Temporal evolution of shear-induced dilatancy of rock fractures: Controls from 1 surface roughness and normal stress

Understanding the shear-induced dilatancy of rock fractures is important for assessing the

Therefore, it is of particular importance to study the transient fracture dilation accompanying slip.
The fracture aperture is known to change during the slip of fractures due to the rearrangement and destruction of asperities (Elsworth & Goodman, 1986;Fang et al., 2017).
When slip occurs on a fracture, the aperture could increase as a result of dilation (Barton et al.,  1985;Ishibashi et al., 2016) or decrease due to compaction (Morrow & Byerlee, 1989;Niemeijer & Spiers, 2007;Fang et al., 2017).In most cases, especially in brittle rocks, the apertures of initially bare fractures tend to increase with fracture slip even at high normal stresses (Barton et al., 1985;Esaki et al., 1999;Ye & Ghassemi, 2018), which is called shear-induced dilatancy (Samuelson et al., 2009).The shear-induced increase of fracture aperture () can be quantitatively described by the increase of slip displacement () and dilation angle () as (Barton et al., 1985), In addition to the slip displacement, the slip velocity (v) also influences the transient evolution of fracture aperture during slip (Fang et al., 2017;Ji et al., 2023, Samuelson et al., 2009;Segall & Rice, 1995).Particularly, when subject to a velocity change from v i-1 to v i , the transient evolution of incremental porosity (ε) of the gouge layer in a gouge-filled fracture can be expressed as (Rudnicki, 2022;Samuelson et al. 2009;Segall & Rice, 1995;Sleep, 1995) where Ψ is the dilation factor scaling the magnitude of dilation in response to the velocity step; i refers to the i th velocity step;  is the porosity of gouge layer; t i is the time since the i th velocity step; D c is the characteristic slip distance over which the frictional contacts are renewed (Dieterich, 1979;Ruina, 1983).The change in porosity in Equation 2 is associated with change in the thickness of gouge layer and could be described as ∆ ≅ ∆/, where  and  are the velocity-induced thickness change and the thickness before velocity change, respectively (Samuelson et al., 2009).For initially bare rock fractures,  could be interpreted as the ratio between the velocity-induced aperture change and the aperture before velocity change.Thus, the transient aperture of initially bare fractures influenced by slip velocity change (b evo ) can be expressed as (Fang et al., 2017), where b is the fracture aperture uncorrected by the velocity change at the i th velocity step.The fracture aperture is commonly linked to permeability by k=b 2 /12 (Snow, 1969;Witherspoon et al., 1980).
The dilation angle () of initially bare fractures in the displacement-dependent aperture model (Equation 1) has been extensively studied (Melin, 2012) and is normally in the range of 0-30°.It tends to increase with increasing fracture roughness and decreasing normal stress (Barton & Choubey, 1977).The dilation factor (Ψ in Equation 2) of gouge layers was measured ranging from 10 -5 to 10 -3 by Samuelson et al. (2009) and Ashman & Faulkner (2023).In a numerical study on dilatancy and friction of gouge-filled fractures, the variation in the dilation factor (Ψ) in such ranges could result in significant change in the drainage state of fractures, affecting its stability (Samuelson et al., 2011).More recently, the velocity-dependent porosity model has also been applied in a large-scale permeable fault to investigate the impact of porosity and permeability evolution on aseismic slip induced by injection (Yang & Dunham, 2021).However, for initially bare fractures, the dilation factor (Ψ) and characteristic slip distance (D c ) in the velocity-dependent aperture model (Equation 2) are still poorly constrained, restricting the incorporation of this important semi-analytical model into analytical and numerical analysis of fracture permeability and stability.
To constrain the constitutive parameters in the displacement-and velocity-dependent aperture models, we performed numerical inversions based on the experimental data collected from 16 sets of normal stress unloading tests on sandstone samples (Yin et al., 2023) to obtain the model parameters and explore their dependence on surface roughness and normal stress.This suite of experimental results allowed us to investigate the temporal evolution of shear-induced dilatancy of rock fractures comprehensively.
