Advances in the Application of Fractal Theory to Oil and Gas Resource Assessment

Abstract

In response to the growing complexity of global exploration targets, traditional Euclidean geometric and linear statistical methods reveal inherent theoretical limitations in characterizing hydrocarbon reservoirs as complex geological bodies that exhibit simultaneous local disorder and global order. Fractal theory, with its core parameter systems such as fractal dimension and scaling exponents, provides an innovative mathematical–physics toolkit for quantifying spatial heterogeneity and resolving the multi-scale characteristics of reservoirs. This review systematically consolidates recent advancements in the application of fractal theory to oil and gas resource assessment, with the aim of elucidating its transition from a theoretical concept to a practical tool. We conclusively demonstrate that fractal theory has driven fundamental methodological progress across four critical dimensions: (1) In reservoir classification and evaluation, fractal dimension has emerged as a robust quantitative metric for heterogeneity and facies discrimination. (2) In pore structure characterization, the theory has successfully uncovered structural self-similarity across scales, from nanopores to macroscopic vugs, enabling precise modeling of complex pore networks. (3) In seepage behavior analysis, fractal-based models have significantly enhanced the predictive capacity for non-Darcy flow and preferential migration pathways. (4) In fracture network modeling, fractal geometry is proven pivotal for accurately characterizing the spatial distribution and connectivity of natural fractures. Despite significant progress, current research faces challenges, including insufficient correlation with dynamic geological processes and a scarcity of data for model validation. Future research should focus on the following directions: developing fractal parameter inversion methods integrated with artificial intelligence, constructing dynamic fractal–seepage coupling models based on digital twins, establishing a unified fractal theoretical framework from pore to basin scale, and expanding its application in low-carbon energy fields such as carbon dioxide sequestration and natural gas hydrate development. Through interdisciplinary integration and methodological innovation, fractal theory is expected to advance hydrocarbon resource assessment toward intelligent, precise, and systematic development, providing scientific support for the efficient exploitation of complex reservoirs and the transition to green, low-carbon energy.

1. Introduction

Hydrocarbon resource assessment, serving as a critical bridge between geological understanding and exploration activities, requires methodological innovation to address the growing challenges of deteriorating resource quality and increasing reservoir complexity [1,2,3]. As global exploration extends into deep-layer and unconventional domains, the multiscale heterogeneity of pore–fracture systems, nonlinear dynamics of hydrocarbon migration, and power-law distributions of resource accumulation have rendered traditional Euclidean geometric frameworks and linear statistical models increasingly inadequate [4,5,6]. For instance, the self-similar distribution of nanoscale pores in shale reservoirs, the chaotic patterns of fracture–vug networks in carbonate rocks, and the scale-dependent seepage processes in tight oil reservoirs are inadequately captured by conventional volumetric methods (based on regular geometric assumptions) or statistical models simplified by normal distributions [7,8]. Within this context, fractal theory has emerged as a cutting-edge approach to overcome the limitations of traditional assessment methodologies, owing to its unique capability to quantify complex systems exhibiting “local disorder yet global order”.

Fractal theory, established by Mandelbrot in the 1970s, fundamentally leverages the principles of self-similarity and power-law scaling to transform the ubiquitous irregular, cross-scale morphologies observed in nature into computable fractal dimensions (e.g., box-counting dimension, capacity dimension, correlation dimension) [9,10,11]. This theoretical framework enables the quantitative characterization of both the spatial filling capacity and structural complexity of intricate geometric systems [12,13]. Fractal theory exhibits inherent compatibility with geological systems hosting hydrocarbon resources, enabling effective characterization of fractal features across multiple scales. At the microscopic scale, pore-size distributions in shale reservoirs demonstrate high capacity dimensions [14,15]; at the macroscopic scale, hydrocarbon distribution in sedimentary basins follows distinct power-law scaling [16]; while at the engineering scale, hydraulic fracture networks display characteristic fractal dimensions [14,17]. Compared to the integer-dimensional constraints of traditional Euclidean geometry, fractal theory provides a more precise mathematical.

Over the past three decades, the application of fractal theory in hydrocarbon resource assessment has expanded from singular pore structure characterization to a comprehensive technical framework encompassing reservoir classification, flow simulation, and fracture modeling [6,18,19]. Reservoir classification: Fractal dimensions derived from high-pressure mercury injection capillary pressure (MICP) data enable precise sweet spot grading of reservoirs [20,21]. Pore structure characterization: Fractal capillary model-based permeability prediction methods demonstrate >20% higher accuracy compared to conventional approaches [22,23]. Fracture system modeling: Fractal discrete fracture network (DFN) models achieve <10% prediction error in production forecasting for shale gas hydraulic fracture networks [24,25]. However, the application of this theory faces persistent challenges in parameter acquisition costs, dynamic process simulation, and multi-scale modeling consistency. For instance: 3D fractal dimension calculations require high resolution Computed tomography (CT) scanning technology. Existing flow models struggle to accurately capture transient flow characteristics in fractal porous media.

Based on a comprehensive review of journal articles, scientific reports, and technical publications, this study systematically synthesizes advancements in fractal theory applications for hydrocarbon resource assessment, providing an in-depth analysis of key technological breakthroughs and representative case studies across four critical dimensions: reservoir classification and evaluation, pore structure characterization, flow behavior analysis, and fracture system modeling. The research further evaluates both advantages and limitations of current fractal-based methodologies while identifying future priorities including: developing inverse methods for fractal parameters through multi-source data fusion, advancing modeling theories for transient flow processes, establishing consistency coupling techniques for cross-scale fractal models, and ultimately creating intelligent assessment systems integrating “data-model-engineering” synergies.

2. Methodology

This review will employ a systematic literature review and analysis of findings to explore the application of fractal theory in oil and gas resource assessment. During the literature search conducted by querying relevant keywords in databases, a search for “fractal theory and oil and gas resource assessment” in Web of Science reveals that the vast majority of related studies focus on microscopic and classification aspects. In Web of Science, searching for the keyword “fractal theory and reservoir” identifies 434 relevant articles published in the last five years; searching for “fractal theory and pore structures” yields 637 articles from the same period; and searching for “fractal theory and fracture” returns 522 articles. The quantities of other related literature will not be elaborated here in detail. By synthesizing these previous studies, this work aims to provide a comprehensive analysis and assessment of the research direction. The main research content will be categorized and discussed according to a proposed thesis outline.

