Pore Structure Characterization of Eocene Low-Permeability Sandstones via Fractal Analysis and Machine Learning: An Example from the Dongying Depression, Bohai Bay Basin, China.

Poroperm analysis, mercury injection capillary pressure (MICP), and nuclear magnetic resonance (NMR) measurements were performed to delineate the pore structures and fractal behaviors of the Eocene low-permeability sandstones in the Dongying Depression, Bohai Bay Basin, China. Three types of pore structures (I, II, and III) have been classified by applying the self-organizing map (SOM) clustering model. Comparative analysis of three different fractal models indicates that the MICP tubular model and NMR model are quite effective for pore structure characterization. The results show that the reservoirs generally exhibit high fractal dimensions, indicative of complex pore structures. The presence of small pore throats is primarily responsible for the heterogeneities and complexities in the Eocene low-permeability sandstones. A modified Winland model was established for the permeability estimation using MICP data. Different from high-permeability reservoirs or unconventional (e.g., shale and tight formation) reservoirs, r 10 is the best parameter for permeability estimation, indicating that the permeability of the Eocene low-permeability sandstones is largely controlled by the large pore systems. Additionally, a porosity model derived from movable fluids using NMR data has been established and provided better prediction effect compared with the classic Coates and Schlumberger Doll Research (SDR) models. Fractal analysis and permeability estimation are shown to be quite effective for investigating microscopic behaviors and in predicting the reservoir quality of low-permeability sandstone reservoirs.

Introduction

The Eocene low-permeability sandstones (permeability: 0.1–50 mD) are the most important reservoir type in the Dongying Depression with great potential for hosting oil and gas.[1−3] The microscopic pore structure features, including the geometric shape, type, size, distribution, and connectivity, determine the reservoir petrophysical properties and thus control the fluid flow in reservoir sandstones. Investigation of microscopic pore structures is thus vital for refined reservoir characterization and improving the efficiency of oil/gas development.[4,5]

The traditional methods such as the Euclidean geometry and experimental techniques are no longer effective in characterizing and evaluating complex and heterogeneous pore structures.[6] Numerous studies have demonstrated that pore structures in porous rocks are self-similar and do not change with investigation scales.[7,8,36] By applying the fractal theory, complex pore systems can be quantitatively investigated,[9−11] and the fractal dimensions obtained can also be applied for reservoir quality evaluation and estimation of petrophysical properties.[12] Fractal analysis has been shown to be especially suitable for characterizing low-permeability terrestrial sandstones that have undergone complex diagenetic modifications.[48]

Methods applied to analyze the fractal behavior of pore structures include thin sections and scanning electron microscopy (SEM),[13] MICP,[14] NMR,[15−17] X-ray computer tomography,[18,19] and gas adsorption measurements.[20,21] Fractal models established using data sets obtained from the above mentioned methods have advantages and limitations due to their different theoretical principles.[22,23] Hence, fractal characterization of pore structures by an integrated method is commonly used lately.[24,25] Several studies also compared fractal dimensions determined using different techniques.[26,27] Evaluation of different fractal models is helpful to reveal the limitations and advantages of various models, which is crucial for promoting the application and development of the fractal theory in pore structure evaluation. In addition, previous pore structure identification methods such as the morphological analysis or regression analysis of MICP/NMR data are no longer effective and adequate.[27,28] In this paper, a multivariate cluster analysis based on multiple measured variables and machine learning was performed to identify the microscopic pore structure types.

Permeability is an essential parameter in reservoir characterization and controls the ability of fluid flow. It is closely related to pores and pore structures. Numerous studies have been involved in extracting essential attribute parameters from MICP and NMR data for permeability estimation.[29−32] Various empirical permeability models have been established, including Kozeny–Carman (KC) equation,[33,34] PaRiS equation,[35] Winland model,[29,36,37] SDR and Coates model,[30,31] and so forth. Previous studies indicated that the optimal pore throat radius and pore type for permeability estimation are variable in different types of reservoirs.[38] Consequently, investigation of pore structures and its heterogeneity on permeability would be conducive to accurately estimate permeability.

In this study, the fractal behavior of low-permeability sandstones from the Dongying Depression was investigated using MICP and NMR data. The SOM model was employed for pore structure classification using 70 groups of core data. The petrophysical features of various pore structures were analyzed by MICP and NMR data. The relationships between pore structure, petrophysical properties, and fluid mobility were also investigated. A comparative analysis of fractal dimensions using three different fractal models was carried out. Additionally, the effects of pore structures on permeability were discussed and empirical equations for permeability estimation have been established based on capillary pressure curves and NMR T2 spectral parameters. This study attempts to provide an essential workflow for characterizing fractal behaviors of pore structures and an effective method for estimating permeability and evaluating reservoirs with low-permeability and strong heterogeneities.

