Pore Structure Characterization of Low-Permeability Gravity-Flow Reservoirs: A Case Study of the Middle Es3 Member in Daluhu Area, the Dongying Depression, China

Abstract

The low-permeability gravity-flow sandstone reservoirs in the Dongying Depression, China, contain substantial oil reserves, yet their development is constrained by complex pore structures. In this study, optical and scanning electron microscopy (SEM) observations were integrated with nuclear magnetic resonance (NMR) measurements to investigate the pore system, pore size distribution, and connectivity of Es3z sandstone. By applying a Gaussian multi-peak fitting algorithm to the NMR T₂ spectra, parameters that directly capture the physical attributes of the rocks were extracted. Based on the correlation between these parameters and permeability, three distinct pore structure types (A, B, and C) were identified. The results demonstrate the effectiveness of using these NMR T₂ spectral parameters for quantitative pore structure characterization and classification, providing a robust framework for evaluating and predicting the quality of low-permeability reservoirs.

1. Introduction

Low-permeability sandstone reservoirs in the Dongying Depression, China, represent an important source of petroleum [1,2,3]. These reservoirs exhibit poor petrophysical properties, intricate pore structures, and pronounced heterogeneities, predominantly resulting from terrestrial depositional conditions and complex diagenetic processes. Detailed and integrated analyses of pore structures are crucial for reservoir quality evaluation, permeability prediction, and identification of reservoir formation mechanisms [4,5,6]. Pore structures encompass the geometry, classification, size, spatial distribution, and connectivity of pore throats, along with the inter-relationships among these characteristics [7,8]. In low-permeability sandstone reservoirs, the complicated pore systems, wide-ranging pore size distributions, and limited connectivity present significant challenges in accurately characterizing their microscopic pore structures.

Various analytical methods have been applied to investigate the pore structures of low-permeability sandstone reservoirs, including optical petrography, scanning electron microscopy (SEM) [9,10], X-ray computed tomography (CT) [11,12], mercury injection capillary pressure (MICP) [13,14], and nuclear magnetic resonance (NMR) [15,16]. Optical petrography and SEM provide two-dimensional images of pore structures; CT scanning offers insights into three-dimensional pore networks; MICP data reflect the size and connectivity of pore spaces; and NMR primarily reveals pore size distribution, playing a key role in pore structure evaluation.

Previous investigations have predominantly relied on indirect analyses of NMR data, encompassing morphological evaluations of pore size distribution [17], capillary pressure curve simulations [18], and estimations of mobile fluid distribution within the pore space [13]. Some researchers have also focused on examining sensitive NMR parameters for pore structure characterization and assessment, such as the NMR porosity, T₂ geometric mean (T_2gm), T2 cutoff time (T_2cutoff), maximum T₂ relaxation time (T_2max), effective mobile fluid porosity, and pore components (S1, S2, and S3) [19,20]. However, in low-permeability sandstone reservoirs, the NMR T2 spectrum often exhibits a single-peak pattern, and its qualitative characterization depends heavily on the experience of analysts and the sample size. Moreover, the numerous parameters derived from NMR experiments may vary in their effectiveness for characterizing pore structures across different sandstone reservoirs, further complicating the evaluation process.

In this study, in order to fully characterize the pore structure of the low-permeability gravity-flow sandstone reservoirs in the middle submember of the third member of the Shashejie Formation (Es3z) in Daluhu area, the Dongying Depression, China, several technical methods were combined, including optical petrography, SEM, and NMR. We attempted to use the Gaussian multi-peak fitting technique to decompose the nuclear magnetic resonance T₂ spectrum and extract parameters with rock physics significance to achieve quantitative characterization of pore structure. On this basis, these parameters were used to classify the types of pore structures. In this paper, the Section 1 describes the significance, aims, and scopes of this study. The Section 2 presents the geological background. The Section 3 presents the sampling information, experiments, and theories. The Section 4 presents the investigation of petrophysical properties and pore structure characteristics using an integrated method. The Section 5 discusses the effectiveness of the Gaussian multi-peak fitting method based on nuclear magnetic resonance T₂ spectra in pore structure evaluation.