2 Materials and Methods

Sample preparation and experimental method
Fracture surfaces with an xy plane projection area of 100 × 200 mm and different joint roughness coefficient (JRC, Barton et al., 1985) values of 3.21, 5.62, 7.36, and 12.16 were prepared and tested in this study.A 3D laser scanner with an accuracy of ±10 μm was utilized to measure the 3D coordinates of the surface morphology (Figure 1), which were used to calculate the surface roughness parameter Z 2 as (Tse & Cruden, 1979), where y i and z i are the coordinates of a two-dimensional fracture profile along the shear direction (i.e., y direction in Figures 1a-1d); M represents the number of sampling points distributed along the profile at a constant interval of 1 mm.Fifty profiles along the shear direction with an equal interval of 2 mm were used to obtain the average Z 2 value of each fracture surface (Yin et al., 2023).The average JRC value was then calculated based on the empirical correlation between JRC and Z 2 as (Tse & Cruden, 1979), Using the 3D scanning data as a reference, precise surface engraving was performed on sandstones at a bit rate of 18,000 r/min to prepare the samples employed in this study.The raw sandstone materials with a density of 2.71 g/cm 3 and a uniaxial compressive strength (UCS) of 49.88 MPa were collected from Wuding County, Yunnan Province, China.Four replicas were made for each fracture and thus a total of 16 sandstone samples were prepared for the normal stress unloading tests (Yin et al., 2023) to investigate the dilation of rock fractures induced by slip under the conditions of unloading normal stress (e.g., landslide, tunnel excavation, and injection-induced seismicity).
The tests were performed on the MIS-233-1-55-03 servo-controlled direct shear apparatus at Nagasaki University (Figure 2) (Jiang et al., 2004;Yin et al., 2023).Two digital load cells were placed on the top of the upper shear box to measure the normal load and one digital load cell was connected to the side of the shear box to measure the shear load (Jiang et al., 2004).Five linear variable differential transformers (LVDTs), each possessing a precision of 0.001 mm, were employed for the measurement of displacements.One LVDT was affixed on the lower shear box for the purpose of quantifying the slip displacement, while the remaining four LVDTs were positioned at the four corners of the upper shear box to monitor the normal displacement.The data acquisition and instrumentation systems used in this study were based on LabVIEW (Han et al., 2020).The acquisition system recorded the normal stress and shear stress at a sampling frequency of 1 Hz, each achieving a precision of 0.001 MPa.In parallel, the acquisition system captured the slip displacement and normal displacement at a sampling rate of 1 Hz, each with an accuracy of 0.001 mm.The increase of normal displacement indicates the compression of the fracture whereas the reduction of normal displacement implies the dilation of the fracture.The positive change of the slip displacement is associated with the activation of the fracture.The slip velocity was computed as the difference between adjacent slip displacements divided by the time interval of the acquisition (1 s).The experimental conditions for all the 16 samples are listed in Table 1.Note that five normal stress unloading tests were conducted repeatedly under each experimental condition but we focused only on the first test to evaluate the roles of surface roughness and normal stress.A more detailed description for the 16 normal stress unloading tests can be found in Yin et al. (2023).
Before each test, the sample was positioned within the shear box, and an initial normal stress ( ) was applied at a loading rate of 0.5 MPa/min until reaching the target value.Note that for each JRC, four different initial normal stresses, i.e., 1, 3, 5, 7 MPa, were tested (see Figure A1 for all testing results).Subsequently, in each test (Figure 3), the whole experimental process can be divided into three stages, consisting of a displacement-driven shear stage (stage Ⅰ), a shear stress adjustment stage (stage Ⅱ), and an unloading-driven shear stage (stage Ⅲ).During the displacement-driven shear stage, the normal stress remained constant, while the lower shear box was actuated by the horizontal jack at a loading rate of 0.3 mm/min to apply shear stress (τ) until reaching the peak shear stress (τ max ).Then, during the shear stress adjustment stage, τ was linearly reduced to 85% of  at a rate of 1 MPa/min and held constant afterwards at an always constant  .Finally, during the unloading-driven shear stage, we unloaded the normal stress at a rate of 0.6 MPa/min to induce fracture opening and slip.We further divided the unloading-driven shear stage into four segments, including a pre-slip segment during which the fracture dilated because of normal stress unloading but slip did not occur, a velocity-fluctuation segment where the fracture exhibited dynamic slip events with fluctuating slip velocities, an accelerating slip segment with constant shear stress where the fracture showed dramatically increasing slip displacement, and an accelerating slip segment with decreasing shear stress where the apparatus could not provide constant shear stress as a result of the high slip velocity.