Geological Background

The Dongying Depression is a dustpan-shaped depression developed in the southwestern Bohai Bay Basin, China.[39] It is composed of five structural units, including the northern steep slope zone, northern sag zone (Lijin and Minfeng sags), central anticline zone, southern sag zone (Niuzhuang and Boxing sags), and southern gentle slope zone (Figure A,B).[40] The Dongying Depression comprises a thick Paleogene sediment sequence, including the Kongdian, Shahejie, and Dongying formations. The Eocene Shahejie (Es) Formation is composed of four sedimentary units from bottom to top: Es4, Es3, Es2, and Es1 (Figure C). The upper Es4 (Es4u) unit is considered as the target interval of the study area, and the burial depth of Es4u varies from 2160 to 3563 m. The Es4u unit was deposited in shore-shallow lacustrine environments and consists mainly of fine-grained sandstone interbedded with thin mudstone (Figure C).[1]

Figure 1

(A) Location of the Dongying Depression in the Bohai Bay Basin, China; (B) cross section showing major structural units and stratigraphic configurations; and (C) generalized stratigraphic column of the Paleogene formations in the study area. Adapted in part with permission from ref (39). Copyright 2020 Elsevier.

Sampling and Methodology

Samples and Analytical Techniques

A total of 70 core plug samples (50 mm long and 25.4 mm in diameter) from different depths were collected from 38 wells located in the Dongying Depression. All samples were subjected to petrophysical property analysis and MICP analysis, among which 32 samples were selected for optical petrographic analysis and 21 samples were selected for NMR measurements (Table ).

Table 1

Information of Samples and Various Measurements

measurements	number of wells	number of sandstone samples
helium porosity	38	70
air permeability	38	70
thin sections	21	32
MICP	38	70
NMR	12	21

Helium porosity and air permeability were performed to investigate the petrophysical properties of the sandstone reservoirs. Pore structure features of 70 samples were detected by MICP analysis, with the maximum mercury inlet pressure up to 116 MPa. NMR measurements were conducted to extract the T2 spectra of 21 samples. First, the NMR T2 spectra at the 100% water-saturated state were measured, with an echo interval of 0.21 ms. The experimental temperature was kept in 25 °C. Then, the samples were centrifuged with a rotating speed of at least 6000 rpm for removing the free water in the plugs and measured again to obtain NMR T2 spectra under a centrifuged state. It should be mentioned that Sample Cn371-1 is not well-consolidated; thus, only the T2 spectra at the 100% water-saturated state were measured.

Fractal Theory

Fractal Models for MICP Data

According to fractal theory, self-similar fractal behavior does not change with the scale of magnification.[10,22,41] Since its introduction by Mandelbrot in 1977, it has been widely used in pore structure evaluation and characterization.[42,43] The self-similarity behavior of pore structures can be illustrated by eq .[7,44]where r is the pore throat radius; N(r) is the number of pores with a radius larger than r; and Df is the fractal dimension. The fractal dimensions calculated using both the tubular model (model I) and spherical model (model II) are discussed.

Fractal Model I

The N(r) can be expressed in eq , when using a tubular model.where j = i + 1, V is the cumulative mercury intrusion volume at a certain pore radius r, and l is the length of a capillary tube. It should be mentioned that N(r) refers to the equivalent number of pores when the pore space is filled by bundles of capillary tubes with radius r.[45,46]Df can be obtained by the slope of the curve of lg N(r) – lg r.

By combining with the Young–Laplace equation (eq ), the fractal model I can also be expressed in another form (eq ).where Pc is the capillary pressure (MPa); σ is the surface tension; θ is the contact angle (°); and SHg is the accumulative mercury saturation, which can be converted with VHg.[46]

Fractal Model II

According to the fractal theory, the saturation of the wetting phase can be expressed aswhere V (

For rmin ≪ r, we get

By combining eqs , 6, and 7 and taking logarithms on both sides of eq , it becomes

Fractal Dimensions Based on NMR Experiments

Lai et al. (2018) proposed a fractal model using NMR data, and it can reflect the whole pore size distribution.[47] The pores detected by NMR measurement were assumed to be spherical and then the N(r) can be described by eq .where Vpi is the cumulative pore volume at a given pore radius ri, ρ is the surface relaxivity, and T2 is the transverse relaxation time.

By combining eqs and 11

Using logarithms for eq where and lop(3ρ) are constants. The fractal dimension (Df) can be calculated from the logarithmic relationships between and T2.

SOM Neural Network Algorithm

The SOM neural network algorithm is a clustering analysis method based on unsupervised machine learning.[48,49] The network structure is composed of an input layer and a competing layer (Figure ). The neuron in the competing layer can automatically compete for the opportunity to respond to the input pattern and adjust the weights W by competitive learning.[50] The neuron with the strongest response is known as the winning neuron and is also known as the best matching unit (BMU). After repeated training and weight adjustment, the derived topological mapping can be used for assigning the best fitting category for each input pattern.

