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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 October 2025 | Viewed by 6506

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Dr. Yang Wang

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School of Resources and Geoscience, China University of Mining and Technology, Xuzhou 221116, China
Interests: shale gas; coalbed methane; pore structure; adsorption mechanism; fractal characterization
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Special Issue Information

Dear Colleagues,

Unconventional reservoirs, such as shale gas, tight oil, and coalbed methane, have become pivotal in meeting the global energy demand. However, their inherent geological complexity and heterogeneity present significant challenges for accurate characterization, modeling, and efficient extraction. Multiscale fractal analysis offers a robust framework to quantify and interpret the intricate spatial patterns and scaling behaviors within these reservoirs. By applying fractal geometry across multiple scales, researchers can gain deeper insights into fracture networks, pore structures, and fluid flow dynamics, ultimately enhancing reservoir performance and hydrocarbon recovery.

This Special Issue, "Multiscale Fractal Analysis in Unconventional Reservoirs", seeks to compile innovative research that leverages fractal and multiscale methodologies to address the complexities of unconventional hydrocarbon reservoirs. We invite contributions that explore theoretical advancements, novel computational techniques, and practical applications of multiscale fractal analysis in various aspects of reservoir engineering. Topics of interest include, but are not limited to, the following:

Fractal characterization of fracture networks and pore geometries;
Multiscale modeling of fluid flow and transport phenomena;
Integration of fractal analysis with seismic interpretation and well logging;
Application of fractal-based methods in reservoir simulation and optimization;
Case studies demonstrating the practical implementation of multiscale fractal techniques;
Synergies between fractal analysis and other advanced technologies, such as machine learning and digital rock physics.

By bringing together diverse perspectives and cutting-edge advancements, the focus of this Special Issue is to foster interdisciplinary collaboration and drive forward the understanding and exploitation of unconventional reservoirs. We encourage researchers and industry professionals to share their latest findings, methodologies, and case studies, contributing to the advancement of sustainable and efficient energy extraction practices.

Dr. Yang Wang
Guest Editor

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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.

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Keywords

multiscale fractal analysis
unconventional reservoirs
fractal geometry
fracture network characterization
pore structure analysis
fluid flow modeling
reservoir simulation
machine learning
digital rock physics

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Published Papers (10 papers)

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Research

24 pages, 16565 KB

Open AccessArticle

Dynamic Characteristics of the Pore Heterogeneity of Longmaxi Shale Based on High-Pressure Triaxial Creep Testing

by Yan Dai, Hanyu Zhang, Yanming Zhu, Haoran Chen, Yao Ge, Qian Wang and Yiming Zhao

Fractal Fract. 2025, 9(9), 564; https://doi.org/10.3390/fractalfract9090564 - 28 Aug 2025

Viewed by 479

Abstract

The dynamic changes in shale pore structure due to tectonic uplift are crucial for understanding the processes of shale gas enrichment and accumulation, particularly in complex tectonic regions. To explore the heterogeneous changes in pore structure and their influencing factors during the last tectonic uplift of Longmaxi shale, triaxial creep experiments were performed under varying confining pressure conditions. In addition, FE-SEM, MIP, LN2GA, and LCO2GA experiments were employed to both qualitatively and quantitatively characterize the pore structure of three distinct groups of Longmaxi shale samples. To further investigate pore heterogeneity, the multifractal dimension was applied to examine the evolution of the shale pore structure under the influence of the last tectonic uplift. The results revealed that the primary pore types in Longmaxi shale include organic matter pores, microfractures, intergranular pores, and intragranular pores. The shale’s mechanical properties and mineral content have a significant impact on the heterogeneity of these pores. Notably, the shale pores exhibit distinct multifractal characteristics, highlighting the complex nature of pore heterogeneity. The singular index (α₀) and ten other multifractal dimension parameters provide valuable insights into the heterogeneity characteristics of shale pores from various perspectives. Additionally, dynamic changes in pore heterogeneity are primarily controlled by the mineral composition. Under identical creep pressure variation conditions, significant differences are observed in the pore rebound behavior of Longmaxi shale with different mineral compositions. Under high-pressure conditions, the content of TOC and quartz plays a dominant role in controlling pore heterogeneity, with their influence initially decreasing and then increasing as pressure decreases. The reduction in creep pressure emphasizes the controlling effect of TOC, quartz, and feldspar content on pore connectivity. This study introduces high-pressure triaxial creep experiments to simulate the stress response behavior of pore structures during tectonic uplift, offering a more comprehensive reflection of pore evolution in organic-rich shale under realistic geological conditions. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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15 pages, 3280 KB