3. Overview of Fractal Theory

As a pivotal branch of nonlinear scientific systems, fractal theory transcends the limitations of traditional Euclidean geometry-which is restricted to regular geometric forms-by establishing a fractional-dimensional descriptive framework [26,27]. This paradigm provides a quantitative characterization methodology tailored to the irregular morphologies ubiquitous in natural systems [9]. The theory is fundamentally defined by its core principle of “fragmented morphology with structural self-similarity,” employing recursive iterative algorithms to unveil the mathematical homology between local structures and global systems [28,29]. This theoretical breakthrough offers a robust foundation for cross-scale analysis of complex geological systems [30].

Self-similarity reveals the statistical or geometric resemblance between parts and the whole in complex systems [31,32]. For instance, the self-similar spiral inflorescences of cauliflower exhibit identical repetition from millimeter to centimeter scales; the branching networks of coral reefs demonstrate consistent bifurcation patterns across centimeter to meter scales; and snowflake crystals display perfect hexagonal fractal growth trajectories from micrometer to millimeter scales [33] (Figure 1). These phenomena enable the inference of global patterns from local features. Scale invariance translates this similarity into mathematical expression through the power-law relationship, where the fractal dimension (D) serves as the core parameter quantifying system complexity and spatial filling efficiency [34,35]. The fractal dimension is defined by the power-law relationship between pore volume and pore size in a double-logarithmic plot, where the slope quantifies the structural heterogeneity. As D approaches the Euclidean dimension, the system’s morphology becomes more regular; greater deviations indicate stronger heterogeneity [36]. The diverse types of fractal dimensions adapt to different geological scenarios: Box-counting dimension characterizes spatial coverage features of fracture networks and pore boundaries through the grid coverage counting principle, enabling quantitative evaluation of fracture system complexity [37]. Capacity dimension focuses on scale-scaling properties of pore volumes, analyzing reservoir heterogeneity through pore-size distribution analysis, and is widely applied in shale nano/micropore studies. Correlation dimension quantifies point-set correlations, achieves connectivity quantification of flow networks, and provides key parameters for permeability prediction [38].

In hydrocarbon resource assessment, fractal theory enables cross-scale analysis linking reservoir microstructure to macroscopic distribution patterns. Shale reservoir micro-nanoscale pores exhibit fractal characteristics in both pore-size distribution and connectivity [38,39]. The capacity dimension shows an inverse correlation with permeability each 0.1 increase in D corresponds to a 20–30% reduction in permeability. Simultaneously, hydrocarbon resource distribution follows fractal principles, where basin-scale resource dimensions correlate with trap dynamics, providing critical insights for potential prediction in unexplored areas [40]. Unlike conventional methods, fractal theory employs non-integer dimensions and power-law scaling to accurately characterize the branching topology of pore networks, the chaotic nature of fracture systems, and the heavy-tailed distribution of resources [41]. Currently, fractal theory has become a cornerstone tool in unconventional resource assessment, with its cross-scale framework providing scientific basis for sweet spot identification, hydraulic fracturing effectiveness evaluation, and development strategy optimization [42]. This paradigm shift is transforming hydrocarbon exploration from qualitative description to quantitative prediction, enabling more reliable resource forecasts and economic evaluations.

4. Fractal Theory and Hydrocarbon Resource Assessment

4.1. Historical Development

Since its formal proposal in 1977, fractal theory has provided a mathematical foundation for characterizing irregular, complex, and self-similar objects ubiquitous in nature [43]. From the late 1980s to the 1990s, fractal theory began to be introduced into geoscience and petroleum engineering. Early researchers (e.g., Katz, Wong, Pfeifer) discovered that many reservoir properties, such as rock fracture surfaces and pore distributions, exhibit significant fractal characteristics [44,45,46].

During 2000–2010, foundational research on fractal theory advanced significantly in hydrocarbon resource assessment. In 2006, Song et al. applied fractal methods to estimate petroleum geological resources based on reservoir size distribution characteristics [47]. In 2009, Guo et al. proposed a two-dimensional fractal model for hydrocarbon spatial distribution using stochastic simulation and Fourier transform power spectrum methods [48]. This model enabled refined resource abundance mapping, exclusion of high-risk areas, and predictive quantification of resource volumes and spatial distribution.

From 2010 to present, fractal theory has continued to deepen its applications and expand into unconventional domains. Dashtian et al. pioneered a fractal model for flow through porous media, laying the groundwork for simulating fluid transport in complex reservoirs [49]. Babadagli et al. systematically elucidated the value of fractal and multifractal methods for characterizing pore structures in unconventional reservoirs [50]. Ji et al. applied fractal theory to microscopically investigate the degree of formation damage caused by fracturing fluid filtration during reservoir stimulation and its impact on gas well productivity [51], Jiang et al. classified fractal dimensions of continental shale pore structures into two distinct types through physical experiments, providing detailed analysis of their characteristics and influencing factors [52]. In the same period. Liu and Ostadhassan established quantitative relationships between fractal dimensions and petrophysical parameters using nano-CT and FIB-SEM techniques [53]. In 2023, Cheng et al. integrated high-pressure mercury injection experiments, fractal theory, and principal factor analysis, utilizing fractal dimension as an effective parameter to characterize reservoir physical properties, pore structure, and heterogeneity, thereby offering a reference for rock type classification [54]. Most recently, Lu et al. proposed an intelligent evaluation method for shale oil reservoirs based on the fractal characteristics of pore systems, overcoming the limitations of parameter contradictions inherent in traditional approaches [55].

The historical development of fractal theory in hydrocarbon resource assessment clearly demonstrates the evolution of human understanding of subsurface petroleum systems-transitioning from subjective empiricism to data-driven paradigms, and from isolated disciplinary approaches to interdisciplinary integration. Each methodological innovation has been accompanied by technological breakthroughs and cross-disciplinary convergence, while simultaneously revealing new theoretical and engineering challenges in increasingly complex geological settings.