Numerical inversion using the combined aperture model
The modelled fracture aperture considering both displacement-and velocity-dependent aperture changes can be expressed as (Ji et al., 2023), where the indexes n and i refer to the n th and i th velocity steps, respectively (i≤n);  is the aperture predicted by the displacement-and velocity-dependent aperture models;  is the aperture computed as the displacement-dependent aperture change (Equation 1) plus the aperture at the onset of stable sliding;  is the incremental porosity/aperture in Equations 2 and 3 (particularly,  =0).When computing  ,  in Equation 2 is calculated as, where  is the slip displacement at the end of the n th velocity step;  is the slip displacement at the start of the i th velocity step; v i is the slip velocity during the i th velocity step.
As the model was derived from velocity-stepping shear tests showing stable sliding at low velocities (Samuelson et al., 2009), the velocity-fluctuation segment characterized by abrupt slip velocity changes was excluded from our study.The accelerating slip segment with decreasing shear stress has also been excluded as the constant shear stress boundary condition cannot be secured in this segment.As the normal stress change of the 16 normal stress unloading tests is less than 5 MPa, the normal displacement change induced by normal stress unloading could be assumed as linearly related to the normal stress change (Ji et al., 2022;Rutqvist et al., 2008).This relationship is fitted based on the measured changes of normal displacement and normal stress during the pre-slip segment in each test to estimate the combined normal stiffness of the shear box, rock matrix, and fracture.The near linear relationship between the normal stress change and normal displacement change confirms that the normal deformation caused by normal stress unloading is mostly elastic (Figure 4).The slopes of the fitted lines were taken as the combined normal stiffnesses (k n ).In addition, note that we assumed a constant dilation angle (Equation 1) in our model as the slip displacement before the accelerating slip segment with decreasing shear stress in all the 16 tests is not more than 2 mm.
To investigate the transient shear-induced dilatancy, we focused on the aperture change caused only by shear dilation and subtracted the deformations of shear box, rock matrix and fracture opening induced by normal stress unloading from the measured mechanical aperture in the accelerating slip segment with constant shear stress.The vertical LVDTs can only measure the average aperture change, and the maximum mechanical aperture without any external load (b max in Table 1) in this study was estimated as the root mean square (RMS) asperity height of the fracture (Fang et al., 2017;Magsipoc et al., 2020).Given the substantially greater capacity for normal deformation of the interstitial gap between adjacent fracture walls compared to that of the shear box and rock matrix, we consider that the normal deformation monitored by the LVDTs is predominantly attributable to the change in the fracture aperture.The mechanical aperture at the onset of each test (b ini ) could thus be estimated as, where  is the initial normal stress.The measured mechanical aperture (b exp ) during the test can then be computed as,  ) where F obj , i.e., root mean square error (RMSE), is the objective function to be minimized;  is the mechanical aperture predicted by the model at the i th velocity step;  , is the measured mechanical aperture at the i th velocity step; n is the total number of data points.There were three independent parameters (i.e., Ψ, D c , and ω) that need to be constrained during the optimization and thus three degrees of freedom were in the numerical inversions.The genetic algorithm (GA) (Holland, 1992;Zbigniew, 1996) was utilized to find the global minimum of F obj over the ranges of 0≤Ψ≤0.5, 10 -5 ≤D c ≤2 mm, and 0≤ω≤20° for all the numerical inversions.The Python package named scikit-opt was used to implement the numerical inversions based on GA (Zhang et al., 2023b).

Results and Discussion
3.1 Contribution of slip velocity to transient aperture evolution displacement with upper and lower bounds of 0.001 and 0.373 mm/s, respectively.The velocity factor characterizes the contribution of slip velocity additional to that of slip displacement to the mechanical aperture change.It is higher than 1 when the slip velocity tends to enhance the modelled aperture increase, while the aperture increase is suppressed by the velocity when it is lower than 1.The velocity factor ranges from -1.95 to 3.66, suggesting a significant contribution of slip velocity to aperture change in the accelerating slip segments with constant shear stress in our experiments.Moreover, the nonlinear evolution of fracture aperture change with evolving slip velocity can only be modelled by combining displacement-and velocity-dependent aperture models.The influence of slip velocity on the nonlinear aperture change could be attributed to the frequent velocity change.Frictional contacts on a fracture would be renewed over a characteristic slip distance after a change in slip velocity (Dieterich, 1979;Ruina, 1983), causing the fracture aperture to nonlinearly evolve.The velocity changes so frequently that the aperture cannot be stabilized between each two adjacent velocity steps.Consequently, the transient change in fracture aperture is affected by the temporal evolution of slip velocity.
