Fluid flow and evolution mechanisms in fractured rocks are fundamental tasks in engineering fields such as geohazards prediction, geothermal resource exploitation, oil and gas exploitation, and geological sequestration of carbon dioxide. This study employed an enhanced X-ray imaging digital radiography to investigate nonlinear flow model of fluid through different roughness fractures. The X-ray images of fluid flow during rock failure were analyzed using a multi-threshold segmentation method applied to the X-ray absorption dose. The result show that a proposed nonlinear flow equation considers the joint roughness coefficient and the uniaxial compressive strength of the jointed rock, enabling a better understanding of the nonlinear flow behavior in fractured rock masses. This modeling approach has important theoretical and practical implications. By accounting for key factors influencing fluid flow behavior, it can help guide monitoring efforts to support early warning of fractured rock mass instability. Additionally, a more mechanistic understanding of flow processes may inform strategies to prevent engineering geological hazards.
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Abbreviations
JRC
Joint roughness coefficient
JCS
Joint compressive strength (MPa)
Q
Flow volume rate (m3/s)
A
Linear coefficient
B
Nonlinear coefficient
α
The proportionality coefficient of pressure gradient
C
The contact ratio of fracture closure when rock failures
Qn
Flow volume in fracture under the nth normal stress (m3)
\(\overline{Q}_{{\text{n}}}\)
The average flow rate of fluid through a deformable fracture (m3)
w
Width of rock fracture (m)
∇P
Flow pressure gradient (Pa/m)
1 Introduction
The fluid flow properties of fractured rock masses play a crucial role in various geotechnical engineering projects, including tunnel excavation, oil and natural gas extraction, carbon dioxide geo-sequestration, and others. Fractured rock masses can have complex geometries and varying degrees of roughness, which greatly influence fluid flow behavior. The interconnected network of fractures within the rock mass can create preferential pathways for fluid flow, leading to seepage-related problems. Fluid flow behavior through roughness fractures and its induced disaster mechanism remain cutting-edge topics in the domain of engineering geology (Zimmerman et al. 2004). Different experimental and numerical modeling techniques are employed to understand the flow behavior through fractures (Barani et al. 2020; Natarajan and Kumar 2011), including laboratory testing (Sun et al. 2021a), field measurements (Chen et al. 2015b), and numerical simulations (Liu et al. 2020; Xiong et al. 2019; Zhou et al. 2023). These studies help engineers and geologists design appropriate measures to mitigate seepage-related issues and prevent potential disasters. The nonlinear fluidic evolution during the dynamic extension and alteration process within rock or rock mass fractures is a fundamental challenge in simulating engineering geological disasters. Fractured rock masses are formed due to long-term geological tectonic forces and historical environmental influences. They consist of a complex structure composed of matrix rock blocks and joint faces (Ma et al. 2023). This complexity makes the simulation calculations of engineering geological disasters a challenging task. The permeability of intact rock blocks is typically very low, but the presence of macro-fractures significantly influences fluid flow and permeability properties. The seepage characteristics of fractured rock masses are primarily controlled by the properties and conditions of the joint faces (Zou et al. 2015). The macro-fractures can act as preferential pathways for fluid migration, allowing fluids to flow through the rock mass more easily. This increased permeability can have significant engineering implications, as it may lead to issues such as groundwater seepage, rock mass instability, or even the sudden release of accumulated fluid pressure, resulting in catastrophic events like landslides or collapses. Early research presumed that fractures were often assumed to be simple fissures with smooth, planar surfaces. The classical cubic law (Zimmerman and Bodvarsson 1996) was commonly used to describe fluid flow through a single fracture based on this assumption. The rough and uneven nature of fracture surfaces in rock masses, along with the presence of protruding contacts, makes the classical cubic law inappropriate for describing the relationship between fracture fluid pressure and flow rate changes. Fluid flow through fractured rock masses is often influenced by inertial effects (Wang et al. 2022), especially at higher flow rates. At high Reynolds numbers, nonlinear flow phenomena can occur, deviating from the assumptions of linear flow in the classical cubic law. The seepage experiments conducted by Chen et al. (2015a) on rough fractures revealed three types of nonlinear flow behaviors. These behaviors were attributed to various factors influencing the flow, including inertia, fracture expansion, and fluid–solid interactions. The nonlinear flow phenomenon caused by fluid inertia in fractures can be described using the Forchheimer equation (Forchheimer 1901) and the Izbash equation (Izbash and Leleeva 1971), but further research has confirmed that the Forchheimer equation can better describe the nonlinear flow behavior of fracture fluids (Javad and Ramezanzadeh 2020; Xiong et al. 2019). The fracture morphology has a significant impact on fracture nonlinear flow and is usually described using parameters such as fracture aperture, roughness, and contact area. Liu et al. (2020) conducted theoretical and numerical simulation studies on the nonlinear flow characteristics within fracture intersections and the nonlinear flow of fracture networks, establishing a seepage characteristic prediction model for fracture networks based on fractal theory. In considering the relationship between shear strength parameters and permeability, Rong et al. (2018) proposed a hydraulic coupling model based on the Forchheimer equation for the shear deformation process in rock fractures. In unaltered conditions, fractured rock masses are commonly subjected to tectonic stress and disturbance loads (Zhang et al. 2021), which directly control the permeability of fractures by affecting their apertures.
