Mechanical, Seepage, and Energy Evolution Properties of Multi-Shaped Fractured Sandstone Under Hydro-Mechanical Coupling: An Experimental Study

Abstract

Rocks with multi-shaped fractures in engineering activities like mining, underground energy storage, and hydropower construction are often exposed to environments where stress and seepage fields interact, which heightens the uncertainty of instability and failure mechanisms. This has long been a long-standing challenge in the field of rock mechanics. Current research mainly focuses on the mechanical behavior, seepage, and energy evolution characteristics of single-fractured rocks under hydro-mechanical coupling. However, studies on the effects of multi-shaped fractures (such as T-shaped fractures, Y-shaped fractures, etc.) on these characteristics under hydro-mechanical coupling are relatively scarce. This study aims to provide new insights into this field by conducting hydro-mechanical coupling tests on multi-shaped fractured sandstones (single fractures, T-shaped fractures, Y-shaped fractures) with different inclination angles. The results show that hydro-mechanical coupling significantly reduces the peak strength, damage stress, crack initiation stress, and closure stress of fractured sandstone. The permeability jump factor (ξ) demonstrates the permeability enhancement effects of different fracture shapes. The ξ values for single fractures, T-shaped fractures, and Y-shaped fractures are all less than 2, indicating that fracture shape has a relatively minor impact on permeability enhancement. Fracture inclination and shape significantly affect the energy storage capacity of the rock mass, and the release of energy exhibits a nonlinear relationship with fracture propagation. An in-depth analysis of energy evolution characteristics under the influence of fracture shape and inclination reveals the transition pattern of the dominant role of energy competition in the progressive failure process. Microstructural analysis of fractured sandstone shows that elastic energy primarily drives fracture propagation and the elastic deformation of grains, while dissipative energy promotes particle fragmentation, grain boundary sliding, and plastic deformation, leading to severe grain breakage. The study provides important theoretical support for understanding the failure mechanisms of multi-shaped fractured sandstone under hydro-mechanical coupling.

1. Introduction

In the fields of rock mechanics and engineering, understanding the mechanical behavior, permeability characteristics, and energy evolution laws of fractured rocks under hydro-mechanical coupling is critical for various engineering applications, including underground mining, hydrogen storage in geological formations, tunnel excavation, and hydropower construction [1,2,3,4]. Among the various types of fractures, factors such as their geometry, size, orientation, and connectivity are fundamental in determining the mechanical behavior of rocks [5,6,7]. These fractures, widely present in nature, have garnered significant attention due to their impact on rock mass stability [8,9,10].

In recent years, as research deepens, scholars have extensively explored the mechanical behavior of rocks under different hydro-mechanical coupling conditions. Ding and Tang found that with an increase in water confining pressure, the peak stress and elastic modulus of saturated rocks increased linearly, while for dry rocks, the peak stress and elastic modulus first decreased and then increased sharply [11]. Zhang et al. showed that fracture shape and inclination have minimal effects on the stress ratio, and that water pressure exerts different effects on local tensile and shear failure in rocks, particularly in multi-shaped fractured sandstones where fracture shape, inclination, and water pressure collectively determine the failure mechanisms of the rock [12]. Liu et al. conducted triaxial tests to study the mechanical properties of saturated sandstone and found that parameters such as strength, elastic modulus, and fracture angle were positively correlated with confining pressure and negatively correlated with pore pressure, while Poisson’s ratio showed the opposite trend [13]. Zhao et al. reviewed the research progress on hydro-mechanical coupling in rocks and proposed new coupling theories and mathematical models [14]. Meng et al. investigated the effects of dry–wet cycles and salinity on the microstructural evolution of sandstone, revealing the close relationship between the rock’s microstructure and its macroscopic mechanical properties [15]. Shao et al. used in situ micro-CT imaging to analyze 3D printed rock-like samples with single defects, finding that the failure modes and behaviors were similar to those of natural rocks [16].

In addition to mechanical properties, the permeability of rocks under hydro-mechanical coupling is also an important research direction. Chen and Zhou revealed the nonlinear permeability behavior influenced by pore pressure and confining pressure through studies on hydro-mechanical coupling, identifying key parameters that differentiate linear and nonlinear flow [17,18]. Liu et al. found that permeation pressure significantly affects rock strength and deformation and established an appropriate coupling constitutive model [19]. Sharifzadeh and Mehrishal conducted hydro-mechanical coupling tests on artificial granite joint samples, revealing the variation in joint permeability and accurately simulating the test results using a new flow model [20]. Teng et al. developed an effective stress-sensitive permeability model that considers the effects of external stress and internal water pressure and validated the model through permeability experiments and numerical simulations [21]. Other researchers have revealed the permeability changes in natural and artificial fractures induced by fatigue loads [22], studied the permeability evolution of tunnel lining cracks under hydro-mechanical coupling [23] and emphasized the complexity of the hydro-mechanical coupling process.

At present, considerable progress has been made in the study of rock energy evolution. Song et al. proposed the energy dissipation and fracture mechanisms of layered sandstone under hydro-mechanical unloading, emphasizing the effects of confining pressure and layering angle on energy evolution. They also highlighted the diversity of rock energy flow and failure mechanisms under complex hydro-mechanical coupling, revealing the attenuation of peak strength and failure degree of sandstone under different loading paths [24]. Wang et al. used triaxial compression tests to study the changes in total strain energy, elastic strain energy, and dissipated energy of water-saturated coal under different moisture and confining pressure conditions, providing a theoretical basis for evaluating the safety and stability of underground engineering [25]. Zhu et al. investigated the energy evolution characteristics of cement-based composites under tensile and compressive conditions, finding that energy dissipation characteristics are crucial for material failure criteria [26]. Xu et al. studied the mechanical properties and energy evolution characteristics of tunnel lining concrete in western mines under hydro-mechanical coupling, proposing new strength decay equations and effective shear strength reduction coefficients, and determining that the energy consumption ratio can serve as an indicator for assessing the strength decay of tunnel lining concrete [27]. Other researchers have studied the energy variation in sandstone during the post-peak cyclic loading and unloading process under hydro-mechanical coupling conditions [28] as well as the energy evolution mechanism and bearing capacity degradation of pre-fractured sandstone under non-uniform cyclic loading [29]. Additionally, Xue et al. conducted shear, tensile, and uniaxial compression experiments to analyze the changes in energy parameters at different loading stages and their relationship with various failure modes, proposing an energy-dissipation-based health assessment technique [30]. Zhang et al. performed direct shear tests on marble samples of different sizes and shapes, proposing a strain energy density calculation method considering shear stress state [31].