The four subplots along the horizontal direction represent the evolution of the results with different initial normal stresses ( ) at the same JRC.With the increase of initial normal stress, the velocity factor under the four JRC values changes from a monotonically increasing trend to a trend showing a reduction followed by an increase, while the slip velocity at the onset of the accelerating slip segment with constant shear stress decreases.This transition may be caused by the wear of asperities under higher normal stresses.Particularly, when the normal stress is low, the deformation of asperities is negligible, and the slip velocity increase could monotonically promote the aperture increase.However, if the normal stress on the fracture is high enough to cause asperity wearing, the aperture increase could be slightly inhibited firstly and then promoted by further velocity increase.The wearing of the surface asperities in this study has been visually and quantitatively demonstrated by Yin et al. (2023), where the ratio of sheared area to the total fracture area was dramatically intensified under high initial normal stresses.One exception to this transition occurs when  is 7 MPa and JRC is 7.36, where the velocity factor decreases monotonically and is always lower than 1.The reason for this exception is probably that the slip displacement change (0.761 mm) and velocity (0.011mm/s) at the end of the accelerating slip segment with constant shear stress is not high enough to lead to the increase of the velocity factor.
The four subplots in the same column present the evolution of the results with different JRC values at the same initial normal stress ( ).With the increase of JRC, the transition of the velocity factor from monotonic increase to a trend showing a reduction followed by an increase is promoted and occurs at lower initial normal stresses, accompanying the generally decreasing velocity at the onset of the accelerating slip segment with constant shear stress.A higher JRC normally indicates a larger maximum asperity height and less real contact area of the asperities on a pair of fracture surfaces (Dieterich & Kilgore, 1996;Hisakado, 1974), which means the asperities are easier to be worn under the same average normal stress due to the locally concentrated stress on the contacted asperities.As a result, the worn asperities associated with increasing JRC could make the transition earlier and more significant, potentially facilitating the occurrence of such transition at lower normal stresses.This mechanism could be evidenced by the increasing ratio of sheared area to the total fracture area with increasing JRC value of fracture surface, especially at higher normal stresses (Yin et al., 2023).while it is 1 to 2 orders of magnitude higher than the dilation factor in synthetic quartz-clay (kaolinite) fault gouge (Ashman and Faulkner, 2023) and 2 to 3 orders of magnitude higher than the dilation factor in fine-grained quartz fault gouge (Samuelson et al. 2009).The smaller dilation factor in gouge-filled fractures compared to our initially bare fractures may be due to the pre-compaction/consolidation of gouge before the tests, resulting in its lower sensitivity to velocity.The values of our best-fit  are generally consistent with the  reported on initially bare fractures ranging from several microns to millimeters (e.g., Marone & Cox, 1994;Fang et al., 2017;Ishibashi et al., 2018;Shen et al., 2022;Im et al., 2018;Ji et al., 2023).The range of dilation angle in our study is similar to that in previous studies, ranging from 0 to 30° (Chen et al., 2000;Melin, 2012;Yeo et al., 1998).
The dilation factor () decreases and converges with the increase of initial normal stress ( ) and JRC (Figures 6a-6b).The decrease of  means the velocity-dependent dilation can be suppressed by the increase of normal stress and JRC.The convergency and decrease of  with  increase in our study is inconsistent with the enigmatic relationship between  and  of gouge-filled fractures (Ashman & Faulkner, 2023;Samuelson et al., 2009), which may be caused by the different responses of initially bare and gouge-filled fractures to normal stress change.