Previous research has indeed confirmed that dilatation, caused by factors such as fluid pressure or shearing, tends to increase fracture apertures. On the other hand, the action of normal stress, which is the stress perpendicular to the fracture surfaces, can result in the closure or reduction of fracture apertures. The coupling of stress and fluid flow in fractured rocks has become a prominent topic of discussion and a challenge for many international scholars. Some of the key issues include:
In fractured media, fractures act as the primary pathways for fluid storage and migration. The presence of fluids can significantly influence the behavior of the rock mass by altering water pressure and inducing physical and chemical changes within the fractures and the surrounding rock matrix. One example of the fluid-mediated alteration of the rock mass is the formation process of karst conduits, which occurs over long periods under the effects of weathering and erosion by fluids.
The internal structure of a rock mass can undergo significant changes under the influence of tectonic stress or engineering activities, leading to alterations in fluid migration channels and the permeability of fractured rock masses. When a rock mass is subjected to tectonic stress, pre-existing fractures may deform or new fractures may be generated, affecting the connectivity and aperture of the fracture network. These changes can impact the pathways through which fluids migrate, resulting in variations in fluid flow patterns.
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These interactions between fractured rock mass and seepage under the situations mentioned above are termed fracture seepage-stress coupling. The analysis of seepage and stress coupling is a crucial characteristic of rock mechanics. High seepage pressure and hydraulic action are key factors inducing catastrophic nonlinear flow in fractured rock mass. The critical hydraulic gradient (Jc) is used to describe the transition from linear to nonlinear fluid flow in fractures (Ovalle-Villamil and Sasanakul 2019). Zimmerman et al. (2004) proposed that when the total energy loss caused by viscous dissipation and inertial dissipation exceeds 10%, the effect of the inertia term cannot be ignored. Hydraulic erosion significantly affects the properties of fluid flow in fractured rocks. Under the action of hydraulic erosion, debris particles of filled rock joints continually lose or redistribute, leading to a sustained rapid development of the fractured rock mass permeability. Ma et al. (2019) studied the relationship between rock porosity and permeability under erosion and obtained the time evolution characteristics of hydraulic parameters under erosion. The hydraulic flow units around the faults and fractures are characterized by low porosity, high permeability, high resistivity and are generally observed to enhance their flow properties (Al-Dujaili 2023). Additionally, when external loads change, the fluid filling the fractures will form a squirt flow effect. Hydraulic splitting and expansion at the single-fracture tip form a fracture network fluid flow, as confirmed by the jet flow model (Tang 2011), Biot–Squirt model (Biot 2005), and pore-fracture microstructure model (Lang et al. 2014). The aforementioned research has achieved progress in the theory of fracture seepage and has been applied to engineering seepage calculations. However, many practical projects have confirmed that the rock failure process and its abrupt seepage behavior induced by engineering disturbance loads are one of the main reasons for large-scale rock mass instability, such as water inrush from the goaf induced by mining or tunnel excavation. Hence, examining the mechanism of nonlinear flow in fractured rock mass under the effect of disturbance stress can further advance the theoretical framework and experimental and numerical simulation studies of hydraulic coupling in fractured rock mass, which bear significant practical value in guiding engineering applications.
The environment and internal structure of fractured rock mass are complex. Currently, there is scant research on the dynamic seepage characteristics of fractured rock mass under stress. In particular, the quantified interpretation of the impact of the normal stress or shear stress on single fracture or fracture network is challenging to obtain. Simultaneously, the varying laws of fracture fluid motion are even more difficult to describe uniformly. Therefore, it is imperative to conduct in-depth investigations into the fluid flow evolution with the fracture closure, deformation, and expansion processes under normal stress. In this study, medical cardiovascular enhancement imaging technology was employed to conduct experiments on the nonlinear flow characteristics of rock fractures at the block scale during the failure process. X-ray digital radiography (DR) images of dynamic fluid migration changes during the rock failure process were obtained. The multi-threshold image segmentation method of X-ray absorption dose was adopted to quantitatively analyze the dynamic evolution process of fracture fluids. Based on this, the Forchheimer equation for the roughness index (JRC) of the fracture, the joint compressive strength (JCS) of the fractured rock, the flow rate (\(\overline{Q}_{i}\)) of fluids in deformable fractures, and the fluid pressure gradient (− ∇P) is established. The research results can deepen the understanding of deformation and failure mechanisms of fractured rock masses and the influencing factors of rock mass stability from the perspective of fracture stress-seepage coupling. The results hold significant application value in engineering geological disaster monitoring, early warning, and prevention.
2 Geological Setting
To reveal the nonlinear flow of fluid in fractured rocks, uniaxial pre-crack single-fracture permeability tests were conducted using brittle rock samples. As shown in Fig. 1a, basaltic rocks is generally distributed in the northern part of Hainan Islands. Basalt rock samples were collected from strata in the Fushan Sag of Hainan Islands, China. The estimated exposed area of basalts in the northern part of Hainan is approximately 4,109 km2. The Fushan Sag contains 78 basalt layers with a total thickness of 380 m. The collected basalt samples in this study were from a depth of approximately 50 m. Relatively intact basalts were prepared as single-fracture permeable test samples. Figure 1a displays the geological map of Hainan Island, China. Figure 1b illustrates the shallow stratigraphy of the sample collection area. Figure 1c depicts the size of the collected basalt sample and the processed single-fractured basalt based on the standard JRC profile. Because basalts exhibit notable brittle mechanical behavior (Liu et al. 2023), their fractures can represent the brittle fracture characteristics of most engineering rocks. Thus, basalt was chosen as the sample for this study. Similarly, intact granite or other brittle rocks could serve as test samples under equivalent conditions.