2. Geological Background

The Bohai Bay Basin is one of the most significant oil and gas basins in East China, covering an area of approximately 200,000 km² and containing substantial oil and gas reserves (Figure 1A) [8]. The Boxing Sag is developed on the north-dip faulted block in the Dongying Depression within the Jiyang Sub-basin, the Bohai Bay Basin (Figure 1B). It is bordered by the Chunhua–Caoqiao Fault to the north, the Luxi Uplift to the south, the Gaoqing Fault to the west, and the Shicun Fault to the east (Figure 1C) [21]. The Daluhu area, situated in the northwest of the Boxing Sag, lies adjacent to the Qingcheng Uplift to the west (Figure 1C) [22].

Drilling data reveal that the Daluhu area is stratigraphically composed of Paleogene, Neogene, and Quaternary deposits, with the Kongdian Formation (Ek), Shahejie Formation (Es), Dongying Formation (Ed), Guantao Formation (Ng), Minghuazhen Formation (Nm), and Pingyuan Formation (Qp) present from bottom to top [23]. The Shahejie Formation is further subdivided into four members: Es4–Es1 (bottom to top). The third member (Es3) is further divided into three submembers: lower (Es3x), middle (Es3z), and upper (Es3sd) (Figure 1D) [23]. The middle submember of the third member of the Shahejie Formation (Es3z), which is the primary focus of this study, is located at a burial depth of 2600–2800 m. Es3z is predominantly composed of gravel sandstone, fine sandstone, siltstone, muddy siltstone, and mudstone. It contains typical gravity-flow reservoirs with porosity values ranging from 6% to 20% and permeability values between 0.1 and 10 mD. It exhibits low-permeability characteristics, but oil and gas are highly enriched in the reservoir.

3. Experiments and Theories

3.1. Samples and Experimental Procedures

A total of 140 core samples were analyzed under a net confining pressure of 363 psi (2.5 MPa). Thin-section samples were impregnated with blue fluorescent epoxy resin to enhance the visibility of porosity. Scanning electron microscopy (SEM) was conducted on freshly broken surfaces (carbon-coated) to examine the pore throat characteristics. Nuclear magnetic resonance (NMR) experiments were performed on 13 selected samples.

A total of 140 sandstone samples from Es3z reservoirs were analyzed in the laboratory of China University of Petroleum (East China). The samples were first cleaned with petroleum ether to remove the oil, and then dried for 24 h under a temperature of 95 °C. Core plug samples (50 mm long and 25.4 mm in diameter) were drilled for helium porosity, air permeability, and NMR measurements. Core chips were used to prepare thin sections for optical microscopic analysis and SEM analyses. The sample preparation procedure conformed to the protocol of GB23561.1-2024 [24], “Methods for Determining the Physical and Mechanical Properties of Coal and Rock”.

Helium porosity and air permeability measurements were conducted on 140 core plug samples at a net confining pressure of 400 psi. Optical petrographic analysis was conducted on 70 thin sections to identify the pore throats and petrological characteristics of the reservoir. SEM analysis was performed on 50 samples to identify pore throat types, sizes, and contact relationships between the pore throat and clay minerals. NMR T₂ spectra of the 13 samples were initially recorded at 100% water saturation, followed by centrifugation for further analysis. Before testing, the samples were saturated with a sodium chloride solution (12,000 mg/L salinity). The experiments were conducted at a controlled temperature of 25 °C and a relative humidity of 50%. The saturation process involved immersing the samples in the sodium chloride solution at a net pressure of 20 MPa for 24 h, after which NMR T₂ spectra were obtained under full water saturation. The NMR analysis was performed with the following parameters: a polarization time (Tw) of 1000 ms, an echo interval (TE) of 0.3 ms, 1024 echoes, and 128 scans. After centrifugation at 6000 rpm to remove free water, the samples were re-examined to obtain post-centrifugation NMR T₂ spectra.