Figure 2

Typical network structure of SOM (A) and competing layer with neurons (B). Adapted in part with permission from ref (39). Copyright 2020 Elsevier; adapted in part with permission from ref (51). Copyright 2015 Wiley-Blackwell.

The process of weight adjustment is shown as followswhere i and j are neurons in the input layer and the competing layer, respectively; D is the Euclidean distance; x are the input variables; g is the winning neuron; P(j) is the winning probability; μ is a constant; and W is the weight between the neuron i and neuron j.

The advantage of the SOM classifier is that the clustering results are not affected by incorrect user-defined information,[51,52] and there are no restrictions on the number of parameters participating in the training process.[53,54] Therefore, the unsupervised neurocomputing algorithm presents an excellent application in solving pattern recognition problems. It provides a way for automatic classification of pore structures.

Results

Pore Structure Characterization

Pore Structure Characteristics from MICP Analysis

Highly variable MICP capillary parameters and pore throat size distributions were derived from 70 core samples in the Es4 sandstones. The pore structure features were investigated by three typical sandstone groups (Figure ). The samples with good physical properties (Group I) exhibit a high mercury withdrawal efficiency and a relatively low displacement pressure, which indicates a good pore throat connectivity. The samples also display a wide pore throat size distribution with a pore throat radius mainly in the range of 0.0063–10.0 μm (Figure B). The pore systems of Group I are dominated by dissolution pores and residual intergranular pores (Figure D). In contrast, the samples with poor physical properties (Group II and Group III) are characterized by high displacement pressures and relatively low mercury withdrawal efficiencies, suggesting that the samples are dominated by small pore throats. The pore throat sizes in samples of Group II and Group III exhibit a narrow distribution, with the pore throat radius being mostly less than 1.0 μm (Figure B). The development of small pore throats is commonly associated with incomplete dissolution of clay minerals and clastic particles (Figure D).

Figure 3

Typical capillary pressure curves (A), pore throat size distribution (B), Pittman’s plot (C), and photomicrographs (D) for Es4 sandstones derived from MICP measurements and petrographic analysis.

By plotting SHg and SHg/Pc, a sharp apex can be identified,[36] and the pore throat radius corresponding to the apex was defined as rapex by Lai and Wang (2015)[14] (Figure C). The pore systems can be separated into large and connected pore systems and small pore systems based on the value of rapex.[14,55] Pore throats with a radius larger than rapex contribute significantly to permeability, whereas the contribution of the small pore throat system (r < rapex) to permeability is small (Figure B). Therefore, rapex can be used as an important index to evaluate reservoir quality and pore throat connectivity.

Pore Size Distribution Obtained by NMR T2 Spectra

Previous studies show that the NMR T2 transverse relaxation time of the hydrogen nucleus is closely related to pore size.[43,56] Samples with short transverse relaxation times indicate the existence of small pores, whereas long transverse relaxation times are often associated with macropores or fractures. The signal amplitude of T2 transverse relaxation time is indicative of the pore volume fraction with different pore sizes. The NMR T2 spectra can provide more comprehensive pore structure information compared with the MICP measurements, and the differences among various samples revealed by NMR are more distinct.[57,58]

Figure shows typical NMR T2 spectral distributions of Es4 sandstones, among which three types of NMR T2 spectral distributions are revealed. Type I T2 spectra exhibit a continuous unimodal behavior, with T2 times in the range of 0.01–1000 ms (Figure A). Type II T2 spectra (bimodal behavior with a higher right peak) display a coexistence of large and small pores, and the large pores account for a great proportion (Figure B). The left peak varies from 0.1 to 10 ms, while the right peak is in the range of 10–1000 ms (Figure B). Type III T2 spectra (bimodal behavior with a higher left peak) show the lack of the long T2 components compared with the second type. The T2 time is mainly concentrated in 0.1–100 ms (Figure C).

Figure 4

Typical NMR T2 spectra with unimodal (A) and bimodal behavior [samples with higher right (B) and left peak (C)].

Fractal Analysis

Fractal Dimensions Obtained from MICP Data

The lg(N(r)) and lg(r) plots based on the MICP data were constructed for all 70 typical sandstone samples (Figure ), and the plots show good fits (R2 > 0.90) (Figure ). The results indicate that the pore structure derived from the MICP data is characteristic of fractal dimensions and can be characterized using the MICP tubular model. In order to determine the fractal dimensions of small pore systems and large pore systems, the lg(N(r)) and lg(r) plots were separated into two linear segments using the same segment threshold values (r = 0.1 μm), and the double-fractal characteristics can be approximated by piecewise regression (Figure B,D). The fractal dimensions of the left segments (DT1) correspond to the fractal behavior of the small pore systems (r < 0.1 μm), ranging from 2.01 to 2.98 (averaging 2.23). For the right segments, the calculated fractal dimensions (DT2) are characteristic of the large pore systems (r > 0.1 μm). DT2 may be greater than 3.0 and shows a broad distribution between 2.32 and 6.54 (averaging 3.53). The presence of microfractures, the use of an oversimplified tubular model, and the skin effect may result in the phenomenon that some samples at the right segments (DT2 are greater than 3.0) may not possess fractal characteristics.[59,60]

Figure 5

Calculated fractal dimensions using the MICP tubular model. (A,C) Total fractal dimensions (DT); (B,D) fractal dimensions of small pore systems (DT1) and large pore systems (DT2), respectively.