Open AccessArticle

Fractal Scaling of Storage Capacity Fluctuations in Well Logs from Southeastern Mexican Reservoirs

by Sergio Matias-Gutierres, Edgar Israel García-Otamendi, Hugo David Sánchez-Chávez, Leonardo David Cruz-Diosdado and Roberto Cifuentes-Villafuerte

Fractal Fract. 2025, 9(8), 548; https://doi.org/10.3390/fractalfract9080548 - 21 Aug 2025

Viewed by 508

Abstract

This study focuses on a hydrocarbon reservoir located in southeastern Mexico. The analysis uses well log data derived from petrophysical evaluations of storage capacity. The structural complexity of the reservoir and observed heterogeneity in Cretaceous units motivate a fractal-based characterization of spatial fluctuations. [...] Read more.

α

(roughness),

β

(growth), and z (dynamic) are obtained through structure function metrics. Data collapse techniques and local Hurst exponent distributions are used to explore long-range memory and spatial heterogeneity across wells. This study aims to classify storage capacity fluctuation records based on Euclidean or fractal geometries. This analysis allows a novel characterization of storage trends in the reservoir. The analysis reveals persistent scaling behavior, indicating long-range correlations in the storage capacity fluctuations. Multiscale patterns and variations in local Hurst exponents highlight the presence of multifractality and regional heterogeneity. Specifically, the spatial distribution of local Hurst exponents obtained in this study enables the inference of statistical properties in synthetic wells, providing key input for the structural and functional characterization of the reservoir’s geological model. This approach aims to identify preferential subsurface flow pathways for hydrocarbons and gas. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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29 pages, 5533 KB

Open AccessArticle

Automated First-Arrival Picking and Source Localization of Microseismic Events Using OVMD-WTD and Fractal Box Dimension Analysis

by Guanqun Zhou, Shiling Luo, Yafei Wang, Yongxin Gao, Xiaowei Hou, Weixin Zhang and Chuan Ren

Fractal Fract. 2025, 9(8), 539; https://doi.org/10.3390/fractalfract9080539 - 16 Aug 2025

Viewed by 542

Abstract

Microseismic monitoring has become a critical technology for hydraulic fracturing in unconventional oil and gas reservoirs, owing to its high temporal and spatial resolution. It plays a pivotal role in tracking fracture propagation and evaluating stimulation effectiveness. However, the automatic picking of first-arrival times and accurate source localization remain challenging under complex noise conditions, which constrain the reliability of fracture parameter inversion and reservoir assessment. To address these limitations, we propose a hybrid approach that combines optimized variational mode decomposition (OVMD), wavelet thresholding denoising (WTD), and an adaptive fractal box-counting dimension algorithm for enhanced first-arrival picking and source localization. Specifically, OVMD is first employed to adaptively decompose seismic signals and isolate noise-dominated components. Subsequently, WTD is applied in the multi-scale frequency domain to suppress residual noise. An adaptive fractal dimension strategy is then utilized to detect change points and accurately determine the first-arrival time. These results are used as inputs to a particle swarm optimization (PSO) algorithm for source localization. Both numerical simulations and laboratory experiments demonstrate that the proposed method exhibits high robustness and localization accuracy under severe noise conditions. It significantly outperforms conventional approaches such as short-time Fourier transform (STFT) and continuous wavelet transform (CWT). The proposed framework offers reliable technical support for dynamic fracture monitoring, detailed reservoir characterization, and risk mitigation in the development of unconventional reservoirs. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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19 pages, 2887 KB

Open AccessArticle

Multifractal Characterization of Heterogeneous Pore Water Redistribution and Its Influence on Permeability During Depletion: Insights from Centrifugal NMR Analysis

by Fangkai Quan, Wei Lu, Yu Song, Wenbo Sheng, Zhengyuan Qin and Huogen Luo

Fractal Fract. 2025, 9(8), 536; https://doi.org/10.3390/fractalfract9080536 - 15 Aug 2025

