreciable pore-size spread. The consistently sub-unity f(α)max (~0.9) implies suboptimal pore-network development, while f(α)min mirrors the spatial configuration of the densest pore clusters—lower values indicate discrete point-like “hot spots”, whereas higher values suggest extended architectures appearing at the right-hand side of the spectrum (Figure 10). The f(α)–α curves exhibit an approximately inverted-bell shape whose variation is systematically divided into four segments (Figure 10): Steep rising limb (α ≈ 0.70–0.80)—f(α) increases sharply, documenting the continuous population of small-scale pores; the broad interval signals high complexity within the microporous matrix that forms the skeletal background of the shale pore network. Gentle rising limb (α ≈ 0.80–1.00)—gradual increase constitutes the dominant signature; f(α)max ≈ D₀ indicates good space-filling capacity and connectivity, evidencing a well-developed, uniformly distributed micro- to mesopore system. Gentle falling limb (α ≈ 1.00–1.65)—declining f(α) marks the departure toward more singular, sparser structures such as larger pores, incipient micro-fractures or inter-particle boundaries. Steep falling limb (α ≈ 1.65–1.72)—rapid descent produces a “plunging” tail reflecting extremely sparse, high-α domains—isolated macropores, open micro-fracture networks and multi-scale fissures. Although volumetrically minor, these features are critical for macro-scale gas migration and hydraulic-fracture stimulability; the width and slope of the tail directly quantify the development degree of large-scale heterogeneity.

3.4. Discussion

Rock composition is the fundamental control on pore architecture (Figure 11). In marine shales, samples with elevated TOC/clay contents commonly exhibit higher NMR porosity and a dominant short-relaxation component in the T₂ spectrum. By contrast, the present LTF transitional marine shales show no significant correlation between TOC/clay content and NMR porosity (Figure 12), chiefly because the organic fraction is dominated by Type III kerogen that generates limited organic porosity. Additionally, the succession experienced deep burial followed by uplift [71] and was affected by regional thermal events during deposition [72], imposing a complex diagenetic overprint that obscures the expected porosity–composition trend. Nevertheless, these compositional end-members act as carriers of contrasting primary pore systems, amplifying local amplitude fluctuations and enhancing overall heterogeneity: (i) Compaction: the Longtan Formation experienced deep burial followed by tectonic uplift, leading to mechanical and chemical compaction that reduced primary intergranular porosity. (ii) Cementation: carbonate cement occludes pore throats, as evidenced by the high bound fluid saturation. (iii) Dissolution: limited dissolution of feldspar and carbonate grains is observed in published SEM studies of the same formation [6]. The weak positive correlation observed between carbonate minerals and NMR porosity is probably linked to sub-micron fractures nucleating along cleavage planes [42,73].

Fluid mobility is universally low and governed by pore architecture. All plugs exhibit high bound fluid saturation (BFN > 70%), diagnostic of a tight reservoir matrix. Flexible fluid saturation (FFN) is tightly linked to T₂ spectral shape (Figure 8): samples whose dominant peak is shifted toward longer relaxation (larger pores), e.g., YJ^#1-3 with FFN = 39.23%, possess elevated flexible fluid saturation. Multifractal theory provides a robust metric for pore heterogeneity; singularity width, Δα, correlates positively with multifractal span, ΔD (Figure 13a), confirming multifractal intensity consistency [60,64]. Figure 13a further reveals a good correlation between Δα and T₂ skewness, evidencing the efficacy of Δα in quantifying heterogeneity. Multifractal spectral width (Δα) and information dimension (D₁) serve as key indicators. Across the sample set (Figure 13b), D₁ and BFS follow the same trend with increasing Δα, indicating that greater pore-size irregularity and stronger heterogeneity result in higher BFS. TOC/clay contents display a robust positive correlation with BFS (Figure 13c), indicating that a fraction of bound water is controlled by adsorption onto organic matter and clay surfaces [15,54]. Employed as a heterogeneity index, Δα correlates well with both BFS and TOC/mineral contents (Figure 13b,d), furnishing compelling evidence that water is immobilised within a heterogeneous pore network. Intense heterogeneity implies poor pore connectivity; fluids (especially the flexible fraction) encounter numerous bottlenecks while migrating from micro- to macropores and are consequently trapped. In summary, rock composition influences fluid mobility both directly and indirectly by imposing heterogeneity-controlled connectivity on the pore structure.

Integrating multifractal metrics, rock composition and petrophysical attributes, the twelve plugs are classified into three reservoir types (

Table 6. Sample characteristics and classification.