In summary, although extensive research has been conducted on the mechanical properties of rocks under hydro-mechanical coupling, the mechanical, seepage, and energy evolution characteristics of rocks under hydro-mechanical coupling remain a complex, systematic issue. This involves various rock materials, loading methods, and engineering contexts. Furthermore, studies on the mechanical behavior, permeability, and energy evolution of multi-fracture shape sandstones under hydro-mechanical coupling are still relatively limited, with most research focusing on the impact of a single type of fracture. A comprehensive analysis of the hydro-mechanical coupling effects of multi-fracture shape rocks is lacking. Therefore, this study conducts hydro-mechanical coupling tests on sandstones with different fracture shapes (single fracture, T-shaped fracture, Y-shaped fracture, etc.) and inclinations, aiming to reveal the mechanical, seepage, and energy evolution characteristics of multi-fracture shape sandstones and their interrelationships and to provide suggestions for future research directions based on the findings.

2. Experiment Preparation and Procedures

2.1. Sample Preparation

The samples used in the experiment were all sourced from the same rock, with no visible defects observed on their surfaces, ensuring consistency between the samples. According to the standards of the International Society for Rock Mechanics (ISRM), the recommended standard diameter for cylindrical rock samples is approximately 54 mm, with a height-to-diameter ratio ranging from 2.5 to 3.0. However, for practical reasons, the samples in this study were processed into cylindrical sandstone specimens with dimensions of ϕ50 mm × 100 mm after drilling, cutting, and polishing. The flatness of the ends of the samples was required to be within 0.02 mm, and the axis of the sample should be perpendicular to the sample ends, with a deviation not exceeding 0.001 rad.

As shown in Figure 1, analysis using a polarizing microscope (ZEISS, Oberkochen, Germany), revealed that the rock has a fine to medium-grained sandstone texture, composed of quartz, amphibole, lithic fragments, and matrix material. The mineral composition is as follows: quartz (80%), occasional amphibole, lithic fragments (5%–10%), and matrix material (10%). The grain size of quartz ranges from 0.06 to 0.50 mm, amphibole from 0.10 to 0.20 mm, and lithic fragments from 0.10 to 0.40 mm. The matrix material contains less than 5% volcanic ash, approximately 5% clay minerals, and opaque minerals with a grain size between 0.05 and 0.25 mm, comprising less than 5%. Zircon is occasionally present with a grain size of 0.10 to 0.20 mm.

As shown in Figure 2, X-ray diffractometer (XRD) and X-ray fluorescence (XRF) tests indicate that the chemical composition of the sandstone samples, in order of content, is as follows: SiO₂ (89.6611%), Al₂O₃ (7.2145%), Fe₂O₃ (1.2425%), TiO₂ (0.738%), K₂O (0.3774%), MgO (0.2666%), CaO (0.1216%), P₂O₅ (0.0947%), WO₃ (0.0717%), MnO (0.0565%), SO₃ (0.0527%), ZrO₂ (0.0354%), Na₂O (0.0265%), Cr₂O₃ (0.0187%), Cl (0.0137%), NiO (0.0041%), SrO (0.0023%), Y₂O₃ (0.0020%).

Additionally, the uniaxial compressive strength of the intact sandstone sample is 97.54 MPa, the Young’s modulus is 17.03 GPa, and the Poisson’s ratio is 0.17. The sandstone samples have an average density of 2240.33 kg/m³, an average P-wave velocity of 2190 m/s, and a porosity of 19.80%, as measured by mercury intrusion porosimetry.

As shown in Figure 3 and Figure 4, cylindrical sandstone samples (ϕ50 mm × 100 mm) were prefabricated with single, T-shaped, and Y-shaped fractures using a water jet cutter and wire cutting equipment. The single fracture measures 20 mm in length with an aperture of 0.3 mm. Both the T-shaped and Y-shaped fractures consist of three fractures, each 10 mm in length and with an aperture of 0.3 mm. The fracture geometries are defined by the angle parameter α, which represents the angle between fractures A, B, F, and the sample’s axis, with values set at 0°, 15°, 30°, 45°, 60°, 75°, and 90°.

2.2. Testing Methods

To examine the mechanical properties, energy evolution, and seepage behavior of sandstone samples with single, T-shaped, and Y-shaped fractures under hydro-mechanical coupling, a confining pressure of 10 MPa and a water pressure of 3 MPa were applied based on the distribution of in situ stresses and the measured static water pressure of the aquifer. The detailed experimental setup is provided in

Table 1. Experimental setup for fractured sandstone samples.

Fracture Inclination Angle α	Sample No.			Confining Pressure/MPa	Water Pressure/MPa
	Single Fracture	T-Shaped	Y-Shaped
0°	SF0	ST0	SY0	10	3
15°	SF15	ST15	SY15
30°	SF30	ST30	SY30
45°	SF45	ST45	SY45
60°	SF60	ST60	SY60
75°	SF75	ST75	SY75
90°	SF90	ST90	SYT90

.

2.3. Test System

During the experiment, as shown in Figure 3, axial and confining pressures were regulated by the hydraulic servo system of the MTS815 rock mechanics testing machine, while water pressure was controlled using the Teledyne ISCO D-Series Pumps system. Permeable plates were positioned at both ends of the sample, which was wrapped in heat shrink tubing. Axial and circumferential strain gauges, as well as an acoustic emission probe, were installed. The sample was placed on a base with the top and bottom connected to the corresponding water pipes. Before testing, the sample was fully immersed in distilled water in a sealed container for 48 h to ensure complete saturation.

In the experiment, the confining pressure was first set at 10 MPa, followed by water pressure application up to 3 MPa, ensuring that the water pressure stayed lower than the confining pressure throughout. Axial loading was performed under two control modes: load and deformation (axial or circumferential). Initially, load control was used at a rate of 300 N/s. Once the load reached about 80% of the peak strength, the system switched to deformation control at a rate of 0.02 mm/min, continuing until the sample failed.

