Introduction

In actual mining operations, mining disturbances induce rock mass damage and fracturing, which in turn leads to drastic changes in the seepage field. This is a critical factor in triggering rock mass instability and mine water inrush disasters1,2. Water inrush severely threatens personnel safety, damages equipment, and interrupts production. Research on the mechanism and evolution law of seepage-stress coupling is central to solving water inrush problems and promoting breakthroughs in mining technology3,4. During excavation in deep mines, the rock mass is subjected to a complex mechanical environment of superimposed loading and unloading, where its internal microstructure undergoes dynamic reconstruction, and permeability characteristics significantly change. This directly affects the long-term stability of wellbore structures and increases risks such as wellbore collapse5,6. Systematically investigating the laws of rock permeability characteristics under cyclic loading and unloading conditions of water pressure and confining pressure is of significant importance for actual mining operations. From the perspective of disaster prevention and control, it enables early prediction of permeability changes, identification of water inrush hazards, and timely implementation of reinforcement and other measures to reduce the probability of disaster occurrence. From the perspective of safety assessment, it can provide a basis for mine safety rating, assist enterprises in rationally arranging production and optimizing resource allocation, ensuring safe and efficient mining operations, and thus possesses significant practical value.

Currently, numerous scholars are dedicated to exploring stress-seepage coupling, with the aim of revealing the intrinsic relationship between permeability and the total stress–strain process. Meng Q7 investigated the strength and deformation characteristics of sandstone under triaxial cyclic loading and unloading conditions, elucidating the inherent laws governing the variation of rock strength with cyclic loading times, loading rate, and confining pressure. Momeni A8 explored the mechanical properties of rocks under uniaxial cyclic loading and unloading, specifically investigating the influence of uniaxial cyclic loading and unloading on key mechanical parameters such as rock strength, elastic modulus, and Poisson’s ratio. Zhao Y9,10,11,12 studied the coupled-damage effects in fractured rock masses. By establishing a sophisticated numerical model and integrating it with laboratory test results and field monitoring data, their research provided robust theoretical support for the stability analysis and disaster prevention of fractured rock mass engineering projects. Cai Q13 employed pore elasticity theory in conjunction with fracture mechanics to investigate the formation, expansion, and closure of hydraulic fracturing cracks within rock formations. In contrast, Fang L14 adopted a linear elastic fracture mechanics framework to elucidate the seepage behavior of fractured rock masses. Shuang H15 investigated crack initiation strength criteria for rock masses under dynamic and water pressure loads. Similarly, Zhengzheng C16 and Li H17 analyzed rock hydraulic splitting mechanisms using fracture mechanics models, while Zheng Y18 employed a fluid–solid coupling model with critical stress fissures to study rock hydraulic fracturing. Fracture mechanics models partially elucidate rock fracturing and instability mechanisms under combined seepage-stress effects. However, these models struggle to describe how damage processes—such as initiation, propagation, and interconnection of random microfractures within rock—influence seepage-stress coupling mechanisms19. Zhang Chunhui et al.20 summarized the strain softening characteristics of rocks under varying confining pressure conditions based on triaxial permeability test results. Although Wang Huini21 employed computed tomography to investigate water’s role in fracture damage evolution at the mesoscale, these findings nevertheless require further experimental validation. Chen Liu Jun22 and Ahrens B.23 analyzed sandstone responses under high stress and high water pressure conditions, but did not comprehensively incorporate temperature variables. Antoine’s work24 focused on permeability variation patterns, with limited discussion of reservoir mechanical parameters. Vitel25 attempted pairwise decoupling solutions for multi-field coupled models but failed to achieve fully coupled analytical solutions. Alzouobi26 developed a mathematical model for seepage incorporating temperature effects; however, in practical applications, temperature exerts a relatively minor influence on seepage behavior. The model proposed by Vallier27 effectively reproduces multi-field coupled logging curves, though its applicability remains somewhat constrained under complex geological conditions.

Based on a comprehensive analysis of existing literature, it is observed that research on rock cyclic loading–unloading hydraulic coupling experiments remains underdeveloped. Hydraulic coupling effects exhibit distinct characteristics in seepage path evolution and stress path response, yet existing studies predominantly focus on single-factor mechanisms, thereby lacking systematic understanding of rock mass damage accumulation under multi-field coupling. Therefore, a comprehensive investigation into the intricate interplay between external stress fields and internal seepage fields, particularly under coupled conditions, holds significant scientific value. To this end, this study employs a French triaxial multi-field coupling test system, utilizing sandstone as the subject material, to conduct experiments under cyclic loading–unloading stress-seepage coupling conditions. The research systematically investigates the evolution of sandstone’s mechanical properties and the mechanism of permeability development within this complex stress-path multi-field coupling environment. Ultimately, this study aims to provide theoretical foundations and data support for the safe design and risk prevention of rock engineering under complex geological conditions involving high stress and high seepage pressure.

