Fractal-Based Modeling and Quantitative Analysis of Hydraulic Fracture Complexity in Digital Cores

Abstract

Hydraulic fracturing in shale reservoirs is affected by microscale structural and material heterogeneity. However, studies on fracture responses to the injection rate across different microstructural types remain limited. To examine the coupled effects of microstructure and flow rate on fracture propagation and mineral damage, high-fidelity digital rock models were constructed from SEM images of shale cores, representing quartz grains and ostracod laminae. Coupled hydro-mechanical damage simulations were conducted under varying injection rates. Fracture evolution and complexity were evaluated using three quantitative parameters: stimulated reservoir area, fracture ratio, and fractal dimension. The results show that fracture morphology and mineral failure are strongly dependent on both the structure and injection rate. All three parameters increase with the flow rate, with the ostracod model showing abrupt complexity jumps at higher rates. In quartz-dominated models, fractures tend to deflect and bypass weak cement, forming branches. In ostracod-lamina models, higher injection rates promote direct penetration and multi-point propagation, resulting in a radial–branched–nested fracture structure. Mineral analysis shows that quartz exhibits brittle failure under high stress, while organic matter fails more readily in tension. These findings provide mechanistic insights into the coupled influence of microstructure and flow rate on hydraulic fracture complexity, with implications for optimizing hydraulic fracturing strategies in heterogeneous shale formations.

1. Introduction

Hydraulic fracturing has become a critical technique for enhancing the development of unconventional reservoirs [1,2,3]. The propagation path and complexity of induced fractures directly affect the formation and connectivity of fluid flow channels within the reservoir, thus determining the overall production performance [4,5]. However, shale reservoirs are characterized by pronounced microstructural heterogeneity, including spatial distribution of mineral grains, micron-scale biogenic remnants, and various organic matter occurrences. These features significantly influence the fracture initiation points, propagation trajectories, and branching behaviors [6,7,8]. Although previous studies have confirmed the influence of microstructure on fracture evolution, most remain at a qualitative level or rely on idealized models, lacking a systematic quantitative understanding of fracture behavior under realistic microstructural conditions.

Numerical simulation has become an essential tool for investigating hydraulic fracturing in heterogeneous rocks. Various approaches have been proposed to quantify rock heterogeneity, including stochastic parameter assignment, particle-based geometric modeling, and modeling of interfacial bonding strength variations [9,10,11,12,13]. Notably, the Realistic Failure Process Analysis (RFPA) method developed by Tang et al. (2002) introduced Weibull-distributed parameters to capture mesoscopic heterogeneity and its effect on fracture evolution [14,15]. Similarly, the bonded-particle model (BPM) proposed by Potyondy and Cundall (2004) significantly improved the accuracy of micromechanical simulations [16]. However, most existing studies still rely on homogeneous or oversimplified structural assumptions, failing to capture the complexity of actual shale microstructures, which limits the fidelity of fracture path prediction and mechanism analysis.

The emergence of digital rock technology has enabled high-resolution characterization of rock microstructures. Through advanced scanning and digital image processing (DIP), internal features such as mineral grains, organic matter, pores, and fractures can be accurately reconstructed to generate high-fidelity digital models [17,18,19]. For example, Yue et al. (2003) applied region growing and edge detection algorithms to improve model realism [20]. Li et al. (2023) combined CT scanning with QEMSCAN mineral analysis and utilized a U-Net deep learning framework for multicomponent shale reconstruction [21]. Li et al. (2017) used SEM images to build digital core models and, by incorporating flow–stress–damage coupling, explored the influence of brittle mineral content on fracture propagation, confirming the positive effect of brittleness on fracture stimulation [22].

Studies have shown that brittle minerals significantly affect fracture trajectories, often causing deflection and promoting fracture network complexity due to their high strength [23,24,25]. Nevertheless, most existing research still focuses on simplified geometries or idealized particle distributions, with limited attention to biogenic structures such as ostracod laminae or the true morphology of mineral grains. Ostracods are common biogenic sedimentary features in shales, are composed primarily of silicates or carbonates, and exhibit clear stratification and mechanical anisotropy [26,27,28]. These unique structures may strongly influence fracture initiation and interaction with surrounding minerals, yet remain underexplored in quantitative studies. Similarly, quartz—one of the most representative brittle minerals in shale—plays a crucial role in controlling fracture deflection, bypassing, or penetration, but its specific influence on path selection under varying injection rates has not been fully addressed [29,30].