The log–log plots of 1 – SHg and pc based on the spherical model are shown in Figure , and the double-fractal characteristics are investigated using the inflection points corresponding to r = 0.1 μm. The detailed fractal dimensions of small pore systems (DS1), large pore systems (DS2), and full range of pore systems (DS) calculated using a spherical model are shown in Figure . The results indicate that the fractal dimensions DS1 of some samples appeared to be less than 2.0, which does not conform to the Euclidean dimension and has no significance for pore structure evaluation. For the large pore throats, DS2 is in the range of 2.16–2.82 (averaging 2.53), showing strong fractal characteristics.

Figure 6

Fractal dimensions obtained using a spherical model. (A,C) Characteristics of total fractal dimensions (DS); (B,D) fractal dimensions of small pore systems (DS1) and large pore systems (DS2), respectively.

Fractal Dimensions Derived from NMR Analysis

The log–log plots of V and T2 are widely used in characterizing the fractal behavior of pore systems.[20,45] However, almost all the researchers indicate that the slope of the linear fitting line in the log–log plot of V and T2 is not a constant for sandstone samples.[12,23] The slope is steeper when approaching the shorter T2 times and becomes gentler at longer T2 times (Figure A–D). Previously, it has been a common practice to use the inflection point to segment the curves of lg V and lg T2 and to define multifractal models.[20,45] However, the criteria for selecting inflection points are not consistent among various published studies, including the T2cutoff,[16]T2a (the lowest connection point between the first and the second peak of NMR T2 spectra) and T2b (nine times of T2a), and T2 values calibrated by mercury intrusion porosity (MIP).[61]

Figure 7

NMR fractal dimensions calculated from the cross-plots of lg N(r) and lg T2, showing the fractal behavior of the entire pores. (A) Sample L752-1; (B) sample F119-1; (C) sample C276-2; and (D) sample F119-2.

In this study, the fractal model introduced by Lai et al. (2018)[47] was adopted to describe the fractal behavior of the entire pores. Figure shows a linear relationship between the lg N(r) and lg T2i, and the total fractal dimensions can be derived by the slope of the best fitting line. In addition, the fractal behavior of the pores with a movable fluid can also be determined by the inflection points of T2cutoff. The fractal dimensions of the entire pores (DNMR) and pores with a movable fluid (Dmov) are summarized in Table , and good determination coefficients are identified (Figure A–D; Table ). DNMR and Dmax are in the range of 2.14–2.60 and 2.67–2.99, respectively (Table ), indicating that the entire pores and large pores hosting a movable fluid are typical of fractal behavior.

Table 2

NMR T2 Spectral Parameters and Fractal Dimensions Derived from NMR Data

					T2 > T2cutoff
sample no	BVI (%)	FFI (%)	Smov (%)	T2cutoff (ms)	Dmov	R2	DNMR	RNMR2
C276-1	7.30	3.35	31.47	34.36	2.88	0.99	2.40	0.98
C276-2	12.28	4.14	25.21	43.45	2.89	0.99	2.20	0.97
G351-1	2.97	0.73	19.76	2.83	2.96	1.00	2.30	0.96
G351-2	9.31	0.98	9.52	8.96	2.97	1.00	2.32	0.98
G890-1	11.45	7.41	39.29	84.02	2.71	0.99	2.15	0.98
G890-2	2.93	0.44	13.06	5.09	2.97	1.00	2.54	0.98
F151-1-1	8.12	0.57	6.56	12.82	2.93	0.99	2.31	0.98
F153-1	7.94	5.09	39.06	91.99	2.82	0.99	2.40	0.98
C141-1	8.17	3.76	31.52	4.72	2.90	0.99	2.22	0.96
C141-2	6.07	5.48	47.45	41.49	2.84	0.99	2.41	0.97
C406-1	9.44	5.42	36.47	7.97	2.90	0.99	2.14	0.93
F143-1	6.90	3.74	35.15	46.93	2.76	0.99	2.38	0.99
L218-3	9.07	5.61	38.22	76.98	2.73	0.99	2.40	0.99
L218-1	9.59	7.52	43.99	85.61	2.67	0.99	2.16	0.99
L218-2	9.27	6.60	41.59	64.88	2.73	0.99	2.30	0.99
F119-1	7.36	2.76	27.27	150.41	2.84	0.99	2.37	0.98
F119-2	5.11	0.13	2.48	3.95	2.99	1.00	2.51	0.98
G351-1	8.53	1.67	16.37	63.97	2.88	0.99	2.36	0.98
L752-2	6.70	3.09	31.56	5.53	2.92	0.99	2.60	0.99
L752-3	7.33	0.26	3.43	9.88	2.99	1.00	2.31	0.96

