Volume 46, Issue 15 p. 8862-8871
Research Letter
Free Access

Disentangling the Simultaneous Effects of Inertial Losses and Fracture Dilation on Permeability of Pressurized Fractured Rocks

Jia-Qing Zhou

Jia-Qing Zhou

Faculty of Engineering, China University of Geosciences, Wuhan, China

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, China

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Yi-Feng Chen

Corresponding Author

Yi-Feng Chen

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, China

Correspondence to: Y.-F. Chen and H. Tang,

[email protected];

[email protected]

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Huiming Tang

Corresponding Author

Huiming Tang

Faculty of Engineering, China University of Geosciences, Wuhan, China

Correspondence to: Y.-F. Chen and H. Tang,

[email protected];

[email protected]

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Lichun Wang

Lichun Wang

Institute of Surface-Earth System Science, Tianjin University, Tianjin, China

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M. Bayani Cardenas

M. Bayani Cardenas

Geological Sciences, University of Texas at Austin, Austin, TX, USA

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First published: 29 July 2019
Citations: 17

Abstract

How fluids flow through pressurized fractured rocks is relevant to many engineering applications and geophysical processes including fault rupturing, hydraulic fracturing, induced seismicity, fluid extraction, and contaminant transport. With increasing fluid pressure and concomitantly elevated hydraulic gradient, the permeability of fractured rock is reduced because of inertial losses within the fluid. There is an accompanying flow regime change when this happens. On the other hand, increasing pressure causes fracture dilation which enhances permeability. In this case, the fracture geometry changes. These two competing consequences of increasing pressure had always been studied independently. Here we present an analytical expression for fractured rock permeability where flow regime and medium geometry simultaneously co-evolve. The theory was applied to core and field flow tests. With continuously increasing fluid pressure, the inertial effect on permeability first dominates over that of fracture dilation and this dominance theoretically reverses at Forchheimer number = 1/3.

Key Points

  • The competition between inertial losses and fracture dilation in controlling fractured rock permeability was analyzed
  • An analytical expression for pressure-dependent permeability considering inertial effect and dilation was derived and tested with experiment
  • Transition from inertial effect-dominated to fracture dilation-dominated permeability evolution phases happens at Forchheimer number = 1/3

Plain Language Summary

Flow through fractured impermeable rock mainly occurs in fractures. These flows are dictated by fracture permeability. Permeability is usually considered to be a sole and intrinsic property of the fractured rock irrespective of the fluid and hydrodynamic conditions. But this may not be the case in high-pressure scenarios. Elevated pressure gradients typically occur when the rock is pressurized, that is, when pressure is generally high. Under such large gradients, significant inertial losses take place in the fluid, which results in flow regime transition where the fracture appears to be relatively less permeable. At the same time, any further increase in fluid pressure can cause the fracture to dilate. This expansion increases permeability. Although inertial losses and fracture dilation are intertwined under continuously increasing fluid pressure, their concurrent action and collective effects are unknown. Here, we present a pressure-dependent theoretical expression for inertial and viscous permeabilities of fractured rock under situations where both flow regime and medium geometry evolve. The competing mechanism of inertial losses and fracture dilation on the permeability was revealed with linked experimental and theoretical analyses. These results deepen our understanding of fluid flow in fractured rock when they are subject to coupled hydraulic and mechanical geophysical processes.

1 Introduction

High-pressure fluid flow in fractured rock is relevant in many industrial and scientific applications, such as geothermal energy and hydrocarbon extraction (Watanabe et al., 2017), geological carbon sequestration (Figueiredo et al., 2015), and high-head hydraulic engineering of infrastructures (Zhou et al., 2018). Such flow is also relevant for understanding and predicting hazards from induced seismicity and the fate of environmental contamination (Birdsell et al., 2015; Cappa et al., 2018; Elkhoury et al., 2006; Ellsworth, 2013; Weingarten et al., 2015). Permeability, the key hydraulic property of fractured rock, is thus of prime importance for properly understanding and accurately predicting fluid flow involved in these high-pressure scenarios.