4.2. Application Context

Traditional resource assessment methodologies such as volumetric methods, statistical analogy, and basin modeling typically rely on homogenization assumptions and regular geometric configurations [56,57]. These approaches struggle to accurately characterize the full spectrum of heterogeneity and complexity inherent in hydrocarbon systems, ranging from microscopic pore structures and spatial distribution of fracture networks to macroscopic reservoir sand bodies and trap geometries, and even extending to organic matter richness in source rocks and spatial distribution of hydrocarbon kitchens.

For instance, in mature exploration areas, remaining hydrocarbons predominantly accumulate in small-to-medium-sized subtle reservoirs, whose size distribution patterns are increasingly challenging to capture using conventional statistical methods. In unconventional resource development, the evaluation of complex fracture networks created by hydraulic fracturing is often compromised when applying traditional Stimulated Reservoir Volume (SRV) methods, as these approaches frequently overlook internal connectivity and density variations within fracture networks, leading to significant estimation biases. These challenges have driven the research community to seek quantitative tools capable of characterizing heterogeneity. Fractal theory, anchored in the principles of self-similarity and scale invariance, addresses this need by employing the D to quantify structural complexity, thereby offering a novel pathway to resolve these longstanding issues [58].

Numerous studies have demonstrated that hydrocarbon reservoir systems universally exhibit fractal characteristics, observed across multiple scales: from nanometer pore distributions in scanning electron microscopy, through extension patterns of fracture networks at various scales, to power-law distributions of reserve sizes in hydrocarbon accumulations [14,16,22,56,59]. Under the modern requirement for petroleum exploration to shift from traditional qualitative description to precise quantitative characterization, parameters such as fractal dimension and multifractal spectrum width have provided reliable quantitative indicators for pore structure and fracture intensity. In unconventional resource development, these metrics serve as a critical bridge connecting geological features with engineering responses for instance, by establishing fitting relationships between 3D fractal dimensions and both brittleness index and fracture toughness to optimize hydraulic fracturing design [25]. In contrast to conventional methodologies, fractal theory transcends the homogeneous assumptions of Euclidean geometry from a nonlinear perspective. Although challenges such as high model complexity and difficulties in multi-scale data integration remain, its strengths in cross-scale characterization and revealing hidden patterns have established it as a core tool for evaluating unconventional reservoirs and complex fracture networks [60] (

Table 1. Comparison between traditional methods and fractal theory in resource assessment.

No.	Dimension	Traditional Methods (Volumetric/Statistical Approaches)	Fractal Theory	References
1	Theoretical Foundation	Euclidean geometry (regular shapes), linear statistical models, reservoir homogeneity assumptions	Fractal geometry (self-similarity, scale invariance), nonlinear theory	Adapted from [26,27]
2	Core Assumptions	Reservoir homogeneity, regular geometric forms (e.g., cubic sand bodies, cylindrical traps)	Reservoir heterogeneity, cross-scale complex structures (pores-fractures-traps)	Adapted from [38]
3	Data Dependency	Limited exploration data (well points, 2D seismic profiles)	Multi-scale digital data (CT scans, FIB-SEM 3D reconstruction)	Adapted from [59,60]
4	Characterization Capability	Qualitative descriptions dominate, limited ability to quantify heterogeneity	Quantitative parameters: fractal dimension (D), multifractal spectrum width (Quantifies the fluctuations in heterogeneity within complex systems) (Δα)	Adapted from [34,35]
5	Typical Limitations	Fails to capture nano-pore complexity in shale; lacks patterns for subtle reservoir distribution	Nano-pore fractal dimension D = 2.5–3.0; Fracture network multifractal spectrum width Δα = 0.3–0.6	Adapted from [38,39]
6	Engineering Application	Empirical fracturing design; SRV evaluation ignores fracture network connectivity	Fractal dimension correlated with brittleness index to optimize fracturing parameters	Adapted from [38]
7	Applicable Scenarios	Conventional reservoirs with strong homogeneity (e.g., high-permeability sandstone)	Unconventional reservoirs with strong heterogeneity (shale gas, tight oil), fractured reservoirs (carbonate)	Adapted from [40]

).

5. Application of Fractal Theory in Hydrocarbon Resource Assessment

5.1. Reservoir Classification and Evaluation

Reservoir classification and evaluation involve selecting representative geological parameters based on the characteristics of a target block and applying appropriate methodologies to categorize reservoir quality grades. Current approaches include macroscopic property evaluation, gray clustering analysis, and fuzzy comprehensive evaluation methods [61,62]. While these techniques-primarily based on statistical averaging or empirical criteria-have provided fundamental support for conventional reservoir assessment, they exhibit significant limitations when addressing the strong heterogeneity inherent in unconventional reservoirs.

The introduction of fractal theory has provided a novel dimension for reservoir classification and evaluation. By leveraging the self-similarity and heterogeneity of reservoir pore structures and permeability distributions, this theory quantitatively characterizes reservoir complexity through fractal dimensions. It transforms traditionally qualitative descriptions of heterogeneity into quantifiable fractal parameters, thereby enabling the establishment of scientifically robust classification and evaluation criteria. For instance, fractal dimensions can be utilized to construct correlation distributions with porosity (Figure 2a) and permeability (Figure 2b). The three reservoir categories (Type I, II, and III) exhibit distinct clustering characteristics in the two-dimensional parameter space: Type I (high-quality reservoirs) concentrate in the low fractal dimension region, corresponding to higher porosity and permeability; Type III (poor reservoirs) distribute in the high fractal dimension zone with significantly reduced porosity and permeability; while Type II reservoirs demonstrate a transitional distribution pattern [54]. This spatial arrangement visually reveals a negative correlation between fractal dimension and reservoir physical properties: higher fractal dimensions indicate stronger pore structure heterogeneity and weaker macroscopic flow capacity, thereby providing quantitative and visual criteria for reservoir classification.