Fig. 1
Distribution of Basaltic rocks in Hainan Island, Geological Map and Samples Preparation. a Basaltic rocks in Hainan Island, b Shallow Geological Profiles of the Fushan Sag, c Samples Preparation
×
3 Methodology
3.1 Theoretical Background
Historical research into the rough fracture seepage problem has predominantly revolved around experimental and numerical methodologies. The parallel plate model has been proposed to describe Darcy flow of fluid in rock fractures. The classic cubic law (Sun et al. 2021b; Wang et al. 2015), widely applied to fracture flow calculation and simulation, has presented a quantitative correlation between fracture flow rate and pressure gradient:
Here, w is the fracture width (Unit: mm), e is the aperture of the idealized parallel smooth fracture (Unit: mm), μ is the dynamic viscosity of fluid, k, is the intrinsic permeability defined as e2/12, and Ah is the cross-sectional area of fluid flow. This equation delineates a direct proportional relationship between the fracture flow rate and the cube of the fracture aperture. However, aperture e has been revised to hydraulic aperture Ah.
The complexity of fracture geometry brings certain challenges. Uneven and rough fracture surfaces may induce early turbulence; thus, the cubic law is only applicable for describing the flow regime of fractures with low-speed and smooth walls. As the pressure of the fracture fluid continues to increase, the fluid exhibits nonlinear flow behavior. With the increase in the Reynolds number of the fracture fluid, the inertia of the fracture fluid increases, making the nonlinear flow phenomenon more pronounced. The Forchheimer model (Forchheimer 1901) (Eq. 2) and Izbash model (Izbash and Leleeva 1971) (Eq. 3) are typically used to describe nonlinear flow phenomena in fractures. Both models establish a quantitative relationship curve between the pressure gradient at both ends of the fracture and the volumetric flow rate, Q.
$$- \nabla P = AQ + BQ^{2}$$
(2)
$$- \nabla P = \lambda Q^{m}$$
(3)
In Eqs. 2 and 3, \(\nabla P\) is the pressure gradient along the flow direction, Q is the volumetric flow rate or discharge, and A and B are the coefficients describing energy losses due to viscous and inertial dissipation mechanisms, respectively. In the Izbash formula, λ and m serve as fitting coefficients.
The Forchheimer number, Fo, is usually introduced to define the critical judgment condition for the termination of linear flow and the initiation of nonlinear flow in fracture fluids. Fo is defined as the ratio of pressure loss in the nonlinear term to the linear term in the Forchheimer equation:
In the above Eq. 4, Fo represents the ratio of the pressure gradient required to overcome inertia to viscous force, which also reflects the relationship between the speed variation of fracture flow and the characteristics of fracture morphology. Zeng and Grigg (2006) defined proportion coefficient “α” of the nonlinear pressure gradient to the total pressure gradient as α = \(\frac{{BQ^{2} }}{{AQ + BQ^{2} }} = \frac{{F_{{\text{O}}} }}{{1 + F_{{\text{O}}} }}\), and they provided a critical threshold value of Fo = 0.11. Ghane et al. (2014) conducted an image analysis experiment of volumetric flow and concluded a critical coefficient of Fo = 0.31. The discrepancy in the critical threshold values of fracture flow state Reynolds number Re and Fo is primarily determined by the geometrical features of porous media and fracture channels. The quantitative relationship between the Reynolds number and critical coefficient Fo is as follows in Eq. 5.
$$Re = \frac{A\rho }{{B\mu w}}F_{{\text{o}}}$$
(5)
3.2 Experimental Procedure
In actual engineering, the catastrophic failure process induced by nonlinear flow in rock fractures is primarily characterized by hydraulic gradients and fracture-breach volumes. However, even more crucial are the connectivity and structural evolution of the surge channels within the rock mass. If a nonlinear flow model can be established that evolves with the rock failure process, it would accurately depict the dynamic evolution of nonlinear flow in fractured rock. To systematically study the characteristics of nonlinear flow during rock failures, a rock fracture fluid visualization experimental system based on enhanced X-ray DR imaging technology has been autonomously designed and developed (Sun et al. 2021a, b). Basalt specimens with a single rough fracture were prepared with dimensions of 200 × 30 × 50 mm. As shown in Table 1, basalt specimens with different roughness levels were prepared according to different standard profile features of rough fractures (JRC = 0–20) (Barton et al. 2023; Zoorabadi et al. 2015), totaling ten specimens. X-ray DR imaging experiments on nonlinear flow of fracture fluid during the basalt rupture process were conducted.