3.2. Nuclear Magnetic Resonance Theory

Nuclear magnetic resonance (NMR) technology uses the transverse relaxation time T₂ as a metric to describe the pore structure within the rock matrix. The T₂ relaxation time provides an insight into the chemical environment of hydrogen protons in a sample, which indicates the binding forces and degrees of freedom of the protons. These properties are closely related to the internal structure of the sample, and provide a method for understanding the internal structure of the sample [25]. The T₂ relaxation time is a fusion of three different relaxation mechanisms: bulk surface relaxation (T_2S), volume relaxation (T_2B), and internal field gradient diffusion (T_2D), each of which contributes to the overall T₂ signal in a unique way [26]. The T₂ relaxation time of porous rocks can be expressed as Equation (1) [27]:

$${\displaystyle \frac{1}{T_2}}={\displaystyle \frac{1}{T_{2B}}}+{\displaystyle \frac{1}{T_{2D}}}+{\displaystyle \frac{1}{T_{2S}}}$$

(1)

Typically, in water-wet rocks, the bulk transverse relaxation time is much greater than the surface transverse relaxation, so the bulk transverse relaxation is negligible [27]. The Carr–Purcell–Meiboom–Gill (CPMG) pulse sequence, combined with a uniform magnetic field and an ultra-short TE (0.3 ms), effectively suppresses diffusion-related transverse relaxation by minimizing the spin diffusion time. This design ensures that internal gradient effects are negligible in sandstone samples. Therefore, the transverse relaxation time can be expressed as Equation (2):

$${\displaystyle \frac{1}{T_2}}\approx {\displaystyle \frac{1}{T_{2S}}}\approx {\rho }_2{\displaystyle \frac{S}{V}}={\rho }_2{\displaystyle \frac{F_S}{r}}$$

(2)

where ρ2 is the T₂ surface relaxivity (the T₂ relaxation intensity at the particle surface), S/V is the ratio of pore surface to fluid volume, FS is the pore shape factor, and r is the pore radius.

Therefore, the relationship between the pore radius r and the relaxation time of T₂ is as follows [28]:

$$r=n{T}_2$$

(3)

where n is related to ρ2, and Fs and is expressed in μm/ms.

3.3. Probability Density Function

A probability density function (PDF) is a mathematical construct that, when integrated over a particular interval, yields the probability that a variable will take on values within that range. The function is inherently non-negative over its entire domain, and the integral of the PDF over all possible values converges to one, ensuring a correct probabilistic interpretation. Probability density functions are primarily used in conjunction with absolutely continuous univariate distributions. Let a random variable X be characterized by a density f_X, where f_X is a non-negative, integrable function [29], if

$$P\left[a\le x\le b\right]={\int }_a^b{f}_X\left(x\right)dx$$

(4)

Therefore, if F_X is the cumulative distribution function of f_X, then

$$F_X\left(x\right)={\int }_{-\mathrm{\infty }}^x{f}_X\left(u\right)du$$

(5)

and (if f_X is continuous at “x”)

$$f_X={\displaystyle \frac{d}{dx}}{F}_X$$

(6)

Intuitively, f_X(x) dx can be thought of as the probability that x falls within the infinitesimal interval [x, x₊ dx].

Expectation and variance are key statistical indicators used to describe the distribution characteristics of random variables. Expectation describes the average value of a random variable, while variance measures the degree of deviation of a random variable from its mathematical expectation. If a random variable X is given, and its distribution allows for a probability density function f, then the expected value of X can be calculated by the following formula:

$$E\left[X\right]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xf\left(x\right)dx$$

(7)

X can be calculated using the following formula:

$$Var\left(X\right)=E\left[{\left(X-E\left[X\right]\right)}^2\right]$$

(8)

3.4. Gaussian Distribution

Gaussian distribution, also known as normal distribution, is an extremely important probability distribution model in mathematics, physics, and engineering [29]. It plays a vital role in many areas of statistics, and has a profound impact on both theoretical and applied statistics. In statistics, Gaussian distribution is a continuous probability distribution of real-valued random variables. The general form of its probability density function is as follows:

$$G\left(x\right)={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{x-\mu }{\sigma }}\right)}^2}$$

(9)

where the parameter μ is the mean or expected value of the distribution, and the parameter σ is its standard deviation. The variance of a Gaussian distribution is σ2.