Permeability is usually considered to be a property of the medium alone in many applications, but it may not be the case in highly pressurized fluid conditions. High fluid pressure is typically necessary to produce elevated hydraulic gradients which lead to significant inertial effects. However, further pressurizing the fluid inside the fracture to continually increase gradients may lead to fracture dilation. Therefore, inertial effects and fracture dilation are intertwined and can either compete against or complement each other under continuously increasing fluid pressure. Because of the connection between pressure and pressure gradients in many situations where high-pressure gradient is accompanied by high pressure, they are used interchangeably below for brevity. That is, high-pressure gradients imply high fluid pressures, and the other way around except for the situation where the fluid pressures at the upstream and downstream portions are both high but with a small pressure gradient. Thus, phenomena which occur under high pressure and pressure gradients, that is, dilation and inertial effects, are considered to potentially overlap and to occur simultaneously.

Permeability is usually determined by applying Darcy's law, which states that the flow rate is linearly proportional to the fluid pressure gradient; this relationship is readily obtained by neglecting the inertial terms in Navier–Stokes equations. However, at elevated hydraulic gradients, non-Darcian flow behavior manifests with significant inertial effects. In this case, the pressure differential increases more than what would accompany a linearly proportional increase in flux under Darcy flow. This phenomenon has been widely observed in fractured rock (Cherubini et al., 2012; Kohl et al., 1997; Quinn et al., 2011; Ranjith & Darlington, 2007; Zimmerman et al., 2004). On the other hand, high fluid pressure can dilate a fracture which results in excess flux in the fractured rock relative to Darcy flow conditions; this phenomenon is typically referred to as hydromechanical coupling (Chen, Hu, et al., 2015; Rutqvist & Stephansson, 2003). Furthermore, very high fluid pressure can result in hydro-shearing and hydraulic fracturing (or fracking) that increases permeability; this makes fracking an effective stimulation technique for extracting oil, gas, and thermal energy from underground (Gischig & Preisig, 2015; Kerr, 2010; Watanabe et al., 2017). Note that the hydro-shearing can happen at a lower pressure condition than the hydraulic fracturing (Gischig & Preisig, 2015).

Several studies have investigated the effects of inertial losses (Cardenas et al., 2009; Friedel & Voigt, 2006; Lee et al., 2014) and fracture dilation (Figueiredo et al., 2015; Vogler et al., 2018; Watanabe et al., 2017) on fractured rock permeability, but almost all of them only focused on one aspect or separately analyzed these two mechanisms. A simultaneous investigation of inertial effects and fracture dilation in fractured rock is required in order to gain more complete understanding. By not doing so, it will remain unclear which factor is more dominant if one even becomes dominant under certain conditions. Do these two factors cancel out each other? Are they synergistic? How much does each matter and under what conditions? All these are open questions.

Zhou et al. (2019) recently reported a universal visco-inertial permeability relationship by synthesizing results from thousands of laboratory and field flow tests and pore-scale simulations. Their analysis showed that the permeability of geologic media can vary over several orders of magnitude during geometry alteration induced by different geophysical processes such as compression, torsion, shearing, and decompression; however, there exists a power law relationship between viscous and inertial permeability across the broad range of permeabilities. This finding provides an important clue for the unresolved issues mentioned above because inertial effects are incorporated into the viscous and inertial components of flow via the Forchheimer equation, while fracture dilation is one of the geophysical processes inducing geometry alteration. The universal relationship found by Zhou et al. (2019) hints at the linkage between inertial effects and fracture dilation in terms of the relationship between viscous and inertial permeabilities when flow regime and medium geometry simultaneously co-evolve.

To address the questions above, we present a pressure-dependent theoretical expression for permeabilities in fractured rock, based on the recently discovered visco-inertial permeability relationship. We compared the theory with results from a series of well-characterized and high-precision laboratory and field flow tests using different types of fractured rock subjected to a wide range of water pressures. For the first time, to our knowledge, the competing contributions from inertial effects and fracture dilation on permeability changes of fractured rock were elucidated with linked experimental and theoretical analyses.

