Predicting NMR T₂ Cutoff in Deep Tight Sandstones via Multifractal Analysis of Fully Water-Saturated Spectra: A Non-Destructive Approach

Abstract

Accurately determining the T₂ cutoff value is critical for evaluating fluid mobility in deep tight reservoirs, yet strong pore structure heterogeneity challenges traditional methods. This study proposes a non-destructive prediction method based on multifractal singularity spectrum analysis of nuclear magnetic resonance T₂ spectra. Using 10 tight sandstone cores from the Denglouku Formation (Songliao Basin), we quantify the intrinsic relationship between multifractal parameters and T₂ cutoff values. Results indicate that the minimum generalized dimension (D_min) and singularity spectrum width (Δα) are not merely mathematical fits but reveal the physical mechanisms controlling fluid binding in micro-throats. A multivariate regression model based on these parameters significantly outperforms traditional methods in accuracy (R² > 0.85). This approach provides a robust, non-destructive tool for identifying reservoir ‘sweet spots’ without compromising core integrity.

1. Introduction

Driven by the escalating global energy demand and the simultaneous decline in newly discovered conventional reserves [1], the exploration and development of unconventional hydrocarbon resources have garnered significant global attention [2,3]. As a critical unconventional resource, deep tight sandstone has emerged as a strategic priority for hydrocarbon exploration and production [3,4,5]. However, distinct from conventional counterparts, these reservoirs have undergone intense compaction and complex diagenetic alterations, resulting in intricate pore–throat networks characterized by a “wide pore size distribution, high tortuosity, and narrow throats” [6,7,8]. Consequently, accurately characterizing their micropore structures and delineating the boundary between movable and bound fluids (i.e., the T₂ cutoff value) constitute the primary challenges in reservoir evaluation.

In geological and mining engineering, Nuclear Magnetic Resonance (NMR) has become an indispensable technique for non-destructive reservoir evaluation. It is widely utilized to characterize pore size distributions, estimate permeability, identify fluid types, and evaluate wettability in complex porous media [9,10]. Despite these broad applications, the accurate determination of NMR T₂ cutoff values remains a specific challenge, typically relying on laboratory centrifugation [11,12]. For conventional reservoirs with relatively homogeneous pore structures, empirical values (e.g., approximately 33 ms for sandstones and 90 ms for carbonates) are often sufficient [13,14]. However, this procedure is labor-intensive, and the requisite high centrifugal forces can compromise the delicate pore skeletons of tight sandstones, resulting in the distortion of petrophysical parameters. To achieve non-destructive prediction, researchers have proposed empirical indices such as the T₂ geometric mean (T_2GM) [15]. Yet, this study identifies a critical limitation: T_2GM represents an averaged statistical metric that often obscures complex distributional features, such as “long tails” or multimodal patterns in pore size distributions [16]. In highly heterogeneous tight sandstones, this averaging effect can cause substantial deviations in predicted T₂ cutoff values. Consequently, there is an urgent need for an automated approach capable of fully capturing and quantifying the multi-scale heterogeneity of these reservoirs. While various prediction-oriented methods, ranging from empirical regressions to advanced simulations, have been employed for reservoir property estimation, the specific challenge in deep tight sandstones lies in the non-linear relationship between pore structure and fluid mobility.

In recent years, the petroleum industry has witnessed a surge in data-driven and machine-learning-based prediction models, which have demonstrated robust performance in estimating complex fluid and reservoir properties [17,18]. However, these methods typically depend on extensive training datasets. To address cases where physical interpretation of pore topology is paramount, Fractal geometry offers a robust mathematical framework for characterizing the space-filling complexity of porous media [19,20]. However, traditional mono-fractal analysis relies on the assumption of constant self-similarity (characterized by a single fractal dimension) across all scales. This limitation hinders the accurate description of deep tight sandstone systems, which exhibit significant heterogeneity due to the coexistence of macropores (dissolution pores) and micro-throats (compaction remnants) [21,22]. In contrast, multifractal theory is capable of decomposing complex porous systems into subsets with distinct probability measures using generalized dimension spectra and singularity spectra [23,24]. For instance, Ge et al. utilized multifractal parameters to classify pore–throat structures in clastic rocks [25], while Zhao et al. observed a strong correlation between the width of the fractal spectrum and the flow unit index [26]. Notably, in NMR studies of unconventional reservoirs, the T₂ spectra of coal and shale have exhibited pronounced multifractal characteristics [27,28,29]. However, deep tight sandstones present distinct challenges. Unlike shale, which typically features relatively concentrated nanoscale pores, the tight sandstones of the Denglouku Formation have undergone intense compaction and dissolution. These processes result in complex structures characterized by a wide pore size distribution, high tortuosity, and narrow throats. Within such structures, fluid mobility is not solely governed by total porosity but is critically restricted by micro-throats. Although these throats generate weak signals in the T₂ spectrum, they act as “bottlenecks” that control overall connectivity. Therefore, applying multifractal analysis to NMR T₂ spectral processing offers a promising avenue for overcoming the limitations of traditional methods, enabling the rapid and accurate prediction of T₂ cutoff values.

This study fills that gap by conducting systematic experiments on Denglouku Formation tight sandstones. We propose a novel prediction method that relies solely on the T₂ spectrum of fully water-saturated samples. Unlike traditional empirical approaches, we demonstrate that multifractal parameters (specifically D_min and Δα) quantitatively reflect the ‘bottleneck’ effect of micro-throats on bound fluids. This provides a physics-based, non-destructive theoretical foundation for identifying ‘sweet spots’ in deep tight reservoirs.