In addition to structural characteristics, injection rate is a key operational parameter that directly impacts fracture morphology and stress disturbance zones [31]. At low injection rates, fracture growth is primarily driven by in situ stress, resulting in relatively stable and linear paths. In contrast, high injection rates cause sharp fluctuations in pore pressure, which promote extensive fracture branching, redirection, and instability, thereby increasing network complexity [32]. Existing studies mainly focus on the macroscopic effects of flow rate, with a limited understanding of how injection rate interacts with microstructure to influence fracture complexity and mineral failure, particularly under realistic structural conditions [33]. This gap hampers theoretical guidance for fracturing design and reservoir zonal stimulation.

In recent years, significant progress has been made in the numerical simulation of hydraulic fracture propagation in shales with complex structural features. For example, Wang et al. (2024) employed the finite-discrete element method (FDEM) to study the effects of stratigraphic architecture on fracture initiation and propagation in shales, revealing that fractures are prone to deflection and branching along weak interfaces [34]. Ran et al. (2023) developed multi-cluster hydraulic fracture propagation models using the extended finite element method (XFEM) and systematically analyzed the influence of stress contrast and cluster interaction on fracture morphology [35]. Chu et al. (2024) conducted a comprehensive analysis of the effects of various injection parameters on fracture propagation using an XFEM model, demonstrating that injection rate has a much greater impact on stress interference between fractures than fluid viscosity and other parameters [36]. Despite these advances, most existing studies have focused on idealized or simplified structural models. There is still a lack of systematic and quantitative understanding of the coupled effects of microscale structural complexity and injection parameters under realistic geological conditions.

To address these challenges, this study integrates SEM images of actual shale samples to construct high-fidelity digital rock models of two representative microstructures: quartz particle structures and ostracod laminae. The models incorporate coordinated representation of structure, mineralogy, and mechanical parameters. Based on the RFPA-Digital-Petroleum platform, hydro-mechanical damage coupled simulations are performed under various injection rates to systematically analyze fracture propagation and mineral failure behaviors. The study aims to elucidate how injection rate influences fracture path evolution and mineral damage in different microstructural contexts. Furthermore, three quantitative metrics, namely stimulated reservoir area (SRA), fracture ratio, and fractal dimension, are introduced to evaluate fracture complexity and stimulation effectiveness. These insights provide a quantitative foundation for optimizing hydraulic fracturing parameters in heterogeneous shale formations.

The novelty of this work lies in the construction of dual-type high-fidelity digital core models directly derived from SEM images of shale samples, representing both quartz grain structures and ostracod laminae, two representative but structurally distinct microstructural types. Unlike previous studies relying on idealized or oversimplified models, this study systematically quantifies the coupled effects of microstructure and injection rate on hydraulic fracture propagation and mineral damage using three quantitative metrics. The main contribution of this work is to provide a realistic and comparative understanding of fracture evolution mechanisms across different shale microstructures under variable operational conditions, which offers new insights for optimizing fracturing design in heterogeneous shale formations.

2. Methodology

2.1. Constitutive Model and Failure Criteria

The Realistic Failure Process Analysis (RFPA) method is a meso-mechanics-based numerical approach designed to simulate damage evolution and failure processes in brittle materials such as rocks under mechanical loading [37,38]. It incorporates statistical heterogeneity (typically modeled using a Weibull distribution) and combines an elastic–brittle constitutive relationship (stress–strain behavior illustrated in Figure 1, where compressive stress is defined as positive and tensile stress as negative) with a Mohr–Coulomb failure criterion truncated by tensile strength. This framework captures the full evolution from elastic deformation to damage accumulation and eventual failure.

The RFPA-Digital-Petroleum method extends RFPA by integrating digital image processing (DIP) techniques with a hydro-mechanical damage (HMD) coupling model, enabling dynamic simulation of hydraulic fracture propagation in geologically realistic rock structures. It has been successfully applied in studies investigating the influence of natural fracture morphology, fracture density, and brittle mineral content on hydraulic fracturing behavior in shale [39,40]. In this study, the constructed numerical models incorporate both the true mineralogical structure and mechanical parameter heterogeneity to more accurately reflect the micromechanical responses of rock. The governing equations of RFPA-Digital-Petroleum are briefly introduced below.

