Relations between hydraulic properties and ultrasonic velocities during brittle failure of a low-porosity sandstone in laboratory experiments

Relations between hydraulic properties and ultrasonic velocities during brittle failure of a... SUMMARY We continuously monitored hydraulic properties and ultrasonic P-wave velocity during triaxial compression of Wilkeson sandstone samples with an initial total porosity of about 9.5 per cent to elucidate the microstructural evolution and its consequences for the relation between these properties during elastic and inelastic deformation. Samples were deformed at drained conditions at a range of effective pressures from below to above the determined crack-closure pressure of about 50 MPa and hydraulic properties were determined using the oscillatory pore-pressure method. As a result of the employed period the evolution of hydraulic properties was resolved for strain increments of 10−4 and thus constrained during imminent localized failure. Except during the development of a localized fault at the lowest imposed effective confining pressures, we found permeability and hydraulic diffusivity to generally increase during progressing brittle deformation associated with dilation. Thus, in situ faulting of fluid-bearing rocks should in general exhibit self-stabilization. Axial P-wave velocity shows a rather uniform correlation with radial strain during inelastic deformation suggesting a prominent role of cracks aligned with the direction of wave propagation, an inference that needs further analyses regarding the varying sensitivity of longitudinal waves for (micro)structural changes in their direction of travel. Hydraulic properties exhibit systematic correlations with radial strain during inelastic deformation, too, a rather expected result for permeability. During nominally elastic deformation, however, stress ratio and mean stress seem to control hydraulic properties. Differences between elastic properties derived from mechanical and hydraulic excitations probably indicate heterogeneity of the microstructures. An additional contribution of anisotropy in hydraulic properties cannot be ruled out that up to now evades experimental quantification. The observed fairly uniform correlation between hydraulic diffusivity and P-wave velocity in the direction of fluid flow suggests that monitoring changes in elastic wave velocities bears some potential to constrain changes in conditions for transient fluid flow. Fracture and flow, Microstructure, Permeability and porosity, Creep and deformation, Elasticity and anelasticity, Acoustic properties 1 INTRODUCTION The derivation of structure and state of rocks from constraints on the spatial distribution of their physical properties represents one of the key objectives in geophysics. Constraints on structure are gained from borehole logging and surface surveys covering spatial scales from centimetres to kilometres. Laboratory experiments with the ability to control the state of rocks are crucial for amending results for structure with information on state. Using information from the propagation of elastic waves in rock volumes of interest probably constitutes the most common approach across all scales. In particular, the sensitivity of wave velocities to structure and state permits monitoring natural or anthropogenically induced subsurface processes using time-dependent variations in their characteristics, for example related to precursory phenomena, earthquake-source mechanisms, and aftershock sequences (e.g. Rice & Rudnicki 1979; Miller et al. 2004; Zaliapin & Ben Zion 2011; Froment et al. 2014; Hillers et al. 2015b), faulting in general (e.g. Faulkner et al. 2010), gas and oil production (e.g. Landrø 2001), or stimulation of geothermal reservoirs (e.g. Charléty et al. 2006). The determination of hydraulic properties of rocks or, more general, the subsurface serves the dual purpose of constraining structure and of providing effective properties for a specific system of interest. Such effective properties permit modelling subsurface fluid flow associated with natural processes and predicting the outcome of pumping operations as associated with, for example, freshwater supply, exploitation of hydrocarbon reservoirs, operation of subsurface liquid-waste repositories, or geothermal energy provision. Furthermore, hydraulic properties are pivotal for the evolution of mechanical states when a system containing fluids is perturbed by either changes in bulk stresses or fluid pressures. The related concept of effective pressure and stress goes back to at least Terzaghi (1936). Among other phenomena, hydro-mechanical coupling, that is, the mutual influencing of bulk stresses and fluid pressures (Rice & Cleary 1976), has been held responsible for aftershock activity (e.g. Miller et al. 2004) and induced seismicity (e.g. Majer et al. 2007; Zang et al. 2014). In situ, wave observations potentially provide better coverage of rock volumes in space and time than hydraulic investigations and thus constraints on correlations between elastic and hydraulic properties may provide the basis for improved subsurface characterization with respect to hydro-mechanical aspects. Clearly, wave velocities and hydraulic properties share their general sensitivity to the structure of pores and fractures. Yet, the specific aspects of the structure that affect these two properties might differ considerably. Consequently, their information content may be complementary rather than redundant. Laboratory experiments permit to elucidate the structural evolution associated with brittle faulting and determine the sensitivity of hydraulic and dynamic elastic properties to bulk elastic and inelastic deformation and the relation among them. In this laboratory study we continuously monitored hydraulic properties and elastic wave velocities of sandstone samples during conventional triaxial compression. We suppressed a potentially biasing contribution of hydro-mechanical coupling by maintaining the samples at effectively drained conditions, that is, the pore-fluid pressure remained constant in the samples throughout the deformation experiments. This approach serves two linked objectives, (1) constraining the interrelation between the evolution in structure and hydraulic properties and (2) investigating the correlation between wave velocities and hydraulic properties during brittle faulting. These objectives address the hold-up of substantial modelling of, for example, aftershock activity and induced seismicity, imposed by the lack of fundamental observations regarding the deformation-related evolution in hydraulic properties and how it affects the space-time characteristics of the pore-pressure field surrounding a failure event and thus the potential spread of failure. Our approach intentionally contrasts and extends previous studies in several respects. In the suite of compression experiments, applied confining pressures cover the range from below to above a pressure level identified as critical for crack closure to control the state of pre-existing cracks. Of the main three hydraulic methods, steady-state flow tests, pulse tests, and oscillatory tests, we selected the latter for their advantages regarding continuous high-resolution monitoring of hydraulic properties during deformation (Fischer 1992; Zhang et al. 1994; Mitchell & Faulkner 2008). Studies on correlations between hydraulic properties and velocities of elastic waves that were restricted to hydrostatic loading (Benson et al. 2005; Fortin et al. 2005) are extended to conventional triaxial compression. 2 MATERIAL AND METHODS The experiments were performed on samples of Wilkeson sandstone (WS) to exploit the extensive mechanical characterization performed by Duda & Renner (2013). In particular, the previously elaborated requirements for drained experiments and the constraints on deformation regimes build the foundation for the current study. We report here the relevant specifics of the used apparatus but also refer to its extensive description in Duda & Renner (2013). 