2. Geological Setting

The Songliao Basin is a major Mesozoic petroliferous basin situated in Northeast China (Figure 1a) [30]. The southern region of the basin comprises four first-order tectonic units: the Central Depression, the Southeast Uplift, the Western Slope, and the Southwest Uplift (Figure 1b) [31,32]. Specifically, the Changling Gas Field is tectonically situated within the central uplift belt of the Changling Fault Depression, flanked by northern and southern sub-sags (Figure 1b). The Lower Cretaceous strata in this area contain, in ascending order, the Shahezi, Yingcheng, Denglouku, and Quantou Formations, which are dominated by fine sandstone, mudstone, and sandy conglomerate (Figure 1c). The target interval for this study is the Denglouku Formation. With a typical burial depth exceeding 3000 m, this formation represents a key successor target for deep natural gas exploration.

During the deposition of the Denglouku Formation, a shallow-water braided river delta system prevailed, with subaqueous distributary channel sandbodies serving as the primary reservoir rock types [33,34]. The lithology predominantly consists of fine sandstone and siltstone intercalated with thin layers of purplish-red mudstone, frequently exhibiting cross-bedding and parallel bedding. Crucially, due to the significant burial depth (with study samples retrieved from 3560 to 3605 m), these reservoirs have undergone intense mechanical compaction and complex diagenetic cementation [35]. This severe diagenetic alteration caused a drastic reduction in primary intergranular porosity; consequently, the present-day pore space is dominated by secondary dissolution pores, micro-fractures, and residual intergranular micropores. This complex, polygenetic pore evolution has imparted extreme microscopic heterogeneity and intricate pore–throat topology to the Denglouku tight sandstones, representing the fundamental geological cause for the failure of conventional petrophysical methods.

3. Samples and Experimental Methods

3.1. Experimental Samples

In this study, ten tight sandstone core samples were retrieved from the Denglouku Formation in the Changling Gas Field, southern Songliao Basin (

Table 1. Physical properties and NMR pore structure parameters of tight sandstone in Denglouku Formation.

No.	Formation	Length (cm)	Diameter (cm)	Dry Weight (g)	NMR Porosity (%)	Helium Porosity (%)	Gas Permeability (mD)
S1	Denglouku	3.67	2.52	46.77	3.07	3.51	0.012
S2	Denglouku	3.46	2.52	43.59	4.19	4.89	0.021
S3	Denglouku	4.95	2.49	61.19	2.74	2.87	0.003
S4	Denglouku	4.62	2.51	58.81	3.03	3.33	0.016
S5	Denglouku	4.79	2.49	57.02	7.91	8.10	0.007
S6	Denglouku	4.73	2.51	57.62	6.20	6.37	0.055
S7	Denglouku	5.31	2.52	67.63	3.02	3.24	0.015
S8	Denglouku	4.97	2.51	62.96	3.42	3.84	0.036
S9	Denglouku	4.85	2.50	60.50	3.42	3.66	0.007
S10	Denglouku	4.52	2.49	57.84	0.47	0.33	0.003

). The samples were collected from two cored wells targeting the 3rd and 4th sand members (the upper section of the four-member Denglouku Formation), with burial depths ranging from 3560 to 3605 m. Lithologically, the samples consist primarily of siltstone and fine sandstone, exhibiting characteristics typical of deep tight reservoirs. To ensure representativeness, the sample selection spanned various depths, lithofacies, and petrophysical grades. All samples were machined into standard cylindrical plugs. The end-face flatness was controlled to be superior to 0.02 mm, and the lateral surfaces were verified to be free of macroscopic fractures. Prior to testing, the plugs were subjected to Soxhlet extraction using a toluene-methanol azeotrope for over 72 h to remove residual hydrocarbons and drilling fluid contaminants. Subsequently, they were dried in a vacuum oven at 105 °C for 24 h until a constant weight was achieved. Porosity and permeability were measured under a net confining pressure of 35 MPa to simulate in situ stress conditions, with a helium pore pressure of 1.5 MPa, followed by vacuum-pressure saturation with water and subsequent NMR measurements.

3.2. Gas Porosity and Permeability Measurements

Petrophysical properties were measured using a PDP-200 overburden porosity-permeability system (Coretest Systems, Morgan Hill, CA, USA) at room temperature (25 °C), utilizing high-purity helium (>99.999%) as the working fluid. The instrument is capable of applying both axial and confining pressures to simulate in situ formation stress conditions. Prior to testing, samples were dried in a vacuum oven at 105 ± 5 °C for 12 h and subsequently cooled to room temperature to ensure the complete removal of residual fluids and volatile components. Before measurement, a system leak test was performed by injecting helium to 0.3 MPa; a pressure decay of less than 0.01 MPa over 30 min was considered acceptable. High-vacuum silicone grease was evenly applied to the O-rings and flange connections of the sample chamber, after which the standard core plugs were loaded into the core holder and sealed. Porosity was determined using the helium expansion method based on Boyle’s law [36], while permeability was measured using the steady-state method [37]. All measurement data were corrected for temperature and pressure variations, ensuring a reproducibility error of less than 5%.

3.3. NMR Experiments

NMR measurements were conducted using a MiniMR-60 low-field NMR analyzer (Niumag Corporation, Suzhou, China), operating with a main magnetic field strength of 0.52 ± 0.05 T and a resonance frequency of approximately 21.78 MHz. T₂ distributions were acquired using the Carr–Purcell–Meiboom–Gill (CPMG) pulse sequence [38]. Key acquisition parameters were set as follows: waiting time (TW) = 3000 ms, echo spacing (TE) = 0.3 ms, number of echoes = 10,000, and number of scans = 32, ensuring a signal-to-noise ratio (SNR) greater than 100. While a TE of 0.3 ms may result in minor signal loss for ultra-fast relaxation components (<0.3 ms) associated with clay-bound water, this duration is significantly shorter than the measured T₂ cutoff values (16.5–80.7 ms), ensuring that the determination of the movable fluid boundary remains unaffected. Centrifugation was performed using a Xiangyi CS-10 high-speed refrigerated core centrifuge with a maximum speed of 10,000 r/min. The experimental procedures, including instrument calibration and data acquisition, were strictly conducted in accordance with the National Oil and Gas Industry Standard SY/T 6490-2014 (Specification for Laboratory Measurement of NMR Parameters in Rock Samples) [39].

