An Improved Model for Evaluating the Hydraulic Behaviour of a Single Rock Joint Considering Contact Area Evolution During Shearing

The mechanical behaviour of a single rock joint has been numerously studied over the decades. Patton (1966) assumed that asperities on the fracture surface were regular saw-tooth, and the dilation angle was described as the inclination of the teeth in an idealized model. They proposed the simplest yet widely used form of shear models for rough rock joints: \(\tau ={\sigma }_{n}\cdot \mathrm{tan}\left({\phi }_{b}+i\right)\). Barton and Choubey (1977) proposed \(\mathrm{JRC}\) and incorporated it in their shear model to describe the shear behaviour of natural rock joints with different roughness. Li et al. (2016) indicated that the shear behaviour of a joint was dominated by critical asperities with the steepest waviness facing the shear direction and the unevenness of the largest base length.

In this paper, the commonly used joint constitutive model—Barton–Bandis model (Barton and Choubey 1977; Barton 1982) is adopted for describing the asperity degradation since it is simple and can be easily upscaled to field scale. After peak shear strength, a mobilized \(\mathrm{JRC}\) (\({\mathrm{JRC}}_{\mathrm{mob}}\)) is used to describe the asperity degradation. The joint model is expressed as:where \({\sigma }_{n}\) is the normal stress, \(\tau\) is the shear stress, \({\phi }_{r}\) is a residual friction angle, \(\mathrm{JCS}\) is joint wall compressive strength. Barton (1982) suggested a table for estimating the ratio \({\mathrm{JRC}}_{\mathrm{mob}}/\mathrm{JRC}\) from the ratio \({\delta }_{s}/{\delta }_{\mathrm{peak}}\), where \({\delta }_{s}\) is the shear displacement, and \({\delta }_{\mathrm{peak}}\) is the peak shear displacement (Asadollahi and Tonon 2010).

The dilation can be calculated accordingly with the Barton–Bandis model. However, since the main objective of this work is to develop a new prediction model for evaluating hydraulic conductivity during shearing, the dilation of the fractures will directly use experimental results. By doing this, the calibration of the shear model for predicting dilation during shearing can be omitted.

The evolution of the contact area during shearing is calculated based on Grasselli’s criterion (Grasselli et al. 2002; Grasselli 2006), which proposed a three-dimensional morphology characterization approach and expressed the variation of the actual contact area \({A}_{{\theta }^{*}}\) as a function of the apparent dip angle \({\theta }^{*}\) of the surface along the shear direction. The equation is expressed as:where \({A}_{0}\) and \({\theta }_{\mathrm{max}}^{*}\) are the maximum possible contact area ratio and the maximum apparent dip angle in the shear direction, respectively. \(C\) is a fitting parameter. The parameters \({A}_{0}\), \(C\) and \({\theta }_{\mathrm{max}}^{*}\) depend on the various fracture surfaces, the specified shear direction, as well as on the three-dimensional surface representation (i.e., triangulation algorithm and measurement resolution).

The concept of the threshold inclination angle is introduced to estimate the effective area involved in the flow process in a single rock joint, which is equivalent to the threshold apparent dip angle \({\theta }_{\mathrm{cr}}^{*}\) for Grasselli’s criterion. It is assumed that only those zones facing the shear direction with the areas of the surface inclined greater than the threshold value are potentially in contact and involved in the flow process. The maximum possible contact area ratio \({A}_{0}\) is calculated at a threshold angle \({\theta }_{\mathrm{cr}}^{*}\) of 0°, indicating that the sum of all the areas of the surface facing the shear direction normalized with respect to the total area of the fracture. The threshold inclination angle corresponds to the applied normal load and will be mobilized as asperity degradation during shearing. Initially, as illustrated in Fig. 2, only those zones of the surface steeper than \({i}_{0}\) (areas of blue in Fig. 2) are involved in shearing. As shearing progresses, the threshold inclination angle will decrease, and then all those areas with a surface steeper than \({i}_{\mathrm{mob}}\) (areas of red in Fig. 2) will be involved in shearing.