2 Methodology

2.1 Analytic Solution of Viscous and Inertial Permeabilities

Hydrodynamically, the effective permeability (ke) calculated from the Darcy's law continuously decreases due to significant inertial effects under increasing flux. Hydromechanically, this permeability continuously increases due to significant fracture dilation with increasing fluid pressure. The hydrodynamical process can be described by the Forchheimer equation, the inertial extension of the classic Darcy's law, where a square term of velocity is added to represent the energy losses due to inertial effects (Forchheimer, 1901; Rust & Cashman, 2004; Zimmerman et al., 2004). It reads as
(1)
where kv (m2) and ki (m) are the viscous and inertial permeabilities, respectively; v = Q/A (m/s) and ∇p = Δp/l (Pa/m) are the specific flux and pressure gradient, respectively; Q (m3/s) is the volumetric flux; A (m2) is the cross-sectional area where flow passes through; Δp (Pa) is the pressure difference between the outlet and inlet; l (m) is the distance along the flow direction; and μ (Pa/s) and ρ (kg/m3) are the fluid viscosity and density, respectively.
With respect to the hydromechanical process, the power law relationship established by Zhou et al. (2019) provides insight on permeability evolution:
(2)
where ω (m2) is a highly generalized constant representing intrinsic characteristics of the medium. Pore or fracture geometry can significantly change for a given geologic medium subject to various geophysical processes (such as fracture dilation caused by high fluid pressure which is of interest here). Equation 2 indicates that the simultaneously evolving kv and ki always follow a power law relationship with a constant exponent of 3/2.
The Forchheimer equation 1 and the scaling relationship equation 2 can be combined to investigate the joint contributions of inertial effects and fracture dilation on permeability; this also eliminates ki (a less constrained or harder-to-constrain parameter). This leads to
(3)
Equation 3 is a cubic polynomial equation with respect to , which has closed-form analytical solutions (Press et al., 1992). The solution for to equation 3 varies with the sign of the discriminant Δ, which is given below (see Text S1 in the supporting information for details):
(4)

Recall that the first term in the bracket (−∇pv) (Pa/s) represents the energy dissipation rate per unit volume (Kundu et al., 2008). As for the second term in the bracket, Zhou et al. (2019) discussed that ω is a constant depending on some intrinsic characteristics of the medium which do not change with any geometry alteration. Clearly, the first term describes the kinetic state of the flowing fluid, while the second term describes some static intrinsic property of the medium. Further analysis combined with the analytical solutions and experimental results show that the second term dictates the ability of the medium to resist deformation, which will be discussed below. Therefore, the sign of Δ, determined by these two terms in the bracket, has a physical meaning—it quantifies the relative importance of the kinetic energy of the flowing fluid causing deformation and a static property of the medium resisting deformation.

Based on the sign of Δ, the solution for kv is (see Text S1 in the supporting information for details):
(5a)
(5b)
(5c)
where ke (m2) is the effective permeability calculated from the Darcy's law (= − μv/∇p) and φ (−) and λ (m) are the functions of (∇p, v) (see Text S1 in the supporting information for expressions). Substituting the expression for kv into equation 2, one obtains the expression for ki (refer to equations (S9), (S12), and (S18) in the supporting information). These equations give the expressions for permeabilities (kv, ki) related to the interactions between the medium and the flowing fluid; it provides a quantitative framework for investigating permeability evolution with increasing pressure.

2.2 Water Flow Test at Laboratory and Field Scales

The analytical solutions above show that both kv and ki are functionally pressure-dependent during pore geometry deformation, that is, fracture dilation. To further investigate the permeability variation mechanism suggested by the analytical solutions, we conducted water flow tests in fractured rock/aquifers at both laboratory and field scales. The flow tests covered the full flow phase spectrum from Darcy, inertial, to hydromechanical flow phases. The last phase refers to the state when the fracture should be dilating due to the elevated pressure. In the laboratory tests, single-fracture (Figure 1a) and cataclastic rock (Figure 1b) samples were used as the elementary unit and laboratory-scale representative of fractured rocks, respectively. In the field tests, a sedimentary and an intrusive fractured rock aquifer (Figure 1c) were chosen. Laboratory water flow tests were conducted in a high-precision triaxial cell (see Figure S1 in the supporting information; Zhou et al., 2015), and the field tests were performed via high-pressure packer tests (HPPT) at a site where an underground tunnel system for a pumped-storage power station was constructed (Chen, Hu, et al., 2015).