Pore Structure Identification Based on SOM and Fractal Dimensions

Preparation of Data

Various pore structures and petrophysical parameters were selected to characterize the microscopic pore structures. The parameters include porosity (Φ), permeability (k), reservoir quality index (RQI), maximum pore throat radius (rmax), median pore throat radius (r50), rapex, displacement pressure (Pd), mercury withdrawal efficiency (WE), sorting coefficient (σ), T2 geometric mean value (T2gm), median T2 relaxation time (T2mid), T2 time corresponding to the highest peak of T2 spectra (T2peak), free fluid index (FFI), and irreducible water saturation (Swi). In addition, fractal dimensions were also selected to identify the pore structures. Due to insufficient NMR data, only the petrophysical parameters and parameters derived from MICP measurements were used as the input of the SOM model, and the NMR T2 spectral parameters were used as auxiliary validation. In order to eliminate the dimensional differences, the selected parameters were standardized prior to feeding into the SOM classifier.

Pore Structure Classification

An unsupervised learning strategy was used in pore structure classification, and a 10 × 10 symmetric network was defined. The initial learning rate was set to 0.5 and the learning process can repeat until the learning rate reduced to 0.0001. The maximum number of iterations was set to 120,000. By assuming three reservoir types, the 70 groups of sandstone samples were clustered into three categories (I, II, and III) by the competitive learning process (Figures and 9). Figure shows the topological self-organized map, showing the clustering result of all sandstone samples. Each neuron represents a specific response of pore structure, and one hundred pore structure features were automatically clustered in the topological map. The rose diagram in each neuron displays a combination of the pore structure and petrophysical parameters (Figure ). The data points located in the same grid reveal similar pore structure characteristics, whereas the differences among various types of pore structures are significant (Figure ).

Figure 8

Topological mapping (left) and the related 3D Sammon mapping (right).

Figure 9

Cluster dendrogram based on the SOM neural network algorithm [2D dimension (left) and 3D dimension (right)], showing the process of pore structure classification.

Petrophysical Features of Various Pore Structures

The clustering results show that the Type I pore structure exhibits a good pore throat connectivity and the strongest microheterogeneity (Figure and Table ). The sandstones dominated by the Type I pore structure are abundant in residual intergranular pores and dissolution pores (Figure A,B). Long T2 components are well-developed in the Type I pore structure, and the r50 and rapex are mainly in the range of 0.05–2.36 μm and 0.40–2.50 μm, respectively (Table ). The Type II pore structure shows a relatively good pore throat connectivity and a strong microheterogeneity. The displacement pressure is relatively high in comparison with those in the type I pore structure (Figure and Table ). The pore systems of the Type II pore structure are dominated by dissolution pores and intergranular pores (Figure C,D). Moreover, the micropores (<0.1 μm) and mesopores (0.1 < r < 1.0 μm) account for a great proportion in the Type II pore structure (Table ). The Type III pore structure is characterized by the highest displacement pressure and the lowest FFI (Figure and Table ), which indicates a poor pore throat connectivity and fluid mobility. The sandstones dominated by the Type III pore structure are characterized by extensive carbonate cementation, with no visible pores under a petrographic microscope (Figure E,F). In addition, the Type II and Type III pore structures with a high proportion of small pores tend to have larger fractal dimensions compared with the Type I pore structures (Figure and Table ), indicating a more complex and heterogeneous pore structure.

Figure 10

Rider chart showing the characteristics of three types of pore structures.

Table 3

Capillary Parameters and Petrophysical Properties for Three Types of Pore Structures