Table 1
Basic physical parameters of rock
JRC
Typical roughness profiles
Weight/kg
Density/kg·m−3
JRC
Typical roughness profiles
Weight/kg
Density/kg·m−3
0–2
0.71
2378
10–12
0.70
2317
2–4
0.75
2486
12–14
0.74
2457
4–6
0.69
2315
14–16
0.75
2486
6–8
0.70
2345
16–18
0.71
2381
8–10
0.71
2351
18–20
0.70
2318
The channel structure of fracture fluid flow is somewhat similar to the flow of blood in the human body. The relatively mature medical X-ray enhancement angiography technology can easily capture real dynamic images of the entire rock failure process and the flow of fracture fluid. Based on quantitative analysis of the multilevel unloading strength of rock and the characteristics of nonlinear flow in fractures via X-ray DR images, a nonlinear flow equation considering the compressive strength of jointed rock (JCS) and the roughness of jointed fractures (JRC) (Barton et al. 2023) have been established. The equation reveals the mechanism of nonlinear flow in fractures with varying degrees of roughness during the rock failure process.
To observe the changes in the X-ray images of fracture fluid during the fracturing of rock at the block scale, an X-ray DR imaging platform (model DSM-80) was selected for the experimental system, as shown in Fig. 1a. A multistage rock loading test device that can match this platform was developed. The maximum loading capacity of the hydraulic jack in this device is 400 tons, the maximum rise height of the jack piston column is 20.0 cm, and the control speed of the electric hydraulic loading is 0.70 mm/s. A DN10 flow meter and seepage meter (1.0 MPa) were used to monitor the hydraulic conduction parameters of single fractures with different roughness levels, see in Fig. 1b. Synchronization of X-ray imaging of fracture fluid during the single-fracture extension process under multistage rock loading conditions was realized. During the test shooting, the distance between the X-ray ball tube and the target plate was kept at 1.5 m. The parameters of the diversion pump used in this experiment are listed in Table 2 and the components of the entire experimental system are shown in Fig. 2.
Table 2
The parameters of the diversion pump
Rated voltage
Maximum current
Motor speed
Temperature
Maximum head
Initial flow
DC 12 V
2.5 A
2800 r/min
80 °C
40 m
2.8 L/min
Fig. 2
Experimental platform for X-ray visualization of fluid flow through the fracture during rock failures. a Experimental system including the multi stress loading machine matched the X-ray DR system of DSM-80; b Rock specimens preparation and testing procedure illustrations
×
In this experiment, rock samples of different fracture roughness levels were first prepared according to the typical JRC profile image features listed in Table 1. X-ray scanning voltage is 130 kV, electric current is 320 mA, and exposure time is 280 ms. Following the preparation, conduction joints were installed and sealed on single fractured rock samples. The rough fractures were subjected to conduction testing using a self-priming pump, yielding the inlet flow rate (QInlet) and outlet flow rate (QOutlet) of the conduction pipeline, which allowed the calculation of the fluid Reynolds number (Re) at the inlet and outlet positions of the fracture conduction. The captured X-ray DR images of rock fracture fluid were categorized on the basis of the magnitude of multilevel normal stress loads. The multithreshold segmentation method was employed to identify and extract the fracture fluid X-ray image features, with the dose of X-ray absorbed in the image features being quantitatively analyzed. This facilitated the calculation of the actual flow rate (Qi) of fluid filling-in real fractures and the study of nonlinear flow problems caused by fracture closure and expansion under multilevel normal stress loads. During the experimental testing process, synchronous data monitoring of flow rate, pressure, and multilevel normal stress loading and unloading was achieved, allowing for the full process of edge loading, conduction, and shooting of rock samples. This provides a basis for further analysis and description of the synchronous evolution relationships of rock damage strength, fracture hydraulic parameters, and fluid properties. The entire experimental system testing process and steps are shown in Fig. 2.
4 Results and Discussion
4.1 Normal Compression Impact on Fracture Roughness with Rock Failures
Fractured rock bodies are comprehensive structural features composed of matrix rock blocks and structural planes, existing in historical stratigraphic geological environments, such as non-continuous, multiphase, multimedium, and anisotropic geological bodies with geostress, geological fluid, and geotemperature. The seepage characteristics of fractured rock bodies are mainly determined by the features of structural planes, which are often influenced by the load of parent rock and disturbance stress. The action of dilatation stress can lead to increase in fracture width, while the action of normal stress can cause fracture closure, thereby reducing the fracture aperture. The interaction between stress and seepage in fractured rock bodies mainly manifests in two aspects. On one hand, fractures are the main channels for fluid storage and migration, and fracture fluid acts on the solid matrix through pressure and hydrophysical chemistry. On the other hand, the external stress load of the rock body intensifies the change in the internal defect structure of the fracture channel, forming fracture deformation, expansion, penetration, and shear slippage, thereby changing the conduction performance of the fracture. The interaction between rock bodies and fluids is referred to as seepage-stress couplings. This study performs radiographic experiments on the dynamic changes of fluid in fractures during fracture closure and expansion under multilevel normal stress loading. The failure process of single fractures is different roughness under the influence of multilevel normal stress loading is clearly defined. The multilevel normal stress loading paths of preprepared fractured rocks with different roughnesses are illustrated in Fig. 3 and Figure S1. Specifically, Fig. 3a–d correspond to JRC = 0–2, JRC = 8–10, JRC = 10–12, and JRC = 18–20, respectively. In this context, the maximum value of the multilevel stress loading is defined as the uniaxial compressive strength (JCS) of the fractured rocks.