Based on elastic damage mechanics, element stress degrades with the damage variable D, following Lemaitre’s strain equivalence principle [41]:

$$E=(1-D)E_0$$

(1)

Under tensile loading, the damage variable is defined as [38]:

$$D=\left\{\begin{array}{ll}0& \overline{\epsilon }>{\epsilon }_{t0}\\ 1-{\displaystyle \frac{{\sigma }_{rt}}{\epsilon E_0}}& {\epsilon }_{tu}<\overline{\epsilon }\le {\epsilon }_{t0}\\ 1& \overline{\epsilon }\le {\epsilon }_{tu}\end{array}\right.$$

(2)

$${\epsilon }_{tu}=\eta {\epsilon }_{t0}$$

(3)

where the equivalent principal strain $\overline{\epsilon }$ is given by:

$$\overline{\epsilon }=-\sqrt{{〈-{\epsilon }_1〉}^2+{〈-{\epsilon }_2〉}^2+{〈-{\epsilon }_3〉}^2}$$

(4)

For compressive or shear modes, the damage variable is defined as [38]:

$$D=\left\{\begin{array}{ll}0& {\epsilon }_1>{\epsilon }_{c0}\\ 1-{\displaystyle \frac{{\sigma }_{rc}}{{\epsilon }_1{E}_0}}& {\epsilon }_{c0}\le {\epsilon }_1\end{array}\right.$$

(5)

Based on the value of D, damage status is classified into three categories: elastic state (D = 0), partial damage (0 < D < 1), and complete failure (D = 1).

The failure criterion includes two modes. The first is tensile failure, which occurs when the minimum principal strain exceeds the tensile strain threshold obtained from uniaxial tensile tests, or when the minimum principal stress exceeds the tensile strength [40]:

$$\left\{\begin{array}{l}{\epsilon }_3\le {\displaystyle \frac{{\sigma }_{t0}}{E}}\\ {\sigma }_3\le {\sigma }_{t0}\end{array}\right.$$

(6)

The second mode is shear failure, which is deemed to occur when the shear stress state satisfies the Mohr–Coulomb failure criterion [42]. The corresponding mathematical expression is given as follows:

$${\sigma }_1-{\sigma }_3{\displaystyle \frac{1+\sin \varphi }{1-\sin \varphi }}\ge {\sigma }_{c0}$$

(7)

Once an element meets either criterion, damage initiates and accumulates, and its mechanical properties degrade progressively with increasing D.

2.2. Hydro-Mechanical Damage Coupling Algorithm

In this study, a coupled hydro-mechanical damage model is employed to simulate the flow behavior of fracturing fluid within fractured porous media. The injected fluid increases the pore pressure, which alters the stress state of elements via the effective stress principle, thereby initiating damage evolution. In turn, the evolving damage feeds back into the system by modifying the mechanical properties of the rock and its permeability characteristics, resulting in a multi-physical field interaction process.

For isotropic, linear poroelastic media, the stress–strain relationship under fluid pressure is expressed as [43]:

$$\sigma =2G\epsilon +{\displaystyle \frac{2\nu G}{1-2\nu }}tr(\epsilon )\mathrm{I}-\alpha p_F\mathrm{I}$$

(8)

The equilibrium equations and the strain–displacement relationships can be expressed as follows:

$$\nabla \sigma +f=0$$

(9)

$$\epsilon ={\displaystyle \frac{1}{2}}\left({(\nabla \mathrm{u})}^T+\nabla \mathrm{u}\right)$$

(10)

Assuming the element has isotropic permeability and that fluid flow follows Darcy’s law, the governing equation can be expressed as [44]:

$${\displaystyle \frac{\rho gk}{\mu }}{\nabla }^{2}H-q=S_s{\displaystyle \frac{\partial H}{\partial t}}$$

(11)

where $S_s$ (m⁻¹) is the storage coefficient, defined as [44]:

$$S_s=\rho g(c_M+n\beta )$$

(12)

Rock compressibility $c_M$ may be determined experimentally or approximated using the uniaxial compressibility coefficient [45]:

$$c_M={(\lambda +2G)}^{-1}={\displaystyle \frac{(1+\nu )(1-2\nu )}{1-\nu }}{\displaystyle \frac{1}{E}}$$

(13)

During the hydraulic fracturing process, permeability is influenced by both stress and damage state. In the elastic stage (D = 0), the variation of permeability with stress is described as follows [46]:

$$k_e=k_0{e^{-b}}^{({\displaystyle \frac{{\sigma }_{ii}}{3}}-\alpha p)}$$

(14)

When an element is partially damaged under tensile conditions (0 < D < 1), microcracks develop and evolve into flow pathways. Fluid flow through these cracks is modeled as laminar flow following the cubic law, and the unit-width discharge is given by [47]:

$$q_0={\displaystyle \frac{d_0^3\rho g}{12\mu }}{J}_f=k_d{\displaystyle \frac{\rho g}{\mu }}A{J}_f$$

(15)

The permeability of partially damaged elements is approximated as [48]:

$$k_d={\displaystyle \frac{d_0^3}{12}}$$

(16)