The permeability test was performed using the steady-state method with pure water as the permeating fluid. Permeability was determined using Darcy’s law, with the following equation:

$$k=-{\displaystyle \frac{\gamma QL}{\Delta PA}}$$

(1)

where k is the permeability (m²), Q represents the volume of water passing through the sample per unit time (m³/s), γ is the specific weight of water (kN/m³), L is the sample height (m), and A is the cross-sectional area of the sample (m²)—calculated as A = πd²/4—where d is the sample diameter. ΔP denotes the pressure difference across the sample (MPa).

3. Results and Analysis

3.1. Mechanical Behavior of Sandstone with Various Fracture Shapes

As shown in Figure 5, the progressive failure process of the SF30 fractured sandstone samples were divided into five stages:

Stage I: Crack Closure Stage

During this stage, microscopic pores or micro-defects are compacted, and the sandstone particles become tightly interlocked, leading to crack closure. The crack closure stress σ_cc represents the peak stress in this stage.

Stage II: Elastic Region Stage

In this stage, the pores or micro-defects in the sandstone sample continue to compact. Due to the minimal impact of hydro-mechanical coupling, micro-cracks have not yet propagated and the sample mainly exhibits elastic deformation, showing typical elastic mechanical behavior. The stress–strain curve is nearly linear, and the crack initiation stress σ_ci represents the peak stress in this stage.

Stage III: Stable Crack Growth Stage

As the loading pressure increases, damage begins to occur inside the sandstone. Plastic deformation gradually becomes apparent, micro-cracks propagate stably, and new cracks gradually converge to form macroscopic fractures. At this stage, the stress–strain curve shows a distinct nonlinear trend, and the damage stress σ_cd represents the peak stress for this stage.

Stage IV: Accelerated Crack Growth Stage

In this stage, micro-cracks rapidly propagate and coalesce into large-scale through-going fractures, forming a macroscopic failure surface. Volumetric deformation transitions from compression to expansion, and the expansion accelerates, with a nearly linear increase in the deformation rate until the sample reaches its peak strength σ_c.

Stage V: Post-Peak Region Stage

In this stage, the cracks interact with each other, and in certain directions, cracks further intersect, merge, and connect to form a macroscopic fracture surface, leading to a decrease in the sample’s bearing capacity. This is followed by a rapid decline in stress. Afterward, the sample undergoes frictional sliding along the failure surface and retains some residual load-bearing capacity.

As shown in Figure 6, the characteristic stress of sandstone samples with different fracture shapes under hydro-mechanical coupling was further investigated, focusing on how these stresses change with fracture inclination. The peak strength values corresponding to the fracture samples with different shapes and inclinations are shown in

Table 2. Peak strength values of fractured sandstone samples.

Fracture Inclination Angle α	Single Fracture Sample	Peak Strength/MPa	T-Shaped Fracture Sample	Peak Strength/MPa	Y-Shaped Fracture Sample	Peak Strength/MPa
0°	SF0	78.73	ST0	63.51	SY0	54.77
15°	SF15	67.70	ST15	64.58	SY15	60.56
30°	SF30	57.04	ST30	52.58	SY30	55.53
45°	SF45	60.92	ST45	59.19	SY45	55.29
60°	SF60	58.15	ST60	51.47	SY60	54.33
75°	SF75	54.32	ST75	55.83	SY75	61.18
90°	SF90	61.99	ST90	62.41	SYT90	54.70

. For easier comparison of the characteristic stresses for single fractures, T-shaped fractures, and Y-shaped fractures, a “normalized characteristic stress” dimensionless parameter was introduced. This parameter is defined as the ratio of the characteristic stress of the multi-fracture sandstone sample under hydro-mechanical coupling to the corresponding characteristic stress of the intact sample without water pressure, i.e., σ_c/σ_c-w, σ_cd/σ_cd-w, σ_ci/σ_ci-w, σ_cc/σ_cc-w, where σ_cc-w, σ_ci-w, σ_cd-w, σ_c-w represent the crack closure stress, crack initiation stress, damage stress, and peak strength of the intact sample without water pressure, respectively.

When the confining pressure is 10 MPa and the water pressure is 3 MPa, the ratios of characteristic stresses for single fractures, T-shaped fractures, and Y-shaped fractures showed distinct patterns with varying fracture inclination. From Figure 6a, it can be observed that the σ_c/σ_c-w ratio for single fractures is lowest at 75° and highest at 0°; for T-shaped fractures, it is lowest at 60° and highest at 15°; and for Y-shaped fractures, it is lowest at 60° and highest at 75°. The average σ_c/σ_c-w values across different inclination angles are 0.643 for single fractures, 0.600 for T-shaped fractures, and 0.581 for Y-shaped fractures.

Figure 6b reveals that the σ_cd/σ_cd-w value for single fractures is lowest at 90° and highest at 15°; for T-shaped fractures, it is lowest at 45° and highest at 90°; and for Y-shaped fractures, it is lowest at 60° and highest at 90°. The average σ_cd/σ_cd-w values at various inclination angles are 0.435 for single fractures, 0.473 for T-shaped fractures, and 0.524 for Y-shaped fractures.

In Figure 6c, the σ_ci/σ_ci-w values are lowest at 30° and highest at 15° for single fractures, lowest at 30° and highest at 90° for T-shaped fractures, and lowest at 60° and highest at 15° for Y-shaped fractures. The average σ_ci/σ_ci-w values for different inclinations are 0.517 for single fractures, 0.568 for T-shaped fractures, and 0.640 for Y-shaped fractures.

Finally, Figure 6d shows that the σ_cc/σ_cc-w values for single fractures are lowest at 30° and highest at 15°, for T-shaped fractures lowest at 45° and highest at 90°, and for Y-shaped fractures lowest at 60° and highest at 90°. The average σ_cc/σ_cc-w values for various inclinations are 0.620 for single fractures, 0.646 for T-shaped fractures, and 0.717 for Y-shaped fractures.