When the random variable X is Gaussian-distributed with a mean μ and a standard deviation σ, it can be written as follows:

$$X~N\left(\mu ,{\sigma }^2\right)$$

(10)

The normal distribution has several different properties. Specifically, if X₁, X₂…, X_n are mutually independent normal random variables, the distribution can be written as follows:

$$X_i~N\left({\mu }_i,{\sigma }_i^2\right)$$

(11)

where μi and σi2 are the expected value and variance of the variable X, i = 1, …, n,

Then, any linear combination of these random variables, as shown in Equation (12), also conforms to the Gaussian distribution shown in Equation (13).

$$Y=\sum _{i=1}^n{a}_i{X}_i$$

(12)

$$Y~N\left(\sum _{i=1}^n{a}_i{\mu }_i,\sum _{i=1}^n{a}_i^2{\sigma }_i^2\right)$$

(13)

3.5. Innovative Technologies

3.5.1. Data Preprocessing

In order to ensure the accuracy and reliability of data analysis, a series of meticulous data preprocessing steps were implemented in this study. Preliminary analysis showed that when the transverse relaxation time T₂ exceeded 1000 ms, tiny pores, mainly corresponding to microcracks, were observed in the sandstone samples [8]. Considering that these microcracks were not the focus of this study and were negligible in quantity, they were excluded from the experimental data processing. In addition, given that the raw data represented the increment of porosity, they were converted into a porosity probability density function for more accurate data analysis. The calculation process was as follows.

The sum of the porosity increments of all sampling points is the porosity of the sample, which can be expressed as follows:

$${\phi }_{NMR}=\sum _{i=1}^n{\phi }_{T_{2i}}$$

(14)

where φ_NMR is the NMR porosity of the sample, and φT_2i is the porosity of the i-th component of the transverse relaxation T_2i (also called incremental porosity).

Therefore, the actual cumulative distribution function of porosity (F*) can be calculated using the following formula:

$$F*=\sum _{T_2\le T_{2i}}{\phi }_{T_{2i}}$$

(15)

Finally, the actual porosity probability density function (f*) can be calculated by the logarithmic differentiation of Equation (15), as follows:

$$f*\left(\log T_2\right)={\displaystyle \frac{dF*}{d\log T_2}}$$

(16)

3.5.2. Gaussian Multi-Peak Fitting

The Gaussian multi-peak fitting algorithm is a widely used technique in data analysis and processing [30]. It can effectively identify and extract multiple peaks in the data and fit each peak with a Gaussian distribution. The algorithm is widely used in various fields, such as earth science, chemistry, medicine, and biology, and is recognized as an efficient and accurate data analysis method. The basic principle of the multi-peak Gaussian fitting algorithm involves using a Gaussian distribution model to describe the characteristics of data distribution.

In the initial stage of the Gaussian peak fitting algorithm, the experimental data are carefully observed to identify potential components or sub-peaks. Subsequently, a series of Gaussian distributions are used as models to accurately characterize these components, each representing an independent component. The algorithm then iteratively optimizes the parameters of the Gaussian model using a nonlinear least squares method, aiming to minimize the deviation between the composite curve fitted by the model and the experimental data. Finally, the R- squared value is used to evaluate the fit between the processed distribution and the actual distribution, to ensure the accuracy and effectiveness of the model [31].

In this study, it is assumed that the distribution of the processed NMR T₂ spectral data is formed by the linear superposition of several Gaussian distributions (Figure 2A), and its mathematical expression is as follows:

$$f\left(\log T_2\right)=y_0+\sum _{i=1}^3{f}_i\left(\log T_2\right)=y_0+\sum _{i=1}^n{A}_i{G}_i\left(\log T_2\right)$$

(17)

where f (logT₂) is the porosity probability density function of the total peak (f) obtained after applying Gaussian multi-peak fitting, representing an accurate fit to the actual porosity probability density (f*); y0 is the baseline, in this case, y0 = 0; fi (logT₂) is the porosity probability density function of the sub-peak (fi); Ai is the area of the different sub-peaks, reflecting the weight assigned to each sub-peak; Gi (logT₂) is the Gaussian distribution of the sub-peaks; and n is the number of sub-peaks.