To simplify the approximation of hydraulic fracture aperture, fractures with a length much greater than width are assumed to be height-contained fractures. Accordingly, the fracture aperture $d_0$ can be expressed as a linear function of net pressure [49]:

$$d_0\approx w={\displaystyle \frac{2p_{net}{h}_{Ff}(1-{\nu }^2)}{E}}$$

(17)

where net pressure $p_{net}$ is defined as the difference between internal fluid pressure and the minimum in situ stress [43]:

$$p_{net}=p_f-{\sigma }_c$$

(18)

The permeability for a fully failed unit (D = 1) can be determined by the following formula [50]:

$$k_f={\displaystyle \frac{d_f^2}{32}}$$

(19)

where the equivalent hydraulic aperture $d_f$ is calculated based on the mesh element size $a$, ${d_f}^2=4a^2/\pi$. In most hydraulic fracturing simulations, damaged zones are dominated by tensile failure. Equations (14), (16), and (19) describe the permeability evolution for elastic, damaged, and failed elements, respectively.

3. Numerical Model Construction

To investigate the effects of representative microstructures on hydraulic fracture propagation and mineral damage characteristics in shale under varying injection rates, high-fidelity digital core models were constructed based on high-resolution SEM images. These models represent two typical microstructural types: a quartz mineral grain model and an ostracod-lamina structure model. Each model incorporates realistic structural geometry, accurate mineralogical assignments, and coupled mechanical–hydraulic parameters, providing a reliable foundation for subsequent HMD coupling simulations.

Considering the geometric and scale differences between the microstructural types, image resolution was selectively adjusted during acquisition. For the quartz model, higher resolution was applied to capture fine details such as grain boundaries, intergranular pores, and crystallographic features. In contrast, the ostracod-lamina model, as a macroscopically identifiable biogenic structure, presents more prominent morphological features and thus requires relatively lower resolution. To balance structural fidelity and representative scale, slight differences in image dimensions were permitted under the constraint of consistent spatial scaling.

Figure 2 shows the original SEM images and corresponding multivalued images of the quartz and ostracod samples. A representative region (highlighted in red) from the central portion of each image was extracted to construct the 2D digital core model, which was then converted into a mechanical simulation mesh incorporating spatially distributed mineral properties. Figure 3 illustrates the final hydraulic fracturing models, where a central circular injection borehole simulates a real fracturing well. A uniform 509 × 509 mesh was used across all models to ensure consistent structural resolution and facilitate comparative analysis.

In terms of dimensions, the quartz model measures 60 μm × 60 μm with a borehole diameter of 1.50 μm, while the ostracod-lamina model is 140 μm × 140 μm with a borehole diameter of 3.50 μm. All relevant mineral components within the models, including organic matter, clay minerals, carbonate minerals, and quartz/silicate minerals, were assigned their respective mechanical and permeability parameters.

The determination of mechanical parameters for each mineral phase followed the workflow shown in Figure 4. Initially, targeted nanoindentation tests were performed. These involved SEM localization of mineral phases, followed by indentation tests at specified locations. The resulting data were analyzed using the Oliver–Pharr method to obtain initial mechanical parameters for each mineral (such as hardness and elastic modulus). Subsequently, large-scale grid nanoindentation was conducted to improve the statistical reliability of the measurements. The mixed indentation data were then deconvoluted to extract the final mechanical parameter distributions and Weibull modulus (m) values for each mineral phase. Detailed data for the mechanical parameters of each mineral are provided in

Table 1. Physical parameters of typical mineral components used in numerical modeling.

Mineral Type	${\mathit{E}}_0$ (GPa)	${\mathit{m}}_{{\mathbf{E}}_0}$	$\mathbf{UCS}$ (MPa)	${\mathit{m}}_{\mathbf{UCS}}$	$\mathit{\nu }$	k (m2)
Qtz	102.44	10.43	507.68	9.84	0.07	6.2 × 1020
Cb	58.45	6.38	315.44	4.21	0.28	6.2 × 1019
Osh	85.13	8.25	471.68	8.31	0.18	6.2 × 1019
Cl	35.86	6.09	143.85	1.73	0.34	6.2 × 1018
OM	8.05	2.15	94.48	2.09	0.14	6.2 × 1017

, and the properties of the fracturing fluid are listed in

Table 2. Fluid parameters used in the hydraulic fracturing simulation.

Fluid Parameter	Symbol	Reference Value	Unit
Fluid density	$\rho$	1000	kg/m3
Storage coefficient	Ss	7.3 × 10−6	m−1
Dynamic viscosity	$\mu$	0.001	$\mathrm{Pa}\cdot \mathrm{s}$
Volumetric source	$\mathrm{Q}$	0.012	m3/s

.