Based on the Barton–Bandis model, the threshold inclination angle, which would be mobilized during shearing, is defined as:

Thus, the relation between the mobilized threshold inclination angle \({i}_{\mathrm{mob}}\) and mobilized potential contact area ratio \({c}_{\mathrm{mob}}\) is calculated as:

In this work, the 3D scanning point cloud data of two rough joint surfaces are processed using MATLAB code (Heinze et al. 2021) to obtain the three Grasselli’s parameters \({A}_{0}\), \({\theta }_{\mathrm{max}}^{*}\) and \(C\). The \(\mathrm{JRC}\) value can be obtained from a statistical parameter formula or back-calculated from the direct shear test. Here, we adopted the back-calculation analysis from shear tests since this provides more accurate \(\mathrm{JRC}\) values for 3D fractures. The geometrical properties of the two fracture surfaces are listed in Table 3.

Both normal and shear stresses change the fracture void geometry that serves as the spaces for water flow. The mechanical aperture of a rock joint, \({e}_{m}\), can be calculated from:where \({e}_{0}\) represents the initial aperture at the given stress environment. \(\Delta {e}_{n}\) is the variation of aperture induced by normal loading, which is equal to \(0\) under constant normal load conditions. \(\Delta {e}_{s}\) is the variation of aperture due to shearing, which is the change in the normal displacement \({\Delta \delta }_{n}\). The initial mechanical aperture \({e}_{0}\) can either be obtained through the cyclic loading–unloading test, or directly using the measured hydraulic aperture before shearing (Xiong et al. 2011).

Many previous works have stated that the flow behaviour is highly sensitive to the aperture distribution (Renshaw 1995; Zimmerman and Bodvarsson 1996; Xiong et al. 2011; Xie et al. 2015; Finenko and Konietzky 2021), which is always described by the mean mechanical aperture and its standard deviation. However, the difficulty is the measurement of the variable joint aperture distribution, especially when considering the evolution of the aperture distribution during shearing. Hakami (1995) introduced several physical techniques to measure fracture aperture, including measuring the surface topography, injecting resin into a fracture and measuring the resin thickness, and casting to make a replica of the void space. But it is generally difficult to measure the aperture distribution changes during shearing using a physical measurement method. Tan et al. (2020) pasted the marking points on the surface of two fracture halves, then scanned and digitized the fracture surfaces and calculated the variable aperture distribution. However, these methods may have limitations to be commonly applied to other cases. Some researchers (Li et al. 2008; Xiong et al. 2011; Huang et al. 2017; Wang et al. 2022) utilized a computational method for calculating the aperture distributions with assumptions that no large damage occurred on asperities under relatively small normal stresses and gouge materials developed during shearing have negligible influence on the fluid flow. It is evident that this method is simplified since the surface damage and gauge materials could exist and have an influence on the fluid flow to a certain extent. However, the computational method can be easily applicable in various contexts. Furthermore, in our work, the influence of asperity degradation is primarily considered within the above contact treatment method.

Here, a modified computational algorithm was adopted based on an open-access source code FSAT in MATLAB (Heinze et al. 2021), which provides a method for measuring the variable aperture distribution with the input mean mechanical aperture. The algorithm was improved by considering the evolution of the mechanical aperture during shearing and calculating the standard deviation of the aperture for merely the open area without contact, as the following steps (Fig. 3): First, align the sides of the upper and lower surfaces according to the specific shear displacement \({\delta }_{s}\). Then, shift the upper surface relative to the lower to match the given mean mechanical aperture, which is calculated from Eq. (5). When the height of the lower surface is higher than the upper, the height of the lower surface will be corrected to match the upper, as illustrated in Fig. 3a to b. Hence, the apertures of these corrected zones are zero. Finally, the standard deviation of the mean mechanical aperture can be calculated from the fractures after being shifted. It should be noted that only those non-zero apertures were considered in calculating the mean mechanical aperture and the standard deviation in our work.

The aperture distributions at three specific shear displacements \(({\delta }_{s}\) = 2 mm, 4 mm, and 8 mm) of two fracture surfaces (J1 and J2) measured by the modified algorithm are shown in Fig. 4. The dark blue areas in it represent the areas of zero apertures due to damage or contact.

Figure 5 shows the comparison of the aperture distribution of the fracture surface J2 at 8 mm shear displacement under \({\sigma }_{n}\)=1.5 MPa obtained from the modified computational procedure and experimental works. Details of the shear-flow experiments will be described in the later section. A thin white paint was sprayed on the surface in this case. It can be seen from Fig. 5b that the areas where the paint is worn off represent the areas of zero apertures due to damage or contact. The zero aperture areas are circled with red dashed lines in both images from the computational measurement (Fig. 5a) and experimental work (Fig. 5b). The position and shape of these areas from the two images are found to be roughly the same, indicating that the modified computational algorithm we adopted is effective for evaluating the aperture distributions.

