• A novel numerical approach has been proposed to retrieve the hydraulic transmissivity of elliptical rock fractures in triaxial shear-flow experiments.
• The accuracy of different methods for estimation of hydraulic transmissivity of elliptical rock fractures has been evaluated and discussed.
• The numerical approach and electrical analogy are recommended for accurate evaluation of hydraulic transmissivity of elliptical rock fractures.
Notes
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Abbreviations
Ψ
Fracture inclination angle with respect to the sample axis
2a
Length along slip-parallel direction
2b
Length along slip-normal direction
B
Geometry constant
d
Distance between borehole and ellipse edge
k
Permeability
L
Length along the flow direction
n
Ratio between b and a
q
Fluid flux
Q
Volumetric flow rate
Qexp
Volumetric flow rate recorded in the laboratory
r
Borehole radius
t
Effective thickness of fracture
w
Fracture width
Δp
Pressure difference between the inlet and outlet boreholes
μ
Dynamic viscosity of fluid
∇p
Pressure drop over a given distance
1 Introduction
Understanding the evolution of hydraulic transmissivity of rock fractures under various hydromechanical conditions is vital to the successful exploitation of geo-energy systems (e.g., Bossart et al. 2002; Ellsworth 2013; Fang et al. 2017; Im et al. 2018; Ji et al. 2019, 2020, 2021, 2022; LongJohn et al. 2018; Ma and Zoback 2017; Passelègue et al. 2018; Shovkun and Espinoza 2017; Shapiro et al. 1997; Shen et al. 2020; Zoback 2010). The hydraulic transmissivity of a rock fracture is defined as the product of the permeability of the material within the fracture with the effective thickness of the fracture, and has a dimension of m3 (Acosta et al. 2020; Passelègue et al. 2020; Rutter and Mecklenburgh 2017, 2018). Considerable experimental effort has been made to investigate the hydromechanical properties of rock fractures. In these studies using different experimental setups, the triaxial shear-flow setup has three major advantages over the other configurations, including the direct shear-flow setup, double direct shear-flow setup, and rotary shear-flow setup (Ji et al. 2021). First, it can be readily deployed in conventional triaxial cells equipped with fluid pumps. Second, the good sealing provided by the jacket material allows for the easier and safer application of relatively high fluid pressures within pores and fractures, which can almost reach but be slightly lower than the confining pressure (e.g., Ji 2020; Ye and Ghassemi 2018; Wang et al. 2020). Third, the hydraulic properties of rock fractures can be evaluated in different directions (i.e., parallel or normal to the slip direction) while the fracture is subjected to mechanical loading (Okazaki et al. 2013; Rutter and Mecklenburgh 2017, 2018). In triaxial shear-flow experiments, an inclined fracture is prepared in a cylindrical rock sample and the fluid gains access to and egress from the fracture normally through small boreholes made to intersect the fracture plane (Ji et al. 2021). However, the rock fracture in triaxial shear-flow experiments on samples fabricated from cylindrical cores is an ellipse with an unknown effective thickness and a variable fracture width along the flow path, posing difficulties for the evaluation of its hydraulic transmissivity. The transmissivity of a rock fracture with a constant fracture width can be estimated based on Darcy’s law (Acosta et al. 2020). Thus, a rectangular approximation, in which a constant fracture width is assumed, has been proposed to provide an estimation of the transmissivity of a rock fracture that is elliptical in shape (e.g., Crawford et al. 2008; Jeppson et al. 2021; Bijay and Ghazanfari 2021; Okazaki et al. 2013; Wang et al. 2021; Ye and Ghassemi 2018). However, the transmissivity of an elliptical rock fracture can also be derived analytically based on the electrical analogy with the flow of electric current in a planar sheet of elliptical shape (Rutter and Mecklenburgh 2017, 2018), so that it is no longer necessary to use a rectangular approximation (Acosta et al. 2020; Passelègue et al. 2020). Nevertheless, given the extensive use of the rectangular approximation and electrical analogy, the relative accuracy of the rectangular approximation and the electrical analogy should be evaluated and discussed for better interpretation and comparison of inter-laboratory results.