2.1 Experimental apparatus and procedure 2.1.1 Setup for triaxial deformation experiments Conventional triaxial compression tests were performed at constant confining pressure pc and nominally constant pore-fluid pressure pf. Here, we refer to their difference as effective pressure, that is, peff = pc − pf, and more generally to any difference between a normal stress and fluid pressure as effective stress. We refrain from introducing a weighting parameter differing from 1 in these differences for two reasons: (1) The entire suite of experiments was performed at the same nominal fluid pressure and thus all pressure and stress levels would simply be shifted by the same constant value; (2) brittle strength seems to obey a scaling with these unweighted differences (see review by Paterson & Wong 2005). The pore-pressure and confining-pressure systems were filled with distilled water and hydraulic oil, respectively. Samples were jacketed by rubber tubes to prevent a connection between the two fluid systems. Porous spacers at the end faces of the samples guaranteed a spatially uniform fluid pressure (Fig. 1). The upper axial loading piston had a central bore to which an external pressure transducer (strain-gauge based, range 100 MPa, accuracy 0.1 per cent) was connected. A flexible high-pressure capillary tube connected the bores of the plug at the sample bottom and of the bottom closure of the vessel enabling connection of a second, identical pressure transducer outside of the vessel that continuously recorded the pore pressure at the lower sample end at hydrostatic conditions (i.e. separated bottom plug and bottom closure) and during deviatoric loading. Figure 1. View largeDownload slide Experimental setup for continuous determination of hydraulic properties and ultrasound measurements during triaxial compression experiments (modified after Duda & Renner, 2013). Distilled water and hydraulic oil were used as pore fluid (blue) and as confining medium (green), respectively. Figure 1. View largeDownload slide Experimental setup for continuous determination of hydraulic properties and ultrasound measurements during triaxial compression experiments (modified after Duda & Renner, 2013). Distilled water and hydraulic oil were used as pore fluid (blue) and as confining medium (green), respectively. Axial piston movement was controlled using a couple of external inductive displacement transducers (range 20 mm, accuracy 0.1 per cent). Axial load was measured with an external load cell (strain gauge based, range 5000 kN, accuracy 0.01 per cent). Confining pressure and pore pressure were applied by servohydraulically controlled pressure intensifiers equipped with inductive displacement transducers (range 300 mm, accuracy 0.1 per cent). The pore-pressure intensifier served as pore-volumometer allowing us to determine pore-volumetric strain ψ = −Δϕcon, that is, the change in connected porosity. Axial strain εax and pore-volumetric strain ψ were determined correcting the corresponding piston displacements for system characteristics as derived from tests on a steel dummy replacing the actual sample. Stress difference Δσ was calculated from loads in excess of the load at hit-point divided by the initial cross-section of the samples, an appropriate approximation for the moderate axial strains achieved. The tangent modulus or apparent Young’s modulus Eapp = dΔσ/dεax representing the local derivative of axial stress with respect to axial strain was calculated for strain increments of 1/100 of axial strain at peak stress. Following the geoscientific convention, compressive stresses and strains are considered positive. 2.1.2 Ultrasonic measurements Two identical, piezo-ceramic ultrasound P-wave transducers (1 MHz centre frequency, 38.1 mm diameter) acting as source and receiver were placed above and below the vessel, respectively, in line with the axial loading pistons (Fig. 1). The transducers were embedded in hollow steel cylinders to prevent them from carrying load. A spring and a coupling medium (graphite lubricant) were used to provide reproducible and constant coupling conditions between transducers and assembly parts. A rectangular signal with an amplitude of 400 V produced by an external waveform generator (Panametrics EPOCH 4B) activated the source to emit 60 mechanical pulses per second into the rock sample. The receiver converted the transmitted elastic wave to an electrical signal. Every 600 s, stacks of 1000 records each covering 400 μs following the excitation by the waveform generator were stored within 17 s with a sampling interval of 8 × 10−9 s using a digital oscilloscope (Tektronix TDS5034B). The first arrival of each signal corresponds to the direct P-wave propagating in axial direction through sample and assembly. Arrival times were determined using a semi-automatic method based on the similarity of the recorded signals. A reference-arrival time was manually picked using the waveform with the highest signal-to-noise ratio of a specific experiment. Arrival times of all other recordings were then determined accounting for the time shift gained from two successive cross correlations performed between consecutively stored signals. The first cross correlation of full time series provided an estimate for the group travel time of the entire coda and allowed for automatically constraining the relevant time interval for the second cross correlation restricted to the first-break wavelet, which provided the finally used time shift. This procedure mitigates some of the ambiguity problems associated with manual picking of signals with variable frequency content and detects subtle changes in arrival time from one transmission to the next. Arrival times were corrected for the travel time in the assembly parts to calculate effective travel times for the samples. The correction values were determined from triaxial tests on a steel sample at the same conditions as the tests on the rock samples. The analysis of the calibration accounted for the contribution of changes in length and P-wave velocity of the steel cylinder with increasing pressure and axial stress (Gerlich & Hart 1984). Velocities were calculated by dividing the current sample length by the determined travel times. Uncertainties in sample geometry, displacement and force measurements, as well as accuracies of Young’s modulus of steel (accuracy 5 per cent), travel and dead time (accuracy 0.3 per cent) lead to absolute errors in P-wave velocity values of less than 110 m s−1. Differences between two determined velocity values (be it at ambient pressure or under triaxial stress) can be resolved to 10 m s−1. 2.1.3 Determination of hydraulic properties Hydraulic properties of the samples were determined using the oscillatory pore-pressure method (e.g. Turner 1958; Stewart et al. 1961; Bennion & Goss 1977). The ends of the saturated porous samples were separately connected to an upstream (composed of upper axial piston, pressure intensifier, and pressure transducer) and a downstream reservoir (composed of bottom plug, capillary tube, bottom closure, and pressure transducer, Fig. 1). Sinusoidal pore-pressure variations were initiated in the upstream reservoir. Amplitude ratio and phase shift between the pressure oscillations in the two reservoirs were derived by Fourier analysis applied to time windows with a duration of four oscillation periods that were successively shifted by 1 s. The standard deviation of amplitude ratio and phase shift gained from 10 successive analyses serves as an uncertainty measure. Response characteristics were inverted to effective hydraulic parameters, that is, permeability k, specific storage capacity s, and hydraulic diffusivity Dhyd = k/(μs), where μ denotes pore-fluid viscosity. Our inversion relies on the analytical solution of the one-dimensional diffusion equation for harmonic flow through an isotropic, homogeneous medium (e.g. Kranz et al. 1990; Fischer 1992; Bernabé et al. 2006), as is true for all currently practiced transient methods