When water flows through rock joints, the complexity mainly comes from the surface irregularity and the tortuosity of the flow path caused by contact areas, as illustrated in Fig. 6.

As summarized above in Table 1, Zimmerman and Bodvarsson (1996) took into account both aperture variation in the void area and contact obstacles in their equation, which adopted \(\left(1-1.5{\sigma }_{\mathrm{apert}}^{2}/{e}_{m}^{2}\right)\) as the aperture correction term, thereafter \(\left(1-2c\right)\) as the contact correction term. However, this equation indicates that when \({\sigma }_{\mathrm{apert}}/{e}_{m}\) values over 0.816, the ratio \({e}_{h}^{3}/{e}_{m}^{3}\) will be less than \(0\). This is unreasonable, since even in a fracture with large roughness that \({\sigma }_{\mathrm{apert}}/{e}_{m}\) is larger than 0.816, the flow may still be present in the fracture. Xiong et al. (2011) improved this term with \(\left(1-1.0{\sigma }_{\mathrm{apert}}/{e}_{m}\right)\) through curve fitting from simulation results. They did not include the contact correction term in their equation since they stated that the contact areas with zero aperture had been taken into account. However, this is still a 2D description of the contact of the fractures, which is limited to fully capturing the complexities of 3D flow behaviour. Also, this term is presented based on a simplified computational method that neglects surface damage. Thus, the contact correction term is still necessary for evaluating hydraulic behaviour, which incorporates a 3D representation of fracture contact and considers the effects of asperity degradation during shear.

Yeo (2001) modified the contact correction term to \(\left(1-2.4c\right)\) through finite element simulations. But this coefficient still had no clear physical meaning. This equation indicates that when \(c\) value approaches 1/2.4, the ratio \({e}_{h}^{3}/{e}_{m}^{3}\) will approach 0, and the flow will be entirely blocked off. Thus, the coefficient before \(c\) should be calculated from the maximum possible contact area in the shear direction, i.e., \({A}_{0}\) as mentioned above in Sect. 3.2. Then the contact correction term can be modified as \(\left(1-\frac{1}{{A}_{0}}c\right)\).

Therefore, an improved equation is proposed by incorporating two correction terms, the aperture correction term and contact correction term, which explain the reduction of flowrate by roughness and contact obstacles, respectively:where \({e}_{h}\) is the hydraulic aperture. \({e}_{m}\) is the mean mechanical aperture, obtained from Eq. (5), and \({\sigma }_{e\_\mathrm{mob}}\) is the standard deviation of the mean mechanical aperture, which will mobilize during shearing and be calculated using the computational procedure described above in Sect. 3.3. \({A}_{0}\) is the maximum possible contact area ratio in the shear direction. Here, \(1/{A}_{0}\) is given values of 2.29 and 2.22 for J1 and J2, respectively. \({c}_{\mathrm{mob}}\) is the mobilized contact area ratio, calculated from Eq. (4). \({c}_{\mathrm{mob}}/{A}_{0}\) can be defined as the normalized contact area ratio, denoted by \({c}_{\mathrm{mob}}^{*}\). Thus, the proposed equation can be expressed as:where the normalized contact area ratio \({c}_{\mathrm{mob}}^{*}\) can be calculated from \({\left(\frac{{\theta }_{\mathrm{max}}^{*}-{i}_{\mathrm{mob}}}{{\theta }_{\mathrm{max}}^{*}}\right)}^{C}\) by substituting Eq. (4) into Eq. (6). Furthermore, Xiong’s equation is adopted as the aperture correction term. However, the improvement is that the contact areas with zero aperture were not considered in the proposed equation when measuring the variable aperture distribution, while the reduction of flow rate by contact obstacles will be corrected by the specific contact correction term.

The proposed equation conforms to the dimensional consistency, since the hydraulic aperture \({e}_{h}\), the mean mechanical aperture \({e}_{m}\) and its standard deviation \({\sigma }_{e\_\mathrm{mob}}\) are all in the dimension of\(\left[L\right]\), \({\mathrm{JRC}}_{\mathrm{mob}}\) and \({c}_{\mathrm{mob}}\) are dimensionless, \({c}_{\mathrm{mob}}\) is a dimensionless ratio, and \(1/{A}_{0}\) is a constant dependent on the geometric characteristic of different fracture surfaces.