From Equations (14)–(17) and the properties of the Gaussian distribution, it can be inferred that the sub-peak area (A_i) and the NMR porosity (φ_NMR) satisfy the following relationship:

$${\phi }_{NMR}=\sum _{i=1}^n{\phi }_{T_{2i}}={\int }_{\log T_{2min}}^{\log T_{2max}}f*\left(\log T_2\right)d\log T_2\approx {\int }_{-\infty }^{\infty }f\left(\log T_2\right)d\log T_2=\sum _{i=1}^n{A}_i$$

(18)

where logT₂min and logT₂max are the minimum and maximum values of T₂ under the logarithm, respectively.

In engineering applications, the Gaussian distribution function of Equation (9) is usually transformed slightly:

$$f_i\left(\log T_2\right)=A_i{G}_i\left(\log T_2\right)={\displaystyle \frac{A_i}{\sqrt{\frac{\pi }{4ln\left(2\right)}}{w}_i}}{e}^{-4\ln \left(2\right){\left({\displaystyle \frac{\log T_2-c_i}{w_i}}\right)}^2}$$

(19)

where w_i is the full width at half maximum (FWHM) of the sub-peak, and c_i is the center of the sub-peak.

Therefore, the properties of each sub-peak (Figure 2B) can be determined by the specific parameters A_i, w_i, and c_i. In this study, the number of Gaussian distributions was objectively determined by first-derivative analysis of the T₂ spectrum, which identified inflection points as potential peak centers.

The R-squared value reflecting the degree of fit can be calculated as follows:

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^N{\left(f_i*\left(\log T_2\right)-f_i\left(\log T_2\right)\right)}^2}{\sum _{i=1}^N{\left(f_i\left(\log T_2\right)-\overline{f_i\left(\log T_2\right)}\right)}^2}}$$

(20)

4. Results and Discussion

4.1. Physical Characteristics of Rocks

The porosity and permeability results of the 140 sandstone core samples from the Es3z sandstone reservoirs indicate that the petrophysical properties are generally unfavorable, with porosity mainly ranging from 8% to 20%, and permeability ranging from 0.1 to 3 mD (Figure 3). The majority of the samples belong to the category of low-permeability sandstone. The porosity and permeability show a good correlation, with a value of 0.46 (Figure 3).

4.2. Pore Systems

As can be seen from the SEM images and thin casting sections, the gravity-flow low-permeability sandstone reservoirs of Es3z mainly develop residual intergranular pores, and the primary intergranular pores are less preserved, indicating that the compaction and cementation during diagenesis are strong (Figure 4A,B). Through the observation of the thin sandstone casting sections, it can be seen that the edges of the primary intergranular pores of the reservoir are straight, there is no obvious dissolution trace on the surface of the particles, the outline is clear, and the pore radius is generally between a few microns and tens of microns. The pore throat is well connected, and it has good porosity and permeability conditions (Figure 4A); some pores are filled with kaolinite and authigenic quartz, and they appear as residual intergranular pores (Figure 4B).

The dissolution pores are predominantly in feldspars and lithic fragments (Figure 4C,D). Some of the intergranular pores and dissolution pores are filled by clay minerals, forming intercrystalline pores (Figure 4E,F). There are a large number of intercrystalline pores within kaolinite, chlorite, and the mixed layer of illite/smectite, and the pore diameter is less than 1 μm (Figure 4F).

The pore throats are sheet-like, bending-flake-like, or tube-like in geometry (Figure 4G–I), and they could be observed under SEM observation.