Fig. 3
Peak values evolutions and JCS of the fractured rocks under multilevel stress loading path. a JRC = 0–2; b JRC = 8–10; c JRC = 10–12; d JRC = 18–20; other data see Figure S1 in the supplementary materials
×
According to the experimental results, the failure of the preprepared basaltic fractures presents typical brittle failure characteristics. The unloading phenomenon of the fractured rock mass caused by a single loading action is apparent, with only a few specimens showing a relatively low level of residual strength. The maximum uniaxial compressive strength of the fractured rock mass is 43.83 MPa and the minimum is 20.17 MPa. The influence of hydraulic conduction pressure on the strength of the intact fractured rock mass is relatively small, making it difficult to form a pronounced local hydraulic fracturing effect. The characteristics of the multilevel normal stress paths of fractured rocks of different roughness can be found in the supplementary materials.
The experimentally obtained X-ray DR images of the dynamic changes in fluid in fractures during the failure process of the rock mass. The dynamic changes of fracture fluid were analyzed according to the size and path characteristics of multilevel normal stress loading, using the multi-threshold segmentation method of X-ray absorption dose to identify and extract the characteristic images of fracture fluid. Based on the threshold range of X-ray absorption dose, the fracture flow rate was analyzed. Figure 4 shows the X-ray analysis results of the distribution characteristics and size changes of fracture fluid flow during the fracture process of fractured rocks of different roughness. The hydraulic gradient − ∇P, fracture flow rate, and multilevel normal stress peak load listed in Table 3 were statistically obtained.
Fig. 4
Flow volume calculation of fluid filling different roughness fractures under multilevel stress loading based on X-ray DR image analysis (JRC = 0–2, 8–10, 10–12 and 18–20; other data see Figure S2 in the supplementary materials.)
Table 3
The hydraulic parameters of fluid flow through the roughness fracture in rocks under different normal stress
JRC
JCS(MPa)
− \(\nabla P\)
(MPa/m)
Peak values of multilevel stress loading (JCS*)/MPa and volumetric flow Qi / × 10−5m3
σ1
σ2
σ3
σ4
σ5
σ6
σ7
σ8
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
0–2
22.00
0.118
3.17
8.00
12.83
19.17
22.00*
18.67
N/A
N/A
1.69
3.95
3.65
1.81
1.01
1.64
N/A
N/A
2–4
43.83
0.122
5.00
7.17
24.00
43.83*
43.50
29.33
N/A
N/A
1.61
3.71
3.86
1.96
3.02
2.93
N/A
N/A
4–6
28.84
0.122
3.84
7.17
13.00
28.84*
20.34
6.50
N/A
N/A
1.45
3.96
3.96
1.47
3.87
3.06
N/A
N/A
6–8
26.67
0.125
3.84
6.00
11.34
21.50
26.67*
14.84
5.34
N/A
0.91
1.58
1.43
1.27
0.49
1.37
1.25
N/A
8–10
20.17
0.131
1.00
3.67
7.67
13.67
20.17*
16.17
N/A
N/A
0.04
1.69
1.99
2.24
1.45
2.01
N/A
N/A
10–12
34.33
0.139
21.17
24.83
28.00
30.50
34.33*
26.17
N/A
N/A
1.24
0.88
1.27
1.03
1.23
1.38
N/A
N/A
12–14
35.00
0.146
3.00
16.33
24.50
35.00*
24.50
13.50
N/A
N/A
0.52
1.60
1.52
0.86
1.49
1.05
N/A
N/A
14–16
21.17
0.165
5.17
13.84
19.67
21.17*
16.5
10.17
N/A
N/A
0.37
0.82
1.28
1.07
1.02
0.75
N/A
N/A
16–18
25.67
0.169
3.84
7.17
19.34
24.17
25.67*
25.00
N/A
N/A
0.48
0.98
1.01
0.82
0.59
0.52
N/A
N/A
18–20
28.33
0.177
2.00
3.67
7.17
15.00
20.67
26.00
28.33*
25.67
0.19
0.28
0.38
0.45
0.34
0.25
0.20
0.33
*Is the maximum value of multilevel stress loading
×
Normal stress causes significant closure or opening of the fracture, while the mechanical behavior of rock failure is mainly manifested in loading and unloading (Malama and Kulatilake 2003; Sun et al. 2019; Tang et al. 2014). Normal stress loading will cause fracture closure, while unloading will lead to local fracture deformation, expansion, and opening. It is difficult to accurately describe the changes in fracture roughness during fracture closure, deformation, and expansion processes under normal stress. The changes in fracture closure contact points can be obtained from the analysis of fracture fluid in the X-ray DR images. The prepared fractures with different roughness inside the rock mass have different closure contact rates under multilevel normal stress conditions. To facilitate the comparison of the differences in closure contact rates of fractures of different roughnesses, this study selects X-ray DR images of fracture closure contact and fracture fluid under the rock mass unloading failure state as the basis. The discontinuous position of the fracture fluid characteristic image is the fracture closure point. The horizontal length contact of the fracture closure contact and the total horizontal length total of the fracture are counted. According to the calculation method of the fracture closure contact rate after the unloading failure of fractured rocks with different roughness under normal stress, as shown in Fig. 5, the closure contact rate of the different roughness fracture in the complete failure state of rocks is calculated as listed in Table 4. It can be seen that the prepared fractured rocks with rougher conditions have a higher closure contact rate in the complete failure state. This is mainly because rougher fractures easily form more compact bite points under normal stress, hence the higher contact rate of rougher fractures after rock failure.