In conclusion, the presence of water pressure and prefabricated fractures increases the porosity of the sandstone samples, significantly reducing their peak strength, damage stress, crack initiation stress, and closure stress. This indicates that with the action of hydro-mechanical coupling, the closure stress required for pore compression and densification gradually decreases and the time for the sandstone to reach crack initiation stress, damage stress, and peak strength shortens. Furthermore, for the single-fracture samples, stress concentration mainly occurs at both ends under hydro-mechanical coupling. According to Coulomb’s criterion, the shear strength of rock is related to the normal stress. When the angle between the fracture surface and the maximum principal stress is at a critical angle (45° + Φ/2, where Φ is the internal friction angle), shear failure is most likely to occur. When the fracture inclination angle approaches this critical angle, sliding failure along the fracture surface is more likely, resulting in lower strength. In contrast, when the fracture surface is parallel or perpendicular to the principal stress direction, the strength is higher. For both T-shaped and Y-shaped fracture samples, there are three fractures, leading to multiple stress concentration points. However, the direction of crack propagation is still related to the inclination angle of the main fracture. When the main fracture is at certain inclination angles (ST60, SY60), wing cracks are more likely to propagate, resulting in lower strength. On the other hand, other inclination angles (ST15, SY75) suppress crack propagation, requiring higher stress. Due to the complex geometry of T-shaped and Y-shaped fractures, further investigation combining fracture mechanics theory is needed to explore the influence of fracture inclination on strength.

As shown in Figure 7, the trends in the elastic modulus and Poisson’s ratio of single-fracture samples at different fracture inclination angles exhibit significant differences. Specifically, for inclination angles in the range of 0° to 30°, the presence of the single fracture causes a continuous decrease in the stiffness of the sandstone as the inclination angle increases. At an inclination of 45°, there is a slight recovery, and in the range of 45° to 90°, the overall stiffness of the sandstone initially decreases, with a slight recovery again at 90°. Meanwhile, the Poisson’s ratio for the single-fracture sample shows notable fluctuations as the fracture inclination angle increases, reflecting the irregular impact of fracture morphology changes on the material’s deformation properties. For T-shaped and Y-shaped fracture samples at different inclination angles, both the elastic modulus and Poisson’s ratio exhibit more pronounced fluctuations. These fluctuations are due to the junctions of fractures at different angles, which cause directional variations in stress transfer and deformation modes, thus affecting the overall stiffness of the material. Consequently, under the combined effect of fracture morphology and water pressure, the mechanical response of T-shaped and Y-shaped fractured sandstones displays nonlinear characteristics.

3.2. Evolution of Permeability

Figure 8 shows the variation of permeability with time for single-fracture, T-shaped fracture, and Y-shaped fracture samples at different inclination angles. To further investigate the characteristics of permeability changes, this study defines the initial permeability S_ini, minimum permeability S_min, and maximum permeability S_max for sandstone samples with different morphologies and inclination angles as characteristic permeability points. The ratio S_max/S_min is defined as the jump coefficient ξ. From the permeability–time curves, it can be observed that the permeability of the multi-fracture samples undergoes a typical sequence of changes: rapid decrease, slow decline, rapid increase, sharp increase, rapid decline, and stabilization. This process is accompanied by a weakening of the cementation ability between particles due to pore water pressure, which not only softens the detrital and clayey components inside the sandstone but also accelerates crack propagation, resulting in an increase in permeability and a subsequent decrease in sample strength. In Stage IV, S_min occurs due to the rapid increase in cracks, which causes the permeability to shift from a slow decline to a rapid increase, creating a distinct inflection point. S_max appears in Stage V, which corresponds to the point of sharp stress drop. Specifically, the ξ values for single-fracture samples are 1.37, 1.54, 1.24, 1.23, 1.28, 1.39, and 1.82; for T-shaped fracture samples, they are 1.87, 1.22, 1.29, 1.22, 1.44, 1.18, and 1.16; and for Y-shaped fracture samples, they are 1.92, 1.29, 1.22, 1.26, 1.37, 1.23, and 1.30. These ξ values characterize the amplitude of permeability changes and can effectively assess the permeability enhancement effect of sandstone samples, serving as key parameters for evaluating hydraulic efficiency enhancement. At the same time, the rapid increase in permeability (characterized by ξ) corresponds to the sharp expansion of fractures and a sudden drop in stress, making it a potential early warning indicator for water inrush in underground engineering. Additionally, the ξ value can serve as a key input parameter for numerical simulations of seepage behavior in fractured rock under hydro-mechanical coupling. By incorporating ξ, the model can better capture the nonlinear changes in permeability during failure, improving the accuracy of predictions for engineering applications.

The experiments also found that at the moment of failure, the jump coefficient ξ values do not show significant differences for different fracture morphologies, indicating that the fracture morphology has a relatively small effect on the permeability increase. While fracture morphology (single, T-shaped, Y-shaped) influences the initial stress distribution and crack initiation points, the final permeability enhancement is determined by the overall network of newly formed cracks, which tends to homogenize the differences caused by initial fracture geometry. This phenomenon can be attributed to the fact that the permeability of sandstone is influenced by various factors, including the number and connectivity of fractures, fracture aperture, and effective stress. However, it should be noted that the experiments in this study were conducted under fixed water pressure and confining pressure conditions. To achieve more significant permeability enhancement, future research should consider varying the water pressure and confining pressure, thereby identifying the main factors and indicators that cause substantial changes in permeability.

3.3. Energy Evolution Analysis of Multi-Shaped Fractured Sandstone

3.3.1. Energy Dissipation Principle

Figure 9 illustrates the relationship between elastic energy and dissipation energy during the progressive failure process of sandstone samples. Throughout the entire hydro-mechanical coupling loading process of the multiform fracture sandstone samples, it is assumed that there is no heat conduction with the external environment during the deformation process, treating the system as a closed system. According to the principle of energy balance, the energy transformation n process of this system can be described by a mathematical model. Numerous studies have deeply explored the deformation and energy evolution of intact rock samples during loading and deformation [32,33,34,35]. Based on this, the energy evolution law of multiform fracture samples before peak stress is further discussed in this paper.

In triaxial compression, the total input energy U of the rock is the sum of the work done by axial stress (U₁) and the work done by confining pressure (U₃), expressed as follows [24,36,37]:

$$U=U_1+U_3$$

(2)

In this study, based on the principle of effective stress and fully considering the effect of pore water pressure, the expression for the total input energy of multi-shaped fractured sandstone samples before peak stress is derived as follows [24,36,37]:

$$U=U_d+U_e$$

(3)

$${\sigma }_e=\sigma -p$$

(4)

where U_e is the elastic energy, U_d is the dissipation energy, σ_e is the effective stress, σ is the total stress, and p is the pore water pressure.

