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Analysis of Geological Pore Structure Based on Fractal Theory

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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: 31 July 2026 | Viewed by 5979

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Special Issue Editors

Dr. Wei Yang

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Guest Editor

Unconventional Petroleum Research Institute, China University of Petroleum, Beijing 102249, China
Interests: pore structure characterization; fractals; organic–inorganic interactions; reservoir capacity evaluation; unconventional oil/gas

Prof. Dr. Mianmo Meng

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Guest Editor

College of Marine Science and Technology, China University of Geosciences (Wuhan), Wuhan 430074, China
Interests: pore structure characterization; fluid occurrence; water–rock interaction; nuclear magnetic resonance; unconventional oil/gas
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Special Issue Information

Dear Colleagues,

Oil and gas account for more than half of global energy consumption, making them essential resources for modern life. With an increasing energy demand and the declining production of conventional oil and gas, utilizing unconventional hydrocarbon resources has become critical to meet our energy needs. However, the efficient development of these resources is still hindered by certain theoretical or technical issues, particularly in pore structure characterization.

Multiple methods are employed to characterize the pore structure at the micro to nano scale, including scanning electron microscopy (SEM), nitrogen adsorption (NA), mercury intrusion porosimetry (MIP), and small-angle neutron scattering (SANS). However, certain factors complicate the high‑resolution characterization of pore spaces. For instance, shale oil within nanopores is challenging to remove due to low pore connectivity and its strong adsorption onto nanopore surfaces, often resulting in the significant underestimation of the pore volume. Of the multiple techniques that are used, each operates based on different theories and models, making accurate comparisons difficult.

Given these challenges, further research is essential for the efficient development of unconventional oil and gas resources. This Special Issue will focus on the analysis of geological pore structures based on fractal theory in unconventional hydrocarbon reservoirs. Potential topics include, but are not limited to, the following:

The efficiency of oil extraction;
Pore–crack observation in two dimensions;
Pore–crack reconstruction in three dimensions;
Pore connectivity and the whole pore network;
Scanning electron microscopy (SEM);
Nitrogen adsorption (NA);
Mercury intrusion porosimetry (MIP);
Nuclear magnetic resonance (NMR);
Small-angle neutron scattering (SANS);
Spontaneous imbibition.

We look forward to receiving your valuable contributions.

Dr. Wei Yang
Prof. Dr. Mianmo Meng
Guest Editors

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Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

fractal theory
fractal analysis
multifractal theory
multifractal analysis
pore structure
pore–crack observation
pore connectivity

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Published Papers (7 papers)

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Research

22 pages, 4914 KB

Open AccessArticle

Characterization Method for the Conductive Response of Shale Based on Multi-Dimensional Fractal Theory

by Weibiao Xie, Qiuli Yin, Xueping Dai, Jianbin Zhao, Jingbo Zeng and Pan Zhang

Fractal Fract. 2026, 10(5), 301; https://doi.org/10.3390/fractalfract10050301 - 29 Apr 2026

Viewed by 112

Abstract

Resistivity is a key parameter in shale reservoir characterization. Diverse micro-pore types and complex conduction mechanisms in shale result in poor accuracy when applying existed conductivity models. Establishing a high-precision conductivity response model requires comprehensive consideration of the pore structure and clay-bound water conduction. The primary novelty of this work lies in replacing macroscopic empirical fitting parameters with a mechanistic, multi-dimensional fractal framework. We develop a novel conductivity response characterization model that explicitly couples multi-dimensional fractal pore structure theory with clay-bound water conduction. Experimental data verification demonstrates the new model’s superior characterization accuracy. Results indicate three distinct zones in the shale conductivity-pore water conductivity relationship: a nonlinear zone, a transition zone, and a linear zone. A higher cation exchange rate on clay surfaces leads to an increase in the nonlinear characteristics of the conductivity for both the shale and the pore water in low-salinity regions. An increase in the values of the conduction path fractal dimension, pore morphology fractal dimension, and pore fractal dimension all contribute to reduced shale conductivity. While sharing clay-induced conductivity terms with conventional dual-water and shale volume models, the new model offers advantages in operational simplicity and parameter accessibility. This research provides a physically rigorous and highly accessible approach for conductivity-based reservoir parameter calculation, offering new technical perspectives for complex shale oil/gas evaluation. Full article