Details are in the caption following the image
Illustration of the (a) single fracture and (b) cataclastic samples used in the laboratory experiments and (c) of the borehole intervals covered by the field tests. The microscopic structures of the single fracture and cataclastic rock were scanned using high-resolution X-ray computed tomography, while the borehole intervals were imaged by an acoustic televiewer.

The single-fracture sample (50 mm in width and 100 mm in length; Figure 1a with a CT image) was generated by the Brazilian technique. A series of water flow tests were then systematically conducted for the single fracture under a specific confining stress (σ3), beginning at the Darcy flow phase, then transitioning to the inertial flow phase, and finally culminating within the flow phase with significant hydromechanical coupling. Five single fractures (named as samples SF1–SF5) were subjected to σ3 = 18, 20, 23, 25, and 28 MPa, respectively. The cataclastic rock sample was prepared via prefracturing an intact cylindrical granite (50 mm in diameter and 100 mm in length) under a specific σ3 by increasing the axial stress until the peak stress condition. The axial stress was increased in a load control mode, and after the rock failure, the control mode was switched to a displacement control mode where the axial displacement was maintained constant to ensure constant sample length during the whole flow test. Numerous microcrack and macrocrack developed during the prefracturing procedure (Figure 1b); these cracks are the flow pathways and together with the rock matrix is a fracture-porous system. Four cataclastic rocks (named as samples CR1–CR4) were prepared, which were prefractured under σ3 of 2, 5, 8, and 10 MPa, respectively, resulting in different degrees of crushing. Similarly, a series of water flow tests were then conducted on the samples. Note that for each cataclastic rock sample, σ3 was kept constant for the whole experiments from sample generation to flow tests. Two circumferential strain sensors were used in the laboratory experiments (see Figure S1 in the supporting information), to observe the possible deformation and prevent the instability of the sample caused by water pressure. Details on the laboratory flow tests are provided in Text S2 in the supporting information.

For the field test, the HPPTs were conducted by a stepwise increase (0.3–0.5 MPa) of injection pressure up to 7.0 MPa, followed by a stepwise release (0.3–0.5 MPa) down to 0.3 MPa. At each step, the injection water pressure and flow rate were measured until the flow was at a steady or quasi-steady state. More details of the testing site and procedures are reported in Chen, Hu, et al. (2015). The results of five test intervals (named as samples Int1–Int5) at the rising stage of the injection pressure were analyzed in this study, taken from three boreholes (ZK124, ZK126, and ZK129–2). The number and the orientation of fractures in the tested intervals were revealed by acoustic televiewer images (Figure 1c). Other information including the depth and formation of the intervals are provided in Table S1 in the supporting information.

3 Results and Discussion

Possible fracture dilation was precisely captured and recorded in the laboratory tests. Figure 2a shows the recorded circumferential strain (ε) and pressure gradient (∇p) of sample CR4 in real time. Decreasing ε was observed during the flow tests, which became more obvious with continuously increased water pressure, indicating fracture dilation caused by excessive water pressure (ε is positive for compression and negative for dilation). Note that ε returned to the initial state once the water pressure was released. Fracture dilation cannot be directly observed in field tests, but it can be inferred from the difference between pressure rise and release stages (Chen, Hu, et al., 2015). The whole HPPT process showed a distinct hysteresis loop between the pressure rise and release stages (Figure 2b, taken from sample Int4), indicating that irreversible fracture dilation and/or propagation occurred during the rising pressure stage. Note that ∇p (negative) in laboratory tests refers to the pressure gradient between the outlet and inlet of the sample, while P (positive) in HPPTs refers to the injection pressure into the interval of the borehole. Since instability could happen during the fracture dilation resulting in fluctuation of pressure or flux, all the data of pressure and flux were measured in a steady state flow condition.