In the laboratory, the hydraulic properties of a sample with a relatively large transmissivity can be measured by the steady state flow method (Scheidegger 1958), which records the volumetric flow rate of fluid through the sample under a constant fluid pressure difference between the two sample ends. However, the steady state flow measurements can become too time-consuming for very low transmissivity samples (Milsch et al. 2016). In such cases transmissivity can be measured by transient flow methods, such as the pulse decay method (Brace et al. 1968) and the oscillating pressure method (Kranz et al. 1990; Fischer 1992; Faulkner and Rutter 2000; Bernabe et al. 2006). In either steady state flow or transient flow measurements of fracture transmissivity, the flow boundary shape is an important factor that substantially influences the accuracy of transmissivity evaluation.
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The objective of this study is to revisit the evaluation of hydraulic transmissivity of elliptical rock fractures in triaxial shear-flow experiments. The significance of this study lies in the proposal of a novel numerical back-calculation method for determining the transmissivity of an elliptical rock fracture, the accuracy evaluation of the two commonly used transmissivity estimation methods, and the final recommendations.
2 Triaxial Shear-Flow Experiments
2.1 Experimental Configuration
Triaxial shear-flow experiments are readily performed in conventional triaxial cells, which can apply axial stress and confining pressure independently. The loading path of shear and normal stresses on the fracture inclined at an angle Ψ (°) to the sample axis can be adjusted by servocontrol of the axial stress and confining pressure (Fig. 1). Particularly, the confining pressure is updated continuously under computer control during the experiments to maintain a constant normal stress on the fracture, taking into account the contact area reduction and jacket deformation with shear displacement (Ji 2020, Sect. 3.3; Tembe et al. 2010, Appendix A). The inclined fracture can be a smooth sawcut fracture (e.g., Ji et al. 2021; Rutter and Mecklenburgh 2017, 2018), a rough tensile or natural fracture (e.g., Byerlee, 1967; Ji and Wu et al. 2020), or may be filled with granular powders with various particle sizes and mineral components (e.g., Marone and Scholz 1989; Tembe et al. 2010).
Fig. 1
Triaxial shear-flow setup with an elliptical rock fracture inclined at Ψ to the sample axis. The axial stress and confining pressure on the sample (left) can be resolved as the shear and normal stresses on the fracture (middle). Two boreholes are drilled near the edges as inlet and outlet boreholes to facilitate fluid communication with the rock fracture (right). a Transmissivity measurement along slip-parallel direction. b Transmissivity measurement along slip-normal direction
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For most rocks, the rate of increase of strength with confining pressure leads to a fresh fault forming at around Ψ = 25°–30° to the maximum compressive stress direction (Sibson 1990). In triaxial shear-flow experiments that employ a pre-cut ‘fracture’, it is not essential to use this ‘optimal’ inclination angle for Ψ. The only constraint is that Ψ should lie within the range of about 20°–55°, depending on the magnitude of the cohesive strength, to avoid the formation of a fresh fault in the rock matrix at the ‘optimal’ angle (Brady and Brown 1993). In previous studies using the triaxial shear-flow configuration the most frequently used pre-cut fracture inclination angles have been 30° (e.g., Ji, 2020; Passelègue et al. 2018) and 45° (e.g., Rutter and Mecklenburgh 2017, 2018).
The transmissivity of the rock matrix in relation to the fracture transmissivity is another major consideration. The rock matrix transmissivity should be low enough to be negligible compared to the fracture transmissivity during the experiments, thus confining the fluid flow within the relatively conductive fracture plane (Ji et al. 2021). To evaluate the hydraulic transmissivity of the rock fracture parallel to the slip direction, two boreholes (inlet and outlet) are drilled from the sample ends to intersect the major axis of the fracture near its ends (Fig. 1a). The slip-normal hydraulic transmissivity of the rock fracture can be measured by drilling two similar boreholes near the edges on the minor axis of the fracture (Fig. 1b).