(This article belongs to the Special Issue Analysis of Geological Pore Structure Based on Fractal Theory)

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25 pages, 25859 KB

Open AccessArticle

Insights into Pore–Throat Fractal Characteristics and Shale-Oil Mobilization by HTHP Imbibition in Lacustrine Calcareous Shale

by Xianda Sun, Qiansong Guo, Yuchen Wang, Chengwu Xu and Ziheng Zhang

Fractal Fract. 2026, 10(3), 156; https://doi.org/10.3390/fractalfract10030156 - 27 Feb 2026

Cited by 1 | Viewed by 354

Abstract

Upper E_s4 lacustrine calcareous shale in the Dongying Depression is characterized by strong pore–throat heterogeneity that limits shale-oil producibility. This study quantifies multiscale pore–throat complexity using high-pressure mercury intrusion-based fractal analysis (segmented fractal dimensions D₁–D₃ and a weighted comprehensive fractal dimension, D_c) and evaluates its control on oil occurrence and mobilization using low-field 2D NMR (T₁–T₂) and confocal microscopy before and after high-temperature, high-pressure spontaneous imbibition. Reservoirs show clear scale segmentation, with micropore fractality governing quality differentiation. Imbibition promotes desorption and redistribution from adsorbed to free oil, but effective mobilization is primarily controlled by pore–fracture connectivity: samples with well-connected networks can mobilize both adsorbed and free oil efficiently, whereas poorly connected systems exhibit desorption without effective production, implying that thermal stimulation alone is insufficient without fracture-assisted flow pathways. Movable-oil saturation decreases systematically with increasing D_c, indicating that higher roughness and tortuosity intensify capillary retention and Jamin trapping. D_c provides an actionable criterion for sweet-spot ranking and for designing stimulation–imbibition coupling and water-based EOR strategies in lacustrine calcareous shale-oil reservoirs. Full article

(This article belongs to the Special Issue Analysis of Geological Pore Structure Based on Fractal Theory)

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24 pages, 3549 KB

Open AccessArticle

Fractional Order Derivative Models of Porosity on Physical Fractal Spaces

by Li Yang, Guangui Zou, Xiaodong Wang, Siyuan Xie and Yajun Yin

Fractal Fract. 2026, 10(2), 118; https://doi.org/10.3390/fractalfract10020118 - 10 Feb 2026

Viewed by 403

Abstract

Rock pore–fracture systems exhibit inherent fractal characteristics, which exert a significant influence on fluid transport. In this study, coal rock is selected as the representative medium. Based on fractional calculus in physical fractal space, and by integrating operator algebra with the force–electric analogy method, a fractional order control equation is derived. To validate the proposed model, porosity measurements of coal and limestone were performed using the two-compartment Boyle’s law method and compared with conventional porosity calculation approaches. The results demonstrate that the fractional order model achieves a coefficient of determination (R²) of up to 0.99 for porosity and 0.98 for pressure, representing an improvement of approximately 0.07 over the exponential model. Moreover, the root mean square error (RMSE) of porosity is as low as 0.0008, while the RMSE of pressure is 0.0715, both significantly lower than those obtained using the exponential model. These results indicate that the fractional order model more effectively captures the non-Darcy flow behavior and the temporal evolution of porosity, providing substantially improved fitting accuracy. Further analysis reveals that the porosity–time relationship is jointly governed by fluid compressibility and pore compressibility under effective stress conditions. Comparative results across different lithologies reveal that the pore compressibility coefficient increases with porosity; for the same rock type, a higher coefficient implies a more complex pore structure and a longer equilibration time. Overall, the proposed fractional order framework provides a more accurate description of the fractal pore structures in rocks, establishing a clear link between microscale fractal geometry and macroscale fractional order response. Full article