To quantitatively evaluate the spatial distribution of mineral phases and the geometric complexity of hydraulic fractures in digital rock models, this study employs the box-counting dimension for fractal analysis, as shown in Figure 5a. The method involves overlaying a series of square grids with various sizes (δ) on the binary images, such as mineral phase distributions or fracture networks. For each grid size, the number of boxes containing the target structure, denoted as N(δ), is counted. The process is repeated as the grid size decreases, and the values of N(δ) are recorded for each scale.

The relationship between the number of boxes and the grid size follows a power law:

$$N(\delta )\sim {\delta }^{-d}$$

(20)

where d s the fractal, or box-counting, dimension.

Taking the logarithm of both sides gives a linear relationship:

$$\log N(\delta )=d\log (1/\delta )+C$$

(21)

where C is a constant.

As shown in Figure 5b, the fractal dimension d is obtained as the slope from the linear regression of logN(δ) against log(1/δ). The fractal dimension quantitatively characterizes the spatial complexity of mineral phases and fracture networks. A higher fractal dimension indicates a more complex structure. This metric can be used to compare fracture complexity in different models or under different conditions. The volume fraction and fractal dimension of each mineral phase in various structural models are listed in

Table 3. Volume fraction and fractal dimension of minerals in numerical models.

Model Type	dOM	VOM (%)	dCl	VCl (%)	dCb	VCb (%)	dQtz/Osh	VQtz/Osh (%)
QPM	1.46	6.94	1.78	56.63	1.73	20.49	1.33	15.94
OLM	1.37	2.56	1.7	23.07	1.72	56.01	1.3	18.36

Note: QPM = Quartz Particle Model; OLM = Ostracod Laminae Model.

.

To systematically assess the influence of the injection rate on fracture morphology, complexity evolution, and mineral response mechanisms, four injection rates were applied: 0.08 m³/s, 0.12 m³/s, 0.16 m³/s, and 0.20 m³/s. The model uses a unit thickness assumption, and all volumetric rates are expressed as total flow per this reference thickness. In all simulations, boundary conditions, material parameters, and numerical strategies were kept constant except for the injection rate. This ensured control of a single variable, enhancing the reliability and interpretability of cross-model comparisons.

4. Simulation Results and Analysis

To investigate the hydraulic fracture propagation behavior and mineral damage characteristics in shale with typical microstructures under varying injection rates, this section focuses on the evolution of fracture morphology, changes in quantitative complexity indicators, and mineral response patterns.

4.1. Quartz Particle Model

4.1.1. Hydraulic Fracture Morphology

Figure 6 illustrates the evolution of fracture morphology in the quartz particle model under different injection rates. The results indicate that the injection rate has a significant impact on fracture paths, network complexity, and interactions with quartz grains. At a low injection rate of 0.08 m³/s, the fracture propagates along the direction of the maximum principal stress in a relatively linear and regular manner, with short length and limited branching. This behavior is attributed to the minimal increase in pore pressure and relatively gentle variation in the internal stress field under low-rate injection, which suppresses nonlinear fracture development. In this case, the fracture path is primarily governed by the in situ stress field and the distribution of weak minerals (e.g., organic matter) within the matrix.

As the injection rate increases to 0.12 m³/s, significant branching appears along the main fracture, extending in multiple directions and forming a more complex network. The enhanced fluid pressure gradient and local stress concentration lead to deflection or bifurcation at grain boundaries or weakly bonded zones, resulting in a more diverse spatial propagation pattern. At 0.16 m³/s, the fracture exhibits rapid propagation and markedly increased complexity, forming a characteristic branched and interwoven network. The fracture length increases, and the tips display a “mushroom-head” shape, indicating that local stress heterogeneity triggers mixed tensile–shear failure mechanisms. Under high fluid velocity, the fracture penetrates some quartz-dominated regions, even showing intergranular propagation.

However, when the injection rate reaches 0.20 m³/s, although the fracture morphology becomes increasingly complex, the fracture length tends to plateau or slightly decrease. This is because the increased number of branches expands the overall stimulated region but also leads to significant energy dissipation, thereby impeding the further extension of the main fracture as energy is diverted into numerous secondary branches.

Further analysis of a local quartz-dominated region (PA1) reveals various interaction behaviors between fractures and grains, including bypassing, deflection, and penetration. When encountering large quartz grains, fractures may either propagate along the grain boundary or penetrate the grain itself. The actual propagation path is controlled by a combination of factors, such as grain spatial distribution, cementation quality, and local stress concentration, resulting in high complexity and nonlinearity.