By combining Equations (3) and (4), Equation (5) can be derived:

$$\begin{array}{l}U={\displaystyle {\int }_0^{{\epsilon }_1}\left({\sigma }_1-p\right)}d{\epsilon }_1+{\displaystyle {\int }_0^{{\epsilon }_2}\left({\sigma }_2-p\right)}d{\epsilon }_2+{\displaystyle {\int }_0^{{\epsilon }_3}\left({\sigma }_3-p\right)}d{\epsilon }_3={\displaystyle {\int }_0^{{\epsilon }_1}\left({\sigma }_1-p\right)}d{\epsilon }_1+2{\displaystyle {\int }_0^{{\epsilon }_3}\left({\sigma }_3-p\right)}d{\epsilon }_3\\ ={\displaystyle \sum _{i=1}^n\frac{1}{2}}\left[{\left({\sigma }_1-p\right)}_i+{\left({\sigma }_1-p\right)}_{i-1}\right]\left({\epsilon }_{1i}-{\epsilon }_{1i-1}\right)+{\displaystyle \sum _{i=1}^n\left[{\left({\sigma }_3-p\right)}_i+{\left({\sigma }_3-p\right)}_{i-1}\right]}\left({\epsilon }_{3i}-{\epsilon }_{3i-1}\right)\end{array}$$

(5)

where ε₁ is the axial strain; σ₁ is the axial stress; ε₃ is the radial strain; σ₃ is the confining pressure; n is the number of calculated segments of the stress–strain curve at any time before peak stress; and i is the segmentation point, where σ_i and ε_i represent the stress and strain at that point, respectively.

The elastic energy U_e in Equation (3) can be expressed as:

$$U_e={\displaystyle \frac{1}{2E}}\left\{{\left({\sigma }_1-p\right)}^2+2{\left({\sigma }_3-p\right)}^2-2v\left[2\left({\sigma }_1-p\right)\left({\sigma }_3-p\right)+{\left({\sigma }_3-p\right)}^2\right]\right\}$$

(6)

where E is the elastic modulus of the fractured sandstone sample and v is Poisson’s ratio.

Thus, the dissipation energy expression is:

$$U_d=U-U_e$$

(7)

3.3.2. Energy Evolution During the Progressive Failure Process Under Hydro-Mechanical Coupling

In the 1980s and 1990s, scholars proposed the “memory effect” of rocks. Specifically, the physical properties of rocks, such as stress–strain relationships and permeability, may “remember” or reflect previous loading histories, especially during multiple loading and unloading cycles [38,39]. This “memory effect” is related to the microstructure of the rock and its stress history, and it may influence the mechanical behavior and permeability characteristics of the rock to some extent. Thus, the energy evolution of rocks shows some common patterns. Using dissipation energy theory, the total input energy, elastic strain energy, and dissipation energy of sandstone samples with single, T-shaped, and Y-shaped fractures under hydraulic coupling were calculated. As shown in Figure 10, Figure 11 and Figure 12, the energy evolution process of multi-shaped fractured sandstone under hydraulic coupling is divided into five stages, similar to the progressive failure process of sandstone samples.

Stage I: In this stage, the rock sample undergoes elastic deformation and the effective stress is relatively low. The energy required for the compaction of micro-pores or micro-defects accumulates gradually, with slow energy changes. The energy is primarily stored as elastic strain energy. At this point, the influence of fractures on the mechanical properties is minimal, with both elastic strain energy (U_e) and dissipation energy (U_d) being low. The total input energy (U) remains at a low level, and the energy changes are relatively smooth. For single-fracture sandstone samples at different inclination angles, the energy values corresponding to the crack closure stress range from 2.679 kJ·m⁻³ to 11.235 kJ·m⁻³ (total input energy), 2.024 kJ·m⁻³ to 8.358 kJ·m⁻³ (elastic energy), and 0.533 kJ·m⁻³ to 3.169 kJ·m⁻³ (dissipation energy). For T-shaped fracture sandstone samples, the corresponding energy values range from 1.899 kJ·m⁻³ to 15.891 kJ·m⁻³ (total input energy), 1.841 kJ·m⁻³ to 9.006 kJ·m⁻³ (elastic energy), and 0.027 kJ·m⁻³ to 6.886 kJ·m⁻³ (dissipation energy). For Y-shaped fracture sandstone samples, the corresponding energy values range from 6.225 kJ·m⁻³ to 11.106 kJ·m⁻³ (total input energy), 3.659 kJ·m⁻³ to 7.074 kJ·m⁻³ (elastic energy), and 1.875 kJ·m⁻³ to 5.075 kJ·m⁻³ (dissipation energy).

Stage II: As the external load progresses, effective stress increases slightly but remains relatively low. The energy required for the compaction of micro-pores or micro-defects continues to rise slowly. The energy growth remains dominated by elastic strain energy. For single-fracture sandstone samples at different inclination angles, the energy values corresponding to the crack initiation stress range from 6.502 kJ·m⁻³ to 18.934 kJ·m⁻³ (total input energy), 3.706 kJ·m⁻³ to 14.225 kJ·m⁻³ (elastic energy), and 2.413 kJ·m⁻³ to 5.150 kJ·m⁻³ (dissipation energy). For T-shaped fracture sandstone samples, the corresponding energy values range from 5.136 kJ·m⁻³ to 27.551 kJ·m⁻³ (total input energy), 3.124 kJ·m⁻³ to 15.111 kJ·m⁻³ (elastic energy), and 2.011 kJ·m⁻³ to 12.440 kJ·m⁻³ (dissipation energy). For Y-shaped fracture sandstone samples, the corresponding energy values range from 11.991 kJ·m⁻³ to 17.544 kJ·m⁻³ (total input energy), 7.569 kJ·m⁻³ to 11.503 kJ·m⁻³ (elastic energy), and 3.862 kJ·m⁻³ to 7.570 kJ·m⁻³ (dissipation energy).