By utilizing high-pressure mercury injection, CT scanning, and field emission scanning electron microscopy to acquire multi-scale pore structure data, and integrating the coupling relationships between fractal dimensions and macroscopic properties such as permeability and porosity, reservoir classification and precise identification of “sweet spots” can be achieved [20,21]. This methodology transcends traditional homogenization assumptions through multi-source data integration combining experimental measurements, image analysis, and logging data to establish an integrated evaluation framework linking microscopic pores to macroscopic reservoir properties. This approach provides critical theoretical support for the efficient development of unconventional reservoirs such as shale gas, tight sandstone, and carbonate formations [24,63,64].

In practical applications, researchers have conducted fractal-driven classification and evaluation studies for various reservoir types, yielding substantial. Shi et al. focusing on carbonate tight oil reservoirs, utilized high-pressure mercury injection data to identify significant self-similarity in pore structures [64]. By correlating pore size distributions with macro-pore porosity ratios, displacement pressure, and permeability, they established a three-tier reservoir classification standard based on microscopic pore-throat characteristics. Cheng et al. integrated fractal theory with factor analysis in their reservoir study, extracting fractal dimensions from high-pressure mercury injection data [54]. Their work confirmed a positive correlation between fractal dimensions and both reservoir physical properties and pore structure complexity (i.e., higher fractal dimensions indicate poorer reservoir quality), leading to the development of a ternary reservoir evaluation system incorporating fractal parameters.

For tight sandstone reservoirs, Cheng et al. integrated cast thin section observation with high-pressure mercury injection (HPMI) technology to classify pores into four categories: micropores, transition pores, mesopores, and macropores [63]. Based on pore proportion and pore-throat parameters, they established a four-tier classification system, providing clear guidance for tight sandstone exploration and development. In a separate study, Dong et al. employed fractal dimensions derived from mercury injection curves to investigate tight reservoirs in the Kuqa Depression. They defined precise pore size boundaries for fractures (>3000 nm), macropores (1000–3000 nm), mesopores (100–1000 nm), and micropores (<100 nm), establishing a standardized classification framework for complex tight reservoir evaluation.

In terms of technological innovation, the application of fractal theory has evolved toward greater precision and multi-scale capabilities. Li et al. utilized nuclear magnetic resonance (NMR) T₂ spectra to calculate D and combined this with a Generalized Regression Neural Network (GRNN) to construct a segmented power function model [65]. This approach classified tight oil reservoirs into four distinct categories, confirming that higher D values correspond to poorer reservoir quality. The method effectively addresses the challenge of quantitatively characterizing heterogeneity in tight oil reservoirs. In a study on deep sandstone reservoirs, Yang et al. employed constant-rate mercury injection techniques to reveal, for the first time, that the fractal dimension of large-scale pores is significantly higher than that of small-scale pores, and that the pore fractal dimension exceeds the throat fractal dimension [66]. Based on these findings, they established a ternary reservoir classification standard: “micro-pores with small throats, small pores with small throats, and large pores with coarse throats.” This work clearly demonstrates a negative correlation between fractal dimension and reservoir permeability.

Despite the breakthroughs facilitated by fractal theory in reservoir classification, the current system still faces two core challenges: First, existing indicators predominantly rely on static geological data such as porosity, permeability, and pore-throat structure; while failing to effectively integrate dynamic production indicators like productivity, pressure decline behavior, and fluid flow characteristics. This limitation hinders real-time coupling between geological features and development performance. Second, the universality of classification standards across different regions remains insufficient, with significant regional variations in the applicability of fractal parameters. There is an urgent need to establish correlation models linking fractal dimensions, multifractal spectra, and other parameters to regional geological contexts, thereby advancing reservoir classification and evaluation from single-well/block scales to regional scales.

5.2. Fractal Characterization of Reservoir Pore Structures

The highly heterogeneous and cross-scale irregular nature of reservoir pore structures makes it difficult for traditional Euclidean geometry-based characterization methods to accurately describe their complex features. Leveraging its strengths in self-similarity and scale invariance, fractal theory converts qualitative characteristics such as pore surface roughness and pore size distribution complexity into quantitative parameters through fractal dimension, offering a novel approach for detailed characterization of reservoir pore structures [54]. Supported by techniques including MICP, nitrogen adsorption/desorption isotherms, scanning electron microscopy (SEM), and CT, this theory-combined with the Menger sponge model-enables quantitative description of pore spatial distribution and connectivity [59,67]. Higher fractal dimensions indicate more complex pore structures, poorer connectivity, and consequently reduced reservoir flow capacity (Figure 3).

In a study on tight sandstone reservoirs, Chen et al. integrated cast thin sections, SEM, HPMI, and constant-rate mercury injection (CRMI) experiments with fractal theory to reveal the diverse pore-throat types and strong heterogeneity characteristic of such reservoirs [69]. Their research demonstrated that the full-aperture fractal dimensions (D₂ and D₄) effectively characterize the heterogeneity of pore structures and are significantly influenced by diagenetic factors such as clay mineral content. By comparing different fractal models, the study proposed a novel method for characterizing full-aperture pore-throat distributions based on fractal dimension splicing points, providing a fractal parameter foundation for evaluating the flow capacity of tight sandstone reservoirs.

Regarding shale reservoirs, Zhang conducted a study on the fractal characteristics of pore structures by applying the Frenkel–Halsey–Hill (FHH) fractal model and method of moments estimation, combined with nitrogen adsorption experiments [70]. The research revealed that shale pores exhibit distinct fractal characteristics. The fractal dimension showed a positive correlation with specific surface area, while demonstrating negative correlations with mean pore size, standard deviation, coefficient of variation, and skewness. These fractal parameters were closely aligned with pore morphological characteristics [71,72]. The study confirmed that the integration of fractal theory and method of moments estimation can effectively characterize shale pore structures. Fractal dimension not only reflects the gas adsorption and transport capacity of shale, but also quantitatively describes the geometric features of pores.