for the determination of hydraulic properties (e.g. Brace et al. 1968; Neuzil et al. 1981; Song et al. 2004; Song & Renner 2006). Specific storage capacity has two contributions, the compressibility of the pore-fluid cf and the pore compressibility in response to pore-pressure variations, cpp, following the notation of Zimmerman et al. (1986) for compressibilities:   \begin{equation} s = \phi _\mathrm{con} c_\mathrm{f} + c_\mathrm{pp} . \end{equation} (1)We used cf for water (Wagner 2009) to quantitatively compare our results to the contribution of the pore-fluid compressibility. Apart from the fundamental contemplation of the validity of the simplified model underlying the employed one-dimensional diffusion equation, the uncertainty of hydraulic parameters also strongly depends on the peculiarities of the analytical solution space. Oscillation period and the size of the downstream reservoir are the dominant factors controlling where test results plot in the solution space (e.g. Faulkner & Rutter 2000; Bernabé et al. 2006; Song & Renner 2006). Our tests were carried out with a downstream-reservoir whose storage capacity is only (1.2 ± 0.1) × 10−15 m3 Pa−1 corresponding to about half the samples’ storage capacity and thus providing significant sensitivity for storage-capacity determination. Before commencing triaxial compression we performed oscillatory tests at a range of periods and amplitudes to optimize the parameters of the harmonic excitation subsequently applied during the axial compression regarding avoidance of the peculiar tail of the solution space with poor resolution of hydraulic parameters. The absolute uncertainty of a single value for each of the hydraulic properties amounts to 10 per cent, yet relative variations in permeability, specific storage capacity, and hydraulic diffusivity as low as 4 per cent, 2 per cent and 2 per cent, respectively, can be resolved in a single experiment. 2.2 Sample material WS forms part of the Pudged Group of the Mid Miocene (Gard 1968); our blocks originate from an outcrop in Pierce County, Washington (USA). The sandstone is composed of approximately 55  per cent quartz, 15  per cent plagioclase, 15  per cent sericite, 10  per cent muscovite, 4  per cent opaque minerals, and 1  per cent microcline. The poorly sorted grains exhibit an average size of about 200 μm. The grain-supported fabric features convex-concave grain and phase boundaries (Fig. 2), which indicate an advanced diagenesis of the formation. Plagioclase grains are partly altered to sericite and saussurite. Some quartz grains contain fluid inclusions and show undulatory extinction as well as recrystallization features. Muscovite grains are partly kinked. Figure 2. View largeDownload slide (a) Optical micrograph (crossed polarizers) and (b) scanning-electron microscope image of a polished thin section of intact Wilkeson sst. Long edges of images are parallel to the symmetry axis of the cylindrical test samples. Figure 2. View largeDownload slide (a) Optical micrograph (crossed polarizers) and (b) scanning-electron microscope image of a polished thin section of intact Wilkeson sst. Long edges of images are parallel to the symmetry axis of the cylindrical test samples. Measurements on the block from which samples were cored indicate an anisotropy of 10 to 15 per cent in ultrasonic P-wave velocities at ambient conditions despite the lack of visible layering. Five cylindrical samples with a diameter of 30 mm were diamond-drilled parallel to the direction of the highest velocity and their end faces were ground square to a final length of 75.00 ± 0.02 mm, that is, the sample geometry complied with a length-to-diameter ratio of 2.5:1 that limits the effect of friction at the interface with the steel pistons on the state of stress in the sample centre (Paterson & Wong 2005). Preparation was conducted with water as coolant. Fundamental physical properties were determined on all samples at ambient conditions to evaluate the homogeneity of the suite considered for triaxial testing. Bulk density ρ of samples was calculated from their masses after drying at 60 °C and geometrical volume determination. The average mineral density $$\bar{\rho }_{{\rm min}} = 2623 \pm 27\,{\rm kg \, m}^{-3}$$ was gained from pycnometer measurements on powder produced by crushing and grinding pieces of the sandstone permitting to derive total porosity according to $$\phi _{{\rm tot}}= 1 - (\rho /\bar{\rho }_{{\rm min}})$$. Connected porosity was calculated from the increase in masses after evacuation and subsequent saturation of samples with distilled water (for more details see Duda & Renner 2013). 2.2.1 Physical properties at ambient conditions The measurements of the basic physical properties of the cylindrical samples confirms the general homogeneity of the sandstone block but the resolution of the individual properties indicates some variability (Table 1). Total and connected porosity differ by about 0.5  per cent, yet, this difference is barely significant considering absolute uncertainties. Dry samples exhibit ratios of P- to S-wave velocities close to 1.5. Upon saturation, P-wave velocities increase by almost 30  per cent. The properties of the samples used in this study deviate only slightly from the ones used by Duda & Renner (2013) that were prepared from a different block. The previously used samples were slightly less porous (8.4  per cent connected porosity compared to 8.9  per cent), but this difference will not affect the validity of results from Duda & Renner (2013) relevant to and for this study. Table 1. Basic physical properties of tested samples as determined at ambient conditions. Sample  ρ (kg m−3)  ϕtot (per cent)  ϕcon (per cent)  vP,dry (m s−1)  vS,dry (m s−1)  vP,sat (m s−1)  WS-4 K  2375 ± 8  9.5 ± 0.7  8.9 ± 1.0  2797 ± 83  1835 ± 219  3974 ± 153  WS-4 L  2364 ± 8  9.9 ± 0.7  9.4 ± 1.0  2687 ± 80  1827 ± 229  3949 ± 150  WS-4 N  2381 ± 8  9.2 ± 0.7  8.9 ± 1.0  2705 ± 80  1860 ± 222  4034 ± 156  WS-4 O  2383 ± 8  9.2 ± 0.7  8.9 ± 1.0  2776 ± 83  1903 ± 222  3937 ± 148  WS-4 R  2370 ± 8  9.7 ± 0.7  9.0 ± 1.0  2738 ± 82  1871 ± 224  3938 ± 150  mean ±SD  2378 ± 8  9.5 ± 0.3  8.9 ± 0.2  2738 ± 46  1860 ± 30  3949 ± 41  Sample  ρ (kg m−3)  ϕtot (per cent)  ϕcon (per cent)  vP,dry (m s−1)  vS,dry (m s−1)  vP,sat (m s−1)  WS-4 K  2375 ± 8  9.5 ± 0.7  8.9 ± 1.0  2797 ± 83  1835 ± 219  3974 ± 153  WS-4 L  2364 ± 8  9.9 ± 0.7  9.4 ± 1.0  2687 ± 80  1827 ± 229  3949 ± 150  WS-4 N  2381 ± 8  9.2 ± 0.7  8.9 ± 1.0  2705 ± 80  1860 ± 222  4034 ± 156  WS-4 O  2383 ± 8  9.2 ± 0.7  8.9 ± 1.0  2776 ± 83  1903 ± 222  3937 ± 148  WS-4 R  2370 ± 8  9.7 ± 0.7  9.0 ± 1.0  2738 ± 82  1871 ± 224  3938 ± 150  mean ±SD  2378 ± 8  9.5 ± 0.3  8.9 ± 0.2  2738 ± 46  1860 ± 30  3949 ± 41  ρ: bulk density; ϕtot: total porosity; ϕcon: connected porosity; vP,dry, vP,sat: P-wave velocity of dry and saturated sample; vS,dry: S-wave velocity of dry sample. Quoted uncertainties reflect accuracies of the measurements. The resolution of measurements relevant for assessing the significance of differences between two bulk densities, total porosities, connected porosites, P- and S-wave velocities of dry sample, and P-wave velocities of saturated samples amount to 0.8 kg m−3, 0.3  per cent, 0.1  per cent, 10 m s−1, 46 m s−1, and 21 m s−1, respectively. View Large 2.2.2 Physical properties at effective hydrostatic pressures up to 160 MPa Axial, pore- and bulk-volumetric strain, ultrasonic P-wave velocity, and hydraulic properties were determined for one sample during hydrostatic loading at effective pressures up to 160 MPa to constrain the crack-closure pressure, basic elastic parameters, and the pressure sensitivity of hydraulic properties (Fig. 3). Confining pressure was increased in steps of 10 MPa starting from 40 MPa while the pore pressure had a nominally constant value of 30 MPa but was continuously oscillated at the upstream reservoir with an amplitude of 1 MPa and a period of 20 s. After each increase in confining pressure we waited until pore pressure was