Details are in the caption following the image
Evidence of fracture dilation in a laboratory experiment and a field test. (a) Representative circumferential strain (ε) variation with water pressure gradient (∇p) during a core flow test, taken from sample CR4. (b) The complete pressure-flow rate curve comprising pressure rise and release stages for the HPPT, exemplified by sample Int4. Dashed line in Figure 2a indicates the point where the circumferential strain ε began to decrease. P in Figure 2b refers to the injection water pressure into the interval of the borehole. In the laboratory experiment, ε was recorded in real time using two circumferential strain sensors, while ∇p was recorded by pressure transducers, as illustrated in Figure S1. A decreasing trend of ε during the flow tests indicates fracture dilation caused by elevated water pressure. Note that ε returned to the initial state once the water pressure was released, and the value of ε dropped to the maximum negative range of the circumferential strain sensor after uninstalling the sample from the test apparatus. In the field test, a hysteresis loop between the pressure rise and release stages indicates that irreversible fracture dilation and/or propagation occurred during the pressure rise stage.

As a result of fracture dilation and the presence of significant inertial effects at higher water pressure, the pressure-flow rate curves (∇p~Q for laboratory tests or P~Q for HPPTs, with the raw data provided in Table S2 in the supporting information) obtained at both laboratory and field scales exhibited three distinct flow phases (Figure 3). The curve begins as a straight-line following Darcy's law for a short duration; then it gradually bends upward due to significant inertial effects; and finally, with excessive injection water pressure, the curve starts bending downward due to remarkable fracture dilation. These three flow phases together shaped the curve into a sigmoidal curve, which becomes more pronounced from the single fracture to the fracture-porous system (Figures 3a and 3b), and from the core to the field scale (Figures 3b and 3c). The lower two phases of the sigmoidal pressure-flow rate curve can be well described by the Forchheimer equation (fitted solid lines in Figure 3), where the fracture dilation caused by water pressure is negligible and thus neglected. It is worth mentioning that different water pressure levels were adopted for these flow tests and some flow tests were conducted under the pressure gradient more than 100 MPa/m. In real geophysical processes, the pressure level causing significant inertial effect and further fracture dilation depends on both the medium structure and its ambient environment (such as in situ stress field), which could be lower than the ones applied here. Even so, the high level of water pressure applied and considered in this study does occur in subsurface applications (Cappa et al., 2019; Celia et al., 2015; Chen, Hu, et al., 2015; Zhou et al., 2018).

Details are in the caption following the image
Pressure versus flow rate curves illustrating flow phases where inertial effects and fracture dilation are relevant. (a) Single-fracture samples SF1-SF5, (b) cataclastic rock samples CR1–CR4, and (c) HPPT intervals Int1–Int5. The plots on the right is a zoomed-in portion of that on the left. The right plots highlight the phase where fracture dilation can be neglected. Solid lines refer to the fitted curves using the Forchheimer equation. The discrete symbols are measurement points.

Focusing on the sigmoidal pressure-flow rate curve, the effective permeability ke (= − μv/∇p) went through a decrease and then increase with progressively higher applied water pressure (Figure 4). This suggests that the reduction of ke caused by inertial effects was compensated by fracture dilation. With further increase in water pressure, this compensation makes ke recover to or even exceed the initial value. This highlights the competing mechanisms of inertial losses and fracture dilation in the evolution of permeability.

Details are in the caption following the image
Evolution in effective, viscous, and inertial permeabilities (ke, kv, and ki) as water pressure increases within the flow phase when fracture dilation is important. (a) Single-fracture samples SF1–SF5, (b) cataclastic rock samples CR1–CR4, and (c) HPPT intervals Int1–Int5. Dashed lines refer to the point of Fo = 1/3, where a rapid increase of the permeabilities is apparent after it. Note that for samples Int1 and Int2, the point of Fo = 1/3 is beyond the range of the measured data. ke went through a process of decreasing and then increasing with continually increasing water pressure, while both kv and ki generally remained unchanged at low water pressure and began to increase due to fracture dilation with further water pressure increase.