In a study of carbonate reservoirs, Cheng et al. utilized MICP experiments and NMR techniques to analyze reservoirs on the right bank of the Amu Darya Basin in Turkmenistan. Their analysis revealed multi-segment characteristics in fractal dimensions [73]. The fractal dimension at the macropore–fracture scale was significantly influenced by fracture development, while the fractal dimension at the mesopore scale exhibited the strongest correlation with permeability and was notably associated with microscopic parameters such as pore shape factor and sorting coefficient.

In the field of digital core and pore network modeling, Li and Zheng utilized conventional MICP method to obtain capillary pressure curves, determining the fractal dimension and self-similarity intervals of the core’s microscopic pore structure [74]. By integrating stochastic distribution theory, they established a pore radius distribution probability density function, thereby constructing a digital core and pore network model that accurately reflects the topological structure of real pore spaces. This method overcomes the reliance on regular geometric assumptions inherent in traditional models, providing more realistic foundational data for seepage simulation in porous media.

Integrated multidisciplinary approaches have significantly advanced the application depth of fractal theory. Guan et al. combined MICP, NMR, gas adsorption, and electron microscopy to systematically analyze the fractal characteristics of pore structures in unconventional reservoirs [75]. Their study revealed a positive correlation between D and reservoir heterogeneity: higher D values indicate more complex pore-throat structures and lower permeability. Movable fluid saturation showed a negative correlation with D, while D exhibited a positive correlation with quartz content and a negative correlation with feldspar content. Wang et al. conducted a systematic study on tight sandstone reservoirs, employing thin section identification, SEM, NMR, HPMI, and X-ray diffraction (XRD) experiments integrated with fractal theory [76]. Their research focused on the fractal characteristics of pore structures and their influence on fluid mobility. Comparative analysis of fractal dimensions across different reservoir types revealed a gradual increase in fractal dimensions from Type I to Type III pores, indicating progressively more complex pore structures and enhanced heterogeneity. Specifically, Type I exhibits the simplest structure; Type II is moderately complex; and Type III shows the most complex structure with the strongest heterogeneity [76] (Figure 4).

Although existing studies have established quantitative relationships between fractal models and pore structure parameters, two critical challenges remain: First, most research relies on static experimental data, lacking simulation of dynamic changes in pore structure caused by mineral dissolution/precipitation during diagenetic evolution and pressure-sensitive effects during development. Second, the advantage of fractal dimensions in characterizing reservoir heterogeneity has not been effectively integrated with dynamic production data (e.g., productivity, pressure decline). There is an urgent need to develop cross-scale coupling models linking fractal parameters to reservoir performance, advancing fractal theory from static characterization to dynamic prediction.

5.3. Fractal Theory and Applications in Flow Behavior Characterization

The multiscale heterogeneity of pore–fracture systems in complex reservoirs poses significant challenges for accurately describing flow behaviors. Fractal theory, leveraging its unique capability to characterize self-similar structures, has emerged as a fundamental tool for constructing mathematical models of fluid flow [77,78]. By quantifying the geometric complexity of pore–fracture networks through the D, and integrating theoretical derivations, numerical simulations, and field validations, this theory establishes quantitative relationships between fractal parameters and key flow properties such as permeability and threshold pressure gradient [79]. This approach effectively addresses the limitations of traditional models in capturing the flow characteristics of complex porous media (Figure 5).

In the study of flow behavior in low-permeability reservoirs, Xia established a research framework integrating fractal geometry theory with laboratory experiments. By characterizing tight sandstone pore structures through fractal dimension, lacunarity parameter (measures the “intermittency” or “patchiness” of fractal patterns), and advancing phase parameter (revealing the underlying heterogeneity of pore physical properties), spontaneous imbibition experiments revealed that variations in imbibition curves are primarily controlled by microscopic pore development characteristics, with pore connectivity and tortuosity being the core factors influencing flow capacity. The permeability formula derived from the fractal capillary model demonstrates that the advancing phase parameter can effectively predict permeability, and gas flow exhibits a significant threshold pressure gradient [80].

Wang conducted a study on a specific oilfield reservoir, establishing an oil–water relative permeability fractal model by integrating capillary bundle theory and the Poiseuille law. This model incorporates the effects of pore structure heterogeneity and water film thickness, with validation showing less than 5% error. Parameter sensitivity analysis revealed: an increase in fractal dimension reduces water-phase relative permeability while enhancing oil-phase relative permeability; a larger pore–throat radius ratio improves water-phase flow capacity but weakens oil-phase flow; and increased water film thickness significantly inhibits water-phase flow [81].

Addressing fluid flow in porous media, Wang et al. proposed an elliptical capillary fractal model (A theoretical model incorporating elliptical pore-throat geometry and fractal scaling to govern pore size distribution complexity) and derived fractal expressions for absolute and relative permeability under both saturated and unsaturated conditions. Theoretical analysis indicates that absolute permeability is a function of the maximum/minimum pore area, fractal dimension, and shape factor ε; while unsaturated relative permeability closely correlates with saturation and microstructural parameters. Comparisons between model predictions and experimental data, as well as existing models, demonstrate strong agreement under varying porosity and wettability conditions, revealing the fundamental influence of pore geometry on flow behavior [82].

In the field of shale gas and coalbed methane flow, the application of fractal theory has further expanded the applicability of flow models in complex media. Wang et al. combined liquid nitrogen adsorption, NMR experiments, and the FHH model in their investigation. Their findings revealed that micropores in the 1–100 nm range serve as the primary storage space, with the fractal dimension exhibiting a negative correlation with average pore size and total organic carbon content, while following a quadratic relationship with Brunauer–Emmett–Teller (BET) specific surface area. By categorizing flow regimes based on the Knudsen number, the constructed permeability model demonstrated that in the transition diffusion regime, permeability positively correlates with fractal dimension and porosity, whereas in the Knudsen diffusion regime, the influence of porosity diminishes. This work provides a theoretical framework for analyzing multi-mechanism flow behavior in shale gas systems [83].