Multiple low-energy secondary branches tend to form around grains, and their propagation paths depend on the local energy–resistance balance. According to the principle of minimum energy dissipation, fractures prefer the least energy-consuming path. If the energy required to penetrate a grain is greater than that needed to bypass it, the fracture will extend along the grain boundary; otherwise, transgranular propagation may occur.

In summary, the diversity and evolution of fracture morphology reflect the fracture’s adaptive path selection in response to complex mineral structures under varying injection conditions. This behavior highlights an intrinsic mechanism of energy optimization during fracture evolution.

4.1.2. Hydraulic Fracture Parameters

Figure 7 presents the variation in hydraulic fracture parameters under different injection rates in the quartz particle model, aiming to quantify the stimulation effectiveness and geometric complexity of the fracture network.

As shown in Figure 7a, the stimulated reservoir area (SRA) increases significantly with the rising injection rate, exhibiting an approximately linear trend. When the rate increases from 0.08 m³/s to 0.20 m³/s, the SRA expands from 141.31 × 10⁻¹² m² to 440.18 × 10⁻¹² m², marking a 215.89% increase (298.87 × 10⁻¹² m²). This indicates that the injection rate has a strong positive effect on the extent of fluid influence, thereby enhancing stimulation performance in the quartz-dominated heterogeneous model.

Figure 7b shows the evolution of the fracture ratio and fractal dimension with the injection rate. Both parameters exhibit a rising trend, reflecting enhanced fracture development and increasing network complexity. Specifically, the fracture ratio increases from 0.0223 to 0.0324 (a 45.28% rise), and the fractal dimension increases from 1.151 to 1.221 (a 6.07% increase). These results demonstrate that higher injection rates not only promote spatial fracture propagation but also intensify geometric irregularity. In terms of flow rate sensitivity, the parameters rank as follows: SRA > fracture ratio > fractal dimension.

4.1.3. Mineral Damage

Figure 8 analyzes the effects of injection rate on mineral damage in the quartz particle model from the perspective of different mineral compositions. As shown in Figure 8a, the damage rate of all mineral types generally increases with the injection rate. Notably, at 0.16 m³/s, the quartz damage rate shows a marked surge, surpassing that of organic matter. This suggests that under specific high-stress conditions, brittle minerals may undergo intense localized failure. Apart from this condition, the order of mineral susceptibility (from lowest to highest) under different injection rates is quartz < carbonate < organic matter < clay, indicating that weaker minerals are more prone to fluid-induced damage. In terms of growth rate, when the injection rate increases from 0.08 m³/s to 0.20 m³/s, the damage rate increases are as follows: quartz (264.47%), organic matter (136.67%), carbonate minerals (104.08%), and clay minerals (83.54%). Although quartz shows a lower absolute damage rate, it is most sensitive to flow rate changes, likely due to stress concentration effects that induce failure in high flow environments.

Figure 8b further shows the evolution of tensile and compressive–shear damage rates in organic matter under different injection conditions. Both modes exhibit increasing trends with rising flow rate, but their sensitivities differ significantly. Specifically, the compressive–shear damage rate increases from 0.11% to 0.15% (a 36.84% increase), while the tensile failure rate increases from 0.40% to 1.05%, representing a 163.38% rise. This indicates that tensile damage is more sensitive to flow rate and is the dominant damage mechanism in organic matter, a mechanically weak and structurally significant component of shale reservoirs.

4.2. Ostracod Laminae Model

4.2.1. Hydraulic Fracture Morphology

Figure 9 presents the evolution of hydraulic fracture behavior in the ostracod laminae model under different injection rates. The simulation results demonstrate that the injection rate significantly affects fracture morphology, path selection, and propagation patterns within and around microscale biogenic structures such as ostracod shells. At a low injection rate of 0.08 m³/s, fluid infiltrates the interior of the ostracod shell at a slow rate, leading to gradual pore pressure buildup. Fractures initiate from the central region and propagate outward; however, the pressure is insufficient to break through the shell, resulting in several short fractures confined within or near the shell boundary. These fractures exhibit limited spatial development and penetration capability. When the injection rate increases to 0.12 m³/s, fractures first penetrate the ostracod shell, indicating that the pore pressure has reached a critical threshold. However, due to the high energy required to break through the shell, further propagation beyond the ostracod boundary is limited, ultimately forming a relatively simple main fracture with few branches.