(This article belongs to the Special Issue Analysis of Geological Pore Structure Based on Fractal Theory)

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28 pages, 12747 KB

Open AccessArticle

Full-Scale Pore Structure and Multi-Scale Fractal Characteristics of the Wufeng–Longmaxi Formations Shales in Sichuan Basin, China

by Taotao Cao, Wenqing Yuan, Jiacheng Zeng, Anyang Pan, Wenquan Xie, Jing Liao, Gaofei Ning and Ye Chen

Fractal Fract. 2026, 10(2), 75; https://doi.org/10.3390/fractalfract10020075 - 23 Jan 2026

Viewed by 286

Abstract

Unique fractal characteristics are significantly controlled by shale lithofacies, mineralogical characteristics, and OM features, which in turn determine reservoir properties and gas-bearing capacity. However, a comprehensive understanding of fractal features has remained insufficient. This study presents a systematic investigation into the full-scale pore size distribution for the Wufeng–Longmaxi shales in Sichuan Basin which employed low-pressure CO₂ adsorption (CO₂GA), N₂ adsorption (N₂GA), and mercury injection capillary pressure (MICP), as well as field emission scanning electron microscope (FE-SEM) techniques. The fractal dimensions of pores across different pressure ranges were revealed by different fractal models. The results demonstrate that the shale pores are dominated by micro- to mesopores and partial extremely larger pores, contributed primarily by organic matter (OM) pores and microcracks, respectively. Fractal dimensions follow a consistent increasing order: D_C < D_N1 < D_N2 < D_M or D_C < D_N1 < D_M < D_N2, suggesting that larger pores with diameters lager than 5 nm are more heterogeneous and complex compared to the pores less than 5 nm (smaller pores). This is because smaller pores are predominantly composed of OM pores, while larger pores comprise a mixture of OM pores, mineral-related pores, and microcracks. Different fractal dimensions, in turn, are influenced by distinct factors. The D_C value exhibits a positive correlation with micropore volume. D_N1 and D_N2 values are positively correlated with the content of brittle minerals and TOC, while they show negative correlations with the content of clay minerals. Notably, D_M values do not demonstrate a significant correlation with shale compositions, primarily owing to the development of microcracks. Fractal dimensions, particularly D_N1 and D_N2, are significantly controlled by the lithofacies of shale. The highest D_N1 and D_N2 values occur in the siliceous shale lithofacies, and the mixed shale lithofacies exhibit moderate D_N1 and D_N2 values, whereas the lowest D_N1 and D_N2 values primarily occur in clay-rich shale lithofacies. Different fractal dimensions show various correlations with shale gas content. The Langmuir volume as well as total gas content exhibit significant correlations with D_N1 and D_N2 values, while they exhibit no obvious correlations with D_C and D_M values. This implies that pores with diameters of 1.8–55 nm serve as primary storage sites for both adsorbed and free gas. The findings can significantly improve the cognition of adsorbed gas and free gas behavior in shale reservoirs. Full article

(This article belongs to the Special Issue Analysis of Geological Pore Structure Based on Fractal Theory)

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28 pages, 9766 KB

Open AccessArticle

Fractal and Fluid Mobility Analysis of Pore-Throat Systems in Sandstone Reservoirs Based on HPMI and NMR: A Case Study from the Nahr Umr Formation, Iraq

by Tang Li, Meiyan Fu, Runze Wang, Ya Deng, Jiacheng Xu and Rui Guo

Fractal Fract. 2026, 10(1), 15; https://doi.org/10.3390/fractalfract10010015 - 25 Dec 2025