To establish a quantitative relationship between NMR signal amplitude and pore fluid content, instrument calibration was performed initially. Varying masses of distilled water were weighed and sealed in non-magnetic vials. These were measured under identical CPMG parameters to construct a standard calibration curve correlating signal amplitude with water content, which was subsequently used for precise core porosity correction. Deep tight sandstone core samples were first dried in an oven at 105 °C for 24 h to completely remove original formation water and volatile components. The dried samples were placed in a vacuum saturation vessel and evacuated at −0.095 MPa for 4 h, followed by pressure saturation with distilled water at 20 MPa for 48 h to ensure a fully water-saturated state. Upon completion, surface free water was gently wiped off with filter paper, and the samples were immediately wrapped in non-magnetic polytetrafluoroethylene (PTFE) film to prevent evaporation or moisture absorption. The wrapped samples were loaded into the NMR probe for CPMG measurement to obtain the T₂ relaxation spectrum of the fully saturated state.

Subsequently, the saturated samples underwent stepwise high-speed centrifugation. Centrifugal pressures were set sequentially at 50, 100, 200, 500, and 600 psi (approximately 0.34, 0.69, 1.38, 3.45, and 4.14 MPa, respectively), with a duration of 2 h for each pressure step. The maximum centrifugal pressure of 600 psi (4.14 MPa) corresponds to a theoretical pore–throat radius of approximately 0.18 um based on the Washburn equation. This pressure is sufficient to displace movable fluids from the effective pore network in these tight sandstones, consistent with cutoff calibrations in similar deep reservoirs. After each centrifugation step, samples were immediately retrieved, wiped to remove any extruded water, re-wrapped in PTFE film, and measured to obtain the T₂ spectrum corresponding to that specific centrifugal force. The fluid signal remaining after centrifugation at the maximum pressure of 600 psi corresponds to immobile bound fluid. The cumulative signal difference between the fully saturated and centrifuged states was utilized to determine the T₂ cutoff value using the conventional centrifugation method. A systematic comparison between T₂ spectra of the saturated and stepwise centrifuged states yielded reliable empirical T₂ cutoff values, serving as a validation standard for the subsequent multifractal prediction model.

3.4. Multifractal Theory

Multifractal theory offers a robust framework for quantifying the local scaling behavior of complex heterogeneous systems and has been widely employed in the characterization of rock pore structures [17,21,22,29]. In contrast to monofractal analysis, which characterizes global self-similarity using a single dimension, multifractal analysis elucidates the heterogeneity across different scales and probability regions through the generalized dimension spectrum $D_q$ and the singularity spectrum [40]. In this study, the box-counting method was applied to the NMR T₂ relaxation spectra of fully water-saturated tight sandstone cores to extract characteristic parameters reflecting pore heterogeneity [23,24]. First, the T₂ spectrum was discretized into a series of signal amplitudes corresponding to specific time points, and the total accumulated signal was normalized to unity. The box-counting method was applied to the logarithmic coordinate of the T₂ spectrum. The time axis (0.1 to 10,000 ms) was partitioned into boxes of size $\epsilon$ equidistant in log-space, ensuring accurate characterization of the multiscale heterogeneity. The linear scaling region for box-counting was determined via the maximum Pearson correlation coefficient (R² > 0.95) in the log-log plots. The box size $\epsilon$ followed a dyadic sequence (i.e., 1/2, 1/4, 1/8, …, 1/2ⁿ of the total span) until the smallest box contained at least one data point $N_i(\epsilon )$ denote the total signal amplitude within the i-th box at scale $\epsilon$. The probability mass function is defined as:

$$P_i(\epsilon )={\displaystyle \frac{N_i(\epsilon )}{{\displaystyle \sum _{i=1}^{N(\epsilon )}{N}_i}(\epsilon )}}$$

(1)

where $N_i(\epsilon )$ represents the cumulative porosity or pore volume within the i-th interval, and $P_i(\epsilon )$ denotes the probability mass function.

For samples exhibiting multifractal characteristics, the probability mass function $P_i(\epsilon )$ follows a power-law relationship with the box size $\epsilon$, expressed as:

$$P_i(\epsilon )\propto {\epsilon }^{{\alpha }_i}$$

(2)

where ${\alpha }_i$ Lipschitz–Hölder denotes the Lipschitz–Hölder singularity exponent, the magnitude of which typically depends on the spatial position i of the box. Let $N_{\alpha }(\epsilon )$ denote the number of boxes sharing the same singularity exponent $\alpha$; this quantity scales as follows:

$$N_{\alpha }(\epsilon )\propto {\epsilon }^{-f(\alpha )}$$

(3)

where $f(\alpha )$ represents the fractal dimension of the subset of boxes characterized by the same singularity exponent $\alpha$. The plot of $f(\alpha )$ versus $\alpha$ constitutes the multifractal spectrum. Typically $f(\alpha )$ manifests as a unimodal curve, reaching its maximum value when:

$${\displaystyle \frac{df(\alpha (q))}{d\alpha (q)}}=0$$

(4)

where $q$ represents the order of the probability moment, theoretically ranging from −∞ to +∞. In this study, $q$ assumes integer values within the interval [−10, 10] with a step increment of 0.5. The range of moment orders q was set to [−10, 10] based on convergence tests. Analysis showed that the generalized dimensions $D_q$ reached stable asymptotic values within this interval, ensuring that the multifractal spectra capture the full range of scaling behaviors without edge effects. Furthermore, the partition function $X(q,\epsilon )$ at scale $\epsilon$ is expressed as:

$$X(q,\epsilon )={\displaystyle \sum _{i=1}^{N_i}{P}_i^q}(\epsilon )\propto {\epsilon }^{-(q-1)D_q}$$