Furthermore, the proposed equation possesses a clear physical significance. The relationships between two correction terms with mobilized aperture distribution and mobilized contact area ratio are illustrated in Fig. 7, where \({f}_{\mathrm{aper}}\) represents aperture correction term and \({f}_{\mathrm{cont}}\) represents contact correction term. It predicts \({e}_{h}={e}_{m}\) at \({\sigma }_{e}=0\) and \(c=0\), which recovers the case assumed in the smooth parallel plane surface. For rough joint surfaces with larger \({\sigma }_{e}\) and \(c\), \({e}_{h}\) will decrease, since the surface roughness produces extra flow resistance and contact obstacles block off water flow, both resulting in the decrease of flow rate. When \(c\) value approaches the maximum possible contact area in the shear direction \({A}_{0}\) of Grasselli’s criterion, the water flow will be completely blocked off, and the ratio \({e}_{h}^{3}/{e}_{m}^{3}\) will approach \(0\).

Figure 8 plots the predicted variation of \({e}_{h}/{e}_{m}\) with the aperture parameter \({\sigma }_{e}/{e}_{m}\) at different values of the contact area ratio \(c\). The result implies that \({e}_{h}/{e}_{m}\) decreases with both the increase of the aperture parameter \({\sigma }_{e}/{e}_{m}\) and the contact area ratio \(c\). The variation in \({e}_{h}/{e}_{m}\) corresponding to an increase of 0.1 in \(c\) when \({\sigma }_{e}/{e}_{m} =0\) is more pronounced compared with that corresponding to an increase of 0.1 in \({\sigma }_{e}/{e}_{m}\) when \(c=0\), indicating that the predicted aperture ratio is slightly more sensitive to the contact area ratio \(c\).

Figure 9 shows a flow chart for implementing the proposed model. Geometrical parameters, including three Grasselli’s parameters \({A}_{0}\), \({\theta }_{\mathrm{max}}^{*}\), \(C\) and the \(JRC\) value, can be readily obtained from 3D scanning point cloud data of joint surfaces by processing with the FSAT toolbox in MATLAB. It should be noted that the \(\mathrm{JRC}\) value utilized in this paper is determined through back-calculation analysis based on direct shear tests. The Barton–Bandis model is adopted for describing the mechanical behaviour of rock fractures. In the post-peak stage, \({\mathrm{JRC}}_{\mathrm{mob}}\) is used to describe asperity degradation. The dilation can also be calculated accordingly with the Barton–Bandis model. However, in this paper, the dilation of the fractures will be directly based on experimental results.

The critical inclination angle is predicted by Eq. (3), which will be mobilized due to asperity degradation through shear. Based on Grasselli’s criterion, Eq. (4) continuously updates the contact area ratio in the shear direction during shearing, acting as the governing parameter in the contact correction term. Additionally, the mean mechanical aperture is updated with dilation as Eq. (5), and its standard deviation is updated as well using the computational method coded in MATLAB, serving as the governing parameters in the aperture correction term. The incorporation of these parameters, which are easily obtained following the treatment method explicated above, ensures the capability of the model to investigate the hydraulic behaviour of rock joints, considering the evolution of the aperture distribution and contact during shearing. Thus, the hydraulic aperture evolution during shearing can be obtained from Eq. (6).

A laboratory shear-flow apparatus was adopted, which consists of a shear-flow box, a control and data acquisition system, and normal and shear actuators that are servo-controlled to apply normal and shear forces with a capacity of 500 kN and 300 kN, respectively (Fig. 10a, b). Four linear variable differential transducers (LVDTs) are placed at four corners of the upper box, and the dilations are measured by the mean values of four LVDTs. An all-around gas-pressure silicon rubber seal was adopted for sealing the water during the experiments, and two rubber side seals were used to ensure water flow through the fracture rather than through the sides of the samples, as shown in Fig. 10c–e. The applied gas pressure was 100 kPa, which was relatively small compared with the applied normal stresses (ranging from 1 to 2 MPa). Hence, the sealing system is considered to have negligible influence on the shear-flow tests. Water inlet and outlet pressure were monitored using two pressure transducers at a precision of 0.25 kPa. The pressure drop was calculated from the differential pressure of the water inlet and outlet. In the following analysis, a density of 1.0 × 10³ kg/m³ and dynamic viscosity of 1.0 × 10^–3 Pa s are taken for water at room temperature, respectively.