Stage III: At this stage, internal damage begins to occur in the sandstone, with plastic deformation gradually emerging. Micro-cracks stabilize and expand and new cracks form, while existing fractures gradually merge, eventually forming macroscopic fractures. In this stage, both elastic strain energy and dissipation energy increase, reflecting the accelerated expansion of cracks and the gradual damage of the system. However, studies indicate that under hydraulic coupling, the energy storage capacity for elastic strain is lower than that under dry conditions, indicating that pore water weakens the ability to store elastic energy. During this stage, the energy curves still show noticeable nonlinear behavior, with both total input energy (U) and dissipation energy (U_d) increasing rapidly, while elastic strain energy (U_e) increases slowly. For single-fracture sandstone samples at different inclination angles, the energy values corresponding to the damage stress range from 20.550 kJ·m⁻³ to 49.575 kJ·m⁻³ (total input energy), 12.383 kJ·m⁻³ to 41.806 kJ·m⁻³ (elastic energy), and 7.770 kJ·m⁻³ to 10.753 kJ·m⁻³ (dissipation energy). For T-shaped fracture sandstone samples, the corresponding energy values range from 16.200 kJ·m⁻³ to 92.898 kJ·m⁻³ (total input energy), 9.196 kJ·m⁻³ to 56.055 kJ·m⁻³ (elastic energy), and 7.003 kJ·m⁻³ to 36.842 kJ·m⁻³ (dissipation energy). For Y-shaped fracture sandstone samples, the corresponding energy values range from 30.914 kJ·m⁻³ to 54.796 kJ·m⁻³ (total input energy), 21.222 kJ·m⁻³ to 48.812 kJ·m⁻³ (elastic energy), and 5.985 kJ·m⁻³ to 14.643 kJ·m⁻³ (dissipation energy).

Stage IV: A large number of newly formed micro-cracks gradually expand and merge under the influence of external load, eventually forming the dominant flow path. According to Griffith’s strength theory, the tips of newly developed micro-cracks are prone to forming high-stress concentration zones, leading to significant energy accumulation. In this stage, elastic strain energy (U_e) increases rapidly with axial strain, while dissipation energy (U_d) transitions from slow to rapid increase. Meanwhile, due to the pressure effect of the pore water, energy release at crack tips becomes slower, leading to a further increase in the proportion of dissipation energy. For single-fracture sandstone samples, the energy values corresponding to the peak strength range from 131.006 kJ·m⁻³ to 263.513 kJ·m⁻³ (total input energy), 76.720 kJ·m⁻³ to 174.554 kJ·m⁻³ (elastic energy), and 45.373 kJ·m⁻³ to 109.334 kJ·m⁻³ (dissipation energy). For T-shaped fracture sandstone samples, the corresponding energy values range from 104.806 kJ·m⁻³ to 248.621 kJ·m⁻³ (total input energy), 67.849 kJ·m⁻³ to 117.898 kJ·m⁻³ (elastic energy), and 33.509 kJ·m⁻³ to 130.722 kJ·m⁻³ (dissipation energy). For Y-shaped fracture sandstone samples, the corresponding energy values range from 93.267 kJ·m⁻³ to 192.572 kJ·m⁻³ (total input energy), 79.862 kJ·m⁻³ to 100.171 kJ·m⁻³ (elastic energy), and 10.748 kJ·m⁻³ to 103.929 kJ·m⁻³ (dissipation energy).

Stage V: Under the sustained action of effective stress, multi-shaped fractured sandstone accumulates significant damage, and macroscopic fractures further intersect, merge, and penetrate, forming a macroscopic failure surface. Meanwhile, the dominant flow path is fully formed, leading to a rapid decrease in elastic strain energy (U_e), a sharp increase in dissipation energy (U_d), and continued rapid increase in total input energy (U). The sharp changes in energy indicate that the rock has reached the critical point of failure. This energy change signifies that under hydraulic coupling, the failure process of multi-shaped fractured sandstone exhibits significant nonlinear characteristics. Additionally, the energy evolution trends in multi-shaped fractured sandstone samples are generally consistent with those of conventional triaxial compression tests, indicating that the combined influence of fracture morphology and hydraulic coupling affects the energy evolution curve’s shape but not the magnitude of energy.

3.3.3. Effect of Fracture Inclination Angle on Peak Energy Distribution

This section investigates the effect of fracture inclination angle on the distribution of peak energy in single-fracture, T-shaped fracture, and Y-shaped fracture sandstone samples by analyzing the conversion relationships of total input energy, elastic strain energy, and dissipation energy at peak strength. The goal is to provide both data and theoretical support for understanding the macro- and micro-mechanisms of failure in multi-shaped fractured sandstone under hydraulic coupling.

As shown in Figure 13, the peak total input energy (U_p) fluctuates with changes in the fracture inclination angles of single-fracture, T-shaped fracture, and Y-shaped fracture sandstone. Under single-fracture conditions, the maximum U_p value (263.513 kJ·m⁻³) occurs at an inclination of 0°, while the minimum U_p value (131.006 kJ·m⁻³) is observed at 30°. Under T-shaped fracture conditions, the maximum U_p value (248.621 kJ·m⁻³) is found at 90° and the minimum value (104.806 kJ·m⁻³) occurs at 30°. For Y-shaped fractures, the maximum U_p value (192.572 kJ·m⁻³) is at 0°, and the minimum U_p value (93.267 kJ·m⁻³) is at 90°. For intermediate angles (15°–75°), the U_p values tend to form a valley-like pattern.

In general, the proportion of peak elastic energy (U_pe) to peak total input energy is higher, and the overall trend of U_pe closely follows the trend of U_p. Under single-fracture conditions, the maximum U_pe value occurs at 0° and the minimum at 75°. For T-shaped fractures, the maximum U_pe value (117.898 kJ·m⁻³) occurs at 90°, while the minimum (67.849 kJ·m⁻³) is at 60°. For Y-shaped fractures, the maximum U_pe value (100.171 kJ·m⁻³) is at 75°, and the minimum (79.862 kJ·m⁻³) occurs at 45°.

The peak dissipation energy (U_pd) shows relatively small fluctuations compared to U_p and U_pe. Under single-fracture conditions, the minimum U_pd value occurs at an inclination of 30° and the maximum U_pd value occurs at 90°. For T-shaped fractures, the maximum U_pd value (130.722 kJ·m⁻³) is at 90° and the minimum (33.509 kJ·m⁻³) occurs at 30°. For Y-shaped fractures, the maximum U_pd value (103.929 kJ·m⁻³) is at 0° and the minimum (10.748 kJ·m⁻³) is at 90°.