(5)

where $D_q$ denotes the generalized fractal dimension corresponding to the moment order $q$. Based on Equation (5), it is given by:

$$D_q={\displaystyle \frac{1}{q-1}}\underset{\epsilon \to 0}{\lim }{\displaystyle \frac{\log {\displaystyle \sum _{i=1}^{N(\epsilon )}{P}_i^q}(\epsilon )}{\log \epsilon }}$$

(6)

Equation (6) yields the distribution profile of the multifractal dimension $D_q$ with respect to the moment order $q$. When $q$ < 0, $D_q$ characterizes regions with low probability density in the pore size distribution; conversely, when $q$ > 0, $D_q$ describes regions with high probability density. Additionally, the mass exponent function $\tau (q)$ of order $q$ is defined as:

$$\tau (q)=-\underset{\epsilon \to 0}{\lim }{\displaystyle \frac{\log X(q,\epsilon )}{\log \epsilon }}=-\underset{\epsilon \to 0}{\lim }{\displaystyle \frac{{\displaystyle \sum _{i=1}^{N(\epsilon )}{P}_i^q}(\epsilon )}{\log \epsilon }}$$

(7)

Combining Equations (6) and (7), $\tau (q)$ can be reformulated as:

$$\tau (q)=(1-q)D_q$$

(8)

Via the Legendre transform, the singularity strength $\alpha (q)$ relates to the moment order $q$ and the mass exponent function $\tau (q)$ as follows:

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

(9)

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

(10)

Generally, the generalized fractal dimensions $D_q$ and the spectral width $\left(a_{\max }-a_{\min }\right)$ serve as critical parameters for characterizing the multifractal features, specifically, the singularity and heterogeneity, of pore structures in porous media [17,23].

4. Experimental Results

4.1. Physical Properties

Table 1 summarizes the fundamental petrophysical properties of the ten deep tight sandstone core samples. Helium porosity ranges from 0.33% to 8.10%, with an average of 4.12% and a standard deviation of 2.31%. Gas permeability (Klinkenberg-corrected absolute permeability) varies between 0.003 and 0.055 mD, averaging 0.021 mD and spanning nearly two orders of magnitude. Generally, porosity and permeability exhibit a positive correlation; however, the significant data scatter reflects the strong heterogeneity of the deep tight reservoirs in the study area. NMR-derived porosity ranges from 0.47% to 7.91%, with a mean value of 4.05% (Table 1).

A comparison between NMR porosity and helium porosity (Figure 2) reveals an exceptionally high linear correlation (R² = 0.9894). This indicates that NMR methodology accurately characterizes the effective pore volume of the samples, with only a negligible systematic deviation (NMR porosity averages 0.07% lower than helium porosity). This minor discrepancy may stem from isolated pores or surface relaxation effects. Consequently, these results validate the reliability of the subsequent pore structure analysis and movable fluid evaluation based on NMR T₂ spectra.

4.2. NMR Characteristics of Tight Sandstone Samples Following Saturation and Centrifugation

The NMR T₂ spectrum characterizes the relaxation behavior of pore fluids. Under fully water-saturated conditions, T₂ values are positively correlated with pore size: shorter T₂ times correspond to smaller pores (characterized by high specific surface area and strong surface relaxation), whereas longer T₂ times indicate larger pores or micro-fractures [41,42]. Figure 3 illustrates the T₂ spectral distributions for the ten deep tight sandstone cores under both fully saturated and stepwise centrifuged conditions.

The saturated T₂ spectra generally exhibit a typical bimodal distribution. The left peak (P₁), centered between 3 and 20 ms (averaging ~8 ms), reflects the dominant micropore components, while the right peak (P₂), positioned between 80 and 300 ms (averaging ~150 ms), corresponds to relatively well-developed macropores or connected throats. While all samples display a distinct P₁ peak, the development of the P₂ peak varies significantly. S1, S3, and S7 are dominated by the P₁ peak, with the P₂ peak being weak or absent, indicating an absolute predominance of micropores. Conversely, samples S5, S6, S9, and S10 exhibit well-developed P₂ peaks with P₁-to-P₂ amplitude ratios approaching unity, suggesting a substantial contribution from macropores. S2, S4, and S8 fall between these two extremes, showing typical bimodal patterns. This variability in spectral morphology further confirms the strong heterogeneity of the pore structure in deep tight sandstones.

Post-centrifugation T₂ spectra reveal a significant signal attenuation in the long T₂ range (>50 ms), whereas signals in the short T₂ range (<20 ms) are largely retained. This indicates that fluids in macropores are more easily displaced by centrifugation, while those in micropores exist primarily as bound fluids. Comparison of cumulative curves shows that after centrifugation at the maximum pressure (600 psi), the residual signal corresponds to a bound fluid saturation ranging from 55% to 85% (average 71%), with a corresponding movable fluid saturation of 15% to 45% (average 29%).

