Fractures are the key to fluid flow, transport, and heat transfer inside natural rock masses with distinct apertures due to the complex in situ stress (confining pressure), and knowledge of these fracture apertures plays an irreplaceable role in the seepage through fractures in many subsurface engineering applications, including seawater intrusion (Sebben et al., 2015), geothermal energy exploitation (Wang et al., 2021b; Li et al., 2022), Water–Silt Inrush Hazard (Ma et al., 2022a) and urban drainage systems (García et al., 2015; Ma et al., 2022b). However, previous research has focused on the relationship between the hydraulic aperture and in situ stress, and there are no theoretical models for predicting the change in the hydraulic aperture under in situ stress. It is meaningful to study the hydraulic aperture based on fracture deformation for performance assessment and optimization design in tunnels, urban underground railways, and many other underground spaces (Zhao et al., 2021).

To characterize the hydraulic aperture, the finite difference method was used to solve the lubrication equation, and the calculated results were displayed as a function of the ratio of the mean mechanical aperture to the standard deviation of the roughness distribution. The third power of the ratio of the hydraulic aperture to the average mechanical aperture is related to the ratio of the standard deviation of the mechanical aperture to the average mechanical aperture (Zimmerman and Bodvarsson, 1996). The ratio of the hydraulic aperture to the average mechanical aperture is not constant, and it can be expressed as an exponential relationship with the standard deviation of the average mechanical aperture (Renshaw, 1995). In tests conducted on gradient samples, the mean mechanical aperture was calculated and found to be considerably broader than the hydraulic aperture (Chen et al., 2000). In rock engineering applications, as one parameter used to estimate the fracture aperture, the hydraulic aperture was used in field hydraulic tests (Li et al., 2013; Cao et al., 2016). It is widely accepted that the hydraulic aperture is relevant to the fracture geometry and is meaningful to the accurate measurement of the hydraulic aperture and the permeability of the fractured rock. Affected by the measurement methods and characterization parameters of the fracture morphology used (Sun et al., 2020), the measurement accuracy of the fracture geometry is difficult in rock engineering.

Previous research has shown that the confining pressure has a non-negligible influence on the permeability and hydraulic aperture, which is affected by the ambient conditions, such as the temperature (Watanabe et al., 2017; Hopp et al., 2019), confining pressure (Chen et al., 2016; Shu et al., 2020), hydraulic pressure (Zhao et al., 2019) and dynamic disturbance (Du et al., 2019; Huang et al., 2022). However, the temperature has less impact on the change in the permeability at higher confining pressures (Ding et al., 2016). Moreover, transient pulse test results have shown that the roughness is no longer critical to the permeability when the confining pressure is greater than a certain value (Zhao et al., 2017). As an irreplaceable parameter affecting the permeability, only the experimental regularity of the confining pressure has been summarized. For the permeability reduction of fractured rocks, the technique of filling fractures with grout was proposed, and the stress-induced changes in the permeability of the infilled fractures were investigated (Zhou et al., 2020). Owing to the construction of buildings and the excavation of underground spaces, the distribution of the stress field in the original rock mass is changed (Chen et al., 2022; Yang et al., 2022). The change in the stress field distribution will inevitably lead to changes in the geometric parameters of the fracture in the rock mass, which will contribute to the change in the permeability of the fracture in the rock mass (Wang et al., 2021a; Ma et al., 2021). The change in the permeability of the rock mass will directly give rise to changes in the stability of the rock mass and the buildings on it. There is not even a reasonable theoretical model that describes the change in the hydraulic aperture under confining pressure.

By reviewing the discussion above, the mechanism of confining pressure on hydraulic aperture required further research. In this study, we investigated how the confining pressure affects the hydraulic aperture based on the fracture deformation constitutive law and how the flow rate is controlled by the confining pressure. Hydromechanical tests were conducted under different confining pressures (P _c =10–38 MPa) with a constant flow rate or seepage pressure. The results of this study enhance our understanding of the interaction between the seepage and the surrounding rock under different in situ stress conditions and offer guidance for rock engineering projects.

We studied the mechanism by which the confining pressure influences the hydraulic aperture (Figure 1). The artificial fracture specimens from a shale block, which was taken from Changning. Figure 1B expresses the fracture of the sample, which is produced by cutting along the diameter. There are conclusions that different lithology has a certain effect on permeability and non-Darcy flow coefficient of water flow in fractured rock (Ma et al., 2013), while the lithology was not focused in this manuscript. For incompressible steady-state fluid flow through a single fracture, the Navier-Stokes equation based on Newton’s second law was used: ρ [ ∂ u ∂ t + ( u ⋅ ∇ ) u ] = − ∇ P + μ ∇ 2 u + F (1) where u is the flow velocity; ρ is the fluid density; ▽P is the fluid pressure gradient; µ is the dynamic viscosity coefficient; and F is the body force vector.