Viewed by 506

Abstract

The dynamic process of water depletion plays a critical role in both surface coalbed methane (CBM) development and underground gas extraction, reshaping water–rock interactions and inducing complex permeability responses. Addressing the limited understanding of the coupling mechanism between heterogeneous pore water evolution and permeability during dynamic processes, this study simulates reservoir transitions across four zones (prospective planning, production preparation, active production, and mining-affected zones) via centrifugal experiments. The results reveal a pronounced scale dependence in pore water distribution. During low-pressure stages (0–0.54 MPa), rapid drainage from fractures and seepage pores leads to a ~12% reduction in total water content. In contrast, high-pressure stages (0.54–3.83 MPa) promote water retention in adsorption pores, with their relative contribution rising to 95.8%, forming a dual-structure of macropore drainage and micropore retention. Multifractal analysis indicates a dual-mode evolution of movable pore space. Under low centrifugal pressure, D₋₁₀ and Δα decrease by approximately 34% and 36%, respectively, reflecting improved connectivity within large-pore networks. At high centrifugal pressure, an ~8% increase in D₀−D₂ suggests that pore-scale heterogeneity in adsorption pores inhibits further seepage. A quantitative coupling model establishes a quadratic relationship between fractal parameters and permeability, illustrating that permeability enhancement results from the combined effects of pore volume expansion and structural homogenization. As water saturation decreases from 1.0 to 0.64, permeability increases by more than 3.5 times. These findings offer theoretical insights into optimizing seepage pathways and improving gas recovery efficiency in dynamically evolving reservoirs. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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30 pages, 8331 KB

Open AccessArticle

Fracture Complexity and Mineral Damage in Shale Hydraulic Fracturing Based on Microscale Fractal Analysis

by Xin Liu, Jiaqi Zhang, Tianjiao Li, Zhengzhao Liang, Siwei Meng, Licai Zheng and Na Wu

Fractal Fract. 2025, 9(8), 535; https://doi.org/10.3390/fractalfract9080535 - 15 Aug 2025

Viewed by 627

Abstract

The geological structural complexity and microscale heterogeneity of shale reservoirs, characterized by the brittleness index and natural fracture density, exert a decisive effect on hydraulic fracturing’s effectiveness. However, the mechanisms underlying the true microscale heterogeneity of shale structures, which is neglected in conventional models and influences fracture evolution, remain unclear. Here, high-resolution scanning electron microscopy (SEM) was employed to obtain realistic distributions of mineral components and natural fractures, and hydraulic–mechanical coupled simulation models were developed within the Realistic Failure Process Analysis (RFPA) simulator using digital rock techniques. The analysis examined how the brittleness index and natural fracture density affect the fracture morphology’s complexity, mineral failure behavior, and flow conductivity. Numerical simulations show that the main fractures preferentially propagate toward areas with high local brittleness and dense natural fractures. Both the fracture’s fractal dimension and the stimulated reservoir volume increased with the brittleness index. A moderate natural fracture density promotes the fracture network’s complexity, whereas excessive densities may suppress the main fracture’s propagation. Microscopically, organic matter and silicate minerals are more prone to damage, predominantly tensile failures under external loading. These findings highlight the dominant role of microscale heterogeneity in shale fracturing and provide theoretical support for fracture control and stimulation optimization in complex reservoirs. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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18 pages, 6449 KB

Open AccessArticle

Analysis of the Microscopic Pore Structure Characteristics of Sandstone Based on Nuclear Magnetic Resonance Experiments and Nuclear Magnetic Resonance Logging Technology

by Shiqin Li, Chuanqi Tao, Haiyang Fu, Huan Miao and Jiutong Qiu

Fractal Fract. 2025, 9(8), 532; https://doi.org/10.3390/fractalfract9080532 - 14 Aug 2025