4.3. Multifractal Analysis of the NMR Data

In the multifractal analysis of NMR data for water-saturated deep tight sandstones, the order q was set from −10 to 10 with an interval of 0.5, where $D_{-10}$ represents the minimum scaling behavior ($D_{\min }$) and $D_{10}$ represents the maximum scaling behavior ($D_{\max }$). Based on multifractal theory, multifractal calculations were performed on the fully water-saturated NMR T₂ spectra of 10 deep tight sandstone samples. The relationship between $D_{\mathrm{q}}$ and q for reservoirs of different grades is illustrated in Figure 4, showing an inverse S-shaped monotonic decreasing distribution. For q < 0, $D_{\mathrm{q}}$ decreases sharply with increasing q; for q > 0, $D_{\mathrm{q}}$ exhibits a gradual decline with increasing q. The multifractal dimension $D_0$ (Hausdorff dimension) at q = 0 represents the one-dimensional distribution dimension, indicating the heterogeneity in the average pore size distribution of the studied object. $D_1$ and $D_2$ are the dimensions at q = 1 and q = 2, respectively, known as the information dimension and correlation dimension. The calculated multifractal dimensions ($D_{\min }$, $D_{-2}$, $D_{-1}$, $D_1$, $D_2$, and $D_{\max }$) for the NMR data of the 10 deep tight sandstone samples are presented in

Table 2. Calculated multifractal spectrum parameters derived from NMR T₂ distributions of the studied samples.

Sample	${\mathit{D}}_{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{D}}_{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{D}}_{-2}$	${\mathit{D}}_{-1}$	${\mathit{D}}_1$	${\mathit{D}}_2$	$\mathbf{\Delta }\mathit{\alpha }$
S1	1.571248	0.640748	1.254886	1.135139	0.887926	0.808838	1.135808
S2	1.560919	0.636101	1.259096	1.138448	0.885252	0.804893	1.116925
S3	1.605829	0.650237	1.256797	1.132724	0.893246	0.816765	1.172403
S4	1.587117	0.645482	1.254139	1.133248	0.890599	0.812807	1.154395
S5	1.561024	0.636055	1.259182	1.138512	0.885315	0.804944	1.116858
S6	1.633484	0.622307	1.294782	1.156603	0.877123	0.79302	1.222354
S7	1.590251	0.626879	1.27867	1.149157	0.879842	0.796982	1.157666
S8	1.564386	0.631497	1.266899	1.143127	0.882561	0.800951	1.115574
S9	1.564295	0.631553	1.266945	1.143088	0.882498	0.800889	1.115622
S10	1.605744	0.650302	1.256863	1.132685	0.893308	0.816822	1.172355

. All samples exhibit $D_1$ > $D_2$, indicating that the fully water-saturated NMR T₂ spectra of deep tight sandstones possess distinct multifractal characteristics. $D_1$ ranges from 0.87 to 0.89 with an average of 0.88, while $D_2$ shows a similar distribution, ranging from 0.79 to 0.81 with an average of 0.80. $D_{\min }$ and $D_{\max }$ exhibit opposing distribution trends: $D_{\min }$ ranges from 1.56 to 1.63 with an average of 1.58, whereas $D_{\max }$ ranges from 0.62 to 0.65 with an average of 0.63. This suggests that pores develop across all scales in deep tight sandstones.

The singularity exponents of the multifractal spectrum effectively characterize the heterogeneity of pore structures. ${\alpha }_{\max }$ and ${\alpha }_{\min }$ denote the maximum and minimum singularity exponents, respectively; $\Delta \alpha$ is their difference; ${\alpha }_0$ indicates the singularity exponent at q = 0; and A represents the skewness of the multifractal spectrum. $\Delta \alpha$ and ${\alpha }_0$ indicate the degree of pore structure heterogeneity, with larger values signifying stronger heterogeneity [43]. The multifractal spectra of water-saturated deep tight sandstones exhibit a typical concave parabolic shape (Figure 5). On the left side, $f(\alpha )$ increases with increasing $\alpha$, while on the right side, $f(\alpha )$ decreases with increasing $\alpha$. The singularity strength $\Delta \alpha$, also known as the multifractal spectrum width, serves as another crucial parameter for characterizing the heterogeneity of porous media. The $\Delta \alpha$ values for the 10 water-saturated tight sandstone samples range from 1.11 to 1.22 (Table 2), which are relatively large and indicate a strong heterogeneity in the pore structures of deep tight sandstones.

The relationship between the mass exponent function $\tau (q)$ and the moment order q (ranging from −10 to 10) for deep tight sandstone samples is depicted in Figure 6. As observed, $\tau (q)$ increases monotonically with q for all samples. More importantly, $\tau (q)$ exhibits a pronounced nonlinear relationship with q, with the curve displaying an overall convex upward morphology that significantly deviates from the linear trend characteristic of monofractals. This deviation from linearity unequivocally confirms that the pore structures of deep tight sandstones exhibit typical multifractal features, indicating a high degree of heterogeneity and complexity in their pore space distribution.

5. Discussion

5.1. Traditional Methods for Determining NMR T₂ Cutoff Values

5.1.1. Determination of T₂ Cutoff Value Using the Centrifugation Method

Accurate determination of the T₂ cutoff value is a critical prerequisite for distinguishing between movable and bound fluids, evaluating the full pore size distribution in reservoirs, and estimating permeability [44,45,46,47]. Currently, the centrifugation method is widely regarded as the standard approach for calibrating reservoir T₂ cutoff values [48]. Initially, NMR tests are conducted on deep tight sandstone samples under fully water-saturated conditions and at optimal centrifugal force to obtain the corresponding T₂ distribution spectra and cumulative porosity curves (Figure 7). Subsequently, a horizontal line is drawn based on the maximum cumulative porosity value of bound water in the post-centrifugation sample, intersecting the fully water-saturated cumulative porosity curve at one point. A perpendicular line is then drawn from this intersection to the abscissa, with the corresponding T₂ value designated as the T₂ cutoff value determined by the centrifugation method (defined as T_2c1).