Cited by 1 | Viewed by 1892

Abstract

The pore architecture of the Nahr Umr Formation sandstone reservoirs is highly complex and heterogeneous, severely limiting efficient oilfield development. Conventional methods often fail to adequately characterize such intricate pore systems, necessitating the application of fractal theory. Focusing on sandstone samples from the Nahr Umr-B Member, this study integrates thin section identification, XRD, HPMI, and NMR to characterize the fractal features of the reservoir pore structure and evaluate fluid mobility. The results indicate that from Type I to Type III reservoirs, displacement pressure and median pressure gradually increase, whereas the average and median pore-throat radius gradually decrease, and the pore-throat sorting coefficient decreases. For instance, Type I reservoirs exhibit an average displacement pressure of 0.15 MPa, a median pressure of 0.81 MPa, an average pore-throat radius of 1.96 μm, and a median pore-throat radius of 2.85 μm; in contrast, Type III reservoirs show averages of 14.43 MPa, 45.32 MPa, 0.02 μm, and 0.03 μm, respectively. These trends reflect a gradual deterioration in pore connectivity, increased resistance to fluid flow, and a reduction in the development of larger pore throats. From Type I to Type III reservoirs, both the total fractal dimension (D_H) and the movable fluid pore fractal dimension (D_N2) show a gradual increasing trend. This indicates that the pore structure becomes increasingly complex and heterogeneous, the complexity of the movable fluid pore space increases, and fluid mobility progressively weakens. Furthermore, higher quartz content and lower cement and clay mineral contents correspond to smaller reservoir pore fractal dimensions and stronger fluid mobility. For example, Sample No. 3 (Type I) has a quartz content of 91.97%, a cement content of 1.64%, and a clay mineral content of 6.4%, with a D_H of 2.4385 and D_N2 of 2.9323. Conversely, Sample No. 4 (Type III) has a quartz content of 49.72%, a cement content of 11.21%, and a clay mineral content of 39.07%, with a D_H of 3.9099 and D_N2 of 2.9762. Compared to D_H, D_N2 reduces the prediction error for dynamic quality by over 70% on average, offering a more reliable prediction of fluid mobility and providing a more precise scale for evaluating reservoir development potential. Full article

(This article belongs to the Special Issue Analysis of Geological Pore Structure Based on Fractal Theory)

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36 pages, 17646 KB

Open AccessArticle

Multifractal Characteristics of Heterogeneous Pore-Throat Structure and Insight into Differential Fluid Movability of Saline-Lacustrine Mixed Shale-Oil Reservoirs

by Wei Yang, Ming Xie, Haodong Hou, Zhenxue Jiang, Yan Song, Shujing Bao, Yingyan Li, Yang Gao, Shouchang Peng, Ke Miao and Weihao Sun

Fractal Fract. 2025, 9(9), 604; https://doi.org/10.3390/fractalfract9090604 - 18 Sep 2025