re-equilibrated within the sample as indicated by a ceasing movement of the volumometer piston to ensure effectively drained conditions. Changes in sample length and corresponding axial strain were derived from monitoring hit-points. Bulk-volumetric strain was estimated from the axial strain assuming isotropy, that is, θ = 3εax. Three pressure settings were revisited on the way back to initial conditions to test for permanent changes induced by the pressurization. Figure 3. View largeDownload slide (a) Bulk-volumetric strain θ = 3εax (black circles) and pore-volumetric strain $$\psi = \Delta \psi + \theta (p_\mathrm{eff}=10\,{\rm MPa})$$ (red triangles), (b) P-wave velocity vP,sat, (c) permeability k, (d) specific storage capacity s, and (e) hydraulic diffusivity Dhyd as a function of effective confining pressure peff during hydrostatic deformation of Wilkeson sst up to 160 MPa. In (a), pore-volumetric strains that were determined starting from $$p_\mathrm{eff}=10\,{\rm MPa}$$ were shifted by the bulk-volumetric strain at 10 MPa assuming isotropic deformation. The pressure range highlighted in grey indicates the crack-closure pressure (see the text for discussion). The dashed black line in (d) represents the contribution from the storage capacity of water alone (eq. 1). Figure 3. View largeDownload slide (a) Bulk-volumetric strain θ = 3εax (black circles) and pore-volumetric strain $$\psi = \Delta \psi + \theta (p_\mathrm{eff}=10\,{\rm MPa})$$ (red triangles), (b) P-wave velocity vP,sat, (c) permeability k, (d) specific storage capacity s, and (e) hydraulic diffusivity Dhyd as a function of effective confining pressure peff during hydrostatic deformation of Wilkeson sst up to 160 MPa. In (a), pore-volumetric strains that were determined starting from $$p_\mathrm{eff}=10\,{\rm MPa}$$ were shifted by the bulk-volumetric strain at 10 MPa assuming isotropic deformation. The pressure range highlighted in grey indicates the crack-closure pressure (see the text for discussion). The dashed black line in (d) represents the contribution from the storage capacity of water alone (eq. 1). Bulk- and pore-volumetric strain, and P-wave velocity exhibit similar relations to increasing pressure (Fig. 3). Strains and velocity exhibit steep nonlinear increases with increasing effective pressure that give way to modest linear increases above about 40 to 50 MPa. The transitional behaviour is commonly attributed to the successive closure of microcracks (Walsh 1966; Mavko & Nur 1978) - the higher the pressure the lower the aspect ratio ξ (ratio between width and length) of closing cracks. The pressure at the beginning of a linear relation between bulk-volumetric strain and confining pressure is commonly referred to as the critical crack-closure pressure pcc. Non-porous, but cracked rocks exhibit a velocity plateau once this critical pressure is exceeded (e.g. Mavko & Nur 1978). The continuous velocity increase exhibited by the tested porous sandstone demonstrates that microstructural changes, for example, modifications of grain contacts, continue to occur up to the highest explored pressure. David & Zimmerman (2012) associated such behaviour with the distribution function of aspect ratios. By somewhat generalizing its original meaning, we will use the term crack-closure pressure to address the observed change in sensitivity to pressure probably associated with the presence of a dominant aspect ratio of microcracks perceptible to closure. During hydrostatic loading permeability and specific storage capacity vary from 5 to 12 × 10 − 18 m2 and 2 to 6 × 10 − 11 Pa − 1, respectively, and both exhibit a peculiar difference between the values determined at truly hydrostatic conditions and at a stress difference as little as 1 MPa applied to determine current sample length from the hit-point (Figs 3c and d). At axial load, values of k and s are smaller than at hydrostatic conditions. These differences are the largest below the closure pressure ( ∼ 50 MPa) up to which k and s at axial load tend to increase whereas values determined at hydrostatic conditions more systematically decrease with increasing effective pressure throughout. Almost all of the determined storage capacity values correspond to apparent porosities exceeding, albeit only slightly, the nominal connected porosity determined at ambient conditions (Fig. 3d) underlining that the pore space dominating storage is rather undeformable as also indicated by the modest pressure dependence of s (Fig. 3d). Hydraulic diffusivity ranges between 1.1 and 1.8 × 10 − 4 m2 s − 1 and exhibits a relation between values determined at hydrostatic conditions and at small load that is opposite to the one for k and s (Fig. 3e). The reproducibility during pressure decrease differs for the various properties (Fig. 3). The match between volumetric strains during pressurization and depressurization excludes significant inelastic deformation during the performed cycle of hydrostatic loading but a slight permanent compaction (reduction in bulk volume by loss of porosity of a few tenths of a percent). Permeability decreases by almost 20  per cent in accord with the indicated loss in porosity. Specific storage capacity is however reproducible. These observations indicate permanent closure of voids to which permeability is sensitive but storage capacity is not. The slope of the linear increase of the approximated bulk-volumetric strain with effective pressure (Fig. 3a) corresponds to a lower bound for the quasi-static drained bulk modulus of $$K_\mathrm{d}=c_\mathrm{bc}^{-1}=15.6 \pm 0.6\,{\rm GPa}$$ (Table 3) assuming isotropy. The pronounced pressure dependence of strains and velocity demonstrates that the deformation is dominated by crack closure. Previous work on cracked samples demonstrated that anisotropy diminishes with crack closure (e.g. Zang et al. 1989; Wang 2002) and thus the degree of anisotropy indicated by the velocity measurements at ambient conditions likely does not hold for the samples at test conditions. This estimate constitutes an upper bound for the true value if the anisotropy in P-wave velocity observed at ambient conditions holds also for static elastic parameters at elevated pressure. Even this upper bound is fairly low compared to typical bulk moduli of rock-forming minerals likely due to the still significant total porosity of about 7–8  per cent at the explored effective pressures, estimated by reducing the total porosity determined at ambient pressure by the change in porosity associated with crack closure gained from the intercept of pore-volumetric strain versus effective pressure, that is, $$\psi _\mathrm{crack} \approx 0.9\,{\rm \hbox{\,per\,cent}}$$ (Fig. 3a). 2.3 Experimental conditions We chose effective pressures of 10, 20, 50 and 100 MPa for the triaxial compression experiments to cover the full range from below to above the identified crack-closure pressure (Fig. 3) and from brittle faulting to cataclastic flow as deduced from our previous experiments (Duda & Renner 2013). The hydraulic diffusivity of the samples determined at hydrostatic conditions just before the triaxial deformation confirmed the validity of our previously determined requirements for drained conditions (Duda & Renner 2013). Strain rates between 1 × 10−7 and 3 × 10−7 s−1 (Table 2) were thus imposed that fall significantly below the estimated critical strain rates of 4 × 10−6 to 1 × 10−5 s−1. The nominal pore-fluid pressure of 30 MPa was continuously oscillated at the upstream end with an amplitude of 1 MPa, that is, a modest perturbation of about 3 per cent, and a period of 20 s. Oscillation characteristics were inverted to hydraulic properties every 10 s, corresponding to axial strain increments of 1 × 10−6 to 3 × 10−6 for the imposed strain rates. Stacked seismograms were stored for axial strain increments of about 6 × 10−5 to 1.8 × 10−4 . Table 2. Conditions of performed deformation experiments and hydraulic properties as determined at hydrostatic conditions before deviatoric loading was commenced with the indicated