Viewed by 508

Abstract

This study focuses on the complex pore structure and pronounced heterogeneity of tight sandstone reservoirs in the Linxing area of the Ordos Basin and develops a multi-scale quantitative characterization approach to investigate the coupling mechanism between pore structure and reservoir properties. Six core samples were selected from the Shiqianfeng Formation (depth interval: 1326–1421 m) for detailed analysis. Cast thin sections and scanning electron microscopy (SEM) experiments were employed to characterize pore types and structural features. Nuclear magnetic resonance (NMR) experiments were conducted to obtain T₂ spectra, which were used to classify bound and movable pores, and their corresponding fractal dimensions were calculated separately. In addition, NMR logging data from the corresponding well intervals were integrated to assess the applicability and consistency of the fractal characteristics at the logging scale. The results reveal that the fractal dimension of bound pores shows a positive correlation with porosity, whereas that of movable pores is negatively correlated with permeability, indicating that different scales of pore structural complexity exert distinct influences on reservoir performance. Mineral composition affects the evolution of pore structures through mechanisms such as framework support, dissolution, and pore-filling, thereby further enhancing reservoir heterogeneity. The consistency between logging responses and experimental observations verifies the regional applicability of fractal analysis. Bound pores dominate within the studied interval, and the vertical variation of the PMF/BVI ratio aligns closely with both the NMR T₂ spectra and fractal results. This study demonstrates that fractal dimension is an effective descriptor of structural characteristics across different pore types and provides a theoretical foundation and methodological support for the evaluation of pore complexity and heterogeneity in tight sandstone reservoirs. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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22 pages, 4428 KB

Open AccessArticle

Pore Structure Characteristics and Controlling Factors of the Lower Cambrian Niutitang Formation Shale in Northern Guizhou: A Case Study of Well QX1

by Yuanyan Yin, Niuniu Zou, Daquan Zhang, Yi Chen, Zhilong Ye, Xia Feng and Wei Du

Fractal Fract. 2025, 9(8), 524; https://doi.org/10.3390/fractalfract9080524 - 13 Aug 2025

Viewed by 534

Abstract

Shale pore architecture governs gas storage capacity, permeability, and production potential in reservoirs. Therefore, this study systematically investigates the pore structure features and influencing factors of the Niutitang Formation shale from the QX1 well in northern Guizhou using field emission scanning electron microscopy (FE-SEM), high-pressure mercury intrusion (HPMI), low-temperature nitrogen adsorption (LTNA), and nuclear magnetic resonance (NMR) experiments. The results show that ① The pore size of the QX1 well’s Niutitang Formation shale is primarily in the nanometer range, with pore types including intragranular pores, intergranular pores, organic matter pores, and microfractures, with the former two types constituting the primary pore network. ② Pore shapes are plate-shaped intersecting conical microfractures or plate-shaped intersecting ink bottles, ellipsoidal, and beaded pores. ③ The pore size distribution showed a multi-peak distribution, predominantly mesopores, followed by micropores, with the fewest macropores. ④ The fractal dimension D₁ > D₂ indicates that the shale pore system is characterized by a rough surface and some connectivity of the pore network. ⑤ Carbonate mineral abundances are the main controlling factors affecting the pore structure of shales in the study area, and total organic carbon (TOC) content also has some influence, while clay mineral content shows negligible statistical correlation. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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19 pages, 5482 KB

Open AccessArticle

Multiscale Fractal Evolution Mechanism of Pore Heterogeneity in Hydrocarbon Source Rocks: A Thermal Simulation Experiment in the Xiamaling Formation

by Yang Wang, Baoyuan Zhong, Liu Yang, Yanming Zhu, Jie Xiang, Tong Zhang and Hanyu Zhang

Fractal Fract. 2025, 9(6), 351; https://doi.org/10.3390/fractalfract9060351 - 27 May 2025

Cited by 1 | Viewed by 590

Abstract

The heterogeneity of shale pore systems, which is controlled by thermal maturation, fundamentally governs hydrocarbon storage and migration. Artificial sequence maturity samples of Xiamaling shale were obtained through a temperature–pressure simulation experiment (350–680 °C, 15–41 MPa). In combination with low-pressure CO₂/N₂ adsorption experiments, mercury intrusion porosimetry experiments and fractal theory, the heterogeneity of the pore size distribution of micropores, mesopores and macropores in shale of different maturities was quantitatively characterized. The results reveal that the total porosity follows a four-stage evolution with thermal maturity (R_o = 0.62–3.62%), peaking at 600 °C (R_o = 3.12%). Multifractal parameters indicate that areas with a low probability density are dominant in terms of pore size heterogeneity, while monofractal parameters reflect enhanced uniform development in ultra-over maturity (R_o > 3.2%). A novel Fractal Quality Index (FQI) was proposed to integrate porosity, heterogeneity, and connectivity, effectively classifying reservoirs into low-quality, medium-quality, and high-quality sweet-spot types. The findings contribute to the mechanistic understanding of pore evolution and offer a fractal-based framework for shale gas reservoir evaluation, with significant implications for hydrocarbon exploration in unconventional resources. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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21 pages, 4797 KB