Experimental procedure

Test specimens and apparatus

Sandstone specimens utilized in this study were obtained from an underground mine at a depth of 370 m. Cores were drilled from intact rock blocks and subsequently prepared into standard cylindrical specimens with a diameter of 50 mm and a height of 100 mm, in accordance with relevant regulations (see Fig. 1). Following grinding and precision machining, the specimens exhibited a uniform and fine texture, with parallel and flat ends, and were free of visible natural cracks. The bulk density of the sandstone was determined to be 2.2 g/cm3, measured according to standard rock density protocols. Given that the pore mechanics properties of rock are significantly influenced by its state of saturation, therefore, to ensure the reliability and comparability of experimental results, all specimens underwent a thorough saturation treatment prior to testing.The rock temperature-stress-seepage coupled triaxial rheometer, developed by Top Industrie (France), is a high-end rock mechanics testing apparatus. In comparison to other advanced rock triaxial testing systems currently in use28, this system’s core advantages lie in its highly integrated multi-field coupling capabilities and its high-precision dynamic monitoring of permeability characteristics29.

Fig. 1
Fig. 1
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Schematic diagram of the triaxial pressure chamber and rock sample encapsulation.

An integral part of the experimental apparatus, the self-balancing rock triaxial pressure system, is illustrated in Fig. 1. This experimental system primarily comprises a high-pressure pump, a deformation measurement system, a microcomputer system, and ancillary devices. To mitigate experimental uncertainties, prior to testing, each rock specimen was encased in a rubber sleeve to protect both the specimen and the pressure chamber from hydraulic oil corrosion. Gaskets were positioned at both the top and bottom ends to secure the rock specimen within the loading platens, thereby preventing uneven stress distribution during axial loading and ensuring stable transmission of axial pressure. The strain sensors were calibrated to enable accurate monitoring of rock strain.

Currently, two main methods are employed for determining rock permeability30: The steady-state method is typically applicable to rock samples with permeabilities ranging from 10−16m2 to 10−18m2, whereas the transient pressure pulse method is more suitable for rock samples with permeabilities below 10−12m2 to 10−16m2. The rock samples investigated in this study are sandstone, with an initial permeability of approximately 0.03 × 10−16 m2, classifying them as having medium-to-low permeability. Consequently, the steady-state method was employed for testing.To simplify the experimental process and satisfy theoretical analysis requirements, the following fundamental assumptions were introduced: (1) Apparent seepage flow is treated as an incompressible fluid; (2) The original distribution of pores and microfractures within the rock is considered relatively uniform, approximating a porous continuous medium; (3) Steady seepage occurring under constant pressure can be regarded as continuous flow; (4) Given the slow fluid flow rate in low-permeability sandstone, its behaviour in triaxial permeability tests conforms to Darcy’s Law.

As shown in Formula 1, the fundamental principle of this experimental method is as follows: water is injected from the bottom of the rock sample (i.e., the upstream end of the permeable channel) while maintaining a constant injection pressure; simultaneously, the downstream valve is maintained open to allow communication with atmospheric pressure.This establishes a steady head difference (denoted as ΔP) across the rock sample. Once flow within the sample stabilises, both the upstream injection flow rate Q (in m3/s) and the pressure difference ΔP remain constant over time. Using Darcy’s Law as a foundation, the permeability of the rock sample can be determined through the application of the following equation:

$${\text{k}} = \frac{{{\text{L}}\mu {\text{Q}}_{{\text{t}}} }}{{\Delta {\text{pA}}\Delta {\text{t}}}}$$
(1)

In the equation, Q denotes the fluid flow rate (m3); Qt represents the volume of water permeating the sandstone sample over time (m3); L and A denote the sample length and cross-sectional area respectively; μdenotes the dynamic viscosity coefficient of the fluid31, with a value of μ = 1.005 × 10−3 Pa·s. During the experimental implementation phase, stress–strain curves are automatically recorded and logged via the computer system. Concurrently, the dynamic variations in pump operating parameters over time during the drainage phase are monitored and stored in real time by automated equipment, thereby enabling the calculation of water flow rate changes throughout the experiment.

Experimental protocol

The stress environment of deep rock masses is markedly distinct from that of shallow strata. Accurate simulation of in-situ conditions necessitates experimental protocols tailored to rock mechanical properties, comprehensively integrating in-situ parameters such as confining pressure and water pressure. Deep rock masses are typically subjected to high stress states, primarily attributed to the self-weight stress of overlying strata and tectonic stress resulting from geological activity32,33,34. Confining pressure conditions applied in this investigation were 10 MPa, 11 MPa, and 12 MPa. Calculations revealed that these vertical stresses correspond to equivalent burial depths of approximately 369 m (10 MPa), 406 m (11 MPa), and 442 m (12 MPa), respectively. In deep geological environments, rock masses commonly experience elevated water pressures. To comprehensively investigate the influence of water pressure on rock mechanical behaviour, this study utilized three water pressure conditions (1 MPa, 2 MPa, and 3 MPa) to represent a range of hydrogeological environments.

Sandstone was selected as the study subject. Experimental investigations were conducted on the mechanical and seepage characteristics of sandstone under water pressures of 1 MPa, 2 MPa, and 3 MPa, with confining pressures of 10 MPa, 11 MPa, and 12 MPa. The procedure for the five-cycle hydraulic coupling seepage test with constant amplitude loading is as follows:

  1. (1)

    Initial testing of sandstone specimens for ultimate strength. To minimise experimental variability, three groups of sandstone specimens with comparable physical and mechanical properties underwent conventional triaxial compression testing. The mean value of their strength characteristics served as the reference benchmark.