Fig. 5
The contact rate counting method of the fracture closure when rock failures
Table 4
The closure contact rate of the different roughness fracture in the complete failure state of rocks
JRC
X-ray DR image of fluid in rock fracture
Lcontact/mm
Contact ratio C (%)
JRC
X-ray DR image of fluid in rock fracture
Lcontact/mm
Contact ratio C (%)
0–2
24.07
13.37
10–12
37.23
20.68
2–4
33.10
18.39
12–14
39.91
22.17
4–6
35.76
19.87
14–16
49.71
27.62
6–8
36.41
20.23
16–18
61.25
34.03
8–10
36.71
20.39
18–20
61.66
34.26
×
Under the influence of normal stress, fractures tend to display normal deformation on their structural surfaces. The relationship between the normal stress (σn) and the corresponding normal displacement (δn) is typically characterized by a curve delineating the law of normal deformation on the structural surface. Publicly available research indicates that the normal displacement caused by normal stress generally shows a nonlinear decreasing trend. The normal displacement is a nonlinear curve that gradually approaches the maximum normal closure. This nonlinear mechanical behavior is mainly due to the elastic deformation of contact microprotrusions in the rock particles, crushing and indirect tensile fractures, and the increase in new contact points and contact areas. Goodman (1976) proposed the following hyperbolic relationship between normal stress (σn) and normal displacement (δn):
In the above formula Eq. 6, σn − σi represents the difference between two normal stresses and δmax is the maximum normal displacement (unit: mm).
The aforementioned studies mainly involve the influence of normal stress on the closure and displacement of fracture openings. In addition to affecting fracture closure and displacement, normal stress directly affects the mechanical strength of rock blocks near the fracture surface, especially in the process of fracture surface damage or fracture extension. Barton et al. (2023) proposed a calculation formula for the shear strength of irregularly undulating structural surfaces, \(\tau = \sigma_{{\text{n}}} \tan \left[ {JRC\lg \left( {\frac{JCS}{{\sigma_{{\text{n}}} }}} \right) + \varphi_{{{\text{jn}}}} } \right]\), where φjn is the basic friction angle of the fracture surface (unit: °), σn is the size of the normal stress load (unit: MPa), and the physical meaning of \(JRC\lg \left( {\frac{JCS}{{\sigma_{{\text{n}}} }}} \right)\) represents the control effect coefficient of normal stress on the roughness of the fracture surface. In order to reveal the control effect of multilevel normal stress on the roughness of the fracture surface, this study defines the uniaxial compressive strength of the JCS as the peak load of the multilevel normal stress, and the quantitative relationship between \(JRC\lg \left( {\frac{JCS}{{\sigma_{{\text{n}}} }}} \right)\) and multilevel normal stress load σn is obtained by fitting analysis, as shown in Fig. 6.
Fig. 6
Relations of different roughness fractures and peak values of multilevel stress loading. a JRC = 0–10; b JRC = 10–20
×
The nonlinear fitting relationship between \(JRC\lg \left( {\frac{JCS}{{\sigma_{{\text{n}}} }}} \right)\) and \(\sigma_{{\text{n}}}\) in the Fig. 6 shows that as the multilevel normal stress load increases, \(JRC\lg \left( {\frac{JCS}{{\sigma_{{\text{n}}} }}} \right)\) and different levels of σn have an exponential mathematical relationship in power, i.e., \(JRC\lg \left( {\frac{JCS}{{\sigma_{{\text{n}}} }}} \right) = \left( \sigma \right)_{{\text{n}}}^{{\text{b}}}\). The controlling effect of multilevel normal stress on the roughness of the fracture surface gradually weakens. This effect is particularly pronounced on increasingly rough fracture surfaces. Therefore, for fracture surfaces with larger JRC indicators under the same level of normal stress load shown in Fig. 6a, b, their corresponding \(JRC\cdot\lg \left( {\frac{JCS}{{\sigma_{{\text{n}}} }}} \right)\) values are generally larger.
4.2 Reynolds Number Variations in Fluid Through Roughness Fracture
The Reynolds number is universally used to predict the cessation of a linear flow and the initiation of non-Darcy flow. It is defined as the ratio of inertial forces to viscous forces. The Reynolds number of fluids in a fracture can be calculated through the following equation (Javadi et al. 2014):
In the aforementioned Eq. 7, v represents the volumetric flow rate of the fluid navigating the fracture. Existing literature has demonstrated that the critical Reynolds number, which delineates the transition from linear to nonlinear flow, ranges from 0.001 to 2300.
The objective of the current experiment is to reveal the mechanism of nonlinear flow in rock fractures and to examine the fluid flow behavior during the closure, deformation, and expansion of rough fractures under multilevel normal stress. Flowmeters were installed at the inlet and outlet of the rough fracture to calculate the changes in the Reynolds number of the fluid at these locations, thereby identifying the critical Reynolds number at which the fluid flow in the rock fracture transitions from linear to non-Darcy flow. Figure 7 shows the changes in the fluid flow behavior in different rough fractures under multilevel normal stress. This reveals that the Reynolds number of the fluid at the inlet of the fracture flow is generally smaller (Re < 1200), whereas it is relatively larger at the outlet. In smooth pipes, the Reynolds number is typically defined as Re < 2100 for laminar flow, between 2100 and 4000 for transition flow, and Re > 4000 for turbulent flow. However, rough fracture surfaces exert significant frictional resistance on the fluid flow. Under normal conditions, fluid flowing over a rough surface would experience a decrease in velocity due to frictional resistance. However, due to the closure of the fracture aperture and the squeezing effect on the fluid caused by the normal load, the fluid velocity at the outlet of the fracture flow is noticeably higher. Thus, the Reynolds number of the fluid at the outlet of the fracture flow is always greater than that at the inlet.