Category	Number	Multifractal Characteristics	Rock Composition Characteristics	Physical Property Characteristics
Type I: weakly heterogeneous—high quality	1-2, 1-7	ΔD and Δα are small, and the spectrum is symmetrical.	High brittle-mineral content, moderate TOC	High porosity, high FFN
Type II: moderately heterogeneous—effective	1-1, 1-3,1-9	ΔD and Δα are small; the spectrum is slightly shifted to the right.	Elevated brittle-mineral content	Moderate porosity and flexible FFN
Type III: strongly heterogeneous—ineffective	1-4, 1-5, 1-8, 1-10, 1-11, 1-12	When ΔD and Δα are large, the spectrum is significantly skewed to the right.	High TOC, high clay content	Low porosity, extremely low FFN

, Figure 14): Type I: weakly heterogeneous—high quality: exhibits comparatively homogeneous pore structure and good connectivity, constituting the prospective sweet spot; Type II: moderately heterogeneous—effective: possesses fair storage and flow capacity; Type III: strongly heterogeneous—ineffective: displays extremely uneven pore structure with abundant yet poorly connected nanopores, rendering gas production impractical; however, their high TOC may serve as a hydrocarbon source for adjacent reservoir facies [74].

The measured data are non-normally distributed, permitting application of the Kruskal–Wallis test [75,76]. Δα, FFN, porosity, TOC, ΔD and brittle minerals were selected for the test; for all six metrics the median of at least one class differs from the others (Figure 15). Upon testing, the six parameters returned H = 1.02, 6.49, 7.50, 2.86, 3.02, 6.01 and p = 0.599, 0.039, 0.024, 0.213, 0.221, 0.049, respectively. FFN, porosity and brittle minerals yield p < 0.05, signifying significant inter-class differences. Excluding Δα, the remaining two parameters (TOC and ΔD) record p > 0.05 yet lie close to the 0.2 threshold, satisfying the classification criterion and indicating noticeable differences. The elevated p-value for Δα likely stems from spectral splitting in certain samples. Post hoc Kruskal–Wallis tests confirm that the reservoir-type classification based on intrinsic sample characteristics is statistically robust. Given the limited sample size (n = 11), the proposed three-type classification should be considered exploratory. The Kruskal–Wallis test shows that FFN, porosity and brittle mineral content differ significantly among the three groups (p < 0.05), whereas Δα does not reach statistical significance. This suggests that, while the classification captures major differences in fluid mobility and storage capacity, the heterogeneity index Δα alone does not uniquely separate the groups. Future work with a larger sample set is required to validate and refine the classification thresholds.

The integrated assessment furnishes a robust basis for sweet-spot forecasting. Compared with static geological-parameter approaches, the combined use of NMR, geochemical and multifractal analyses more profoundly elucidates the intrinsic controls on reservoir quality. Multifractal parameters condense intricate pore-structure information into concise quantitative indices and can be coupled with log or seismic data in forthcoming exploration and development campaigns to identify high-quality reservoirs at the regional scale.

4. Conclusions

(1)

NMR analyses confirm that the Longtan shale cores exhibit low porosity and high irreducible-water saturation, with pronounced disparities in fluid mobility and strong matrix heterogeneity.

(2)

Multifractal analysis successfully quantifies pore-structure heterogeneity; the generalised-dimension spectrum q–Dq and the singularity spectrum α–f(α) deliver insights beyond the reach of traditional monofractal metrics. Spectral width Δα and degree of multifractality ΔD emerge as the pivotal indices for appraising heterogeneity.

(3)

Rock composition constitutes the fundamental control on pore heterogeneity. TOC and clay-mineral contents display significant positive correlations with multifractal heterogeneity indicators (Δα), acting as the primary heterogeneity drivers. While composition directly influences fluid mobility, it concurrently modulates pore-network connectivity—thereby exerting an additional, heterogeneity-mediated control on movable fluid volumes.

(4)

Integrating rock composition, NMR-derived pore-structure parameters, multifractal metrics and fluid-mobility data furnishes a robust theoretical framework and a practical workflow for accurately evaluating transitional shale reservoirs and for predicting developmental sweet spots. This integration is essential because rock composition dictates the intrinsic pore-forming potential and heterogeneity sources; NMR quantifies porosity and fluid mobility; multifractal indices capture scale-dependent pore irregularity. Only by combining these can one distinguish direct compositional effects from heterogeneity-mediated controls on fluid flow, enabling reliable sweet-spot forecasting.