  2. (2)

    Following peak strength determination, specimens were placed in triaxial apparatus under predetermined confining pressure conditions, with predetermined water pressure applied. Using stress control mode, axial stress was incrementally increased at a loading rate of 0.05 MPa/s up to 50 MPa. During this process, cyclic loading and unloading were implemented. At each loading and unloading cycle, permeability was measured at five stress levels: 10, 20, 30, 40, and 50 MPa. The specific method involved determining the seepage time required for water to pass through the specimen under constant water pressure conditions, thereby calculating the permeability at the corresponding stress level. This step was repeated throughout subsequent cycles to ensure comparability of results.

  3. (3)

    By further modifying the conditions of confining pressure and water pressure, the same experimental procedure was employed to conduct hydraulic coupling tests under cyclic loading, analysing the relationship between confining pressure and seepage characteristics.

  4. (4)

    Throughout the cyclic loading-hydraulic coupling triaxial test, stress–strain curves, seepage times, and related parameters were recorded in real time. Statistical results of the experimental programme are presented in Table 1, with the specific loading path illustrated in Fig. 2.

Table 1 Seepage test programme for sandstone fluid–solid coupling under cyclic loading.
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Fig. 2
Fig. 2
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Cyclic loading and unloading path.

Analysis of mechanical characteristics in cyclic loading and unloading fluid–structure interaction tests

Stress–strain curves for sandstone under varying confining pressures and water pressures

As depicted in Fig. 3, the stress–strain curves derived from cyclic loading and unloading tests on sandstone under different confining pressures and water pressures demonstrate pronounced hysteresis effects. This phenomenon is attributed to inherent structural discontinuities within the rock matrix, such as microfractures and joints, which cause the material’s behavior to deviate from ideal elastic behavior. Confining pressure substantially impedes volumetric expansion by constraining microcrack propagation, consequently limiting radial strain variations within the specimens. Consequently, stress and strain evolution are not perfectly synchronous during cyclic laboratory testing. Macroscopically, it is observed that the unloading path does not retrace the loading path but rather lies beneath the loading curve, thereby forming a closed hysteresis loop. This hysteresis phenomenon not only reflects plastic deformation resulting from the closure and opening of internal fractures but also functions as a critical metric for evaluating rock energy dissipation characteristics. During the initial loading cycles, before the rock fully enters the elastic regime, the area of the hysteresis loop in the first cycle is typically substantial. During subsequent cycles, the hysteresis loop progressively migrates in the direction of increasing strain and gradually converges as the number of cycles increases. This phenomenon is primarily attributed to the relatively low upper stress limit applied during initial cycles, where the rock specimen primarily undergoes elastic deformation with minimal damage to its internal fracture network. As the upper stress limit progressively increases, leading to the fracture propagation stage, both the area and the migration amplitude of the hysteresis loop exhibit a significant increase, signifying a transition from elastic to plastic deformation.As cycling progresses, the hysteresis loop progressively contracts. The pronounced hysteresis characteristics commonly observed in rock stress–strain curves can be interpreted as follows: During initial loading, pores within the rock undergo primary compression and gradual closure. As the rock enters the unstable deformation stage, fractures initiate, propagate, and progressively interconnect. Throughout this process, the rock specimen exhibits significant plastic deformation characteristics near the unloading point, whereas the intermediate loading and unloading stages are dominated by rapid elastic responses. For a systematic analysis of failure progression, this process is conventionally divided into five distinct stages based on stress–strain characteristics. As an illustrative example, the axial stress–strain curve obtained under a confining pressure of 10 MPa and a water pressure of 2 MPa is presented in Fig. 4:

  1. (1)

    Consolidation stage (OA): At this point, the axial stress is 4.2 MPa, below the unloading stress. The curve exhibits a concave-down trend. This occurs because the initial load consolidates the rock’s internal pores and microcracks, causing the rock to become denser and forming a curve with an increasingly steep slope.

  2. (2)

    Linear stage (AB): As loading continues, previously closed and compacted pores and micro-fractures within the rock undergo micro-fracturing. The curve approaches a straight line with a slope approximately equal to a specific value, representing the rock’s average tangent modulus;

  3. (3)

    Plastic deformation stage (BC): The curve’s slope gradually diminishes as applied stresses disrupt structural planes. High stresses exacerbate this disruption, causing generated cracks to propagate continuously until the plastic phase is reached.

  4. (4)

    Failure stage (CD): Peak stress of 57.1 MPa is reached, with strain increasing markedly. The curve exhibits a concave profile as the rock specimen becomes permeated by fractures, resulting in fragmentation.

  5. (5)

    Post-peak stage (DE): The rock is completely fractured, with macroscopic fracture surfaces gradually becoming apparent. The rock mass exhibits macroscopic displacement, ultimately leading to instability and collapse of the specimen.

Fig. 3
Fig. 3Fig. 3
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Stress–strain curves for sandstone under varying confining pressures and water pressures.

Fig. 4
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Phases of the sandstone stress–strain curve.