Open AccessArticle

Multifractal Characterization of Pore Heterogeneity and Water Distribution in Medium- and High-Rank Coals via Nuclear Magnetic Resonance

by Huan Liu, Shasha Zhang, Yu Qiao, Danfeng Xie and Long Chang

Fractal Fract. 2025, 9(5), 290; https://doi.org/10.3390/fractalfract9050290 - 28 Apr 2025

Cited by 1 | Viewed by 547

Abstract

Comprehensive assessment of pore structure and multiphase water distribution is critical to the flow and transport process in coalbed methane (CBM) reservoirs. In this study, nuclear magnetic resonance (NMR) and multifractal analysis were integrated to quantify the multiscale heterogeneity of nine medium- and high-rank coals under water-saturated and dry conditions. By applying the box-counting method to transverse relaxation time (T₂) spectra, multifractal parameters were derived to characterize pore heterogeneity and residual water distribution. The influencing factors of pore heterogeneity were also discussed. The results show that pore structures in high-rank coals (HCs) exhibit a broader multifractal spectrum and stronger rightward spectrum than those of medium-rank coals, reflecting micropore-dominated heterogeneity and the complexity induced by aromatization in HCs. The vitrinite content enhances micropore development, increasing the heterogeneity and complexity of pore structure and residual water distribution. Inertinite content shows opposite trends compared to vitrinite content for the effect on pore structure and water distribution. Volatile yield reflects coal metamorphism and thermal maturity, which inversely correlates with pore heterogeneity and complexity. Residual water mainly distributes to adsorption pores and pore throats, shortening T₂ relaxation (bound water effect) and reducing spectral asymmetry. The equivalence of the multifractal dimension and singularity spectrum validates their joint utility in characterizing pore structure. Minerals enhance pore connectivity but suppress complexity, while moisture and ash contents show negligible impacts. These findings provide a theoretical reference for CBM exploration, especially in optimizing fluid transportation and CBM production strategies and identifying CBM sweet spots. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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20 pages, 5784 KB

Open AccessArticle

Pore Structure Evolution of Coal After Supercritical CO₂–Water–Rock Treatment: A Multifractal Analysis

by Sijian Zheng, Yanzhi Liu, Fansheng Huang, Shiqi Liu, Shuxun Sang, Xuguang Dai and Meng Wang

Fractal Fract. 2025, 9(3), 144; https://doi.org/10.3390/fractalfract9030144 - 25 Feb 2025

Cited by 3 | Viewed by 817

Abstract

The evolution of coal’s pore structure is crucial to the efficient capture of carbon dioxide (CO₂) within coalbeds, as it provides both adsorption sites and seepage space for the adsorbed- and free-phase CO₂, respectively. However, the conventional single fractal method for characterizing pore structure fails to depict the intricacies and variations in coal pores. This study innovatively applies the low-temperature N₂/CO₂ sorption measurement and multifractal theory to investigate the evolution of the microporous structure of coals (e.g., from the Huainan coalfield) during the supercritical CO₂(ScCO₂)–water–rock interaction process. Firstly, we observed that the ScCO2–water–rock interaction does not significantly alter the coal’s pore morphology. Notably, taking the ZJ-8# sample as an example, low-temperature N₂ sorption testing displayed a stable pore volume following the reaction, accompanied by an increase in specific surface area. Within the CO₂ sorption testing range, the ZJ-8# sample’s pore volume remained unchanged, while the specific surface and pore width performed displayed a slight decrease. Secondly, by introducing key parameters from multifractal theory (such as D_q, α(q), τ(q), and f(α)), we assessed the heterogeneity characteristics of the coal’s pore structure before and after the ScCO2–water–rock reaction. The N₂ sorption analysis reveals an increase in pore heterogeneity for the ZJ-8# sample and a decrease for the GQ-13# sample within the sorption testing range. In the context of low-temperature CO₂ sorption analysis, the pore distribution complexity and heterogeneity of the GQ-11# and GQ-13# samples’ pores were escalated after ScCO2–water–rock interaction. The experimental and analysis results elucidated the dual roles of precipitation and dissolution exerted by the ScCO2–water–rock interaction on the micropores of coal reservoirs, underscoring the heterogeneous nature of the reaction’s influence on pore structures. The application of fractal theory offers a novel perspective compared to traditional pore characterization methods, significantly improving the precision and comprehensiveness of pore structure change descriptions. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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