Fig. 7
Reynolds number variations of fluid through roughness fracture. a JRC = 0–2; b JRC = 8–10; c JRC = 10–12; d JRC = 18–20; other data see Figure S3 in the supplementary materials
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A comprehensive analysis of flow patterns in fractures with different degrees of roughness reveals that as the roughness of the fracture surface increases, the Reynolds number characterizing the fluid flow behavior gradually decreases. The primary reason for this is that the undulations in a rough fracture are prone to induce turbulent behavior in the fracture flow. However, due to the impact of normal stress on the closure and connectivity of the fracture, the fluid in the fracture experiences unsteady flow with lower velocities. Therefore, the overall Reynolds number of the fluid in the fracture gradually decreases with increasing roughness.
To quantitatively describe the unsteady fluid flow phenomenon caused by the diversion of fractures with different roughness, the changes in the Reynolds number of the fluid at the inlet and outlet during the closure, deformation, and extension of rough fracture under normal stress to rock fracture are analyzed. According to the historical path of the multilevel stress load of the rock, the critical time node for the transition from linear to nonlinear fluid flow during the closure of the rough fracture to the destruction of the rock is clarified. The variation law of the critical Reynolds number for the transition from linear to nonlinear flow of fracture diversion with different roughness is analyzed (as shown in Fig. 8). The results indicate that the Critical Reynolds number and JRC exhibit a cubic polynomial functional relationship, consistent with the conclusions of regarding the non-linear flow behavior of rough fractures (Zoorabadi et al. 2015). According to the trend of the JRC index of the typical rough fracture profile given by (Barton et al. 2023), it is found that the critical Reynolds number of the nonlinear fluid flow in the process of the rougher fracture diversion is lower. The critical Reynolds number of the fluid in the fracture changes in the range of JRC = 0–10 is Rec = 1200–1400, and the critical Reynolds number of the fluid in the fracture changes in the range of JRC = 10–20 is Rec = 650–850. Therefore, the roughness of the fracture surface is the main reason for the reduction in the critical Reynolds number of the fracture fluids. At the same time, the changes in fluid pressure (P) and hydraulic gradient (− ∇P) at the inlet and outlet of different rough fracture diversions when the normal stress is zero are analyzed. The results confirm that the internal fluid pressure significantly increases during the diversion process of the rough fracture with JRC > 10 (as shown in Fig. 9). The relationship between the hydraulic gradient of fluid flow − ∇P and the JRC index are obtained from the fitting:
$$- \nabla P = 0.12 + A \cdot JRC + B \cdot JRC^{2}$$
(8)
Fig. 8
Critical Reynolds determination of fluids flow through different roughness fracture
Fig. 9
Hydraulic properties of fluid flow through different roughness fracture. a fluid pressure of inlet and outlet flow; b hydraulic gradient of fluid flow
×
×
In the above Eq. 8, − ∇P is the change in hydraulic gradient of a single rough fracture diversion (unit: MPa/m), A is the loss of hydraulic gradient due to the frictional resistance caused by the rough fracture in linear flow (dimensionless constant term), and B is the loss of hydraulic gradient due to the frictional resistance caused by the rough fracture in nonlinear flow (dimensionless constant term).
4.3 A Nonlinear Flow Model of Fluid Under Normal Stress for Considering JRC
The closure, opening, and expansion of the fracture caused by normal stress directly affect the permeability characteristics of the fracture. By introducing roughness index JRC and normal stress as important parameters into the fracture nonlinear flow model, the dynamic hydrodynamic characteristics of fracture diversion during the rock failure process can be more accurately described. The Forchheimer model is often used to describe the nonlinear flow characteristics of fracture fluids. The nonlinear relationship between the volume flow rate of fracture fluid and the pressure gradient is the key content of this study, but the nonlinear flow mechanism caused by the roughness and normal stress of the fracture has not been fully revealed. In order to study the influence of normal stress and fracture surface roughness on fracture permeability, based on the X-ray DR shadow images of fracture fluid obtained from the experiment, the fitting relationship between different normal stress loads and fracture permeability shown in Fig. 10 is calculated.
Fig. 10
Permeability variations of fracture under multilevel stress loading. a JRC = 0–2; b JRC = 8–10; c JRC = 10–12; d JRC = 18–20; other data see Figure S4 in the supplementary materials
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Figure 10 illustrates the changes in the permeability characteristics of rock fractures during the fracturing process. The findings reveal a quadratic relationship between variations in normal stress and fracture permeability. Normal stress governs the closure, deformation, and expansion of fractures, and when it exceeds the uniaxial compressive strength of the rock, it also controls the fracture propagation process. The permeability of the fracture exhibits a trend of initial increase followed by a decrease, indicating significant variability in the seepage characteristics of the fracture under multilevel normal stress.