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2.2 Experimental Details
To illustrate the comparability between the numerical estimation of hydraulic transmissivity and the value obtained using the electrical analogy, we have used the steady state flow measurement data in the triaxial shear-flow experiments reported by Ye and Ghassemi (2018) as the source data, including the inlet and outlet pressures and volumetric flow rate. The purpose of their study was to investigate the slip-parallel transmissivity evolution of rock fractures with cumulative injection-induced shear displacement. Cylindrical samples with a diameter of 50 mm and a length of 100 mm were cored from the Sierra White granite. A fracture inclined 30° to the sample axis was prepared in the core sample. Two small boreholes were drilled near the edges on the major axis of the fracture to measure the slip-parallel hydraulic transmissivity (Fig. 1a). Distilled water was used as the pore fluid. Four fractures, including two sawcut fractures and two tensile fractures, were used in their study. The JRC (joint surface roughness) values of the four fractures ranged from 1.19 to 15.32, which means the steady state flow measurement data was harvested from fractures with a wide range of surface roughnesses, considering the fact that the full range of JRC was from 0 to 20 (Barton 1973). On each rock fracture, a suite of tests was performed under a total confining pressure of 30 MPa and a near-critical shear stress under the constant axial displacement control. The inlet pressure was increased step-wise at a rate of 0.03 MPa/s from 5 to 28 MPa to induce fracture slip, and then decreased step-wise from 28 to 8 MPa at the same rate to check whether the hydraulic transmissivity was retained after the injection-induced shear slip. The outlet pressure was maintained constant at 5 MPa throughout the tests. At each inlet pressure step (i.e., 8, 12, 16, 20, 24, and 28 MPa), when the volumetric flow rate of the inlet pump was similar to that of the outlet pump (difference less than 5%), the fluid flow was considered to be steady state and the volumetric flow rate was recorded. More details of these tests and the raw data can be found in Ye and Ghassemi (2018).
3 Methods for Hydraulic Transmissivity Evaluation
Here we detail the three methods for hydraulic transmissivity evaluation, including the numerical back-calculation proposed in this study, the electrical analogy derived by Rutter and Mecklenburgh (2017, 2018), and the rectangular approximation used in e.g., Crawford et al. (2008), Jeppson et al. (2021), Bijay and Ghazanfari (2021), Okazaki et al. (2013), Wang et al. (2021), and Ye and Ghassemi (2018). In these three methods, the inlet and outlet fluid pressures as well as the volumetric flow rate measured directly using the steady state flow method in the laboratory are used for the transmissivity estimation based on Darcy’s law. The numerical back-calculation and electrical analogy consider the exact elliptical shape of the fracture, while the third method makes a rectangular approximation to the elliptical fracture.
3.1 Numerical Back-Calculation
The COMSOL multi-physics software has been proved as a powerful finite element analysis tool for investigating fluid flow through rock fractures (e.g., Li et al. 2009; Ji et al., 2020; Wanniarachchi et al. 2018). In this study, we built a numerical model using the COMSOL multi-physics software to back-calculate the hydraulic transmissivity of the elliptical rock fracture based on the measured inlet and outlet pressures, volumetric flow rates and fracture geometry. The elliptical rock fracture (Fig. 2a, left) was simulated by an ellipse with the same major and minor axes as the elliptical fracture. Two boreholes with a radius of r [mm] located at d [mm] from the respective edge on the major axis were introduced as the inlet and outlet boreholes. Fig. 2a (left) shows a numerical model of an elliptical rock fracture tested in Ye and Ghassemi (2018) with major and minor axes of 100 mm and 50 mm, respectively. The borehole radius r is 1 mm and the distance d between the borehole and the ellipse edge is 5 mm. The elliptical boundary is set as no-flow boundary. A highly accurate solution can be obtained by using an extra-fine triangular mesh with maximum and minimum element sizes of 2 and 0.0075 mm, respectively. We used Darcy’s law to describe the fluid flow through the simulated fracture:
$$q = - \frac{k}{\mu }\nabla p,$$
(1)
where q is the fluid flux, i.e., volumetric flow rate per unit area [m/s], k is the permeability [m2], μ is the dynamic viscosity of fluid [Pa·s], \(\nabla\)p is the fluid pressure drop over a given distance [Pa/m].