Cited by 3 | Viewed by 1163

Abstract

The root causes forcing the differential pore-throat performances and crude oil recoverability in heterogeneous shale lithofacies of saline-lacustrine fine-grained mixed sedimentary sequences are still debated. Especially application cases of fractal theory in characterizing pore-throat heterogeneity are still lacking and the significance of differential multifractal distribution patterns on reservoir assessment remains controversial. This present study focuses on the shale-oil reservoirs in saline-lacustrine fine-grained mixed depositional sequences of the Middle Permian Lucaogou Formation (southern Junggar Basin, NW China), and presents a set of new results from petrographical investigation, field-emission scanning electron microscopy (FE-SEM) imaging, fluid injection experiments (low-pressure N₂ adsorption and high-pressure mercury intrusion porosimetry (HMIP)), nuclear magnetic resonance (NMR) spectroscopy and T₁-T₂ mapping, directional spontaneous imbibition, as well as contact angle measurements. Our results demonstrated that the investigated lithofacies are mainly divided into a total of five lithofacies categories: felsic siltstones, sandy dolomitic sandstones, dolarenites, micritic dolomites, and dolomitic mudstones, respectively. More importantly, the felsic siltstone and sandy dolomitic siltstones can be identified as the most advantageous lithofacies categories exhibiting the strongest movable oil-bearing capacity owing to an acceptable complexity and heterogeneity of mesopore-throat structures, as evidenced by the corresponding moderate fractal dimension of mesopores (D₂) from HMIP and apparently lower fractal dimension of movable fluids’ pores (D₂) from NMR results. Particularly noteworthy is the relatively poor shale-oil movability recognized in the dolarenites, micritic dolomites, and dolomitic mudstones due to heterogeneous and unfavorable pore-throat systems, even though an acceptable micro-connectivity and a more oleophilic interfacial wettability prevails in crucial dolomitic components. Finally, a comprehensive and conceptual model is established for an effective and characteristic parameter system for assessing differential reservoir petrophysical properties, interfacial wettability, and shale-oil movability concerning heterogeneous lithofacies categories. Our achievements can serve as an analog for investigating saline-lacustrine mixed shale-oil reservoirs to gain a more comprehensive understanding of differential recoverability of dessert reservoir intervals, and to guide the assessment of “sweet spots” distribution and optimization of engineering technique schemes for commercial exploitation. Full article

(This article belongs to the Special Issue Analysis of Geological Pore Structure Based on Fractal Theory)

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26 pages, 7464 KB

Open AccessArticle

Pore Structure and Multifractal Characteristics of the Upper Lianggaoshan Formation in the Northeastern Sichuan Basin, China

by Jingjing Guo, Guotao Luo, Haitao Wang and Liehui Zhang

Fractal Fract. 2025, 9(7), 430; https://doi.org/10.3390/fractalfract9070430 - 30 Jun 2025

Cited by 2 | Viewed by 1005

Abstract

The Upper Lianggaoshan (LGS) Formation in the northeastern Sichuan Basin, composed of shale with interbedded siltstone, is a promising target layer for shale oil. Accurate evaluation of pore structures is essential for effective exploration of shale oil. This study investigated pore structures of siltstone and shale samples from the Upper LGS Formation using low-pressure CO₂ adsorption (LTCA), low-temperature N₂ adsorption (LTNA), high-pressure mercury intrusion (HPMI), and nuclear magnetic resonance (NMR) methods. The single-exponent and multifractal dimensions of samples were determined, and the relationships between fractal dimensions and pore structures were explored. Results show that the pore size distribution (PSD) of siltstone and shale samples exhibits multi-peak characteristics, with mesopores (2–50 nm) being dominant in the total pore volumes. The multi-scaled pores in shale and siltstone samples exhibit fractal characteristics. The average values of single-fractal dimensions (D₁, D₂) obtained by LTNA data are 2.39 and 2.62 for shale samples, and 2.24 and 2.59 for siltstone samples, respectively. Compared to siltstones, the pore structures of shale samples exhibit greater complexity, indicated by larger fractal dimensions. The samples from subsections Liang 2 and Liang 3 exhibit greater heterogeneity compared to subsection Liang 1. The single-fractal dimensions of micropores and mesopores show positive correlations with specific surface area (SSA) and pore volume (PV), while the fractal dimension of macropores shows a negative correlation with average pore diameter and median radius. The average values of single-fractal dimension D₃ obtained from HPMI data are 2.9644 and 2.9471 for shale and siltstone samples, respectively, indicating more complex structures of macropores in shale samples compared to siltstone samples. The average value of

Δ D^{N M R}

and singularity strength range Δα obtained by a multifractal model for core samples from subsection Liang 1 are 1.868 and 2.155, respectively, which are the smallest among all of the three subsections, indicating that the heterogeneity of pore structures of subsection Liang 1 is the weakest. This research provides valuable guidance for shale oil development in the northeastern Sichuan Basin, China. Full article

(This article belongs to the Special Issue Analysis of Geological Pore Structure Based on Fractal Theory)

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