Despite significant progress in theoretical and applied aspects of fractal-based flow models, two major challenges persist: First, most existing models rely on steady-state flow assumptions, making it difficult to characterize the dynamic coupling between pore structure and flow parameters during transient processes such as periodic water injection and gas–liquid two-phase flow in actual reservoirs. Second, multiphase flow simulations using fractal approaches involve solving high-dimensional nonlinear equation systems, whose computational complexity increases exponentially with higher fractal dimensions. Limited by current hardware capabilities, there is an urgent need to develop efficient numerical simulation methods leveraging dimensionality reduction algorithms and machine learning. Future research should focus on integrating fractal theory with transient flow mechanisms and multi-physics coupling effects to advance seepage prediction in complex reservoirs from static characterization to dynamic evolution simulation.

5.4. Fractal Modeling and Evaluation of Fracture Systems

The multiscale complexity and spatial heterogeneity of fracture systems represent a fundamental challenge in understanding flow mechanisms in complex reservoirs. Fractal theory provides a critical tool to address this challenge. Based on self-similarity and scale invariance, fractal parameters such as box-counting dimension and correlation dimension quantitatively characterize the power-law distributions of fracture length, density, and orientation, effectively capturing the cross-scale coupling relationships within fracture networks-from microscopic microfractures to macroscopic fracture zones [84,85]. As early as 1989, Hirata pioneered the application of fractal theory to fault systems, revealing that fracture density follows a power-law relationship with observation scale and establishing fractal dimension as a key indicator of fracture network complexity [86]. This foundational work laid the theoretical groundwork for subsequent fractal modeling of fracture systems (Figure 6).

In the field of fractal modeling of fracture systems, Li developed a roughness-based fracture model grounded in fractal geometry to characterize the rough surfaces of hydraulic fractures [88]. By quantifying fracture surface roughness through the fractal dimension, the study revealed the influence mechanism of fractal fracture structures on proppant transport processes. This model advances beyond traditional idealized parallel-plate fracture assumptions and provides a quantitative basis for optimizing proppant selection and pumping parameters during fracturing operations. However, limitations such as the neglect of fracture branching patterns and proppant distribution heterogeneity necessitate further research for comprehensive model refinement [88].

Su constructed a multi-scale fractal modeling framework: 1. For matrix systems, a fluid–solid coupling fractal model was established, revealing that pore permeability is controlled by the fractal distribution of capillaries and tortuosity, while stress sensitivity shows significant correlation with rock mechanical parameters (Young’s modulus, Poisson’s ratio). 2. For dissolved vuggy reservoirs, a dual-scale fractal model was proposed. Monte Carlo simulations demonstrated that permeability is primarily influenced by matrix pore size, vug size, and their matching relationships, with tortuosity being the core controlling factor. 3. For fracture–vug coupled systems, a dual-scale model incorporating the fractal dimension of fracture tortuosity was developed. It confirmed that the development degree of fracture networks determines reservoir connectivity, while the fractal matching relationship between fracture aperture and vug size dominates flow capacity [89].

He utilized true triaxial hydraulic fracturing experiments and CT-based 3D reconstruction technology, integrated with fractal theory and topological structure analysis, to investigate the influence of fracture network complexity on flow behavior. The study revealed that under conditions of low stress ratio and moderate fracturing fluid viscosity, the rate of change in fractal dimension of fracture networks reaches its maximum, facilitating the formation of complex fracture networks. Spherical radial permeability variation showed a strong positive correlation with fractal dimension, with the impact of fractal dimension on flow behavior significantly surpassing that of topological structure parameters [90].

Significant advances have been made in topological modeling of fracture networks and production prediction. Zhou et al. introduced an L-system fractal algorithm to construct a multi-stage fracture network model incorporating parameters such as fracture bifurcation angles, iteration, and fractal distances [84]. This model achieved, for the first time, quantitative characterization of the topological features of fracture networks. Hu et al. developed and applied a coupled model that integrates a fractal discrete fracture network with matrix fractal permeability. By incorporating flow mechanisms such as slip flow and Knudsen diffusion, they numerically investigated gas transport phenomena in shale gas reservoirs [87].

Despite substantial achievements in fractal-based modeling and evaluation of fracture systems, two critical challenges remain: First, most existing models rely on idealized assumptions such as parallel-plate fractures and regular bifurcation patterns, which inadequately match the irregular geometric morphologies of fracture networks in actual reservoirs (e.g., randomly distributed branching angles and non-uniform surface roughness). This limitation restricts the accuracy of flow simulations in complex fracture networks. Second, the time-varying fractal characteristics of fracture propagation, closure, and aperture evolution under dynamic stress fields have not been effectively characterized. The lack of dynamic inversion models for fractal parameters that account for variations in situ stress and fluid pressure fluctuations hinders the accurate representation of real-time fracture network evolution during fracturing operations and production. Future research should focus on fractal characterization methods for complex geometries and dynamic fracture modeling under multi-physics coupling conditions to advance the engineering application of fractal theory in fracture system evaluation (

Table 2. Applications of fractal theory in oil and gas resource assessment progress and challenges.

No.	Assessment Aspect	Key Progress	Outstanding Challenges
1	Reservoir classification and evaluation	Quantified correlations between fractal dimension and porosity/permeability enable robust reservoir typing and sweet spot identification. Higher fractal dimensions reliably indicate stronger heterogeneity	Limited integration of dynamic production data; absence of universal classification standards across different geological regions
2	Pore structure characterization	Fractal dimensions from multi-source data (e.g., MIP, SEM) effectively quantify pore complexity and connectivity, improving digital rock model accuracy	Models are predominantly static; lack of dynamic coupling between fractal parameters and reservoir performance during production
3	Seepage behavior analysis	Fractal capillary models accurately predict key parameters (e.g., permeability), with errors <5%, and clarify flow mechanisms in nanoscale pores	Poor performance for transient multiphase flow; high computational cost for complex fractal model solutions
4	Fracture network modeling and evaluation	Fractal dimensions (box-counting, etc.) quantify network complexity; DFN models reveal links between fractal geometry, productivity, and stimulation design	Idealized model assumptions mismatch real fracture geometry; inability to simulate dynamic fracture evolution under changing field stresses

).