Fig. 2
Numerical back-calculation of hydraulic transmissivity of the elliptical rock fracture. a Left: Finite element model of the elliptical rock fracture with 100 mm major axis and 50 mm minor axis in the COMSOL multi-physics software with an extra-fine mesh. The boreholes with a radius of 1 mm are introduced on the major axis at 5 mm from the edge. Right: Example of steady state fluid pressure distribution with inlet and outlet pressures of 8 MPa and 5 MPa, respectively. The grey lines with arrows are the flow lines. b Flow chart showing the numerical back-calculation process of fracture transmissivity using the COMSOL multi-physics software
×
We made use of the optimization module in the COMSOL multi-physics software to estimate the hydraulic transmissivity through minimizing the relative error between the volumetric flow rates in the numerical model and the laboratory test using the Nelder–Mead method. Specifically, we considered a uniform fracture with an average effective thickness of t [m], and thus the volumetric flow rate Q [m3/s] in the numerical model is the product of the thickness t and the line integral of fluid flux q along the circumference of the inlet borehole. Thus, the constraint of the optimization can be mathematically described as:
where Qexp [m3/s] is the volumetric flow rate recorded in the laboratory. We input the volumetric flow rate, inlet and outlet pressures recorded in the laboratory into the numerical model and the hydraulic transmissivity can be obtained directly through optimization. Fig. 2a (right) demonstrates the steady state fluid pressure profile in the fracture with an inlet pressure of 8 MPa and an outlet pressure of 5 MPa, i.e., the first steady state flow measurement data point in Ye and Ghassemi (2018). The grey lines with arrows are the flow lines. The numerical back-calculation process of fracture transmissivity using the COMSOL multi-physics software is summarized in Fig. 2b.
3.2 Electrical Analogy
The electrical analogy relates the hydraulic and electrical conductivity of rock fractures, which has been used to derive the analytical solution for the transmissivity of an elliptical rock fracture. The flow lines between the inlet and outlet boreholes in Fig. 2a are analogous to the electric field lines between two electrodes in Fig. 3. For an elliptical rock fracture, the lengths in the slip-parallel and slip-normal directions are respectively 2a [mm] and 2b [mm], and the ratio b/a is defined as n (Fig. 3). For convenience, the source and sink holes (i.e., inlet and outlet boreholes) are assumed to lie on the elliptical boundary. The analytical solution for the transmissivity based on the electrical analogy is (Rutter and Mecklenburgh 2017, 2018):
where Q is the volumetric flow rate [m3/s], Δp is the pressure difference between the inlet and outlet boreholes [Pa], r is the borehole radius, B is the geometry constant equal to 2/πarctan(2n). It may be noted that the equation that follows Eq. A1 in Rutter and Mecklenburgh (2018) contains an error, in which the pressure gradient is mistakenly shown instead of pressure difference.
Fig. 3
Evaluation of the hydraulic transmissivity of an elliptical rock fracture using the electrical analogy (
adapted from Rutter and Mecklenburgh 2018) and the rectangular approximation (adapted from Ye and Ghassemi 2018). 2a and 2b are the lengths along slip-parallel and slip-normal directions, respectively. L and w are the length and width of the assumed rectangular flow region (delineated by red dashed lines), respectively. L = 2a if the boreholes are assumed to lie on the elliptical boundary
×
3.3 Rectangular Approximation
For the fluid flow in a fracture, the hydraulic transmissivity can be derived from the integral form of Darcy’s law (Eq. 1) as:
$$kt = \frac{Q\mu L}{{w\Delta p}},$$
(4)
where w is the fracture width [m], L is the length along the flow direction [m].
To estimate the transmissivity of a rock fracture using Eq. 4, the fracture width w should be maintained constant along the flow path. In such case, the fluid is injected as a line source along the fracture width w, and similarly withdrawn over a line sink of the same width. However, in the case of an elliptical rock fracture, the fracture width w changes along the flow direction and the fluid is injected and withdrawn via a hole of negligibly small radius compared to other physical dimensions. Hence, for this case, a rectangular approximation was proposed by Ye and Ghassemi (2018), in which the flow region was assumed to be a rectangle with the same area as the elliptical rock fracture and with a length equal to the distance between the inlet and outlet boreholes (i.e., L = 2a) (Fig. 3). The fracture width w can thus be calculated from the length of axis normal to the flow direction 2b [m] as w = πb/2 if once again the boreholes are assumed on the boundary. Thus, the transmissivity of the elliptical fracture that is estimated based on the rectangular approximation can be expressed as:
$$(kt)_{R} = \frac{4Qa\mu }{{\pi b\Delta p}}.$$
(5)
4 Comparison of Methods
Fig. 4a shows that the transmissivity obtained from the solution based on the electrical analogy is almost identical to that from the numerical back-calculation. The experimental data used here were collected from Ye and Ghassemi (2018). It, therefore, presents the simplest route to the accurate determination of hydraulic transmissivity from flow measurements through an elliptically shaped fracture.