Particularly, the increase of normal stress tends to cause the closure of initially bare fractures and thereby inhibit the dilatancy, and  is thus decreased.However, the influence of normal stress on gouge-filled fractures may be complicated by the grain packing frameworks (Kohli & Zoback, 2013) changing with clay contents (Ashman & Faulkner, 2023).In addition, the slight dependency of  of gouge-filled fractures on normal stress could be complicated by the development of shear fabrics associated with accumulated shear strain (Logan et al., 1992;Samuelson et al., 2009).The characteristic slip distance ( ) decreases with the increase of  (Figure 6c).Note that, as  of the three nonconvergent results in Table 1 reaches the upper bound (2 mm), the nonconvergent results are neglected when investigating the variation of  .
also decreases with the increase of JRC (Figure 6d).The reduction of characteristic slip distance may indicate that the fracture can reach steady state more easily after the change of slip velocity at a higher normal stress and JRC.
Unlike the dilation factor () and characteristic slip distance ( ), the dilation angle () does not converge with the variation of neither initial normal stress ( ) nor JRC.The dilation angle was considered negatively correlated with the effective normal stress within the range of ~2-36 MPa (Barton & Choubey, 1977;Chen et al., 2000;Kim & Jeon, 2019).However, no obvious relationship between dilation angle and initial normal stress could be observed in our study (Figure 6e).In addition, the dilation angle is not sensitive to  , which may be caused by the limited range of  (from 1 to 7 MPa) and uncertainties in numerical inversions (indicated by the RMSE values in Table 1 and point sizes in Figure 6) in our study.Nonetheless, the variation of dilation angle decreases with the increase of  .This decrease could be caused by the more severe wear of fracture asperities under higher normal stresses, which could be verified by the increasing ratio of sheared area to the total fracture area with increasing normal stress (Yin et al., 2023).The dilation angle generally increases with JRC (Figure 6f) primarily because of higher asperities, which is consistent with previous experimental results (c.f., Barton & Choubey, 1977;Rafek & Goh, 2012).Therefore, under otherwise similar conditions, the displacement dependence is more significant for rougher fractures with higher JRC due to the positive correlation between the dilation angle and displacement-dependent fracture aperture (Equation 1).
Furthermore, rougher fracture surfaces tend to experience arrested ruptures due to the significant stress barriers induced by higher asperities (Fryer et al., 2022), causing limited slip velocity.The influence of slip velocity on a fracture aperture may be thereby weakened.

Limitations and Suggestions
We constrained the key constitutive parameters, including the dilation angle, dilation factor and characteristic slip distance, in displacement-and velocity-dependent aperture models.
Incorporating the role of slip velocity change in controlling the aperture change, in addition to the slip displacement, has successfully replicated the nonlinear evolution of shear-induced aperture change of fractures.The normal stress applied in our laboratory experiments ranges from 2% to 14% of UCS of the host rock.A much higher normal stress may inhibit both displacement-and velocity-induced aperture increase (Ashman & Faulkner, 2023;Y. Li et al., 2022).The fractures with initially bare surfaces in the laboratory here are fresh, dry, and tightly matched.It is inferred that weathered and unmated fractures in the field may experience less significant dilation as the dilation factor of gouges is lower than that of initially bare surfaces, primarily depending on the degree of weathering (Barton et al., 1985) and the joint matching coefficient (Zhao, 1997).In this study, we used a linear displacement-dependent aperture model (c.f.Barton et al., 1985;Cappa et al., 2018;Zhang, Jeffrey, et al., 2019), which is reasonable in relatively small displacement ranges as explored in our experiments.The nonlinear aperture dependency on displacement should be incorporated for large slip displacement cases (c.f., Fang et al., 2017).We assumed the dilation factor, characteristic slip distance, and dilation angle as constants in our numerical inversions.These parameters, however, may change with the slip of the fracture (e.g., Marone & Cox, 1994).Also note that the aperture here in this study is the mechanical aperture, which is not equivalent to the hydraulic aperture (Chen et al., 2000) that was used by Cappa et al. (2018), Fang et al. (2017) and Ji et al. (2023).However, the congruence between alterations in hydraulic aperture and mechanical aperture may remain steadfast in permeable fractures within controlled laboratory settings (Fang et al., 2017;Ji et al., 2023) and natural field conditions (Cappa et al., 2022).Moreover, the results obtained from the accelerating slip segments with constant shear stress in this study, characterized by a slip velocity up to 0.373 mm/s, cannot extend to the slip regime with extremely high slip velocities, indicating that our results are applicable mostly to slow slipping fractures/fault segments showing no flash heating and associated thermal pressurization (Acosta et al., 2018;Rice, 2006;Tanikawa et al., 2010).