For the 10 deep tight sandstone samples selected in this study, the measured T_2c1 values range from 16.5 ms to 80.7 ms. This range is significantly larger than the shale T₂ cutoff values calculated by Liu et al. (2018) (0.45–2.98 ms) [47]. The possible reason for this discrepancy is that, compared to shale reservoirs, deep tight sandstones, despite their density, exhibit relatively larger average pore–throat radii and better connectivity, resulting in a larger critical pore size (i.e., cutoff value) for differentiating movable from bound fluids. Although the centrifugation method possesses clear physical significance and high acceptance, it also has certain limitations: on the one hand, the operational procedures for a series of NMR centrifugation experiments are cumbersome and time-consuming [49]; on the other hand, due to the application of high-intensity centrifugal forces, this method is unsuitable for samples with loose or fragile structures (e.g., those with developed microfractures), as it may damage the skeletal framework and compromise measurement accuracy.

5.1.2. T₂ Spectrum Morphology Method

Shao et al. proposed a method for determining reservoir T₂ cutoff values based on the morphology of saturated T₂ spectra by analyzing the relationship between the T₂ spectrum morphology of sandstones under fully saturated conditions and the T₂ cutoff values determined by the centrifugation method [50]. This method posits that the saturated T₂ spectra of typical sandstones generally exhibit bimodal characteristics, with the left peak corresponding to small pores occupied by bound fluids and the right peak to large pores occupied by movable fluids, while the T₂ cutoff value is typically located at the relaxation time corresponding to the valley (inflection point) between the two peaks (Figure 7). For samples with T₂ spectra displaying typical bimodal distributions, the inflection point is easily identifiable, rendering the method highly applicable; however, when T₂ spectra exhibit multimodal or complex morphologies, the inflection point is difficult to determine accurately, leading to method failure. Therefore, in this study, the spectrum morphology method was employed to calculate T₂ cutoff values (denoted as T_2c2) only for deep tight sandstone samples whose saturated T₂ spectra exhibit typical bimodal distributions.

Significant differences exist between the T_2C2 values determined by the spectrum morphology method and the T_2C1 values determined by the centrifugation method (Figure 3). For samples S1 and S2, T_2C2 is markedly greater than T_2C1, primarily because the spectrum morphology method assumes that the centrifugation process removes only the movable fluids corresponding to the right peak, while the left peak entirely represents non-centrifugable bound fluids. However, the post-centrifugation T₂ spectra of S1 and S2 in Figure 3 show a significant decrease in the left peak amplitude, indicating that fluids in some left-peak pores can still be removed under centrifugation conditions, thereby causing T_2C2, based on the valley inflection point, to overestimate the cutoff value. Conversely, for sample S10, T_2C2 is significantly smaller than T_2C1, possibly due to residual movable fluids in the right peak not being fully removed after centrifugation. Furthermore, the saturated T₂ spectra of samples such as S3 and S5 do not exhibit clear bimodal structures, making the valley inflection point difficult to identify and thus rendering the spectrum morphology method inapplicable for determining T_2C2.

The T₂ spectrum morphology method is only applicable to conventional sandstones whose saturated T₂ spectra exhibit typical bimodal distributions. For deep tight sandstones, however, the complex pore structures and strong heterogeneity often result in T₂ spectra displaying multimodal or asymmetric features, leading to frequent occurrences such as movable fluids in parts of the left peak or immobile fluids remaining in the right peak. Consequently, this method tends to produce substantial deviations when determining T₂ cutoff values, limiting its application in deep tight reservoirs.

5.1.3. T_2GM Method

T_2GM is the average transverse relaxation time parameter for different pore radii and serves as an important indicator of reservoir pore structure [51]. The specific calculation formula is as follows:

$$T_{2\mathrm{GM}}=\exp \left[{\displaystyle \sum _{T_{2\mathrm{S}}}^{T_{2\max }}\frac{A_{\mathrm{i}}}{A_{\mathrm{T}}}}\ln (T_{2\mathrm{i}})\right]$$

where $T_{2\max }$ = 10,000, ms; $A_{\mathrm{i}}$ is the signal amplitude at transverse relaxation time $T_{2\mathrm{i}}$, p.u., $A_{\mathrm{T}}$ is the total signal amplitude of the NMR spectrum, p.u.

In this study, T_2GM was applied to determine T₂ cutoff values in deep tight sandstone reservoirs, with the aim of providing a simple empirical method for NMR logging interpretation. This method has been widely validated in conventional medium- to high-permeability sandstone reservoirs, typically assuming that the T₂ spectra under fully water-saturated conditions exhibit distinct bimodal distributions, with bound fluids corresponding to short relaxation components and movable fluids to long relaxation components. In such cases, the T₂ geometric mean closely approximates the boundary between bound and movable fluids, serving as a reasonable estimate for the T₂ cutoff value. However, the experimental results from this study on deep tight sandstone reservoirs indicate that this method is not applicable. There is a lack of evident correlation between the T₂ cutoff values measured by the centrifugation method (T_2C1) and T_2GM (Figure 8). The scatter plot reveals highly dispersed data points with no overall trend; notably, when T_2GM is in the lower range (approximately 6–8 ms), the T_2C1 of some samples remains as high as over 80 ms, even approaching 100 ms. This phenomenon starkly contrasts with the expected pattern in conventional sandstones, where T_2GM is typically slightly higher than or close to the T₂ cutoff value, indicating that direct application of the T₂ geometric mean method can severely overestimate or underestimate movable fluid content, leading to substantial deviations in bound water saturation calculations. The fundamental reason for this failure lies in the complexity of pore structures in deep tight sandstones. Due to great burial depths and intense diagenesis, such reservoirs exhibit highly uneven pore size distributions, with developed small and micropores, narrow throats, and poor connectivity. Consequently, the NMR T₂ spectra under fully water-saturated conditions are no longer limited to typical bimodal morphologies but may exhibit unimodal, pseudo-bimodal, or multimodal distributions. The presence of multimodal features, particularly long relaxation tails, significantly elevates the T₂ geometric mean, causing it to deviate from the actual boundary between bound and movable fluids, thereby introducing substantial uncertainty.