4.3. NMR T₂ Spectrum Distribution Characteristics

In addition to conventional core analysis and microscopic observation, characterizing pore structure also involves determining the distribution characteristics of pore size. Nuclear magnetic resonance (NMR) T₂ spectroscopy provides important information for analyzing the physical properties, pore size distribution, and fluid state of rock [32]. In brine-saturated porous media, the intensity of the NMR signal is proportional to the number of hydrogen protons or liquid content. An increase in signal intensity indicates a higher liquid content in the T₂ value, reflecting a higher porosity [28]. According to Equations (2) and (3), the T₂ value is proportional to the pore size, making the T₂ spectrum an effective tool for reflecting the pore size distribution characteristics.

The results of the NMR experimental analysis are shown in

Table 1. NMR T₂ spectrum parameters of 13 sandstone samples.

Sample	Well	Depth (m)	NMR Experimental Parameters				Peak Parameters
			φNMR (%)	K (mD)	BVI (%)	FFI (%)	A1	c1	w1	A2	c2	w2
1	F154-1	2656.5	16.7	0.30	9.25	7.42	10.01	0.32	1.14	6.73	1.14	1.33
2	F154-1	2662	16.4	0.39	8.79	7.59	8.80	0.30	1.19	7.59	1.25	1.24
3	F154-1	2675.5	19.9	0.56	9.46	10.41	10.96	0.29	1.13	8.90	0.83	1.26
4	F154-1	2679	14.8	0.11	11.03	3.80	11.07	0.00	0.90	3.86	0.60	0.70
5	F154-7	2732	13.5	0.08	11.74	1.80	11.02	0.07	1.02	2.35	0.62	0.84
6	F154-7	2752.1	13.6	0.11	10.91	2.66	10.54	0.10	1.00	3.38	0.57	1.25
7	F154-7	2754.7	9.7	0.01	7.61	2.04	7.52	0.12	1.00	1.93	0.71	0.88
8	F154-8	2721.5	15.2	0.03	12.12	3.04	11.87	0.18	1.06	2.89	0.65	1.07
9	F154-8	2726.7	13.0	0.02	10.72	2.31	11.01	0.10	0.96	2.02	0.70	0.89
10	F154-8	2730	18.1	0.47	8.95	9.17	9.15	0.24	1.11	8.98	0.98	1.33
11	F154-8	2736.35	11.3	0.01	10.23	1.03	9.20	0.13	1.05	2.09	0.92	1.00
12	F154-8	2727.2	18.6	2.10	12.50	6.10	12.40	0.36	1.28	6.04	0.94	1.41
13	F162-X7	2734.2	21.2	6.74	12.07	9.13	11.49	0.41	1.28	9.68	1.31	0.96

. Similarly to the physical properties shown in Figure 3, the porosity and permeability of the NMR experimental samples also show a strong correlation (Figure 5), with an $R^2$ value of 0.88. The porosity is mainly distributed between 9% and 21%; the permeability is mainly concentrated between 0.1 and 1 mD, and two samples are also partially distributed in the 1–6 mD range.

4.4. Decomposition of NMR T₂ Spectra

4.4.1. Rationality of Gaussian Multi-Peak Fitting

The distribution of NMR T2 Spectra is actually the sum of multiple distributions, each corresponding to the size distribution of a different pore type. Reasonable decomposition of the T₂ spectrum can extract detailed information for each pore class. Gaussian multi-peak fitting was selected to decompose and characterize the NMR T₂ spectrum for the following reasons.

First, various experimental studies have shown that the logarithm of pore size in rock samples appears to be approximately Gaussian in distribution [33,34,35]. Genty presented four log-normal probability plots, all of which exhibited nearly linear behavior, indicating a log-Gaussian distribution of pore size [36]. Given that T₂ relaxation times are proportional to pore size for pores of similar shape, this implies that T₂ relaxation times should exhibit a log-Gaussian distribution type similar to that of pore size.

Secondly, nuclear magnetic resonance measures the relaxation signal of hydrogen atoms, and the T₂ time is determined by its decay process. The T₂ spectrum is obtained by multi-exponential inversion fitting of the nuclear spin echo sequence. Zhong proposed that the measurement and inversion process of the T₂ spectrum follows statistical principles, indicating that it has a Gaussian distribution [37].