Nevertheless, the dilation factor and characteristic slip distance values constrained in this study provide reference magnitudes for inputs in the velocity-dependent aperture model in slow slip regimes.
The characteristic slip distance, dilation angle, and dilation factor in this study were constrained in a laboratory setting.Therefore, they should only be reference values in laboratoryscale modeling.For field-scale models, it might be possible to determine the characteristic slip distance using D c =ζT, where ζ is a constant estimated as 10 -2 and T is the thickness of the shear band where shear strain occurs (Marone & Kilgore, 1993).The dilation angle could be upscaled base on the experiment results in laboratory.For example, Barton and Bandis (1982) proposed a set of upscaling equations that could link the field-scale dilation angle to the JRCs, joint compressive strengths, lengths, and effective normal stresses of field-scale fractures.In addition, the dilation factor could be discretely set for each fracture segment in field-scale models to simulate shear-induced dilatancy.
The reproduction of the nonlinear aperture change using the combined displacement-and velocity-dependent aperture model is of great importance for various engineering settings.For instance, in the case of injection-induced fracture reactivation and permeability evolution, the nonlinear aperture evolution may complicate the change of pore pressure, effective normal stress and thereby slip response of the fracture, which in turn alter the aperture change and pore pressure diffusion by changing the slip displacement and velocity, especially in the rupture nucleation stage (Cappa et al., 2018;Ciardo & Lecampion, 2019;Heimisson et al., 2022;Yang & Dunham, 2021).Particularly, even an extremely low dilation factor on the order of 10 -5 to 10 -3 can cause drastic nonlinear changes in pore pressure and shear behavior of gouge-filled fracture (Samuelson et al., 2011;Rudnicki, 2022).Besides, the proper assessments of aperture changes and permeability evolutions are important for anticipating and managing the injection-induced seismic hazard (Ishibashi et al., 2016;Li et al., 2022).Therefore, it is critical to incorporate the velocity-dependent aperture model in future analytical and numerical models involving hydromechanically coupled processes.
The mechanical aperture in this study is the orientation-independent average aperture, which discounts the anisotropic aperture evolution.However, the permeability anisotropy associated with the development of aperture anisotropy, which may influence the propagation of the seismic front in different directions with respect to the principal shear direction, is ignored in this study and needs to be investigated in the future (Ji, Zhang, et al., 2022;Shapiro et al., 1997).
Flux-driven particle mobilization accompanied by asperity wearing in injection related activities could also impact the permeability evolution (Candela et al., 2014) and thus clogging and unclogging of particles in worn fractures also necessitates further studies.In addition, in this study we investigated only sandstones.It is thus necessary to further explore the surface roughness and normal stress controls on fractures in other rock types relevant to geoenergy systems.

Conclusions
We combined displacement-and velocity-dependent aperture models to reproduce the temporal evolution of shear-induced dilatancy of fractures in sandstone based on 16 suites of experimental data.Our results suggest the following: The combined aperture model dependent on both slip displacement and velocity can satisfactorily reproduce the nonlinear aperture evolution during the accelerating slip of fractures induced by normal stress unloading better than the displacement-dependent aperture model.
(2) As the initial normal stress in the 16 tests increases, the velocity-dependent mechanical aperture shifts from monotonic increase to reduction followed by increase with the enhancement of slip velocity, and this shift is promoted by higher surface roughness.This transition indicates that slip velocity could enhance the aperture increase on smoother fractures at lower normal stresses and higher slip velocities.
(3) Both the dilation factor and characteristic slip distance decrease and converge with the increase of both initial normal stress and JRC, signifying the reduced contribution of slip velocity to transient shear-induced fracture dilatancy at higher initial normal stress and JRC.
(4) The dilation angle increases with JRC, but this effect becomes less significant at higher initial normal stresses, primarily due to the more severe wear of fracture asperities under higher normal stresses.
The results in this study demonstrate the influence of slip velocity on shear-induced dilatancy of fractures, which could also provide reference input values for the combined With the increase of normal stress, the velocity factor changes from a monotonically increasing trend to a trend showing a reduction followed by an increase.This transition is promoted by higher JRC values.Note that we used different scales except for the velocity factor for better presentation of the results.The gray dashed line indicates the velocity factor with a value of 1.     (Barton et al., 1985).828 b) RMSE is the root mean square error between the predicted and measured mechanical apertures.