Furthermore, the nonuniformity of surface relaxivity and the influence of diffusion relaxation further exacerbate the complexity of T₂ spectra. The T₂ geometric mean method, derived from experience with conventional sandstones, is not suitable for determining T₂ cutoff values in deep tight sandstone reservoirs. In practical logging interpretation, priority should be given to centrifugation calibration, capillary pressure curve conversion, or other more reliable methods to ensure accurate evaluation of bound water saturation and effective porosity.

5.2. NMR T₂ Cutoff Value Determination Method Based on Multifractal Theory

The modeling workflow proceeded as follows: (1) Normalization of raw T₂ spectra; (2) Extraction of multifractal features ($D_{\mathrm{q}}$, $q$, $f(\alpha )$) via the box-counting method; (3) Feature selection based on Pearson correlation with measured T_2c1; and (4) Construction of the multivariate regression model. The multifractal analysis results demonstrate that the water-saturated NMR T₂ spectra of deep tight sandstones exhibit pronounced multifractal characteristics (Figure 9). The generalized fractal dimension $D_{\mathrm{q}}$ varies with q in a typical monotonically decreasing trend: in the strongly negative $q$ region (near ${\mathrm{q}}_{\min }$), $D_{\mathrm{q}}$ values are higher (1.55–1.70), reflecting higher local dimensions and strong heterogeneity in low-probability event regions (small pores corresponding to short T₂); in the positive q region, $D_{\mathrm{q}}$ approaches lower stable values (approximately 0.65–0.88), indicating relative uniformity in high-probability event regions (large pores corresponding to long T₂). This feature aligns with the pore–throat structure of tight sandstones, which is dominated by small pores and exhibits strong heterogeneity.

To quantitatively establish the relationship between multifractal parameters and the T₂ cutoff values determined by the centrifugation method, this study systematically examined the linear correlations between $D_{\mathrm{q}}$ at different q values and T_2C1 (Figure 9). The results indicate that $D_{\mathrm{q}}$ in the negative q region positively correlates with T_2C1, with the correlation coefficient R² reaching up to 0.664, significantly higher than in the positive q region (R² < 0.051). This phenomenon can be physically interpreted as follows: negative q amplifies the contribution of low-signal regions (small pores); when the heterogeneity of small pores increases ($D_{\mathrm{q}}$ enlarges), the proportion of bound fluids rises, reducing the relaxation time boundary required for movable fluid expulsion and thereby decreasing the T₂ cutoff value. The weak correlation in the positive q region suggests limited control of large pore regions over the movable-bound fluid boundary. Furthermore, the T₂ cutoff values in deep tight sandstone reservoirs positively correlate with the multifractal spectrum width $\Delta \alpha$ (Figure 10), with a correlation coefficient of 0.7023, indicating that $\Delta \alpha$ is also an intrinsic controlling factor influencing the T₂ cutoff values in deep tight sandstone reservoirs.

Upon further introducing derived parameters from positive and negative q, the correlations are markedly improved (Figure 11). Among these, the difference between the minimum and maximum generalized dimensions ($D_{\min }$ − $D_{\max }$) exhibits the strongest correlation with the T₂ cutoff value (R² = 0.712), followed by $D_{\min }$/$D_{\max }$ (R² = 0.559), while other ratios such as $D_{-2}$/$D_2$ show lower correlations (R² = 0.228). $D_{\min }$ − $D_{\max }$ essentially quantifies the overall width of the multifractal spectrum; larger values indicate stronger heterogeneity in pore size distribution. The positive correlation with the T₂ cutoff value further confirms the controlling role of heterogeneity in bound fluid occurrence. Compared to individual $D_{\mathrm{q}}$, these derived parameters integrate differential information from positive and negative q regions, significantly enhancing predictive sensitivity.

Based on the optimal parameters selected through correlation analysis, a multiple linear regression prediction model was established. The final model exhibits an excellent fit on the full dataset (R² = 0.9416), with predicted values closely aligned along the 45° line relative to actual values, and residuals randomly distributed around the zero line without systematic bias (Figure 12a). To mitigate the risk of overfitting given the sample size, Leave-One-Out Cross-Validation was performed. The model achieved a cross-validated Root Mean Square Error of 4.23 ms and a Q² (cross-validated R²) of 0.89, demonstrating stable predictive performance on unseen data. Feature weight analysis (Figure 12b) reveals that $D_{\min }$ contributes the most (negative weight of −22.18), indicating that enhanced small-pore heterogeneity significantly reduces the T₂ cutoff value; $\Delta \alpha$ and $D_{\min }$ − $D_{\max }$ exhibit positive contributions, reflecting increases in the T₂ cutoff value with greater generalized dimension differences and amplified probability disparities between large and small pores, respectively. Outliers were identified and removed based on a statistical criterion of standardized residuals >2.5. This procedure ensured that anomalous data points, likely resulting from experimental irregularities, did not bias the regression model.

$$T_{2C3}=5.63\left({\displaystyle \frac{D_{\min }}{D_{\max }}}\right)+13.54D_{\min }-22.79(\Delta \alpha )-22.18(D_{\min }-D_{\max })-14.1829$$

To further assess the model robustness, a diagnostic analysis of the prediction residuals was conducted (Figure 12c). The prediction residuals reflect not only model error but also uncertainty in the reference T_2c1 derived from centrifugation, including minor variability in saturation completeness, handling/evaporation during wiping and re-wrapping, and potential incomplete displacement of movable water in some pore networks at the maximum centrifugation step. Because replicate runs were not performed, we quantify dispersion using LOOCV RMSE (4.23 ms) and the residual distribution; RMSE provides an estimate of residual standard deviation under near-zero-mean residuals. The results show that the residuals are randomly and disorderly distributed on both sides of the zero line, primarily concentrated within the [−4, 4] interval, without exhibiting heteroscedasticity varying with predicted values or evident nonlinear trends, indicating that the model passes statistical tests and lacks significant systematic errors. Furthermore, Figure 12d illustrates the distribution of samples ultimately used for modeling after outlier removal (red points), with the actual T_2C1 values spanning a broad range from low-value zones (<20 ms) to high-value zones (>80 ms), and retaining some high-value samples. This ensures that the regression model maintains good generalization capability and applicability when handling complex reservoir samples with strong heterogeneity and large spans in pore structure.