Finally, Adams demonstrated that each pore type has a specific and relatively consistent shape parameter, indicating that the T₂ relaxation time of each pore type is log-Gaussian distributed [33]. This is why the actual porosity probability density function in data preprocessing is obtained by the logarithmic differentiation of the actual cumulative distribution function of porosity (Equation (16)).

Therefore, considering the above points, Gaussian multi-peak fitting is an effective and economical method for characterizing NMR T₂ distribution. By observing the fitting results and analyzing the parameters of different pore types, the pore size distribution can be accurately and quantitatively characterized.

4.4.2. Gaussian Multi-Peak Fitting Results

Taking sample No. 13 as an example, Figure 6 shows the results of Gaussian multi-peak fitting. After data processing, the porosity content representing microcracks greater than 1000 ms was excluded (Figure 6A), and the original T₂ spectrum was converted into a logarithmic (T₂) spectrum (Figure 6B). In this spectrum, the horizontal axis represents the logarithmic value of the T₂ relaxation time, and the vertical axis represents the probability density function of the porosity.

A comparison between Figure 7A,B shows that the logarithmic (T₂) spectrum maintains a similar form to the original T₂ spectrum. This similarity is due to the fact that the intervals of the T₂ spectrum data points in the logarithmic coordinates are approximately equal, resulting in consistent relative amplitudes of the data points after differentiation.

At 100% brine saturation, the NMR T₂ spectrum of sample No. 13 showed obvious single-peak characteristics (Figure 6A). After Gaussian multi-peak fitting, the sample was decomposed from a single peak into two sub-peaks (Figure 6B). As shown in Figure 6B, there is a high degree of consistency between the original NMR T₂ spectrum data and the total peak after fitting. The R² value is 0.997, indicating that the fitting curve effectively reflects the characteristics of the original data. It can be seen from the previous Formula (17) that the total peak is composed of two sub-peaks linearly superimposed, and each sub-peak represents a different pore type. Peak 1 is compared with the centrifugal porosity increment curve in Figure 6A, and the results show that the two are highly similar. The logarithmic (T₂) spectrum not only fully retains the information of the original NMR T₂ spectrum, but also effectively decomposes the original data.

This study identifies a bimodal pore structure in the low-permeability gravity-flow reservoirs of the middle Es3 member in the Daluhu area (Dongying Depression) through Gaussian multi-peak fitting of NMR T₂ spectra, where the two sub-peaks were determined via first-derivative analysis. However, the recognition of bimodality is method-dependent: alternative mathematical approaches (e.g., second-derivative or wavelet-based decomposition) or threshold criteria (e.g., minimum peak amplitude) may yield divergent interpretations. Furthermore, the observed bimodal distribution is specific to the studied sandstone samples. In reservoirs with contrasting mineralogy (e.g., iron-rich or clay-rich strata) or stronger heterogeneity, the T₂-derived pore classification should be cautiously validated against petrographic and regional geological data.

4.4.3. Physical Meaning of Peak Parameters

From Equation (19), it can be seen that each sub-peak in the Gaussian distribution can be accurately quantified by three peak parameters: A, w, and c, as shown in the calculation results of all samples in Table 1.

A represents the area of the sub-peak, reflecting the porosity of different types of pores, and a larger area means a higher porosity. In the previous section, it was mentioned that the centrifugal porosity increment curve of sample No. 13 and the porosity probability density curve of sub-peak 1 have a high degree of similarity. In fact, by comparing the immovable water saturation (BVI) obtained by accumulating the centrifugal porosity increment curves of all samples and the area A1 of sub-peak 1, and the free fluid index (FFI) and the area A2 of sub-peak 2, the results show (Figure 7) that the two have high similarity.