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Downloaded from https://academic.oup.com/gji/advance-article/doi/10.1093/gji/ggae156/7659842 by guest on 11 May 2024 where h stands for the measured normal displacement during the test; h 0 refers to the measured normal displacement when the test starts.After the reactivation of the preexisting fracture, the measured mechanical aperture excluding the normal deformation induced by normal stress unloading but including the shear dilation by slip ( ) could be calculated with the following equation, where  is the normal stress after fracture reactivation;  , is the normal stress at the onset of fracture reactivation.After the computation of the measured mechanical aperture change induced by shear dilation, we conducted numerical inversions based on the data collected from the 16 sets of normal stress unloading tests to constrain the model parameters in the combined aperture model and explore their dependences on surface roughness and normal stress.Particularly, we optimized the parameters in the model described by Equations 1-3 and Equations 6-7 to match the mechanical aperture predicted by the model (b evo ) with the measured mechanical aperture ( ) by searching the minimum root mean square error (RMSE) between the mechanical apertures measured in the experiments and predicted by the combined aperture model.That is, the following objective function has been minimized during the optimization,

Figure 5
Figure 5 presents the evolution of measured ( ), displacement-and velocitydependent ( ), and displacement-dependent ( ) mechanical aperture, velocity factor ( , where b 0 is the measured aperture at the onset of the accelerating slip segment with constant shear stress), and slip velocity during the accelerating slip segment with constant shear stress used for numerical inversion.The maximum slip displacement change and aperture change

Figure 6 .
Figure 6.The optimized parameters in displacement-& velocity-dependent aperture model, including (a-b) dilation factor (Ψ), (c-d) characteristic slip displacement (D c ) and (ef) dilation angle (ω), for the 16 tests conducted on fractures with different JRC at different initial normal stresses (   ).The open data points represent the results from the three nonconvergent inversions and the solid data points show the converged results.The point sizes are scaled according to the corresponding RMSE values inTable 1 (larger points indicate higher

Figure A1 .O
Figure A1.Time histories of normal stress, shear stress, slip displacement, slip velocity, and normal displacement of the 16 normal stress unloading tests.The computations and inversions in this study utilized the data of the first shear cycles of each test (a total of 5 cycles for each test (Yin et al., 2023)), except for the three tests under the conditions of (a) JRC=3.21, =1 MPa (the 2nd cycle), (i) JRC=7.36, =1 MPa (the 3rd cycle), and (j) JRC=7.36, =3 MPa (the 2nd cycle).The three exceptions were picked for better data quality and may not impact our analysis and results.The slip velocity is computed as the difference between adjacent slip displacement divided by the time interval.Positive changes in normal displacement indicate normal compaction of the fracture.The gray dashed lines divide the test into three stages, consisting of (Ⅰ) a displacement-driven shear stage, (Ⅱ) a shear stress adjustment Downloaded from https://academic.oup.com/gji/advance-article/doi/10.1093/gji/ggae156/7659842 by guest on 11 May 2024 Downloaded from https://academic.oup.com/gji/advance-article/doi/10.1093/gji/ggae156/7659842 by guest on 11 May 2024 Downloaded from https://academic.oup.com/gji/advance-article/doi/10.1093/gji/ggae156/7659842 by guest on 11 May 2024 Downloaded from https://academic.oup.com/gji/advance-article/doi/10.1093/gji/ggae156/7659842 by guest on 11 May 2024 are 2.141 mm (in the test with a  of 3 MPa and a JRC of 7.36) and 0.315 mm (in the test with a  of 5 MPa and a JRC of 12.16), respectively.The slip velocity generally increases with slip Fang et al. (2017)tps://academic.oup.com/gji/advance-article/doi/10.1093/gji/ggae156/7659842 by guest on 11 May 2024  of all convergent cases ranges from 0.0004 to 0.2118,  from 0.0042 mm to 1.6763 mm, and  from 0.55° to 12.93°.The dilation factor of initially bare fractures in sandstone in our study is similar to that reported byFang et al. (2017)on initially bare fractures in shale, Ji et al.