Here we provide a quantitative analysis of permeabilities (kv, ki) when the competition is present. First, the initial permeabilities ( ), present before any medium deformation takes place, are determined by fitting the lower two phases of the sigmoidal pressure-flow rate curve with the Forchheimer equation 1. The constant ω is then determined by equation 2 with the fitted ( ). Meanwhile, ke at any water pressure can be calculated by Darcy's law using the measured pressure-flow rate data point. Based on the determined constant ω and the calculated ke, kv at any water pressure can be finally calculated by the analytical solutions presented in equation (5). Subsequently, ki can be determined by equation 2.

The analytical solutions in equation (5), with its calculation procedures introduced above, were directly applied to the laboratory tests which represented one-dimensional flow. But for the HPPTs, a transformation was required since the test represents three-dimensional spherical flow from the point source in the borehole. This transformation was implemented by a Forchheimer's law-based analytical model developed by Chen, Liu, et al. (2015). Details of the transformation and calculation procedures for HPPTs are provided in Text S3 in the supporting information.

The determined kv and ki generally remain unchanged at low water pressures (Figure 4). This is ascribed to the negligible effect of low water pressure on medium structure, given that a specific geometry corresponds to a specific kv and ki. With further increase of water pressure, kv and ki begin to increase and exhibit pressure-dependent behavior due to significant fracture dilation. The increase of kv is a manifestation of the expanded medium structure, while the increase of ki indicates a possible flow regime transition during this process. Therefore, the Forchheimer equation, describing this complex flow process where both medium structure and flow regime co-evolve, has varying coefficients. Compared with the single fracture (Figure 4a), the rapid enhancement of kv and ki occurred earlier (lower pressures) for the cataclastic rock samples (Figure 4b) and fractured rock aquifer (Figure 4c). This is because the cataclastic rock and fractured rock aquifers have more fractures, resulting in more significant fracture dilation behavior under water pressure. Note that ke varies nonmonotonically with the applied water pressure (Figure 4), so it is not straightforward to directly relate ke to pressure. Unlike ke, kv and ki have an explicitly monotonic relationship with the applied water pressure which allows for the estimation of kv and ki at any given water pressure. Moreover, ke is a function of kv and ki (Zimmerman et al., 2004). Therefore, the functional relationship between ke and the applied pressure can be practically obtained by determining the relationship between (kv, ki) and the applied water pressure as proposed in this study.

The estimated permeability evolution above is intuitively consistent with the physical processes occurring in reality. Equation (5), with its discriminant Δ, provides insight on these processes. For Case 1 (equation 5a), the change of kv is very limited (the same with ki), considering that kv ≤ (4/3)ke and since the square of the trigonometric function is less than 1. This case corresponds to Δ < 0, where the dissipation rate of the flowing water (described by the first term (− ∇p ⋅ v) in the bracket of Δ) is lower than the intrinsic characteristics of the medium (described by the second term ). This is consistent with the fact that the medium can be regarded as nondeformable under low water pressure. Conversely, in Case 3 (equation 5c), the expansion of kv is no longer restricted by the trigonometric function in equation 5a. This situation corresponds to Δ > 0, where the kinetic energy of the flowing water is relatively high and the role of fracture dilation in permeability gradually matters. Moreover, Case 2 represents a critical threshold; that is, the tendency of the fracture to dilate due to pressurization is balanced by the ability of the fractured rock to resist deformation. The sign transition of the discriminant Δ related to the medium property (ω) and the fluid state (∇p and v) thus indicates that the deformation caused by fracture dilation becomes important.