At 0.16 m³/s, fractures initiate simultaneously at multiple weak points along the ostracod shell and penetrate it at several locations, forming a more complex, branched, or radially distributed fracture network. This suggests that a moderately high injection rate can induce cooperative structural failure, enhancing the likelihood of multi-path fracture formation. When the injection rate increases further to 0.20 m³/s, stress rapidly accumulates within the ostracod structure and is violently released, leading to extensive fracture propagation through the laminae in multiple directions. The fracture front exhibits a distinctive “mushroom-shaped” morphology, reflecting rapid evolution, branching, and local tensile damage within the complex structure. The increase in injection rate significantly enhances both the fracture penetration capacity and propagation complexity in the ostracod model. From non-penetration to partial breakthrough and ultimately to multi-path, geometrically complex propagation, the fracture behavior demonstrates a strong dependence on injection rate.

4.2.2. Hydraulic Fracture Parameters

Figure 10 illustrates the variation in hydraulic fracture parameters under different injection rates in the ostracod laminae model, aiming to analyze how stimulation effectiveness and geometric complexity respond to flow rate changes. As shown in Figure 10a, the SRA increases significantly with increasing injection rate. When the rate rises from 0.08 m³/s to 0.20 m³/s, the SRA grows from 812.76 × 10⁻¹² μm² to 2363.63 × 10⁻¹² m²—an increase of 1550.87 × 10⁻¹² m² or 191.00%. This indicates that in shale models dominated by complex biogenic structures, increasing the injection rate greatly enhances the reservoir stimulation range.

Figure 10b further reveals the response of the fracture ratio and fractal dimension to changes in he injection rate. In the range of 0.08–0.16 m³/s, both parameters remain relatively stable, indicating limited geometric variation. This suggests that fracture propagation is still constrained by the structural integrity of the ostracod shells during this stage. However, when the rate increases from 0.16 m³/s to 0.20 m³/s, both the fracture ratio and fractal dimension show abrupt increases from 0.0270 to 0.0605 (123.96% increase) and from 1.185 to 1.329 (12.17% increase), respectively. This transitional behavior suggests that once the fracture overcomes the structural barrier of the shell, a significant increase in geometric and topological complexity occurs, leading to more connected and nonlinear fracture networks under high flow conditions.

4.2.3. Mineral Damage

Figure 11 examines how the injection rate influences the damage behavior of various mineral types in the ostracod laminae model from the perspective of mineralogical response.

As shown in Figure 11a, the damage rates of major mineral constituents exhibit distinct trends under different injection rates. The damage rate of the ostracod shell composed primarily of carbonate minerals shows a consistent increase with rising flow rate, indicating its sensitivity to localized stress concentrations and its tendency to undergo brittle failure at higher injection rates. In contrast, the other three mineral types—organic matter (OM), clay (Cl), and carbonate (Cb)—display a general decrease in the damage rate in the low-to-moderate flow range (0.08–0.16 m³/s), likely due to the limited penetration of fractures into their core regions. However, when the injection rate increases to 0.20 m³/s, the damage rates of all mineral types rise sharply, suggesting that once a threshold flow rate is exceeded, fracture propagation becomes more extensive, triggering structural failure across multiple mineral phases. It is noteworthy that at 0.08 m³/s, the damage rate of carbonate minerals is higher than that of the ostracod shell, whereas at 0.20 m³/s, the damage rate of organic matter surpasses that of carbonates. This indicates that different minerals possess distinct sensitivity thresholds to injection rate. Overall, under increasing flow rates, the damage rates of minerals tend to follow the order organic matter > carbonate grains > ostracod shell > clay minerals.

Figure 11b further analyzes the failure mode evolution of organic matter under different injection conditions. The compressive–shear damage rate of organic matter fluctuates between 0.08 and 0.20 m³/s without a clear trend, possibly due to local stress redistribution and unstable shear failure induced by fracture deflection. In contrast, the tensile damage rate of organic matter shows a more defined trend—gradually decreasing between 0.08 and 0.16 m³/s, followed by a marked increase at 0.20 m³/s. This suggests that under high injection rates, tensile stress becomes the dominant mechanism driving structural failure in organic matter.

5. Discussion

This study utilized digital rock models extracted from SEM images of two representative shale microstructures, namely quartz grain and ostracod lamina. We systematically analyzed the differences in fracture response patterns, geometric complexity evolution, and mineral damage mechanisms between these structures. The results show that fractures in the quartz grain structure display deflection, branching, bypassing, and penetration. Their propagation paths are governed by the combined influence of grain distribution and stress perturbation, leading to highly nonlinear evolution. These findings are consistent with the conclusions of He et al. [51], who observed multiple fracture behaviors when hydraulic fractures encounter hard mineral grains (see Figure 12a). The results also agree with the observations of Lei et al., who reported that fractures tend to propagate along weak interfaces under tensile stress conditions (see Figure 12b) [25]. This further validates the accuracy and applicability of our model in capturing the controlling mechanisms of microstructures.