As shown in Fig. 3(a), (d), (g), the experimental results indicate that due to constant water pressure, the curves still follow the phenomenon where peak stress increases with increasing confining pressure. Under three different conditions, the strain values corresponding to their attained peak stress are basically consistent; simultaneously, in the elastic deformation stage before reaching peak stress, the linear slope of the stress–strain curve, which reflects the material’s elastic modulus (E) , also exhibits similarity.Moreover, the values display a stepwise upward trend, attributable to the increased lateral confinement of the specimen as confining pressure rises, thereby enhancing its bearing capacity. As depicted in Fig. 3(a), (b), and (c), when confining pressure remains constant and water pressure increases, peak stress gradually decreases. As evident from Fig. 3(g), (h), and (i), after five cycles of loading and unloading, specimen failure occurs. The stress–strain curve progressively shifts towards the strain axis, with the corresponding peak strain progressively increasing. Upon reaching peak stress, residual stress gradually decreased. This phenomenon arises from the accelerated expansion of internal pores within the specimen due to the increased water pressure.

Axial peak stress–strain analysis

Figure 5 illustrates that the peak stress of sandstone exhibits a pronounced increasing trend with increasing confining pressure. For instance, under a water pressure of 1 MPa, increasing confining pressure from 10 to 12 MPa elevated peak stress from 59.5 to 64.3 MPa, representing an 8.1% increase. This suggests that heightened confining pressure effectively enhances the sandstone’s compressive strength, thereby improving its resistance to failure.

Fig. 5
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Peak stress–strain response of sandstone under cyclic loading and unloading with fluid–solid coupling.

Further analysis reveals that the influence of water pressure on peak stress is also significant. Under a fixed confining pressure, the peak stress exhibits a slight decline as the water pressure increases. For example, when subjected to a confining pressure of 10 MPa, increasing the water pressure from 1 to 3 MPa reduced peak stress from 59.5 to 49.2 MPa, a 17.3% decrease. This phenomenon can be attributed to the fact that water pressure induces the expansion and connectivity of microfractures within the rock mass, thereby diminishing its overall strength.

Consistent with peak stress observations, the peak strain of sandstone also diminishes as the confining pressure rises. In an experiment with a water pressure of 2 MPa and confining pressure increasing from 11 to 12 MPa, the peak strain decreased from 1.15 to 1.1%, indicating that deformation becomes more difficult under high confining pressure conditions, reflecting the constraining effect of confining pressure on rock deformation. However, the influence of water pressure on peak strain is more complex. Under certain conditions, such as a water pressure of 3 MPa and confining pressure of 12 MPa, peak strain actually increases compared to when water pressure is 2 MPa. This increase may be attributed to the fact that elevated water pressure promotes the expansion and interconnection of internal fractures within the rock mass, enabling greater plastic deformation as the rock approaches failure.

The peak stress and strain characteristics of sandstone under Fluid–Solid Coupling under Cyclic Loading and unloading are synergistically influenced by water pressure and confining pressure. Increased confining pressure significantly enhances the compressive strength and stiffness of sandstone, while water pressure exerts a complex effect on its mechanical properties by influencing the internal structure of the rock mass. Particularly under high water pressure conditions, the failure mode of the rock mass may transition from brittle to ductile failure.

Elastic modulus analysis

Figure 6 illustrates that, under fixed pore water pressure conditions, the E of sandstone generally exhibits an increasing trend with increasing confining pressure. This suggests that elevated confining pressure enhances the sandstone’s resistance to deformation, thereby promoting greater stability in high-pressure environments. However, beyond a certain confining pressure threshold, the rate of increase in E progressively diminishes, indicating a saturation effect in the sandstone’s E enhancement. Under fixed confining pressure conditions, variations in water pressure significantly influence the E of sandstone. As water pressure increases, the E of sandstone consistently exhibits a reduction to varying degrees. This is primarily because an increase in water pressure elevates the internal pore pressure, thereby weakening the cohesive forces between particles and consequently reducing the E. Notably, the extent of water pressure’s influence on the E varies under different confining pressure conditions, highlighting a complex interaction between confining pressure and water pressure. During cyclic loading and unloading, the E of sandstone exhibits pronounced hysteresis effects and fatigue characteristics. In the initial loading phase, the modulus increases rapidly; however, as loading cycles progress, the rate of increase gradually diminishes. During unloading, while some modulus recovery is observed, it fails to fully return to its initial state. This hysteresis and fatigue phenomenon collectively reflect internal structural damage and energy dissipation processes within the sandstone under cyclic loading conditions.

Fig. 6
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Cyclic loading and unloading fluid–solid coupled sandstone elastic modulus (E).