In Case 2, interestingly, the Forchheimer number Fo equals to 1/3 (see Text S1 in the supporting information); the Fo is defined as
(6)
Fo is an important dimensionless number representing the ratio of inertial to viscous forces resisting fluid flow (Chen, Zhou, et al., 2015; Cherubini et al., 2012; Zeng & Grigg, 2006). This indicates that for fluid flow with Fo below 1/3, the variation of (kv, ki) develops in a restricted manner, but above this Fo, the development is less restricted. This is consistent with the laboratory and field results (Figure 4), where a rapid increase of the permeabilities is apparent after Fo = 1/3 (marked with dashed lines). It is worth mentioning that the point where the circumferential strain ε began to decrease in Figure 2 (also marked with a dashed line) generally coincides with Fo = 1/3 (Figure 4b), which both occurred near −∇p = 10 MPa/m.

The point of Fo = 1/3 thus demarcates a critical turning point in the competition between inertial effects and fracture dilation on determining changes in permeability. Before this critical point, inertial effects dominate, which results in limited changes of kv and ki according to the equation (S8) in the supporting information. Above the critical point, fracture dilation dominates resulting in nonrestricted expansion of kv and ki, signifying higher conductivity of the fractured rock. This critical point of Fo = 1/3 is only theoretically based on the two competing mechanisms considered in its development. In reality, the significant level of the changes in permeability beyond this point is comprehensively determined by factors which include other hydraulic and mechanical properties of the fractured rock as well as its ambient environment. These factors could include fracture stiffness and roughness, rock type, elastic modulus, rock quality designation, and in situ stress level, to name a few examples. This study circumvents these aspects and their related traditional constitutive models, by developing a pressure-dependent analytical solution for permeability. Further investigations combining these factors and incorporating the traditional constitutive models of hydromechanical coupling will provide an even more comprehensive view of this topic.

The analytical solution in equation (5) was implemented via the pressure-flow rate data spanning the full flow spectrum from Darcy, inertial, to hydromechanical flow phases. However, it is worth mentioning that this analytical solution can be potentially applied in the absence of or sparsity of flow data. Specifically, if one only knows the effective permeability of a certain fractured rock, one can still approximate its kv and ki from equation (5). The main challenge lies in determining the constant ω, which requires knowing the initial kv and ki prior to medium deformation. Zhou et al. (2019) conducted a nearly exhaustive investigation of flow information in the literature and compiled the data of (kv, ki) for diverse geologic porous media (Figure 1 in their study). This compilation provides a basis for roughly estimating ω of a specific fractured rock by analogy of similar medium type. Through a series of computational fluid dynamics simulations, Zhou et al. (2019) found that the local surface roughness and void distribution pattern may contribute to ω. Here for the problem of geometry alteration caused by fracture dilation, it indicates that the constant ω could also be related to the mechanical properties of the fractured rock, as it dictates the ability of the fractured rock to resist fracture dilation (refer to Δ in equation (5)). These clues serve as the foundation for further systematically evaluating the highly generalized constant ω and parameterizing it, which would expand the application range and deepen the understanding of the derived analytical solutions.

4 Summary and Conclusion

This study presents a pressure-dependent analytic expression for viscous and inertial permeabilities of fractured rock, by combining the Forchheimer equation and a recently established visco-inertial permeability model. This combination integrates the effects of inertial losses and fracture dilation which have competing impacts on how permeability evolves. By analyzing pressure-flow rate curves of different fractured rocks under laboratory and field conditions, the analytic solutions were applied and shown to be physically meaningful. The competition between inertial effects and fracture dilation on dictating permeability changes of fractured rock was theoretically and quantitatively elucidated. Additional analysis showed that the threshold for when inertial effects dominate and when dilation becomes more important is at Forchheimer number = 1/3. This work provides a novel framework for permeability evaluation under the joint impact of inertial effects and fracture dilation. The findings advance the understanding of how fluids flow through fractures subject to coupled geophysical processes.

Acknowledgments

The authors gratefully thank Editor-in-Chief Harihar Rajaram and the two anonymous reviewers for their valuable and constructive comments in improving this manuscript. This work was supported by the National Key R&D Program of China (2018YFC0407001) and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan). All original data used for plotting and analyzing can be found in the supporting information.