6. Development Trends

Future advancements in fractal theory for hydrocarbon resource assessment must overcome persistent challenges, including high parameter acquisition costs, limited dynamic simulation fidelity, and inconsistencies in multi-scale model integration. To this end, research should pivot toward intelligent, dynamic, and practical applications by deeply integrating fractal theory with emerging digital technologies. A key pathway is to strengthen the synergy between fractal theory and artificial intelligence. For instance, deep learning models especially convolutional neural networks (CNNs) and generative adversarial networks (GANs) can automatically interpret pore-scale fractal characteristics from micro-CT images, significantly improving feature extraction efficiency and objectivity. Simultaneously, reinforcement learning and Bayesian optimization algorithms can be employed to invert fracture network parameters, intelligently calibrate fractal dimensions, and optimize structural models under multi-physics constraints, thereby enhancing predictive accuracy in complex geological environments (Figure 7). Furthermore, the introduction of digital twin technology will enable the construction of dynamic fractal-reservoir models capable of continuously assimilating real-time data such as production history, distributed fiber-optic sensing, and 4D seismic. Such models move beyond traditional steady-state assumptions by incorporating time-varying fractal descriptors, which support the simulation of non-equilibrium multiphase flow and the reproduction of dynamic behaviors such as fracture propagation and pressure transients. Ultimately, these intelligent fractal modeling approaches will form a core part of next-generation reservoir evaluation, enabling more reliable production forecasting, optimized hydraulic fracturing designs, and improved recovery rates in complex unconventional reservoirs.

At the technical characterization level, it is essential to develop low-cost methods such as fractal feature inversion techniques based on spectral data or Unmanned Aerial Vehicle (UAV) acquired information to reduce reliance on high-precision experiments and minimize data acquisition costs [91]. Simultaneously, cross-scale modeling technologies should be explored to achieve seamless transfer of fractal characteristics from nanoscale pores to basin-scale systems [92]. This will address multi-scale consistency issues and establish a unified descriptive system for fractal features across pores, fractures, sedimentary units, and entire hydrocarbon-bearing basins.

In engineering applications, it is critical to establish a standardized interpretation system for fractal parameters. For instance, in fracturing design, fractal dimensions of fractures can be utilized to optimize operational parameters, precisely guiding parameters such as fracturing fluid volume, proppant distribution, and fracture propagation direction [93,94]. In reservoir classification, integrating fractal dimensions with production capacity data enables the formation of a comprehensive evaluation standard, providing quantitative basis for efficient development. Additionally, promoting the synergistic application of fractal theory with traditional evaluation methods will enhance the reliability and applicability of resource estimation [13].

The “fractal index” model for fluid identification is poised to become a critical research direction. Future efforts should focus on developing multi-attribute fractal mapping technology based on distance units. By integrating multi-source well logging data including electrical, acoustic, and nuclear magnetic resonance measurements a comprehensive fractal index can be constructed that reflects pore structure, fluid saturation, and seepage capacity. This technology aims to extend the single fractal dimension into a multi-dimensional fractal indicator system, enabling graphical delineation of spatial distributions of different fluid units such as gas-rich zones, light oil zones, and heavy oil zones in both 2D cross-sections and 3D reservoir models. This advancement will enable the critical transition from reservoir structure characterization to fluid property prediction, ultimately providing more precise decision support for well placement optimization and development planning.

Confronted with the demands of novel resource exploration and carbon neutrality, fractal theory can intersect and integrate with geophysics, rock mechanics, and other disciplines. In the evaluation of deep hydrocarbon and natural gas hydrate reservoirs, as well as in the field of CO₂ geological storage, fractal dimensions can be employed to quantify the impact of complex structures on flow and storage capacity, providing theoretical support for the development of new energy resources and carbon emission reduction. Overall, fractal theory must overcome methodological bottlenecks through technological integration, drive innovation via engineering needs, and transition from singular structural characterization to intelligent full-process evaluation. This will establish a synergistic fractal evaluation system integrating data, models, and engineering, offering solid theoretical and technical foundations for efficient hydrocarbon exploration and sustainable energy development.

7. Discussion

7.1. Advantages of Fractal Theory in Petroleum Applications

In the field of hydrocarbon resource assessment, fractal theory demonstrates significant application advantages. Its core strength lies in its ability to characterize reservoir properties across multiple scales. Utilizing the fractal dimension, it precisely quantifies the heterogeneity and complexity of reservoirs from microscopic pore structures to macroscopic geological features. In reservoir classification and evaluation, the fractal dimension effectively represents the complexity of pore structures, enabling the establishment of classification criteria and accurate identification of “sweet spots.” This provides a critical theoretical foundation for the efficient development of unconventional reservoirs. For flow behavior studies, fractal-based mathematical models of fluid flow deeply reveal the influence of pore–fracture fractal characteristics on seepage patterns. They establish quantitative relationships between fractal dimensions and parameters such as permeability, thereby improving the prediction accuracy of flow properties. In fracture system characterization, fractal theory leverages self-similarity and fractal dimensions to quantitatively describe the complex geometric features and scale invariance of fracture networks. This overcomes the limitations of traditional Euclidean geometry in representing complex fractures and provides a theoretical basis for productivity prediction and fracturing design in fractured reservoirs.

Another unique advantage of fractal theory stems from its principles of self-similarity and scale invariance, which make it particularly well-suited for characterizing the distribution of hydrocarbon resources. The spatial distribution of oil and gas resources often follows fractal patterns. By utilizing resource dimensions at the basin scale, trap dynamics can be correlated, enabling the prediction of resource potential in unexplored areas. This approach broadens the perspective of hydrocarbon exploration and provides scientific guidance for decision-making. It is worth emphasizing that, with ongoing technological advancements, the integration of fractal theory with cutting-edge technologies such as machine learning and big data analytics continues to deepen. This synergy is driving the evolution of hydrocarbon resource assessment systems toward greater intelligence, opening new pathways and methodologies for addressing the challenges of evaluating complex reservoirs.