The shear-flow tests are conducted under constant normal load and low inlet pressure from 5 to 40 kPa. The mechanical behaviours of fractures J1 and J2 in coupled shear-flow tests, including the evolution of shear stress with shear displacement and dilation with shear displacement, are illustrated in Fig. 11. For each of the fracture surfaces, tests were performed under 3 different normal stresses \({\sigma }_{n}\) (1 MPa, 1.5 MPa, and 2 MPa), and at each normal stress, an average of eight hydraulic tests with various pressure gradients at about seven different shear displacements \(d\) (ranging from 0 to 12 mm) were carried out. Therefore, a total of 336 experimental data in the form of the pressure gradient (− \(\nabla\) P) versus flow rate (Q) were collected in this study, which correspond to different shear displacement \(d\) for two fractures J1 and J2 at different constant normal load \({\sigma }_{n}\), as shown in Fig. 12. The relationship between − \(\nabla\) P and Q exhibits clear nonlinear characteristics. In general, with the increase of shear displacement \(d\), the slopes of the − \(\nabla\) P–Q curves become less steep, i.e., as \(d\) increases, Q becomes larger at the same hydraulic gradient due to fractures dilation during shear. It can be seen from Fig. 12 that curves of \(d\hspace{0.17em}\)= 0 mm in some cases do not conform to the general trend. This is because the fractures were not ideally well-matched at the initial shearing, and water still can be measured at the initial shearing and the shear contraction stage. These experimental data are then used to validate the performance of the proposed model, as described below.

At sufficiently low flow rates, the well-known cubic law is applicable to describe fluid flow through a single rock fracture since the inertial forces are negligible compared with viscous forces, as:where \(Q\) is the flow rate, \(w\) is the fracture width, \({e}_{h}\) is the hydraulic aperture, \(\mu\) is the fluid viscosity, and \(\nabla P\) is the pressure gradient, defined as the ratio of the pressure drops in the flow direction to the fracture length, that is \(\nabla P=\Delta P/L\).

It can be seen from Fig. 12 that the flow behaviour deviates from the linear relationship with the increase of the flow rate. This nonlinear relationship can be well described by Forchheimer’s law as Eqs. (9), (10), with the values of the correlation coefficient \({R}^{2}\) for all curves much close to 1, as listed in Fig. 12.where \(A\) and \(B\) are the coefficients representing viscous and inertial effects, respectively, and \(\alpha\) represents the inertial resistance.

Furthermore, the Reynolds number (\(\mathrm{Re}\)), which represents the nonlinearity in the fluid flow, is defined as the ratio of inertial forces against viscous forces and is given by (Zimmerman et al. 2004):where \(\rho\) is the fluid density, \(\overline{v }\) is the average flow velocity, and \(D\) is the characteristic dimension of the flow system.

The transmissivity (\(T\)) is used to describe the conductivity of a fracture. \({T}_{0}\) represents the transmissivity when fluid flow in the linear regime. Thus, the normalized transmissivity (\(T/{T}_{0}\)) is commonly used to describe the nonlinear flow behaviour in fractures, defined as:

The normalized transmissivity (\(T/{T}_{0}\)) can also be described as a function of \(\mathrm{Re}\) as:where \(\beta\) is the Forchheimer coefficient. The relationships between \(T/{T}_{0}\) and \(\mathrm{Re}\) for two fractures under different normal loads are shown in Fig. 13. It can be seen that \(T/{T}_{0}\) decreases with an increase in \(\mathrm{Re}\), which reveals the nonlinearity of the flow. \(T/{T}_{0}=0.9\) indicates that the nonlinear term (\(B{Q}^{2}\)) contributes to 10% of the pressure drop, the Reynolds number at this point is referred to as the critical Reynolds number (\({\mathrm{Re}}_{\mathrm{c}}\)) (Yu et al. 2017), as shown by the dashed lines in Fig. 13. The flow in our tests shows obvious nonlinear behaviour. In addition, the \(T/{T}_{0}\)-\(\mathrm{Re}\) curves generally shift downwards as the shear displacement \(d\) increases, indicating that the nonlinearity of fluid flow is stronger at larger shear displacement. However, the investigation of the nonlinear flow equation is out of scope in this study.