Macroscopic failure characteristics of sandstone

Figure 7 illustrates that the relevant diagrams of sandstone failure modes reveal that during seepage tests conducted under varying confining pressures and water pressure cycles until failure, the majority of specimens exhibited shear failure patterns. During failure, no concentrated crushing zones formed, nor did significant swelling deformation occur on the specimen surfaces. The overall behaviour exhibited typical characteristics of brittle fracture, with specimens retaining relatively good integrity post-failure. Under identical osmotic pressure conditions, increasing confining pressure resulted in a greater angle between the fracture plane and the direction of maximum principal stress. A minority of specimens exhibited shear surfaces with a ‘Y’-shaped double crack morphology. These specimens featured vertically continuous fracture surfaces that were relatively smooth and flat. However, upon increased water pressure, the smoothness of the fracture surface diminished compared to low-water pressure fracture, with numerous fine, nearly parallel cracks appearing around the fracture surface. Near the primary fracture plane, some specimens exhibited flaking of fragmented blocks, with the fracture surface becoming slightly rougher. Under constant confining pressure, as permeation water pressure gradually increased, hydraulic seepage penetrated deeper into the specimen. At the microscopic level, this promoted the initiation and continuous propagation of internal cracks. Near the primary fracture plane, tensile cracks intersecting the main fracture plane emerged. These cracks accelerated the fracture progression of the rock specimens, reinforcing the characteristics of tensile failure. In summary, following failure in seepage tests involving cyclic loading and unloading under varying confining pressures and water pressures, the deformation and failure characteristics of the sandstone specimens remained predominantly shear-brittle. The crack orientations remained largely parallel to the maximum principal stress plane, showing no significant alteration.

Fig. 7
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Sandstone failure characteristics.

Analysis of seepage characteristics in cyclic loading and unloading fluid–structure coupling tests

Variation in sandstone permeability under different confining pressure conditions

As illustrated in Fig. 8a, taking a water pressure of 1 MPa as an example, the permeability of rock exhibits distinct differences during loading and unloading processes at the same axial pressure level. Moreover, throughout each loading–unloading cycle, the permeability during the loading phase consistently falls below that of the unloading phase. Taking the second cycle under 10 MPa confining pressure as an example, when the axial pressure reaches 60 MPa, the permeability during loading is 73.1 × 10−18m2, whereas it increases to 120.4 × 10−18m2 during unloading. The behaviour of permeability with axial pressure also varies under different confining pressure conditions. At 10 MPa confining pressure, during loading, permeability first decreases then increases with rising axial pressure; conversely, during unloading, permeability continuously increases as axial pressure decreases. As illustrated in Fig. 8b, at confining pressure of 11 MPa, permeability gradually decreased with increasing axial pressure during loading, while it increased with decreasing axial pressure during unloading. For instance, during the third loading cycle at 80 MPa, permeability was 37.6 × 10−18m2, and at 60 MPa, it was 40.1 × 10−18m2. At lower confining pressures, permeability exhibits greater variability and the rock remains less consolidated; once consolidation is complete and microfractures develop, permeability begins to increase. Under high confining pressures, however, the rock remains in a consolidated state, and permeability continues to decrease. With increasing number of loading–unloading cycles, rock permeability generally exhibits an upward trend. From the third cycle onwards, permeability rises markedly, particularly under medium-to-low confining pressure conditions. As shown in Fig. 8c, at 12 MPa confining pressure, the fifth cycle demonstrates significantly greater permeability fluctuations due to cumulative damage and irregular fissure expansion, whereas the first four cycles exhibit relatively gradual permeability increases.

Fig. 8
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Sandstone permeability under varying confining pressures.

As the number of cyclic loading applications progressively increases, the controlling influence of axial stress on rock permeability intensifies. However, under high confining pressure conditions, its constraining effect effectively inhibits further crack propagation, consequently attenuating the direct influence of axial stress on permeability enhancement. Repeated application of cyclic loading and unloading induces irreversible fatigue damage within the rock mass. This damage disrupts the rock’s original stable microstructure, manifesting specifically as significant joint dilation and particle fragmentation. These microstructural alterations collectively foster the generation and interconnection of pores with varying scales and morphologies, thereby providing more extensive seepage pathways for fluid and consequently enhancing the rock’s permeability significantly. Furthermore, cyclic loading also contributes to the thinning of the bound water layer on pore surfaces, which effectively enlarges the dimensions of seepage channels and accelerates the process of permeability increase.

Variation in sandstone permeability under different osmotic pressures

Hydraulic coupling tests involving five cycles of equal-amplitude loading and unloading were conducted on sandstone specimens under various combinations of confining pressure and water pressure. Based on the steady-state method principle, permeability under constant confining pressure was calculated, and permeability variation curves for sandstone under different water pressure conditions were plotted. The results are shown in Fig. 9

Fig. 9
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Sandstone permeability under varying water pressures.

Following five cycles of equal-amplitude loading and unloading, the confining pressure was maintained constant while the evolution of sandstone permeability was investigated by varying the water pressure conditions (Fig. 9a–c). Experimental results indicate that the change in sandstone permeability under cyclic loading can be divided into three distinct phases: a stable permeability phase, an increasing permeability phase, and a fluctuating permeability phase following the peak. Prior to fracture initiation, the permeability remained consistently high and relatively stable due to the vacuum-saturated preparation of the specimens. Under identical confining pressure conditions, permeability increased with rising permeating fluid pressure throughout each stage, exhibiting a positive correlation. Following peak intensity attainment, permeability exhibited pronounced fluctuations.