Through this analysis, it can be concluded that the fracture inclination angle and shape primarily influence the rock’s ability to store energy. By comparing the energy dissipation behavior at peak strength for multi-shaped fractured sandstone samples, it is observed that the average Up_e value at peak strength is highest for the single-fracture sample (111.318 kJ·m⁻³), followed by the T-shaped fracture sample (92.027 kJ·m⁻³), and lowest for the Y-shaped fracture sample (88.081 kJ·m⁻³). The reason for this trend is that the simpler geometry of the single-fracture sample causes more intense stress concentration at both ends of the pre-existing fracture, delaying crack propagation and allowing for greater elastic energy storage before failure. In contrast, the Y-shaped fractures facilitate stress redistribution and earlier crack interactions, effectively dissipating energy through the branching mechanism and reducing the retention of elastic energy. The T-shaped fractures exhibit intermediate behavior, balancing local stress concentration and crack interactions. These findings highlight how fracture geometry controls energy distribution, with simpler geometries favoring elastic energy storage, while more complex geometries accelerate energy dissipation.

3.3.4. Analysis of Peak Energy Ratios

To elucidate the energy conservation in the hydro-mechanical coupling experiments, the ratio of peak elastic energy to peak total input energy is defined as the elastic energy ratio (α), and the ratio of peak dissipation energy to peak total input energy is defined as the dissipation energy ratio (β). The formulas for these ratios are as follows:

$$\alpha =U_{pe}/U_p$$

(8)

$$\beta =U_{pd}/U_p$$

(9)

The distribution of energy ratios is shown in Figure 14. As seen in Figure 14a, with the increase in the fracture inclination angle of single-fracture sandstone samples, α follows an “M”-shaped variation pattern. When the fracture inclination angle is 15°, α reaches its maximum value; when the fracture inclination angle increases to 45°–60°, a clear turning point occurs; and when the fracture inclination angle is 90°, α reaches its minimum value. Although α fluctuates with increasing fracture inclination angle, the overall trend is a decline.

Figure 14b shows that for T-shaped fracture sandstone samples, within the range of 0° to 45°, α increases with the fracture inclination angle. After that, it fluctuates. The maximum α value occurs at 15°. Beyond 45°, α follows an “N”-shaped pattern, showing an overall decreasing trend, with the minimum value occurring at 90°.

Figure 14c reveals that for Y-shaped fracture sandstone samples, α generally exhibits a wave-like increasing trend as the fracture inclination angle increases. The minimum value of α is observed at 0°, and the maximum value occurs at 90°.

In conclusion, both the elastic energy ratio (α) and the dissipation energy ratio (β) exhibit a clear inverse relationship: when α increases, β decreases, and vice versa. The pre-existing fracture shape and angle have a significant impact on the energy ratios of the samples, influencing the storage capacity of elastic energy and showing an energy distribution adjustment pattern, although with some fluctuations.

4. Discussion

4.1. Exploration of the Energy Competition Relationship

There is significant energy competition between elastic strain energy and dissipation energy. Therefore, the energy competition coefficient (γ) is defined as the ratio of elastic energy density to dissipation energy density under axial strain. In the progressive failure process of multi-shape fractured sandstone samples, the evolution of γ over the five stages is shown in Figure 15, Figure 16 and Figure 17. The overall trend for single-fracture, T-shaped fracture, and Y-shaped fracture sandstone samples all follows a pattern of decrease-increase-decrease.

In Stages I–II, γ rapidly decreases, indicating that in this stage, dissipation energy (U_d) dominates. Compared to elastic energy (U_e), the release rate of dissipation energy is faster. This reflects the gradual compaction of the initial pores within the sandstone and the closure of microcracks, followed by elastic deformation, where the accumulation of elastic energy is relatively weak.

In Stages III–IV, plastic deformation occurs and γ sharply increases, reaching a peak before rapidly decreasing. This shows that elastic energy (U_e) gradually replaces dissipation energy (U_d) as the dominant form, with the accumulation rate of elastic energy significantly higher than the release rate of dissipation energy. This stage reflects the initiation, development, and coalescence of new fractures, as well as the gradual formation of dominant flow paths. The energy evolution at the peak stage confirms the validity of Griffith’s strength theory.

In Stage V, due to the formation of macro-fracture surfaces, the accumulated elastic energy continues to rapidly decrease, while dissipation energy sharply increases. In this stage, dissipation energy again dominates, reflecting the through-fracturing and the occurrence of macro-failure.

Overall, the evolution curve of γ visually demonstrates the energy competition relationship of multi-shaped fracture specimens at different failure stages. It reveals the transition of the dominant role of elastic energy and dissipation energy during the progressive failure process.

As shown in Figure 18, the peak value of γ in Stage IV is defined as γ_max. The “pseudo-peak” of γ in Stage I or II for single-fracture, T-shaped fracture, and Y-shaped fracture sandstone samples at different angles is not analyzed in detail. The degree of fracture is determined by the distribution range of γ_max. Specifically, when γ_max ∈ [0, 2], it indicates weak fracture; when γ_max ∈ (2, 6], it indicates moderate-to-strong fracture; and when γ_max ∈ (6, +∞), it indicates strong fracture. The γ evolution curves of the single-fracture, T-shaped fracture, and Y-shaped fracture sandstone samples show certain differences based on the fracture inclination angle. When the angle is small (e.g., 0°, 15°), γ reaches a higher value in the peak stage, indicating a significant accumulation of elastic energy before failure. Conversely, when the fracture inclination angle is large (e.g., 75°, 90°), the peak of γ is significantly reduced, suggesting that the fracture shape and angle affect the dynamic changes of energy competition. Overall, the evolution of γ visually reflects the competition between elastic energy and dissipation energy at different stages of failure as well as the dynamic evolution of fracture propagation.

4.2. Impact of Energy on the Microstructural Characteristics of Crystals Post-Failure in Multi-Shaped Fractured Sandstone

Figure 19, Figure 20 and Figure 21 present the scanning electron microscope (SEM) images of multi-shaped fractured sandstones. As shown in these figures, under the influence of hydro-mechanical coupling, the connections between mineral grains in single-fracture, T-shaped fracture, and Y-shaped fracture sandstone samples become looser, with phenomena such as cracking and displacement occurring. After failure, the fracture surfaces of the samples are uneven and the surface roughness of the mineral grains increases significantly, characterized by protrusions, exfoliation, and wear marks. This roughness is associated with frictional sliding during the fracture propagation process, highlighting the important contribution of dissipation energy to fracture expansion.