strain rates $$\dot{\epsilon }$$ in comparison to properties of sample WS-4 R exclusively tested at hydrostatic conditions between 10 and 160 MPa effective pressure (see Fig. 3). peff (MPa)  Sample  k (m2)  s (Pa−1)  Dhyd (m2 s−1)  vP (m s−1)  $$\dot{\epsilon }$$ (s−1)  10  WS-4 K  (5.0 ± 0.3) × 10 − 18   (4.3 ± 0.1) × 10 − 11   (9.1 ± 0.4) × 10 − 5   4283 ± 11   1 × 10−7     WS-4 R  (1.0 ± 0.1) × 10 − 17   (5.6 ± 0.1) × 10 − 11   (1.4 ± 0.1) × 10 − 4   4240 ± 11   –  20  WS-4 O  (3.8 ± 0.3) × 10 − 18   (4.1 ± 0.2) × 10 − 11   (7.2 ± 0.2) × 10 − 5   4398 ± 11   1 × 10−7     WS-4 R  (1.1 ± 0.1) × 10 − 17   (5.3 ± 0.1) × 10 − 11   (1.5 ± 0.1) × 10 − 4   4400 ± 11   –  50  WS-4 L  (1.2 ± 0.1) × 10 − 17   (5.1 ± 0.1) × 10 − 11   (1.8 ± 0.1) × 10 − 4   4562 ± 11   3 × 10−7     WS-4 R  (1.1 ± 0.1) × 10 − 17   (5.6 ± 0.1) × 10 − 11   (1.5 ± 0.1) × 10 − 4   4620 ± 11   –  100  WS-4 N  (4.1 ± 0.3) × 10 − 18   (4.3 ± 0.2) × 10 − 11   (7.3 ± 0.6) × 10 − 5   4696 ± 11   3 × 10−7     WS-4 R  (8.4 ± 0.1) × 10 − 18   (4.6 ± 0.1) × 10 − 11   (1.4 ± 0.6) × 10 − 4   4710 ± 11   –  peff (MPa)  Sample  k (m2)  s (Pa−1)  Dhyd (m2 s−1)  vP (m s−1)  $$\dot{\epsilon }$$ (s−1)  10  WS-4 K  (5.0 ± 0.3) × 10 − 18   (4.3 ± 0.1) × 10 − 11   (9.1 ± 0.4) × 10 − 5   4283 ± 11   1 × 10−7     WS-4 R  (1.0 ± 0.1) × 10 − 17   (5.6 ± 0.1) × 10 − 11   (1.4 ± 0.1) × 10 − 4   4240 ± 11   –  20  WS-4 O  (3.8 ± 0.3) × 10 − 18   (4.1 ± 0.2) × 10 − 11   (7.2 ± 0.2) × 10 − 5   4398 ± 11   1 × 10−7     WS-4 R  (1.1 ± 0.1) × 10 − 17   (5.3 ± 0.1) × 10 − 11   (1.5 ± 0.1) × 10 − 4   4400 ± 11   –  50  WS-4 L  (1.2 ± 0.1) × 10 − 17   (5.1 ± 0.1) × 10 − 11   (1.8 ± 0.1) × 10 − 4   4562 ± 11   3 × 10−7     WS-4 R  (1.1 ± 0.1) × 10 − 17   (5.6 ± 0.1) × 10 − 11   (1.5 ± 0.1) × 10 − 4   4620 ± 11   –  100  WS-4 N  (4.1 ± 0.3) × 10 − 18   (4.3 ± 0.2) × 10 − 11   (7.3 ± 0.6) × 10 − 5   4696 ± 11   3 × 10−7     WS-4 R  (8.4 ± 0.1) × 10 − 18   (4.6 ± 0.1) × 10 − 11   (1.4 ± 0.6) × 10 − 4   4710 ± 11   –  peff: effective confining pressure; k: permeability; s: specific storage capacity; Dhyd: hydraulic diffusivity; vP: P-wave velocity. Individual values are amended by the resolution relevant for comparison of the results for the tested samples. The absolute accuracies of the measurements are quoted in the text. View Large 3 RESULTS The homogeneity of the used block, already indicated by the close match between the physical properties of the prepared samples determined at ambient pressure (Table 1), was further confirmed by the measurements performed after pressurization of the samples and before commencing axial deformation. In detail, we find, however, the properties of the two samples with the highest total porosity, WS-4 R tested at hydrostatic conditions and WS-4 L axially deformed at 50 MPa effective pressure, to deviate slightly from those of the other three samples (Tables 1 and 2). Their permeability and diffusivity are initially about twice as high as those of the other samples and their ultrasonic P-wave velocities are the lowest of the sample suite. Yet, these differences are small enough to safely attribute the differences in results of the four deformation tests to the differences in imposed effective pressure rather than to sample variability. 3.1 Deformation characteristics The stress-strain curves are characteristic for the pressure-induced transition from dilative brittle failure to compactive cataclastic flow (Fig. 4) as confirmed by the observed macroscopic failure modes. Here, we use ‘compaction’ and ‘compactive’ or ‘dilation’ and ‘dilative’ to indicate a relative decrease or increase in current sample volume, respectively. Furthermore, ‘low’ and ‘high’ pressure will be meant relative to the determined crack-closure pressure of 40 to 50 MPa. Up to effective pressures of 50 MPa, initial compaction reverses into significant dilation accompanying a drop in axial stress to constant, pressure-dependent residual-stress levels (Fig. 4a). The magnitude of the drop after peak stress is the lower the higher the effective pressure. Likewise, the degree of dilation systematically decreases with increasing effective pressure; the sample tested at 100 MPa deforms at almost constant pore volume beyond the modest peak in stress (Fig. 4b). The stress sensitivity of the initial pore compaction seems unaffected by pressure whereas the slope of axial stress-axial strain curves systematically increase with pressure. The latter are never really linear but exhibit an S-shape as commonly observed for porous rocks (e.g. Duda & Renner 2013). Figure 4. View largeDownload slide Axial stress difference Δσ as a function of (a) axial strain εax and (b) pore-volumetric strain ψ during triaxial compression experiments on saturated Wilkeson sst at indicated effective confining pressures. Diamonds and squares give maxima in P-wave velocity and onset of dilation as deduced from pore-volumetric strain, respectively. Figure 4. View largeDownload slide Axial stress difference Δσ as a function of (a) axial strain εax and (b) pore-volumetric strain ψ during triaxial compression experiments on saturated Wilkeson sst at indicated effective confining pressures. Diamonds and squares give maxima in P-wave velocity and onset of dilation as deduced from pore-volumetric strain, respectively. The morphological features of deformed samples retrieved from the vessel almost fully cover the transition from a single shear fault to pervasive cataclasis barely localized on macroscopic scale (see for example Hadizadeh & Rutter 1983). The angle of the fault to the cylinder axis decreases from about 70° to about 50° and the fault zone widens from about 2 to almost 10 mm with increasing effective pressure up to 50 MPa; the sample tested at peff = 100 MPa exhibits only a hint of a localized fault in a pervasively deformed zone that covers half of the sample. These observations on fault morphology match well with the characteristics of the stress-strain curves exhibiting a transition from significant stress drop to modest softening after peak stress with increasing effective pressure (Fig. 4). 3.2 Evolution of physical properties during triaxial deformation 3.2.1 Ultrasound P-wave velocities During triaxial compression, P-wave velocity evolves qualitatively similar for all investigated effective pressures but quantitative differences are significant (Fig. 5a). The evolution with increasing strain resembles inverted parabolas: P-wave velocity increases significantly right upon the onset of deviatoric loading and subsequently exhibits a maximum well before peak stress. The higher the pressure is the higher the velocities are but the less relative changes they exhibit during the course of an experiment. The axial strains at the maxima in velocity differ only slightly between the four experiments and thus fall increasingly short of the onset of dilation as determined from pore-volumetric strains with increasing effective pressure. At an effective pressure of 100 MPa, the strain at the onset of dilation is more than twice the one at the velocity maximum. The stress drop likely associated with localized brittle failure is accompanied by a distinct drop in velocity whose magnitude decreases with increasing effective pressure. At residual stresses, P-wave velocity falls below its initial value determined on intact samples at hydrostatic conditions for all effective pressures but 10 MPa. Figure 5. View largeDownload slide P-wave velocity vP as a function of (a) axial strain εax, (b) deviatoric radial strain with respect to its value at maximum P-wave velocity, εrad − εrad(vP,max), (c) effective mean stress σm,eff = Δσax/3 + peff, and (d) effective stress ratio λ = (Δσax + peff)/peff during (c) hydrostatic (circles) and deviatoric (lines) loading at indicated effective confining pressures. In (b,d) P-wave velocities were normalized by their maxima vP, max and by the values measured at hydrostatic conditions before triaxial loading was commenced (vP,0), respectively. Squares and diamonds indicate maxima in P-wave velocity (amended by