		PS-I		PS-II		PS-III
pore structure type		range	average	range	average	range	average
porosity (%)		10.85–24.06	16.26	9.33–14.74	11.63	2.72–13.80	8.47
k (mD)		0.76–54.30	9.81	0.14–1.26	0.53	0.01–0.38	0.10
RQI (μm)		0.24–1.77	0.67	0.11–0.30	0.20	0.04–0.22	0.10
rmax (μm)		1.0–8.23	3.39	0.25–3.56	1.31	0.03–0.63	0.20
r50 (μm)		0.05–2.36	0.83	0.02–0.67	0.27	0.01–0.15	0.04
rapex (μm)		0.40–2.50	1.42	0.1–1.17	0.54	0.01–0.25	0.08
σ MSE		0.25–1.73	0.91	0.06–0.65	0.28	0.01–0.23	0.06
Pd (MPa)		0.1–0.6	0.23	0.21–10.23	0.73	0.7–20	4.63
WE (%)		22.54–58.91	39.05	18.84–50.25	30.96	14.83–46.79	31.17
T2gm (ms)		6.56–48.74	26.85	7.57–28.62	15.79	1.00–2.99	2.03
T2peak (ms)		3.65–155.52	93.29	6.37–155.22	57.67	1.12–3.65	2.05
T2mid (ms)		4.83–71.80	41.37	7.49–48.32	21.38	0.99–3.07	1.89
Swi (%)		52.55–63.53	59.13	64.85–83.63	72.91	68.44–97.52	87.66
FFI (%)		5.09–7.52	6.16	1.67–4.14	3.13	0.13–3.09	0.89
DT		2.15–2.62	2.36	2.35–2.83	2.56	2.47–2.98	2.77
pore volume fraction (%)	macropores	2.62–64.43	39.58	0–20.55	5.64	0–0	0
	mesopores	12.81–60.69	31.30	9.65–80.11	49.79	0–47.33	9.51
	micropores	9.06–64.01	29.12	11.51–90.35	44.57	52.67–100	90.49

Figure 11

Microscopic characteristics for various types of pore structures. Type I: (A,B) Sandstones with abundant residual intergranular pores and dissolution pores, Well C371, 2691.95 m, and Well G890, 2597.7 m; Type II: (C,D) Sandstones with intergranular pores and dissolution pores, Well F119, 3292.55 m, and Well F153, 2818.4 m; Type III: (E,F) Extensive carbonate cementation, Well G351, 2464.79 m, and Well G890, 2621.1 m.

Discussion

Comparison of Fractal Dimensions from MICP and NMR Data

As can be seen from Section , the MICP tubular model is effective for fractal characterization. Nevertheless, for large pore systems, the calculated fractal dimension DT2 is not in conformance to the Euclidean dimension.[14,62] The MICP spherical model is only effective when rmin ≪ r.[45,61] The model works well for the samples with large pore throats, but it is not suitable for fractal analysis of small pore systems (DS1 < 2.0). Moreover, the spherical model is more suitable for wetting phase intrusion measurements. It is effective for fractal analysis based on NMR data because water is a wetting phase commonly used in NMR experiments.

The comparative results also indicate that fractal dimensions determined from NMR and MICP data are different for given samples. The discrepancy may be caused by differences in theoretical principles and computation models of the two different methods.[22,62] MICP fractal models are based on the pore throat sizes but cannot account for large pore throats. Due to the limitation of the maximum mercury injection pressure, the smallest part of the pore throat cannot be detected by the MICP measurements.[63] In contrast, NMR fractal dimensions can reflect the fractal behavior of entire pores (pore body sizes).[24] In addition, our investigations suggest that the sandstone samples with a high proportion of small pore throats often show larger fractal dimensions than those with abundant large pore throats (Figure ). Therefore, the content of small pore throats may significantly control the pore throat heterogeneity.

Figure 12

Relationships between fractal dimensions (DT) and pore volume fraction of small pore throats (r < 0.1 μm).

Relationships between Pore Structure and Petrophysical Properties

Analysis of Fractal Dimension and Reservoir Parameters

Data from conventional core analysis of 70 samples reveal a positive correlation between porosity and permeability (R2 = 0.71; Figure A). However, sandstone samples with similar porosities show wide variations in permeability (Figure A), suggesting that permeability of the low-permeability sandstones is not only determined by porosity but also influenced by their microscopic behaviors or other factors.

Figure 13

Relationships between fractal dimensions and petrophysical parameters. (A) Porosity vs permeability; (B) DT vs helium porosity; (C) DT vs air permeability; and (D) DT vs RQI.

The relationships between petrophysical parameters, pore structure parameters, and fractal dimensions (e.g., porosity, permeability, RQI, rmax, r50, sorting coefficient, T2gm, T2peak, and Swi) were analyzed to illustrate the effect of fractal dimension on pore structure and reservoir quality evaluation. DT exhibits a weak negative exponential correlation with helium porosity (R2 = 0.45; Figure B), suggesting that higher DT is often associated with lower porosity. In addition, regression analysis suggests that the DT shows a relatively strong negative correlation with permeability (R2 = 0.65; Figure C) and RQI (R2 = 0.64; Figure D), implying an overall poor reservoir quality for reservoir sandstones with complex and heterogeneous pore structures.

As shown in Figure A,B, fractal dimension DT is negatively correlated with rmax and r50 (R2 > 0.70), suggesting that large pore throats and a concentrated pore throat size distribution would be helpful to reduce the complexity and heterogeneity of the pore structure.[14] This is also evident on Figure C,D, where both the T2gm and T2peak exhibit negative correlations with Dmov. Good correlations among sorting coefficients and fractal dimensions DT are also observed (R2 = 0.74; Figure E), indicating the effectiveness of fractal dimensions in describing pore throat size distribution. Dmov shows positive correlations with Swi with R2 being 0.67 (Figure F). The larger the fractal dimension, the more complex the pore structure would be, leading to a poor pore throat connectivity and high irreducible water saturation.