At a confining pressure of 10 MPa, the permeability of sandstone demonstrates a pronounced variation with escalating damage. From the peak stage, through the post–peak phase, to complete failure, the permeability continuously rises, remaining at its lowest under an water pressure of 1 MPa. The positive correlation between permeability and water pressure suggests that, at this confining pressure, an elevated water pressure exacerbates the damage to the rock’s permeability characteristics. Although the permeability exhibits minor fluctuations after the peak phase, it generally remains stable. This implies that the permeabilities at 2 MPa and 3 MPa have a limited impact on the rock’s permeability, with the damage caused by a 2 MPa permeability being slightly more severe than that at 3 MPa. This phenomenon reflects that, under relatively low confining pressures, while the effect of permeation pressure on rock permeability is positively correlated with the magnitude of permeation pressure, the dynamic variation among different permeation pressures is relatively gradual. The influence of permeation pressure on the evolution of the fracture network follows a relatively stable pattern of change.

When the confining pressure increases to 11 MPa, the evolution of permeability displays complex non-monotonic behavior. At an water pressure of 1 MPa, the permeability remains the lowest; however, the relationship between permeability and water pressure at 2 MPa and 3 MPa varies with the damage stage. During the peak stage, the permeability is slightly higher at 2 MPa; in the post–peak stage, the permeability at 3 MPa exceeds that at 2 MPa; and in the complete failure stage, the permeability at 2 MPa once again becomes the highest. This indicates that the influence of permeation pressure on the evolution of the fracture network dynamically changes across different damage stages. Under high confining pressure, permeation pressure exerts more complex effects on the development, expansion, and interaction of internal rock fractures. The shaping role of different permeation pressures on the fracture network varies across damage stages, thereby inducing dynamic variations in the permeability relationships.

At an overburden pressure of 12 MPa, the permeability progressively increases with rock degradation (from the expansion stage to the complete failure stage), and the difference in permeability between different pressure levels significantly widens as the damage intensifies. During the expansion stage, the permeability difference between 1 and 3 MPa is 213.1 × 10⁻18 m2; this increases to 712.2 × 10⁻18 m2 in the post—peak stage; and further expands to 961 × 10⁻18 m2 in the complete failure stage. The initial permeability shows a positive correlation with permeation pressure: at the same damage stage, a higher permeation pressure corresponds to greater rock permeability. This indicates that under high confining pressure, water pressure exerts a more pronounced influence on the rock’s permeability characteristics. As rock damage progresses, water pressure not only affects the initial permeation state but also amplifies the impact of pressure differentials on permeability throughout the damage evolution process. This reflects that the cumulative effect of water pressure on internal structural degradation becomes more pronounced under elevated confining conditions.

Permeability Increases with Damage Degradation: The underlying mechanism for this phenomenon is attributed to the fact that as rock damage degradation progresses, its internal microstructure undergoes significant changes. Specifically, joints begin to dilate, and previously relatively closed joint surfaces progressively open, providing increased flow space for fluid.Influence of Water Pressure on Permeability: In most cases, higher water pressure correlates with greater rock permeability at the same damage stage. This is attributed to the fact that the presence of water pressure further interacts with the rock’s microstructure: elevated water pressure exacerbates joint dilation, causing joints to open wider, and simultaneously promotes more intense particle fragmentation, generating more and larger inter-particle voids. Collectively, these effects enhance the connectivity and conductivity of seepage channels, thereby leading to an increase in permeability. Furthermore, as the rock degradation process continuously advances.Specifically, at a confining pressure of 11 MPa, the influence of 2 MPa and 3 MPa water pressures on permeability exhibits a non-monotonic behavior. This implies that the relative magnitude of their impact on permeability changes across different damage stages. This observation strongly indicates that the evolution of rock permeability is not solely dependent on the magnitude of water pressure, but is intimately linked to the rock’s stress state (confining pressure) and its degree of damage. When water pressure is appropriately increased, the pressure acts upon the rock microstructure, promoting further dilation of certain joints and the fragmentation of particles, which facilitates crack extension and enhances the patency of seepage channels, thereby increasing permeability.

The effect of coupled confining and seepage pressure on permeability

The Pearson correlation coefficient is a key statistical metric used to evaluate the strength of the linear association between two continuous random variables35. It is defined as the quotient of the covariance between the two variables and the product of their standard deviations, with values ranging from − 1 to 136. A coefficient value of 1 denotes a perfect positive linear correlation between the variables, while -1 indicates a perfect negative linear correlation. A coefficient of 0 implies the absence of any linear relationship. The closer the absolute value of the correlation coefficient is to 1, the stronger the linear relationship between the two variables. Application of the Pearson correlation coefficient requires certain preconditions: both variables must be continuous data, and their joint distribution should approximately follow a bivariate normal distribution. Based on these characteristics, the Pearson correlation coefficient can be used to analyse linear relationships such as those between temperature, confining pressure, and permeability. Based on the aforementioned characteristics, the Pearson correlation coefficient can be utilized for analyzing linear correlations such as temperature, confining pressure, and permeability. Specifically, x corresponds to confining pressure, y to water pressure, and z to permeability. The specific calculation formula is given in Eq. 2:

$$r = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {x_{i} - \overline{x}} \right)\left( {y_{i} - \overline{y}} \right)\left( {z_{i} - \overline{z}} \right)}}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{n} \left( {x_{i} - \overline{x}} \right)^{2} } \sqrt {\mathop \sum \nolimits_{i = 1}^{n} \left( {y_{i} - \overline{y}} \right)^{2} } \sqrt {\mathop \sum \nolimits_{i = 1}^{n} \left( {z_{i} - \overline{z}} \right)^{2} } }}$$
(2)