uncertainty range) and onset of dilation as deduced from pore-volumetric strain, respectively. The arrow in (b) indicates the progression of deformation. Figure 5. View largeDownload slide P-wave velocity vP as a function of (a) axial strain εax, (b) deviatoric radial strain with respect to its value at maximum P-wave velocity, εrad − εrad(vP,max), (c) effective mean stress σm,eff = Δσax/3 + peff, and (d) effective stress ratio λ = (Δσax + peff)/peff during (c) hydrostatic (circles) and deviatoric (lines) loading at indicated effective confining pressures. In (b,d) P-wave velocities were normalized by their maxima vP, max and by the values measured at hydrostatic conditions before triaxial loading was commenced (vP,0), respectively. Squares and diamonds indicate maxima in P-wave velocity (amended by uncertainty range) and onset of dilation as deduced from pore-volumetric strain, respectively. The arrow in (b) indicates the progression of deformation. The relation between the P-wave velocities recorded for the four triaxial compression experiments and the ones observed for hydrostatic conditions exhibit a peculiar change with increasing effective mean stress, σm,eff = σm − pf = Δσax/3 + peff (Fig. 5c). Below the crack-closure pressure (i.e. at peff = 10 and 20 MPa) velocities recorded in the triaxial experiments immediately exceed their hydrostatic counterparts until the maximum is reached that closely matches the hydrostatic reference value. Above the crack-closure pressure, velocities at triaxial conditions fall persistently short of the ones found for hydrostatic conditions. 3.2.2 Hydraulic properties The reported hydraulic properties refer to hypothetical, equivalent, homogeneous and isotropic media that were to give the same phase shift and amplitude ratio as the observed ones. Although samples obviously become heterogeneous during faulting (and likely develop anisotropy that even changes during the course of deformation), such an equivalent medium is found, that is, observed phase shift and amplitude ratio actually fall into the solution space of the diffusion equation. We can determine effective hydraulic properties, but for two strain sequences. The oscillation characteristics observed for the samples tested at effective pressures of 10 MPa and 50 MPa temporarily fall outside of the analytical solution space during the stress drop and during the softening phase, respectively (Fig. 6). Figure 6. View largeDownload slide Graphical representation of the solution space for the one-dimensional pressure-diffusion equation in the domain of phase shift φ and amplitude ratio δpp. The grid is spanned by isolines of dimensionless storage capacity ξ and dimensionless permeability η (increasing in the indicated directions). The coloured lines represent phase shift-amplitude ratio pairs determined during triaxial deformation of Wilkeson sst at indicated effective confining pressures (the black arrow indicates deformation progress). Figure 6. View largeDownload slide Graphical representation of the solution space for the one-dimensional pressure-diffusion equation in the domain of phase shift φ and amplitude ratio δpp. The grid is spanned by isolines of dimensionless storage capacity ξ and dimensionless permeability η (increasing in the indicated directions). The coloured lines represent phase shift-amplitude ratio pairs determined during triaxial deformation of Wilkeson sst at indicated effective confining pressures (the black arrow indicates deformation progress). Our a priori analysis for the drainage conditions of the experiments remains valid in the light of the a posteriori knowledge of the range in diffusivity determined during triaxial compression. We estimate a penetration depth of $$\,r_\mathrm{p}\sim \sqrt{D_\mathrm{hyd}T}$$ of about half the sample length (the longest distance to a reservoir), even for the lowest observed diffusivity values of 5 × 10−5 m2 s−1 on the timescale of the imposed oscillation period of T = 20 s. Also, we did not observe any of the characteristic features pointing to a potential lack in drainage as described in Duda & Renner (2013). We initially restrict to comparing the evolution of physical properties of the experiments performed at 20 and 100 MPa effective pressure (Fig. 7) exemplifying the two distinct failure regimes covered by our experiments, localized brittle failure and incipient cataclastic flow. The relative changes in hydraulic properties during the course of the deformation are larger for lower effective pressure, for example, one and a half orders of magnitude increase in permeability at 20 MPa compared to one order of magnitude increase at 100 MPa. The variations in specific storage capacity and hydraulic diffusivity cover smaller ranges than the ones in permeability. The moderate variability of diffusivity is an expression of the general tendency for permeability and specific storage capacity to covary. Figure 7. View largeDownload slide (a,b) Permeability k, (c,d) specific storage capacity s (in comparison to the contribution of the compressible pore fluid alone as indicated by the horizontal dashed lines calculated employing the connected porosity ϕcon at ambient conditions), and (e,f) hydraulic diffusivity Dhyd during triaxial deformation of Wilkeson sst at 20 and 100 MPa effective confining pressure as a function of axial strain εax (left) and pore-volumetric strain ψ (right). The grey curves indicate the ranges of absolute uncertainty. Where these are not visible, uncertainties do not exceed the thickness of the coloured curves. Squares, diamonds, and triangles indicate maxima in P-wave velocity, onset of dilation as deduced from pore-volumetric strain, and peak stress, respectively. Figure 7. View largeDownload slide (a,b) Permeability k, (c,d) specific storage capacity s (in comparison to the contribution of the compressible pore fluid alone as indicated by the horizontal dashed lines calculated employing the connected porosity ϕcon at ambient conditions), and (e,f) hydraulic diffusivity Dhyd during triaxial deformation of Wilkeson sst at 20 and 100 MPa effective confining pressure as a function of axial strain εax (left) and pore-volumetric strain ψ (right). The grey curves indicate the ranges of absolute uncertainty. Where these are not visible, uncertainties do not exceed the thickness of the coloured curves. Squares, diamonds, and triangles indicate maxima in P-wave velocity, onset of dilation as deduced from pore-volumetric strain, and peak stress, respectively. The evolution of the hydraulic properties changes notably when the maximum in P-wave velocity occurs and when dilation sets in (Fig. 7). At low effective pressure, the two conditions are simultaneously reached and mark the onset of a significant increase in all hydraulic properties; at high effective pressure, the maximum in P-wave velocity occurs first marking the minimum in hydraulic properties while the onset of dilation coincides with an inflection point of their subsequent increase with axial strain. The stress drop associated with macroscopic brittle failure at low effective pressure is accompanied by a sudden, counterintuitive, transient decrease in permeability and specific storage capacity, yet in combination still resulting in an increase in diffusivity. Hydraulic properties seem to approach fairly constant values when frictional sliding on the developed fault prevails at residual stresses. Permeability and specific storage capacity reach their highest levels at residual stress that however correspond to a diffusivity that falls below values reached earlier during triaxial deformation. During the slight post-peak softening associated with a continuous increase in pore volume observed for cataclastic flow at high effective pressure, permeability stays constant and specific storage capacity exhibits a modest decrease resulting in an increase in diffusivity. Over the course of a deformation experiment, the variations of the three hydraulic properties, permeability, specific storage capacity, and hydraulic