As shown in Figure 19, after failure, the single-fracture sandstone sample exhibits wide cracks, indicating that significant stress release accompanies the failure process, with a higher local energy density. The difference in crack length reflects the heterogeneous strength of the local lattice and the uneven energy release in different areas. Additionally, new micro-cracks emerge near the wide cracks, with these micro-cracks being densely distributed and complex, some of them penetrating through individual grains. This is due to the expansion and coalescence of micropores and micro-cracks. In Figure 20, after failure of the T-shaped fracture sandstone sample, cracks of varying depths appear on the surface of the mineral grains, with relatively smooth crack surfaces. This may be due to the complex changes in the crystal structure caused by the competition between elastic and dissipation energy under hydro-mechanical coupling. The elastic energy release drives the fracture expansion, and cracks propagate along grain boundaries or penetrate the crystals. The dissipation energy promotes the plastic deformation and fragmentation of grains, leading to increased crystal fracture intensity and larger frictional forces between mineral grains, resulting in deeper contact and compression between mineral crystals. In Figure 21, after failure of the Y-shaped fracture sandstone sample, the mineral grain surfaces are smoother compared to the single-fracture and T-shaped fracture samples. At the same time, energy release influences the surrounding grains, causing local reorganization or rearrangement of the grain structure, forming larger branching cracks. This reflects the uneven spatial distribution of energy release during the failure process.

In summary, the analysis of the failure of multi-shaped fractured sandstones shows that dissipation energy promotes the fragmentation of grains, grain boundary sliding, and plastic deformation, resulting in significant grain breakdown. The competition and uneven release of energy directly impact the complexity of the grain structure and the fracture propagation path, manifesting as a complex fracture network and uneven changes in particle sizes. These microstructural characteristics indicate that the distribution and release of energy during the rock failure process are crucial factors in determining the evolution of the grain structure. The dominant role of dissipation energy is especially evident in the post-peak stage. Elastic energy mainly drives fracture propagation and elastic deformation of the lattice, evidenced by cracks propagating along grain boundaries and evenly distributed.

5. Conclusions

This study investigated the mechanical, permeability, and energy evolution characteristics of multi-shaped fractured sandstones under hydro-mechanical coupling. By altering the fracture shape (single fracture, T-shaped fracture, Y-shaped fracture) and fracture inclination angle (0°, 15°, 30°, 45°, 60°, 75°, 90°), the effects on the mechanical properties, permeability, and energy evolution of the sandstone were analyzed, elucidating the competition between elastic and dissipation energy and revealing the impact of energy on the microstructural characteristics of the grains after failure. The main conclusions of this study are as follows:

(1) By introducing normalized characteristic stresses, it was found that for different fracture angles, the average values of σ_c/σ_c-w for single fractures, T-shaped fractures, and Y-shaped fractures were 0.643, 0.600, and 0.581, respectively; the average values of σ_cd/σ_cd-w were 0.435, 0.473, and 0.524, respectively; the average values of σ_ci/σ_ci-w were 0.517, 0.568, and 0.640, respectively; and the average values of σ_cc/σ_cc-w were 0.620, 0.646, and 0.717, respectively. In general, the fracture inclination angle and shape significantly influence the variation of characteristic stresses, and Y-shaped fractures exhibit higher normalized stress ratios in most cases.

(2) During the permeability change process, all samples went through a pattern of rapid decrease, slow decline, rapid increase, sharp rise, and stabilization. The jump coefficient ξ was used to represent the permeability enhancement effect, with the ξ values for single fractures, T-shaped fractures, and Y-shaped fractures ranging from 1.3–1.82, 1.16–1.87, and 1.22–1.92, respectively. The differences in ξ values for different fracture morphologies were not significant, indicating that the fracture shape has little impact on the permeability enhancement effect.

(3) Analysis of the peak energy distribution characteristics indicates that fracture inclination angle and shape primarily affect the rock’s energy storage capacity. The U_p value fluctuates with changes in fracture inclination angle, with the maximum U_p occurring at 0° for single fractures and Y-shaped fractures, and at 90° for T-shaped fractures. The proportion of U_pe in peak total input energy is relatively large and follows a trend similar to that of U_p. U_pd values show little variation, with maximum values for single fractures and T-shaped fractures at 90° and for Y-shaped fractures at 0°.

(4) Based on the distribution range of γ_max, the degree of fracture is classified as weak (γ_max ∈ [0, 2]), moderate-to-strong (γ_max ∈ (2, 6]), or strong (γ_max ∈ (6, +∞)). The γ_max value reflects the competition between elastic and dissipation energy as well as the dynamic evolution of fracture propagation.

(5) The impact of energy distribution on the grain structure after failure in multi-shaped fractured sandstone samples was explored. Results show that after failure, single fractures lead to wider cracks with higher local energy density, and the variation in crack length reflects the inhomogeneity of local strength and energy release. In T-shaped fractures, the competition between elastic and dissipation energy leads to complex changes in crystal structure, increasing frictional forces between grains. In Y-shaped fractures, the surfaces of mineral grains are smoother, and cracks take on a branching morphology.

Future research can be deepened in the following directions to promote its practical application in tunnel design, underground mining stability, reservoir engineering, and underground energy storage projects.

(1) It is recommended to study the fracture response mechanism under complex stress paths. Multistage confining pressure and cyclic loading–unloading coupled tests should be conducted to clarify the evolution of fracture morphology under repeated stress and reveal the relationship between energy accumulation–release thresholds and rock instability.

(2) It is suggested to focus on multi-field coupled modeling, developing a coupled seepage-stress-damage numerical model and integrating 3D fracture network reconstruction techniques to reveal the formation mechanisms of fracture seepage channels.

(3) It is recommended to integrate intelligent monitoring technology; explore intelligent correlation algorithms for microseismic, stress, displacement signals, and seepage parameters; and develop an optical fiber grating distributed sensing system to achieve real-time dynamic inversion of the fracture expansion process and seepage path.