Figure 14

Cross-plots of fractal dimensions vs pore structure parameters. (A) DT vs rmax; (B) DT vs r50; (C) Dmov vs T2gm; (D) Dmov vs T2peak; (E) DT vs sorting coefficient; and (F) Dmov vs Swi.

The correlation analysis between the fractal dimensions (DT and Dmov) and the reservoir parameters confirms a link between complex pore structure and petrophysical properties (Figures and 14). Generally speaking, fractal dimensions can be used as important indicators for pore structure and reservoir heterogeneity evaluation. The complex process of pore structure characterization can therefore be simplified using fractal dimensions.[27,64] Furthermore, our studies have found that the fractal dimension DS calculated using the MICP spherical model has no correlation with reservoir petrophysical properties and cannot reflect the complex and heterogeneous characteristics of the Es4 low-permeability sandstone reservoir.

Effect of the Pore Structure on Permeability

As shown in Figure A, there is a strong correlation between permeability and pore volume fraction with a pore throat radius larger than 0.1 and 1.0 μm. A negative correlation between permeability and pore volume fraction of micropores (r < 0.1 μm) is also observed in Figure B, indicating that micropores are generally disconnected or poorly connected in the pore network, consistent with the finding of Lai and Wang (2015).[14] The analytical results indicate that the large pore systems have significant contribution to permeability of the Es4 sandstone reservoirs (Figure A). Therefore, micropores have little contribution to reservoir permeability. Consequently, reservoir microscopic characteristics, especially the pore volume of large pores, can exert more significant effect on the permeability of the low-permeability sandstone reservoir.

Figure 15

Relationships between permeability and pore volume fraction with different pore throat sizes. (A) permeability vs pore volume fraction for macropores and mesopores and (B) permeability vs pore volume for micropores.

As shown in Figures and 17, the contribution of pore throats with different sizes to permeability has also been investigated. The result indicates that the large pore throats have great contribution to permeability, even though those pore throats usually account for a small proportion (Figure ). The permeability contribution is significantly different in samples with different pore throat size distribution and permeability levels. The absolute permeability contribution increases with increasing pore throat radius (Figure ), which is the main reason for strong positive correlation between large pore systems and permeability (Figure A). As shown in Figure , the sandstone samples with higher permeability (>10 mD) always developed the pore throats with a radius greater than 2 μm. For samples with relatively low permeability (<0.1 mD), the rmax is commonly less than 1 μm.

Figure 16

Pore throat size distribution and the corresponding permeability contribution for two typical samples from the Es4 sandstone reservoirs. (A) Sample F119-1, φ = 12.30%, k = 0.77 mD and (B) sample F153-1, φ = 10.85%, k = 4.40 mD.

Figure 17

Cross-plot of pore throat radius and absolute permeability contribution obtained by MICP measurement, showing that the large pore systems contribute significantly to permeability.

Effect of the Pore Structure on Fluid Mobility

The fractal dimensions of movable fluid pores (Dmov) show positive correlations with porosity for a movable fluid (φm), saturation for movable fluids (Smov), and permeability, with R2 being 0.77, 0.64, and 0.75, respectively (Figure A–C). The larger the fractal dimension, the more complex the geometry of movable fluid pores, resulting in a poor pore throat connectivity and fluid mobility. In addition, permeability as the macroscopic performance of microscopic pore structures also show positive correlations with saturation for movable fluids (R2 = 0.75; Figure D), indicating that with the improvement of petrophysical properties, the fluid mobility becomes better. Strong positive correlations were also observed between r50, permeability, and saturation for a movable fluid (Figure E,F), confirming that the strong correlations between petrophysical properties and fluid mobility are greatly controlled by the large pore throats. The pore throat size distribution and pore connectivity have great influences on the permeability and fluid mobility in the pore network.

Figure 18

Relationship between pore structures and fluid mobility. (A) Dmov vs porosity for movable fluids; (B) Dmov vs saturation for movable fluids; (C) Dmov vs permeability; (D) permeability vs saturation for movable fluids; (E) r50 vs permeability; and (F) r50 vs saturation for movable fluids.