diffusivity, significantly exceed the changes due to hydrostatic loading at comparable effective mean stresses (Figs 8a, c and e). Notably, the initial trends with mean stress do not match the ones with pressure in particular for the two experiments performed at 10 and 20 MPa, that is, below crack-closure pressure. The immediate decrease-contrasting the counterintuitive increase of the property values with increasing hydrostatic pressure below the crack-closure pressure- is followed by an increase exceeding the ones observed for hydrostatic conditions. Compared to k and s, diffusivity exhibits the least deviation from the hydrostatic reference curve before dilation sets in. Figure 8. View largeDownload slide Permeability k, specific storage capacity s, and hydraulic diffusivity Dhyd as a function of effective mean stress σm,eff = Δσax/3 + peff (a,c,e) and effective stress ratio λ = (Δσax + peff)/peff (b,d,f) during hydrostatic (circles) and deviatoric (lines) loading at indicated effective confining pressures. For (a,c,e) we shifted the results for the onset of deviatoric loading to match with the reference values gained from hydrostatic loading to remove the slight sample-to-sample variability (Table 1) and thus facilitate comparison of the mean-stress dependence. For (b,d,f) we normalized the hydraulic properties by their respective magnitude at the onset of deviatoric loading (subscript 0) to eliminate the effect of confining pressure on their absolute level. Squares and diamonds indicate maxima in P-wave velocity and onset of dilation as deduced from pore-volumetric strain, respectively. Figure 8. View largeDownload slide Permeability k, specific storage capacity s, and hydraulic diffusivity Dhyd as a function of effective mean stress σm,eff = Δσax/3 + peff (a,c,e) and effective stress ratio λ = (Δσax + peff)/peff (b,d,f) during hydrostatic (circles) and deviatoric (lines) loading at indicated effective confining pressures. For (a,c,e) we shifted the results for the onset of deviatoric loading to match with the reference values gained from hydrostatic loading to remove the slight sample-to-sample variability (Table 1) and thus facilitate comparison of the mean-stress dependence. For (b,d,f) we normalized the hydraulic properties by their respective magnitude at the onset of deviatoric loading (subscript 0) to eliminate the effect of confining pressure on their absolute level. Squares and diamonds indicate maxima in P-wave velocity and onset of dilation as deduced from pore-volumetric strain, respectively. 4 DISCUSSION Our study exceeds previous ones in terms of comprehensiveness of constraints on effective hydraulic properties. Compared to studies that also used the oscillatory method we achieve a higher resolution of permeability and storage capacity during deformation (Fischer 1992; Zhang et al. 1994; Mitchell & Faulkner 2008). Studies employing pulse techniques (e.g. Zoback & Byerlee 1975; Zhu & Wong 1997; Renner et al. 2000) yield only permeability and typically suffer from relaxation effects when deformation is paused for the hydraulic measurement. These effects could potentially be mitigated by adjusting the confining pressure or the axial load (Keaney et al. 1998), but only at the expense of an alteration of the deformation process. The instances during which the oscillation characteristics do not fall into the solution space (Fig. 6) coincide with localization of deformation and thus bear the potential to characterize the development of heterogeneity. Exploitation of this potential requires derivation of solution spaces for more advanced models, a subject that goes beyond the scope of the current presentation. The chosen strain rate ensured internal drainage. One should, however, not confuse the timescale resulting from the analysis of the critical strain rate with a requirement for the oscillation period to be employed in an oscillatory test. The scaling relation between timescale and penetration depth corresponds to a significant pressure perturbation, typically on the order of 10  per cent or more depending on flow geometry (e.g. Weir 1999). The pore-pressure method, in contrast, yields information regarding the entire sample length independent of chosen oscillation period because Fourier analysis allows for the detection of rather weak signals. Here, the lowest amplitude ratios analysed are on the order of 10−2 (Fig. 6). The gained effective hydraulic properties represent the path(s) connecting the sample ends along which pressure diffusion is most effective. Oscillation period affects to what extent subsidiary branches of the pore and crack network (including dead ends) contribute to transport and storage in a heterogeneous sample. In the following, we first discuss the microstructural state of the Wilkeson sandstone samples before and then indicators for changes in microstructure during deviatoric deformation, such as anisotropy and onset of inelastic deformation. We proceed by analysing the interrelation between the monitored properties before concluding with implications for contributions of hydro-mechanical coupling to in situ processes. 4.1 Microstructural state of samples before deformation as derived from their physical properties Wilkeson sandstone exhibits a peculiar combination of the characteristics of cracked and porous media. The pronounced changes in the pressure dependences of axial and pore-volumetric strain, and P-wave velocity under hydrostatic conditions (Figs 3a and b) demonstrate the presence of microfractures that close in the explored range of effective pressures. Yet, the modest pressure dependence of the hydraulic properties and the absolute value of specific storage capacity, which deviates little from what is expected from the fluid compressibility alone (Fig. 3d), suggest a well connected rather stiff pore space. Thus, the microfractures closing under pressure have apparently restricted relevance for the interconnectivity of the conduit network and storage dominantly occurs in pores that are not sensitive to pressure. The absolute value of specific storage capacity and its insensitivity to pressure suggest about 8  per cent stiff pores that are complemented by deformable microfractures with ϕcrack ∼ 1  per cent as indicated by the intercepts of the linear relations of bulk and pore-volumetric strain to pressure (Fig. 3a). These fractions appear in good agreement with the visual impression gained from SEM imaging (Fig. 2b). The conceptual notion of a combination of pores and cracks can also explain the contrasting observations of hysteresis in the examined physical properties during the exerted cycle of hydrostatic pressurization and depressurization (Fig. 3). Likely, cracks oriented parallel to the cylinder axis contribute significantly to bulk permeability and remain partly closed during unloading with no major impact on axial strain or axial P-wave velocity, and only subordinately affect specific storage capacity. Obviously, the small increase in permeability with increasing pressure and the curious difference between hydraulic properties at hydrostatic conditions and at small deviatoric loads are also related to the ‘closable’ fractures since this phenomenon essentially vanishes above the identified crack-closure pressure (Figs 3c–e). Intrinsic anisotropy and its development during hydrostatic loading -not accounted for in the analyses- may cause artefacts in the followed determination of hydraulic properties that result in these peculiar observations. This behaviour may, however, also be a result of the shift in density and thus relevance of the two void features involved, fractures with large aspect ratios and fairly equant pores as representative of a granular medium. The closing of the fractures may lead to a subtle change towards a less fracture-affected pore network for which the averaging involved in transient hydraulic tests actually yields an increase in permeability. The mechanical response of an isotropic, homogeneous, linear poroelastic medium to hydrostatic loading is