The inertial forces are negligible compared to the viscous forces. Thus, the N-S equation reduces to the Stokes equation, and the cubic law is obtained based on the assumption that the single fracture consists of two smooth parallel plates. The cubic law is as follows (Witherspoon et al., 1980): Q = − k A h μ ∇ P = − w e h 3 12 μ ∇ P (2) where Q is the discharge; k is the intrinsic permeability; A _h is the cross-sectional area; w is the fracture width; and e _h is the hydraulic aperture. The hydraulic aperture e _h can be calculated from the cubic law (Watanabe et al., 2008; Quinn et al., 2020) as follows: e h 3 = Q P s l 12 μ w (3)

I. Goodman proposed the hyperbolic model of the fracture deformation constitutive law (Goodman, 1974): Δ V f = V m − V m P i 1 P c (4) Δ V f = e h i − e h (5) where Δ V f is the fracture closure deformation; V m is the maximum hydraulic aperture closure deformation; P i is the initial confining pressure; P c is the confining pressure; and e h i is the initial hydraulic aperture. By combining Eqs 4, 5, the hydraulic aperture can be expressed as e h = e h i − V m + V m P i 1 P c (6)

By combining Eqs 3, 6, under a constant seepage pressure P _s , the flow rate Q can be expressed as Q = P s w 12 μ l ( e h i − V m + V m P i 1 P c ) 3 (7)

II. Eq. 4 can be replaced by the hyperbolic model proposed by Bandis (Bandis et al., 1983): Δ V f = P c V m P c + K n i V m (8) where K n i is the initial normal stiffness of the fracture. The hydraulic aperture and flow rate, respectively, can be written as e h = e h i − P c V m K n i V m + P c (9) Q = P s w 12 μ l ( e h i − P c V m K n i V m + P c ) 3 (10)

III. Sun proposed the exponential model (Sun and Lin, 1983): Δ V f = V m − V m ⁡ exp ( − P c / K n ) (11) where K n is the normal stiffness of the fracture. The hydraulic aperture and flow rate, respectively, can be written as e h = e h i − V m + V m ⁡ exp ( − P c K n ) (12) Q = P s w 12 μ l ( e h i − V m + V m ⁡ exp ( − P c K n ) ) 3 (13)

IV. Rong proposed the g−λ model (Rong et al., 2012): Δ V f = V m − V m ( λ P c K n i V m + 1 ) − 1 λ (14) where λ is a parameter associated with the fracture weathering, roughness, fluctuation degree, the matching of fracture surface and strength of the wall of the rock fracture. The hydraulic aperture and flow rate, respectively, can be written as e h = e h i − V m + V m ( λ P c K n i V m + 1 ) − 1 λ (15) Q = P s w 12 μ l [ e h i − V m + V m ( λ P c K n i V m + 1 ) − 1 λ ] 3 (16)

In this theoretical model, the hydraulic aperture is used to characterize the aperture of the fracture, and the fracture deformation constitutive law is used to describe the effect of the confining pressure on the hydraulic aperture. This model gives the theoretical expression for the confining pressure P _c and the hydraulic aperture e _h related to the interaction between the flowing fluid and the surrounding rock. It provides a quantitative framework for investigating the evolution of the seepage characteristics with confining pressure.

The above models show the mechanism by which the confining pressure P _c influences the hydraulic aperture e _h based on the fracture deformation constitutive law (fracture deformation constitutive law models of Goodman, Bandis, Sun, and Rong), and the relevant equations are proposed. To further validate the equations relating P _c to e _h or Q, we conducted hydromechanical tests on a shale sample cut by a single fracture (100 mm in width and 100 mm in length; Figure 1B) in the laboratory. The tests were performed using a device consisting of a confining pump (2J-X plunger metering pump), a syringe pump (ISCO 65D), pressure gauges, and a sample holder. The hydromechanical tests were conducted at a constant flow rate Q (2–14 ml/min) under confining pressure P _c (10–28 MPa), and at a constant seepage pressure P _s (400–2000 kPa) under confining pressure P _c (20–38 MPa).

The fluid mechanic’s test was carried out using laboratory instruments (Figure 2). The fluid mechanic’s experiment processes are as follows:

(1) The prefabricated single fracture rock sample was put into the holder. After placing the rock sample, connect the high-pressure pump, ring pressure pump and holder with capillary iron pipes to combine the fluid mechanic’s experimental system.

In this study, we investigated the mechanism by which the confining pressure influences the hydraulic aperture based on the fracture deformation constitutive law using a combination of hydromechanical tests and a theoretical model, and regression analysis was performed on the test data using the proposed model. The main conclusion of this study are as follows.

(1) Four types of models for the relationship between the hydraulic aperture or flow rate and the confining pressure were obtained, and the hydromechanical tests were performed under different confining pressures.