In practical engineering, the impact of engineering disturbance on the permeability of rock masses is exceedingly complex. Under hydrostatic pressure, rock fractures may exhibit hydraulic splitting phenomena. After hydraulic splitting occurs, the local hydrostatic pressure inside the rock mass tends to decrease. However, the induced microcracks, joints, and fractures alter the properties of these discontinuous surfaces, leading to increased permeability in the engineered rock mass and enhanced risk of seepage disasters. Therefore, fluctuations in seepage pressure and flow rates have become key indicators for revealing the potential for seepage disasters in engineered rock masses. This experiment, based on the Forchheimer nonlinear flow model, introduces the average flow rate \(\overline{Q}_{i}\) caused by multilevel stress loading and the roughness index of the fracture into the Forchheimer equation (as shown in Fig. 11a).
Fig. 11
Nonlinear flow model of fluid through the fractured rocks
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The research findings demonstrate a satisfactory linear fitting relationship between the \(\overline{Q}_{{\text{n}}}\), and the joint roughness coefficient (JRC). The rougher the fracture, the lower \(\overline{Q}_{{\text{n}}}\). Forchheimer model shown in Fig. 11b is derived based on experimental data:
In the above Eq. 9, JRC is the roughness index of the fracture (JRC = 0–20); JCS is the joint compression strength of the fractured rocks under normal stress (in MPa); \(\overline{Q}_{{\text{n}}}\) is defined as the average flow volume of fluid through a rock fracture under multilevel stress loading (in × 10−5 m3). The calculation formula for \(\overline{Q}_{{\text{n}}}\) is \(\overline{Q}_{{\text{n}}} = \sum \frac{{Q_{{\text{n}}} }}{{\text{n}}}\), where n is the total number of loading times for multilevel normal stress. The calculated parameters for different rough fracture nonlinear seepage experiments are listed in Table 5.
Table 5
Nonlinear flow parameters of fluid filling different roughness fractures during rock failures
This study focused on prefabricated fracture basalt and, by proposing an enhanced X-ray imaging experimental visualization system for fluid flow through fracture in basaltic rocks during failure at the block scale, it employed a combination of theoretical analysis and laboratory tests. This approach has shed light on the mechanism of nonlinear flow in rock failure processes. The primary conclusions derived are as follows:
(1)
Under multilevel normal stress loading, the fracture process in fractured rock primarily involves fracture closure, deformation, and expansion. There is an exponential power relationship between JRC·lg (JCS/σn) and the multilevel normal stress load (σn). The multilevel normal stress directly affects the degree of fracture opening/closing and changes in the flow state of fracture fluid. The Reynolds number of fracture fluid and the fracture roughness index JRC show a cubic polynomial relationship, with the Reynolds number gradually decreasing with increasing roughness.
(2)
There exists a quadratic relationship between the flow pressure gradient of fracture and the roughness index. Therefore, with multilevel normal stress, the permeability of the fracture first increases and then decreases. Rougher prefabricated fractured rock under a complete failure state has a higher closure contact rate. Rougher fractures after closure and expansion are more prone to form denser meshing points. These mechanical behaviors have a significant fluctuating impact on the seepage characteristics of the fracture. Therefore, it is necessary to introduce the average flow rate of fluid through a rock fracture \(\overline{\user2{Q}}_{{\varvec{i}}}\) to quantify the flow behavior changes in fracture.
(3)
When fluid flows through a fracture in a rock, the behavior of the flow is influenced by the aperture, roughness and flow pressure, etc. The normal stress loading on a fracture can induce closure or expansion of the fracture, and this behavior is closely related to rock failures. Traditional linear flow models may not adequately capture the complexities associated with fluid flow in deformable fractures during rock failures. By incorporating the fracture roughness index (JRC) and the joint compressive strength index (JCS) into the nonlinear flow model, i.e. \(- \nabla {\varvec{P}} = \left( {\frac{{\mathbf{A}}}{{{\varvec{JRC}}\log \left( {{\varvec{JCS}}/{\varvec{\sigma}}_{{\mathbf{n}}} } \right)}}} \right)\overline{\user2{Q}}_{{\mathbf{n}}} + \left( {\frac{{\mathbf{B}}}{{{\varvec{JRC}}^{3} \left( {\log^{2} \left( {{\varvec{JCS}}} \right)/{\varvec{\sigma}}_{{\mathbf{n}}} } \right)}}} \right)\overline{\user2{Q}}_{{\mathbf{n}}}^{2}\), the proposed model aims to capture the effects of deformable fracture roughness and strength on fluid flow behavior during rock failures. This model can provide a more accurate representation of the fluid flow characteristics in deformable fractures and improve our understanding of fluid flow in fractured rock masses under normal stress conditions.
Acknowledgements
This study was supported by the National Natural Science Foundation of China (Approval Nos. 52109120 and 42267021), the Hainan Province Science and Technology Special Fund (Approval Nos. ZDYF2024SHFZ060 and DYF2022SHFZ106), the Hainan Provincial Natural Science Foundation of China (Approval Nos. 421RC487 and 521QN203), the Research Project of Collaborative Innovation Center of Hainan University of China (Approval Nos. XTCX2022STC17 and XTCX2022STC18), Open Research Fund Program of State key Laboratory of Hydroscience and Engineering. (Approval No. sklhse-2024-D-01)
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The authors declare that they have no conflicts of interest in this study. The authors certify that they have no commercial or associative interest that represents a conflict of interest in connection with the paper submitted.
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