5.3. Geological Implications and Insights for “Sweet Spot” Identification

Because the T₂ cutoff defines the movable-bound fluid boundary, its value directly controls reservoir-scale calculations of movable-fluid saturation, bound-water saturation, and effective porosity from NMR measurements [47]. However, centrifugation-based calibration is difficult to apply consistently at field scale due to its time-consuming workflow and potential damage to fragile tight-sandstone pore skeletons. Therefore, predicting T₂ cutoff directly from the fully water-saturated T₂ spectrum provides a scalable, non-destructive route for reservoir-wide evaluation and sweet-spot screening. In the deep tight sandstones of the Denglouku Formation in the Songliao Basin, The reservoir has undergone intense mechanical compaction and dissolution. Thin section and SEM analyses (Figure 13) reveal a complex pore network dominated by secondary dissolution pores and residual intergranular micropores. The failure of traditional statistical prediction methods [46], fundamentally stems from overlooking the complex pore–throat topological structures formed under intense diagenetic modifications [44,45,51,52,53,54]. The multifractal analysis results from this study indicate that the minimum generalized dimension $D_{\min }$ is a sensitive parameter for indicating variations in the T₂ cutoff value, revealing the decisive controlling role of strong heterogeneity in fluid mobility.

From a physical perspective, an increase in $D_{\min }$ signifies a marked enhancement in the heterogeneity of low-probability density regions in the NMR T₂ spectrum, namely the micropores and fine throats [55]. Geologically, such high $D_{\min }$ values typically correspond to zones of intense compaction or areas with complex intercrystalline pores in clay minerals [40]. In these regions, the microscopic pore structures exhibit extremely complex and irregular features, leading to increased specific surface areas that trigger pronounced surface relaxation effects and capillary binding capacities [56]. This directly causes the T₂ cutoff value to shift toward longer relaxation times (i.e., higher values), thereby substantially reducing movable fluid saturation. On the other hand, an increase in the multifractal generalized dimension difference ($D_{\min }$ − $D_{\max }$) indicates an expansion of the pore size distribution range, which geologically often reflects secondary dissolution superimposed on a tight background [57]. Although the development of dissolution pores broadens the pore size distribution and thereby increases overall heterogeneity, it also provides crucial effective seepage pathways.

Based on the aforementioned mechanisms, the multifractal parameter model established in this study provides quantitative microscopic criteria for identifying “sweet spots” in deep tight sandstones: intervals with high $D_{\min }$ values (e.g., greater than 1.60) are typically “non-sweet spot zones” or barrier layers controlled by intense compaction, where fluids predominantly exist in bound forms, whereas intervals with relatively low $D_{\min }$ values and larger generalized dimension differences indicate reservoirs with relatively minor compaction damage and the development of dissolution pores on a certain scale, representing “sweet spot” zones with optimal fluid mobility. This finding extends the application of NMR logging from mere rock physics parameter calculations to the identification of microscopic diagenetic facies, enabling exploration personnel to utilize logging data for a deeper understanding of the genetic mechanisms underlying pore–throat structures and to non-destructively and efficiently identify premium reservoir intervals with high mobility.

While the proposed NMR-multifractal method offers a powerful, non-destructive tool for evaluating fluid mobility in tight reservoirs, certain limitations exist. Paramagnetic minerals in complex lithologies may accelerate relaxation, potentially distorting pore structure characterization. Additionally, 1D T₂ spectra cannot fully distinguish between fluid types in mixed-saturation states. Future research should focus on integrating this method with 2D NMR (T₁–T₂) techniques or digital rock physics (e.g., Micro-CT) to enhance fluid identification accuracy and extend applicability to diverse geological settings.

6. Conclusions

This study addresses the challenge of accurately determining NMR T₂ cutoff values in deep tight sandstone reservoirs due to strong heterogeneity by proposing and validating a prediction method based on a multifractal analysis of fully water-saturated T₂ spectra. Through systematic experiments and analyses of tight sandstone samples from the Denglouku Formation in the Songliao Basin, the following main conclusions are drawn:

(1) Deep tight sandstone T₂ spectra exhibit distinct multifractal behavior. The minimum generalized dimension (D_min) and singularity spectrum width (Δα) show strong intrinsic correlations with measured T₂ cutoff values, quantitatively revealing how micro-throat heterogeneity controls bound fluids.

(2) The proposed multivariate regression model, incorporating these multifractal parameters, achieves high prediction accuracy with an R² of 0.9416. This significantly outperforms traditional single-parameter methods by effectively capturing complex multimodal pore structures. While the proposed multifractal framework is applicable to other tight reservoirs, the specific regression coefficients presented here are calibrated for the Denglouku Formation. Application to different lithologies requires local re-calibration.

(3) Microscale pore heterogeneity is identified as the decisive factor governing fluid mobility. The quantitative characterization of ‘small-pore complexity’ provided by this method offers a robust, non-destructive criterion for identifying reservoirs with superior connectivity.