7.2. Limitations and Challenges of Fractal Theory in Petroleum Applications

The application of fractal theory in hydrocarbon resource assessment also presents certain limitations. On one hand, its effectiveness highly depends on extensive high-quality data, including experimental measurements, image analysis results, and well-logging data. However, acquiring such data is often costly, and issues such as data incompleteness or inaccuracy frequently arise in practice, inevitably affecting the reliability and precision of fractal models. For instance, in the fractal characterization of reservoir pore structures, most current studies establish quantitative relationships between fractal models and pore parameters based on static experimental data. Yet, there is a widespread lack of dynamic simulation capability to capture pore structure evolution during diagenesis or reservoir development, significantly limiting the practical applicability of fractal models.

On the other hand, the construction and solution processes of fractal models are relatively complex, demanding substantial computational resources and specialized expertise from researchers. In the fractal modeling of fracture systems, most models are based on idealized assumptions (e.g., parallel-plate fractures), which inadequately represent the intricate geometric morphologies of actual fracture networks in reservoirs. Moreover, simulating fracture evolution under dynamic stress fields remains largely unexplored, making it difficult to capture the time-varying fractal characteristics of fractures during hydraulic fracturing processes. Additionally, existing models predominantly rely on steady-state flow assumptions, failing to represent transient flow characteristics in real reservoirs, such as periodic water injection or gas–liquid two-phase flow. The high computational complexity of multiphase flow simulations with fractal approaches, coupled with limitations in hardware resources, further restricts the widespread application of fractal theory in hydrocarbon resource assessment.

7.3. Controversial Studies and Divergent Interpretations

While fractal theory is widely applied, some of its conclusions remain contentious within academia, primarily due to the complexity of geological systems and methodological differences.

Regional Contradictions in Fractal Dimension–Property Relationships: The fractal dimension (D) is generally believed to be negatively correlated with porosity and permeability. However, in reservoirs such as intensely dissolved carbonates, high heterogeneity (corresponding to a high D value) can conversely be an indicator of high-quality reservoir development. This suggests that the geological interpretation of the fractal dimension must be integrated with specific genetic contexts and cannot be universally applied.

Influence of Models and Calculation Methods on Results: Employing different fractal models (e.g., Menger model, FHH model) or calculation methods (e.g., box-counting dimension, capacity dimension) to process the same dataset can yield fractal dimensions with varying numerical values and physical interpretations, making direct comparisons between studies challenging.

Cautious Perspectives on “Fractal Universality”: Some studies note that the fractal characteristics of reservoir properties often exist only within a limited “scale-invariant interval.” Forcibly applying fractal theory beyond this interval for extrapolation may lead to prediction biases.

Consensus and Future Directions: To reconcile these contradictions, future efforts should focus on: (1) Deeply coupling fractal parameters with specific geological contexts (e.g., sedimentary facies, diagenetic facies); (2) Promoting comparative studies and standardization of different methods; and (3) Incorporating the fractal dimension into a multi-parameter integrated evaluation system for cross-validation.

7.4. Discussion on Method Universality and Research Sufficiency

Current research is not yet sufficient to establish fractal characterization as a universal method for reservoir description. This field remains in a transitional stage, from principle verification to industrial application.

Insufficient Research Breadth and Diversity: Existing case studies are predominantly concentrated in hotspot areas like shales and tight sandstones, while research on the fractal characteristics of complex reservoirs (e.g., marine carbonates, volcanic rocks) and under different fluid systems remains relatively weak. The lack of a global “diversity map” hinders the establishment of broad classification standards.

Challenges in Establishing Universal Classification Systems: The Class I, II, and III classifications discussed, while valuable and based on specific regional practices, face challenges for universal application. Significant differences in the “critical values” of fractal dimensions across different basins and genetic reservoir types complicate the creation of a unified classification standard.

Need for Future Research: Establishing the universal status of fractal theory requires more systematic work: The primary task is to create a globally standardized database of fractal parameters; secondly, “mapping-style” research on understudied reservoir types is needed; most critically, large-scale correlative analysis between static fractal parameters and dynamic production data is essential to verify their stability in predicting productivity.

In summary, fractal theory is a powerful and innovative tool. However, achieving its status as a mature industrial standard depends on the accumulation of future research that is more diverse, systematic, and closely integrated with production practices.

8. Conclusions

Hydrocarbon resource assessment serves as a critical bridge connecting geological theory and exploration practice. Traditional Euclidean geometry and linear statistical methods fall short in characterizing complex geological systems that exhibit local disorder yet global order. Fractal theory transcends the limitations of integer dimensions by employing parameters such as fractal dimension to quantitatively describe the spatial structure and irregularity of geological bodies with fractional dimensions. This provides an innovative tool for modeling multi-scale heterogeneity in hydrocarbon reservoirs.

Fractal theory continues to deepen its application in hydrocarbon resource assessment. It is widely employed in methodologies such as reservoir classification and evaluation, pore structure characterization, flow behavior analysis, and fracture system modeling. While fractal theory has driven breakthroughs in assessment techniques and spurred continuous improvement and innovation in practical applications, it still exhibits certain limitations that require further addressing.

The application of fractal theory offers significant advantages by enabling precise quantification of reservoir heterogeneity and revealing resource distribution patterns from pore to basin scales through parameters such as fractal dimension, substantially enhancing the accuracy of reservoir evaluation and the effectiveness of exploration efforts. Its integration with cutting-edge technologies is driving the field toward intelligent development. However, the application of fractal theory is constrained by its heavy reliance on high-quality data and the complexity of constructing high-fidelity models. The high cost and frequent incompleteness of data acquisition compromise model reliability, while models based on idealized assumptions struggle to accurately represent the transient flow and multiphase behavior characteristic of actual reservoir dynamics.

Future development of fractal theory in the petroleum field must overcome challenges related to cost, dynamic characterization, and multi-scale consistency. Leveraging artificial intelligence and digital technologies, intelligent dynamic modeling should be achieved to enable unified cross-scale methodologies. Furthermore, standardized engineering application frameworks must be established, and the scope of fractal theory should expand into emerging fields such as carbon neutrality. Ultimately, an integrated intelligent evaluation system synergizing data, models, and engineering practices will provide critical support for efficient hydrocarbon development and sustainable energy transition.