By analyzing the Pearson correlation coefficients between permeability and confining pressure, and water pressure at different cyclic loading and unloading stages, Fig. 10 reveals the dynamic variation of dominant controlling factors with the number of cycles: During the first cycle of loading and unloading, permeability exhibited a strong positive correlation with water pressure (coefficient of 0.85) and a very weak negative correlation with confining pressure (coefficient of − 0.043), indicating that the influence of water pressure on sandstone permeability was significantly greater than that of confining pressure at this stage. In the second and third cycles, the correlation coefficients for water pressure and confining pressure in the second cycle were 0.9 and − 0.21, respectively, while in the third cycle, they were 0.61 and − 0.61. These data indicate that the influence of water pressure gradually decreased during these stages, while the influence of confining pressure gradually increased, and by the third cycle, their degrees of influence were comparable (with absolute values both at 0.61). In the fourth and fifth cycles, the correlation coefficients for water pressure and confining pressure in the fourth cycle were 0.078 and − 0.89, respectively, and in the fifth cycle, they were − 0.18 and 0.76. In subsequent fluid–solid coupled cyclic loading and unloading, the influence of confining pressure on sandstone permeability significantly strengthened and surpassed that of water pressure, with the water pressure correlation coefficient reaching 0.76 in the fifth cycle, becoming the dominant factor in permeability change. Overall, as cyclic loading and unloading progressed, the dominant controlling factor for sandstone permeability shifted from water pressure dominance to confining pressure dominance.

Fig. 10
Fig. 10Fig. 10
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Fluid–solid coupled permeability diagram for five cycles of loading and unloading.

Fitting analysis of permeability variations in sandstone under combined confining and hydraulic pressure during different loading–unloading stages revealed that from the first to third cycles, permeability exhibited an exponential relationship with temperature and confining pressure, achieving an R2 value exceeding 0.93. Fitting parameters and correlation coefficients for different loading and unloading stages are presented in Table 2, where M denotes permeability, W represents confining pressure, S signifies hydraulic pressure, and a, b, c, d denote parameters. The fitting formula is:

$${\text{M}} = {\text{a*exp}}\left( {{\text{bW}} + {\text{cS}}} \right) + {\text{d}}$$
(3)
Table 2 Parametric values and correlation coefficients for the coupling of confining pressure and hydraulic pressure with permeability.
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Numerical validation of seepage characteristics

Experimental data reveal that the seepage characteristics of rock specimens are dynamically variable under various experimental conditions. However, the specific mechanisms governing the interaction between stress and seepage in influencing sandstone permeability remain incompletely understood. Consequently, numerical simulations of seepage rates within sandstone specimens under coupled stress-seepage conditions were conducted. This study utilized the COMSOL Multiphysics software, coupling the Solid Mechanics and Darcy Flow modules to achieve its analytical objectives. The numerical model was constructed using the built-in Duncan-Chang constitutive model. The geometric model strictly adhered to the dimensions of actual saturated sandstone specimens, establishing a two-dimensional axisymmetric configuration. Model parameters were specified in strict adherence to the experimental protocol, including fundamental mechanical properties of the rock sample, geometric dimensions, boundary conditions, and critical parameters such as water content. Axial loading was applied using interpolation functions to precisely represent the cyclic loading path. Vertical cyclic loading was imposed at the upper boundary, fixed constraints were applied at the lower boundary, and lateral confining pressure boundary conditions were specified. The seepage field boundaries were defined with an inlet at the bottom and an outlet at the top, thereby establishing a complete seepage pathway. Figures 11 and 12 present, respectively, the simulated Darcy seepage velocity distribution within the specimen following five cycles of loading and unloading, visually illustrating the evolution of seepage characteristics under dual-field coupling.

Fig. 11
Fig. 11
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Confining pressure = 10 MPa, Darcy seepage velocity at water pressures of 1, 2, and 3 MPa.

Fig. 12
Fig. 12
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Darcy seepage velocity at water pressure = 2 MPa and confining pressures = 10, 11, 12 MPa.

Figure 11 illustrates that, taking an encircling pressure of 10 MPa as an example, with hydraulic gradients of 1 MPa, 2 MPa, and 3 MPa, the arrow lengths increase significantly with rising hydraulic pressure, indicating a positive correlation between seepage velocity and hydraulic pressure. According to Darcy’s Law, increased hydraulic pressure directly amplifies the pressure gradient, driving a linear increase in seepage velocity. Simulation data reveal that when hydraulic pressure rises from 1 to 3 MPa, the expansion effect of pores occurs, increasing the effective pore radius and boosting seepage velocity by approximately 150%. This aligns with the linear relationship of Darcy’s Law under laminar flow conditions. Figure 12 illustrates that, with confining pressure gradients of 10 MPa, 11 MPa, and 12 MPa, the variation in arrow length becomes more pronounced, indicating a greater influence of confining pressure on seepage velocity. The lateral confinement effect of confining pressure and its rock-strengthening effect are well reflected in the seepage simulation. At 10 MPa confining pressure, the seepage velocity is highest; at 11 MPa, it is intermediate; and at 12 MPa, it is lowest with a significant reduction. This is broadly consistent with experimental observations. High water pressure induces pore expansion, thereby enhancing permeation rates. Conversely, when confining pressure increases, the pore compression effect is counteracted by the expansion force exerted by water pressure, resulting in a blunted response in permeability variation.