Therefore, it can be considered that peak 1 mainly represents immobile water associated with micropores, while peak 2 mainly represents movable water associated with mesopores. According to Equation (18), the sum of the areas of the sub-peaks is approximately equal to the NMR porosity. As shown in Figure 6, the micropore and mesopore porosities in sample No. 13 are 11.49% and 9.68%, respectively. Their total porosity is 21.17%, which is close to the NMR porosity of 21.11%. The difference between the two is determined by the accuracy of the Gaussian multi-peak fitting.

w is the full width at half maximum (FWHM) of the different sub-peaks, which indicates the width of the Gaussian distribution and the range of pore sizes within each pore type. In this article, the peak parameter width refers to the full width at half maximum. Narrower peaks indicate a more concentrated pore size distribution, while wider peaks indicate a more dispersed distribution.

c is the center of the different sub-peaks, representing the center of each pore type. A lower peak center corresponds to smaller pores, while a higher peak center represents larger pores.

Since w and c are calculated on a logarithmic scale, their values cannot be directly compared to contrast the range of different pore types, especially across different scales of pore types. Micropores may have a higher w than mesopores, but because their pore sizes are much smaller than that of mesopores, the pore size distribution range of micropores is still lower than that of mesopores.

4.4.4. Pore Structure Classification Based on Multi-Peak Fitting Parameters

In order to further clarify the relationship between permeability and peak parameters, a cross-plot of permeability K and six peak parameters was drawn (as shown in Figure 8). Figure 8A–C shows that there is no significant correlation between the micropore peak parameter and permeability. This analysis quantitatively demonstrates the minimal contribution of immobile water to permeability, and although the micropore peak areas and peak centers were significant in some samples, no significant permeability levels were demonstrated. Figure 8D–F shows that there is a significant correlation between the peak area of the mesopores representing movable water and permeability, while there is no apparent correlation between the peak center, peak width, and permeability.

Therefore, considering the A₂ value as the basis for pore structure classification, a value greater than 9 is defined as a Type A pore structure, 4–9 indicates a Type B pore structure, and a value less than 4 is a Type C pore structure. As shown in Figure 9, the pore structure classification results of all samples are mapped to the intersection diagram of porosity and permeability. Statistical analysis shows that the porosity range of Type A samples is 18.6–21.2%, and the permeability range is 2.1–6.73 mD. The porosity range of Type B samples is 16.38–19.87%, and the permeability range is 0.298–0.56 mD. The porosity range of Type C samples is 9.65–15.16%, and the permeability range is 0.006–0.11 mD.

Obviously, the permeability and porosity of Type A samples are the highest, followed by those of Type B, and those of Type C are the worst. The three show obvious boundaries in the porosity–permeability cross-plot (Figure 9).

Representative samples of three types were selected for comparison, as shown in Figure 10. From C to A, the peak area and peak center of peak 2 increase successively, reflecting that the pore center and porosity in the mesopores increase successively, which is also the main reason for the clear boundaries of the three permeabilities. Since the BVI has almost no contribution to the permeability, the distribution of peak 1 has almost no effect on the permeability.

5. Conclusions

The petrophysical properties and pore structures in pore space were analyzed by integrated optical and SEM petrographic and petrophysical analyses, including poroperm and NMR measurements. The pore structures of the Es3z sandstone reservoirs in the Daluhu Area were characterized and evaluated. Some of the key findings are as follows:

(1)

Petrographic analysis revealed that the pore systems in Es3z sandstone are composed mainly of residual intergranular pores, dissolution pores, intercrystalline pores, and sheet-like, bending-flake-like, and tube-like pore throats.

(2)

Gaussian multi-peak fitting revealed that mesopores, associated with movable water, are the primary contributors to permeability, while micropores, associated with immovable water, have minimal impact.

(3)

This study effectively combined NMR and Gaussian multi-peak fitting to characterize the pore structure of low-permeability gravity-flow sandstone reservoirs. The results confirm that mesopores control permeability, and the pore classification system (Type A, B, C) provides a valuable tool for reservoir evaluation. The methodology offers a more accurate and reliable approach for understanding low-permeability reservoirs.

(4)

While this study establishes a bimodal pore structure model for low-permeability gravity-flow reservoirs, its applicability is currently limited to sandstones in the Daluhu area. Future research should validate and adapt this model to reservoirs with contrasting lithologies (e.g., clay-rich shales, carbonate-cemented sandstones) to assess its broader geological relevance.