In contrast, the ostracod lamina structure, due to its spatial layering and biological features, tends to promote local penetration, multi-point extension, and complex radial or nested fracture fronts at high injection rates. This highlights the strong guiding effect of layered structures, while high-strength components such as quartz grains primarily contribute to fracture path dispersion and perturbation.

The two structures exhibit significant differences in their sensitivity to injection rate. The ostracod model shows a much higher increase in stimulated reservoir area, indicating that complex biological units have a greater potential for fracture activation and reservoir stimulation. In the quartz model, both fractal dimension and fracture ratio increase almost linearly with injection rate, suggesting that structural complexity facilitates the development of fracture networks. Further analysis indicates that quartz grains are prone to localized brittle failure under high injection rates. The sensitivity of their failure rate to flow rate changes even exceeds that of weak organic matter. This suggests that high-strength grains are not always stable and may experience abrupt failure under critical stress conditions.

These results not only provide qualitative insights into how structural heterogeneity controls fracture behavior but also demonstrate the flexibility of digital rock modeling in studying the coupling between rock microstructure and fracture evolution. Although only a single realization was analyzed for each type, the modeling framework can be extended to other lithologies by incorporating their specific microstructural features. Future studies should include multiple realizations and shale types to further evaluate the robustness and universality of the observed mechanisms.

This work primarily focused on injection rate and microstructural type, while other parameters such as fluid viscosity, in situ stress anisotropy, and temperature were not systematically investigated. Although these variables can significantly affect fracture propagation in practical operations, they were kept constant here to identify dominant mechanisms. Future research should expand the parameter space and conduct multifactor coupling and sensitivity analyses to fully reveal the synergistic effects of different physical variables on fracture evolution. The current numerical model is also based on 2D sections, which cannot fully capture stress transmission and fracture network topology in 3D. This may lead to an underestimation of the mechanical disturbances and connectivity complexity arising from vertical heterogeneity. Future work will involve 3D high-resolution image reconstruction and voxel-based modeling to build digital rock models with spatial lamination, grain inclusions, and intersecting fracture networks, thereby improving physical realism and applicability.

Compared with previous studies that mainly used idealized grain distributions or simplified laminations [52,53,54,55], this work reconstructs high-fidelity digital rock models from real SEM images at the two-dimensional scale. By integrating a quantitative parameter system, we systematically reveal the coupled control of microstructure, injection rate, and mineral failure. This broadens the understanding of fracture formation mechanisms and provides theoretical guidance for hydraulic fracturing design optimization.

These findings have important implications for field-scale fracturing design. The observed differences in fracture patterns between quartz-dominated and ostracod-dominated structures indicate that microstructural heterogeneity should be fully considered when selecting fracturing intervals. For example, high injection rates in quartz-rich zones can help expand the stimulated reservoir area, while layered ostracod zones may require customized strategies such as variable rate injection or staged isolation. Mineral damage characteristics also suggest that fracture geometry control should be balanced with mineral integrity to prevent fine migration and permeability reduction. Therefore, high-resolution rock fabric analysis should be incorporated into pre-fracturing geological evaluation and operational planning.

6. Conclusions

This study reconstructed high-fidelity digital rock models of two representative shale microstructures, quartz grains and ostracod laminae, based on high-resolution SEM images. Using the RFPA-Digital-Petroleum platform, a series of hydro-mechanical damage coupled simulations was conducted to systematically investigate the influence of varying injection rates on hydraulic fracture propagation and mineral damage behavior. The main conclusions are as follows:

(1)

Fracture morphology is strongly structure-dependent. In the quartz grain model, fractures exhibit frequent deflection, bypassing, and multi-path branching, forming complex fracture networks. In contrast, the ostracod laminae model displays through-fracturing, radial propagation, and “mushroom-shaped” fracture fronts at high injection rates, highlighting the role of its shell-like architecture in directing fracture evolution.

(2)

Fracture complexity increases with injection rate, with structure-specific response patterns. While the quartz model shows gradual and steady fracture development, the ostracod model exhibits nonlinear, abrupt increases in complexity at higher flow rates, suggesting that intricate biological structures are more prone to triggering topological transitions in fracture geometry.

(3)

Different mineral types respond distinctly to injection rate variations. Hard minerals such as quartz tend to fail in a brittle manner under high stress, whereas weak components like organic matter are more prone to tensile failure, often serving as primary pathways for fracture propagation.

(4)

Fracture evolution and mineral response are jointly governed by the interaction between microstructure and injection rate. Quartz grains influence fracture trajectories primarily through energy dissipation optimization, while ostracod laminae exhibit a transition from resistance to cooperative failure with increasing flow rate, reflecting a flow-driven shift in fracture behavior.