Discussion

While research7 primarily focused on the strength and deformation characteristics of sandstone under triaxial cyclic loading and unloading conditions, the multi-field coupling method adopted in this study offers a superior capability for simulating complex geological environments. In natural geological environments, rock is often subjected to the combined effects of multiple physical fields, such as stress fields, seepage fields, etc. However, triaxial cyclic loading and unloading only considers a specific form within the stress field (triaxial stress) and a cyclic loading pattern, which inherently limits their ability to accurately simulate the actual in-situ environment of the rock. In contrast, the multi-field coupling approach comprehensively accounts for the interactive influences of multiple physical fields, thereby providing a more realistic and complete representation of the rock’s stress state within its original geological environment.

This study observed that the influence of confining pressure and water pressure dynamically varies with the number of cycles, with water pressure dominating in the initial stage, and confining pressure becoming the dominant factor in the later stage. Through quantitative analysis methods such as the Pearson correlation coefficient, this research for the first time systematically revealed and quantified the evolution law of the dominant influence of confining pressure and water pressure on sandstone permeability at different stages under cyclic loading and unloading stress paths, deepening the understanding of seepage mechanisms under fluid–solid coupling.

In deep engineering environments, rock systems, in addition to being affected by fluid–solid coupling, may also be simultaneously influenced by the combined action of chemical fields. The influence of hydrochemical actions (such as ion concentration) on sandstone pore structure has not been fully incorporated. For example, the water absorption and expansion of clay minerals in fracture fillings can change pore connectivity, but existing research largely simplifies water as an ideal fluid, ignoring the long-term impact of chemical corrosion on permeability.

Conclusion

This study conducted cyclic loading and unloading stress-seepage coupled tests on sandstone by systematically varying confining pressure and water pressure conditions, leading to the following main conclusions:

  1. (1)

    Influence of Confining Pressure and water Pressure on Sandstone Mechanical Characteristics: Under a water pressure of 1 MPa, when the confining pressure increased from 10 to 12 MPa, the peak stress increased by 8.1%, which effectively suppressed volumetric expansion and micro-crack propagation. Conversely, when the confining pressure was 10 MPa, and the water pressure increased from 1 to 3 MPa, the peak stress decreased from 59.5 to 49.2 MPa, representing a 17.3% reduction. Elevated water pressure reduced the strength of the rock mass by promoting micro-crack propagation. Sandstone specimens in the cyclic loading and unloading seepage tests under confining pressure and water pressure predominantly exhibited shear brittle failure, maintaining good integrity after failure. As confining pressure increased, the angle between the failure plane and the maximum principal stress also increased; a few specimens presented "Y"-shaped double cracks. Upon an increase in water pressure, the flatness of the failure plane decreased, with the appearance of fine parallel cracks and fragmented blocks, and tensile cracks accelerated failure, though overall, shear brittle failure remained dominant.

  2. (2)

    Influence of Confining Pressure and water Pressure on Sandstone Permeability: Under a water pressure of 1 MPa, when the confining pressure increased from 10 to 12 MPa, permeability decreased by 4.6%. Conversely, under a confining pressure of 10 MPa, when the water pressure increased from 1 to 3 MPa, permeability increased by 20%. Permeability generally exhibited an increasing trend with the increasing number of cyclic loading and unloading cycles, with permeability during the loading stage being lower than that during the unloading stage. The influence of confining pressure and water pressure on permeability dynamically varied with the number of cycles; in the initial stage, water pressure exerted a greater influence on permeability, but as the number of cycles increased, confining pressure gradually became the dominant factor in permeability change.

  3. (3)

    Coupled Influence Law: The coupled influence of confining pressure and water pressure on sandstone permeability demonstrated distinct phased characteristics. At lower cycle numbers, the correlation coefficient between permeability and water pressure was 0.85–0.90, indicating a greater influence of water pressure on permeability. However, with an increase in the number of cycles, the influence of confining pressure on permeability gradually strengthened, and it became the dominant factor at higher cycle numbers, with its correlation coefficient ranging from 0.60 to 0.89. This coupled influence law reflects the evolution mechanism of seepage characteristics in deep rock masses under coupling.

  4. (4)

    Numerical simulation validation indicated that, with constant confining pressure, increasing water pressure from 1 to 3 MPa resulted in a permeability increase of approximately 150%, which was consistent with the linear relationship of Darcy’s Law. At constant hydraulic pressure, increasing confining pressure from 10 to 12 MPa caused a decrease in seepage velocity. At high confining pressures, the balancing effect of pore compression and hydraulic expansion led to a blunted change in permeability, indicating that confining pressure governed the evolution of permeability in sandstone.