Permeability Estimation

Permeability Estimation Based on the Modified Winland Model

Due to the intrinsic strong correlation between permeability and pore throat size distribution, the permeability estimation models based on the pore throat radius usually show a better prediction effect.[65,66] The Winland model described the relationship between permeability, pore throat radius, and porosity.[30] The model indicates that permeability of high-permeability reservoirs is mainly controlled by the r35 pore throat value. Several authors further extended the concept and found that the optimal pore throat radius for permeability estimation varies for different types of reservoirs.[37,67,68] Through the multiple regression analysis of permeability, porosity, and pore throat radius (r5–r50), we demonstrated that the r10 is the best value for permeability estimation of the Es4 low-permeability sandstone reservoirs (Figure ). The r10 parameter was selected for permeability estimation, which was dominated by the pore throat size mainly concentrated between 0.1 and 5.0 μm, corresponding to the large pore systems. The empirical equation (eq ) is shown as followswhere k is the measured permeability (mD); φ is the helium porosity (%); and r10 is the pore throat radius corresponding to 10% of the cumulative mercury saturation (μm).

Figure 19

Determination coefficients corresponding to different pore throat radii (r5–r50). Note that r10 is strongly correlated with measured air permeability.

As shown in Figure , the predicted permeability (k-predicted) is in good correlation with the measured permeability (k-measured) (R2 = 0.87), indicating the effectiveness of the permeability estimation model. Moreover, the pore throat radius can also be predicted by permeability and porosity based on the established empirical equation.

Figure 20

Cross-plot of k-measured vs k-predicted (R2 = 0.87). The black solid line is the 1:1 line, while the red-dashed line marks the best fitting line between the k-measured and k-predicted.

Permeability Estimation Based on the Porosity Model Derived from Movable Fluids

As shown in Section , the disconnected or poorly connected pores in sandstone have little contribution to permeability. Hence, the porosity for movable fluids (φm) is recommended for permeability estimation rather than the total porosity,[16,23] and φm can be calculated using eq . Regression analysis shows that the φm exhibits a strong exponential correlation with permeability (R2 = 0.90; Figure ). The permeability estimation model based on φm is shown in eq .where φN is the NMR porosity and FFI and BVI are the free fluid index and bulk volume of the immovable fluid, respectively.

Figure 21

Relationship between porosity for movable fluids and measured permeability, showing an excellent fitting (R2 = 0.90).

The predicted permeability of the sandstone samples is calculated using eq . As shown in Figure B, the predicted values are in good agreement with measured values with little difference (Figure B). We also shed light on the permeability estimation using the classic (Coates and SDR) models (Figure A). The porosity model derived from movable fluids provides a better estimation effect compared with the classic models (Table ; Figure ), implying the effectiveness and applicability of the new model for permeability estimation.

Figure 22

Cross-plots of k-measured vs k-predicted. (A) Coates and SDR models and (B) porosity model derived from movable fluids.

Table 4

Fitting Equations and Determination Coefficients for Three Different Permeability Estimation Models Using NMR Data

permeability estimation model	fitting equation	determination coefficient
Coates model	k = (φN/0.0675)4 × (FFI/BVI)2	0.75
SDR model	<keep-together>k = 14.901 × φN4T2g2</keep-together>	0.55
porosity model derived from movable fluids	<keep-together>k = 0.0311 × e0.843φm</keep-together>	0.90

Conclusions

The microscopic pore structure of the Eocene low-permeability sandstones was investigated based on NMR and MICP data in combination with fractal theory and self-organizing map (SOM) neural network algorithm. Two permeability estimation models were established based on the pore structure information obtained from MICP and NMR measurements. Some key findings are as follows:

SOM, an unsupervised neural network clustering algorithm, was adopted to investigate the pore structures. Three types of pore structures had been identified using an established SOM clustering model. Type I and II pore structures are dominated by large pore throats with good pore throat connectivity and contribute significantly to the permeability. Type III pore structures are characterized by abundant small pore throats and a poor pore throat connectivity and are unfavorable for the fluid flow.

The MICP tubular model is effective for fractal characterization, and the fractal dimensions calculated using the tubular model exhibit strong correlations with petrophysical parameters and pore structure parameters. The fractal dimensions derived from the MICP spherical model are not recommended for pore structure and reservoir quality evaluation. The spherical model is more suitable for fractal analysis based on NMR data because water is a wetting phase commonly used in NMR experiments.

The fractal dimensions of the reservoirs investigated, DT and DNMR, are in the range of 2.15–2.98 and 2.14–2.60, respectively. The discrepancy may be caused by the difference in theoretical principles and calculation models of the two different types of measurements. Samples with high proportions of small pore throats often show larger fractal dimensions, indicating that the pore volume of the small pore throats significantly controls the pore throat connectivity and heterogeneity of the sandstone reservoirs.

Regression analysis indicates that r10 is the best parameter for permeability estimation, which has a concentrated distribution of 0.1–5.0 μm, indicating that the large pore systems have a significant control on the fluid flow in the Eocene low-permeability sandstones. Another permeability estimation model based on φm has also been established, which provides a better prediction effect than the classic models. These findings are vital for investigating the fluid flow mechanism and predicting the reservoir quality of the Eocene sandstone reservoirs.