Fig. 4
Comparison of methods. a Hydraulic transmissivity (kt) obtained from the numerical back-calculation (N) as a function of that from the electrical analogy (E) using the experimental data of Ye and Ghassemi (2018). The 1:1 correlation between the results of the two approaches demonstrates their agreement. b Ratio of hydraulic transmissivity (kt) obtained from the rectangular approximation (R) and the electrical analogy (E) as a function of inclination angle with respect to the sample axis in the favorable angle range (Eq. 6). The two directions are indicated in the insets
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We have shown above that our numerical model and the electrical analogy can both yield accurate transmissivity of elliptical rock fractures with a large JRC range, indicating that the accuracy of these two methods is nearly independent of the fracture surface roughness. The ratio between the transmissivity obtained from the rectangular approximation (Eq. 5) and the electrical analogy (Eq. 3) can indicate the accuracy of the rectangular approximation. Here, we extended it to include the effect of changing fracture inclination angle in both slip-parallel and slip-normal directions. The ratio is expressed as:
where n = sinΨ and 2a = D/sinΨ along the slip-parallel direction, while in the slip-normal direction, n = 1/sinΨ, 2a = D, where D is the sample end diameter.
Fig. 4b shows that when the inclination angle ranges from 20° to 50°, the transmissivity ratio in the slip-parallel direction reduces from 0.89 to 0.79, suggesting that the accuracy of the rectangular approximation is acceptable with an order-of-magnitude-accuracy. The experimental data of Ye and Ghassemi (2018) with an inclination angle of 30° show a transmissivity ratio in the slip-parallel direction of 0.87, consistent with the value predicted by Eq. 6. However, the transmissivity ratio in the slip-normal direction increases from 0.16 to 0.30 with increasing inclination angle from 20° to 50°, which means the rectangular approximation leads to a transmissivity approximately 1/4 of the true value. The transmissivity ratio (kt)R/(kt)E in both directions decreases with larger n (i.e., the ratio between the lengths normal and parallel to the flow direction, n = b/a), consistent with the fact that the approximation of a point pressure source using a linear pressure source becomes more valid with increasing length parallel to the flow direction relative to the dimension normal to the flow direction. Moreover, the difference between the slip-normal transmissivity and the true value is always much larger than the slip-parallel transmissivity. This is also because the n value in the case of slip-normal transmissivity is always larger than the case of slip-parallel transmissivity.
5 Conclusions
We revisited the evaluation of the hydraulic transmissivity of elliptical rock fractures in triaxial shear-flow experiments. A COMSOL numerical model was constructed to obtain the transmissivity of elliptical rock fractures based on laboratory data. We estimated the transmissivity of fractures with JRC values ranging from 1.19 to 15.32 in a series of triaxial shear-flow experiments using the numerical back-calculation method, the electrical analogy and rectangular approximation. We compared the fracture transmissivity obtained from the three methods and concluded the following:
1.
The transmissivity obtained from our numerical model and the electrical analogy are identical and represent the true transmissivity value. The rectangular approximation leads to a slightly underestimated slip-parallel transmissivity (0.79–0.89 times the true value) and a substantially underestimated slip-normal transmissivity (0.16–0.30 times the true value).
2.
The underestimation of fracture transmissivity using the rectangular approximation is mainly attributable to the unmodified use of a point pressure source instead of a correct linear pressure source in the approximation.
3.
Our study suggests that both the numerical back-calculation and the electrical analogy can be used for the accurate transmissivity determination of elliptical rock fractures, regardless of the fracture surface roughness. However, the rectangular approximation only provides a rough estimation of the slip-parallel transmissivity to an order-of-magnitude-accuracy.
Acknowledgements
Yinlin Ji is supported by the Research Fund of the State Key Laboratory of Coal Resources and Safe Mining, CUMT (SKLCRSM21KF002). This work has also been supported by the Helmholtz Association’s Initiative and Networking Fund for the Helmholtz Young Investigator Group ARES (contract number VH-NG-1516). Wei Zhang (Penn State University) is appreciated for fruitful and insightful discussion.
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