fully described by three bulk moduli (compressibilities), for example, the drained (also addressed as dry or skeleton) modulus $$K_\mathrm{d}(=c_\mathrm{bc}^{-1})$$, the average mineral modulus $$K_\mathrm{r}(=c_\mathrm{r}^{-1})$$, and the fluid modulus $$K_\mathrm{f}(=c_\mathrm{f}^{-1})$$. Our experiments allow us to derive three independent elastic parameters (Kd and cpc from the linear sections of the corresponding strain records, Fig. 3a, and s from the hydraulic testing, Fig. 3d) and thus -considering fluid compressibility to be known- we can evaluate their consistency with linear poroelasticity for a homogeneous and isotropic medium. Specific storage capacity derived from compressibility values is twice as large as the value actually observed above the closure pressure (Table 3, Fig. 3). This apparent discrepancy may in principle point to violated assumptions regarding isotropy and linearity. However, we suppose that the quantitative mismatch reflects structural heterogeneity as also indicated by a storage capacity below the contribution of the pore fluid alone during parts of the deformation suggesting hydraulically preferred pore features (Figs 7c and d). Observed storage capacity is very much expected to be smaller than the one predicted from static compressibility (moduli) measurements since only the conducting fractures contribute to it while the static moduli are sensitive to all compliant pore features. Table 3. Measured and derived dynamic and quasi-static mechanical parameters for Wilkeson sst.   Parameter  Relation  Value    Results from ultrasonic wave transmission at ambient conditions    Drained bulk modulus  $$\tilde{K}_{{\rm d}} = \rho _{{\rm dry}}(v_{{\rm P,dry}}^2-\frac{4}{3}v_{{\rm S,dry}}^2)$$  (6.9 ± 0.7) GPa    Undrained bulk modulus  $$\tilde{K}_{{\rm ud}} = \rho _{{\rm sat}}(v_{{\rm P,sat}}^2-\frac{4}{3}v_{{\rm S,sat}}^2)$$  (27.5 ± 0.3) GPa  dynamic  Shear modulus  $$\tilde{G} = \rho _{{\rm dry}}v_{{\rm S,dry}}^2$$  (8.2 ± 0.3) GPa    Measured parameters at hydrostatic conditions (peff > 40 MPa)    Drained bulk compressibility  $$c_{{\rm bc}}= 3\mathrm{d}\epsilon _\mathrm{ax}/\mathrm{d} \Delta \sigma |_{p_\mathrm{f}}$$  (6.5 ± 0.1) × 10 − 11 Pa − 1    Drained bulk modulus  $$K_{{\rm d}}=c_{{\rm bc}}^{-1}$$  (15.6 ± 0.6) GPa    Pore compressibility  $$c_{\mathrm{pc}}=\mathrm{d}\psi /\mathrm{d} \Delta \sigma |_{p_\mathrm{f}}/\phi _\mathrm{con}$$  (5.1 ± 0.1) × 10 − 10 Pa − 1    Specific storage capacity  s  (4.1 ± 0.3) × 10 − 11 Pa − 1  quasi-static  Derived parameters at hydrostatic conditions (peff > 40 MPa)    Average bulk modulus of minerals  $$K_{\mathrm{r}}(c_{\mathrm{bc}},c_{\mathrm{pc}}) = c_{\mathrm{r}}^{-1} = (c_{\mathrm{bc}}-\phi _\mathrm{con}c_{\mathrm{pc}})^{-1}$$  (51.7 ± 4.1) GPa    Biot–Willis parameter  α(cbc, cpc) = ϕconcpc/cbc  (0.70 ± 0.04)    Specific storage capacity  s(cbc, cpc) = ϕcon[cf − cbc + (1 + ϕcon)cpc]  (8.2 ± 0.4) × 10 − 11 Pa − 1    Parameter  Relation  Value    Results from ultrasonic wave transmission at ambient conditions    Drained bulk modulus  $$\tilde{K}_{{\rm d}} = \rho _{{\rm dry}}(v_{{\rm P,dry}}^2-\frac{4}{3}v_{{\rm S,dry}}^2)$$  (6.9 ± 0.7) GPa    Undrained bulk modulus  $$\tilde{K}_{{\rm ud}} = \rho _{{\rm sat}}(v_{{\rm P,sat}}^2-\frac{4}{3}v_{{\rm S,sat}}^2)$$  (27.5 ± 0.3) GPa  dynamic  Shear modulus  $$\tilde{G} = \rho _{{\rm dry}}v_{{\rm S,dry}}^2$$  (8.2 ± 0.3) GPa    Measured parameters at hydrostatic conditions (peff > 40 MPa)    Drained bulk compressibility  $$c_{{\rm bc}}= 3\mathrm{d}\epsilon _\mathrm{ax}/\mathrm{d} \Delta \sigma |_{p_\mathrm{f}}$$  (6.5 ± 0.1) × 10 − 11 Pa − 1    Drained bulk modulus  $$K_{{\rm d}}=c_{{\rm bc}}^{-1}$$  (15.6 ± 0.6) GPa    Pore compressibility  $$c_{\mathrm{pc}}=\mathrm{d}\psi /\mathrm{d} \Delta \sigma |_{p_\mathrm{f}}/\phi _\mathrm{con}$$  (5.1 ± 0.1) × 10 − 10 Pa − 1    Specific storage capacity  s  (4.1 ± 0.3) × 10 − 11 Pa − 1  quasi-static  Derived parameters at hydrostatic conditions (peff > 40 MPa)    Average bulk modulus of minerals  $$K_{\mathrm{r}}(c_{\mathrm{bc}},c_{\mathrm{pc}}) = c_{\mathrm{r}}^{-1} = (c_{\mathrm{bc}}-\phi _\mathrm{con}c_{\mathrm{pc}})^{-1}$$  (51.7 ± 4.1) GPa    Biot–Willis parameter  α(cbc, cpc) = ϕconcpc/cbc  (0.70 ± 0.04)    Specific storage capacity  s(cbc, cpc) = ϕcon[cf − cbc + (1 + ϕcon)cpc]  (8.2 ± 0.4) × 10 − 11 Pa − 1  View Large The decrease in permeability with increasing effective pressure above the closure pressure corresponds to a permeability modulus $$K_k=\mathrm{d}p_\mathrm{eff}/\mathrm{d}\ln k = 221 \pm 2\,{\rm MPa}$$. An average aspect ratio on the order of $$\bar{\xi }_k \sim K_\mathrm{r}/3 K_k\sim 10^2$$ results when employing the permeability model presented by Gavrilenko & Gueguen (1989). This average aspect ratio is about an order of magnitude smaller than the estimate (see Walsh 1966; Mavko & Nur 1978) of ξcc ∼ Kr/pcc ∼ 103 corresponding to the observed crack-closure pressure. The relation between these two estimates of aspect ratios is, however, consistent. The cracks controlling permeability above the identified ‘closure pressure’ have to exhibit a smaller aspect ratio than those that are closed when this pressure level is reached. The observed continuous velocity increase with increasing pressure (Fig. 3b) is also suggestive of an ensemble of fractures with a range of geometrical properties (see David & Zimmerman 2012). 4.2 Microstructural evolution during deviatoric loading as deduced from the monitored physical properties Deformation of rocks at temperatures at which crystal plasticity is insignificant, like here, is inherently related to the processes involving pre-existing microcracks, that is, their closure, frictional sliding, and eventually their growth. Frictional processes on pre-existing microfractures are intrinsically irreversible and lead to distinct asymmetry and hysteresis during loading and unloading cycles at stress levels too low to initiate microfracturing (e.g. Walsh 1965; David et al. 2012). Thus, rather than addressing the reversibility of deformation -in accord with the fundamental thermodynamic definition of elastic behaviour- the separation of ‘elastic’ versus ‘inelastic’ deformation of rocks by brittle mechanisms often refers (loosely) to the onset of microfracturing, that is, permanent changes in the defect inventory. We will also use this terminology here. Closure and growth of pre-existing microfractures cause anisotropy in physical properties when rocks are subjected to deviatoric loading (e.g. Johnston et al. 1979; Toksöz et al. 1979; Crampin et al. 1980; Jones 1995; Sharma 2005). Previous experimental and numerical studies provided ample evidence that the initial response of rocks to deviatoric loading is controlled by the preferential closure of pre-existing cracks oriented perpendicular to the maximum principal stress. When stresses exceed the elastic limit, brittle deformation is induced on grain scale in the form of directionally dependent extension of existing and formation of new cracks. Anisotropy -well documented for elastic velocities during elastic as well as inelastic deformation (e.g. Guéguen & Schubnel 2003; Schubnel & Guéguen 2003)- may bias the absolute values of the reported effective hydraulic parameters derived relying on an analytical solution assuming isotropy. Yet, up to now the anisotropy in hydraulic parameters is hardly accessible in experiments (but see Dautriat et al. 2009). Thus, we do not have direct evidence for anisotropy in either elastic nor hydraulic properties (measured only in one direction). However, differences between observations for hydrostatic and deviatoric loading serve as an indicator for anisotropy. 4.2.1 Evolving anisotropy The relation between observations for hydrostatic and deviatoric loading is controlled by the identified crack-closure pressure. For experiments conducted at effective confining pressures below this level, pore-volumetric strain, hydraulic properties, and P-wave velocity tend to deviate from their hydrostatic reference curves immediately at the onset of deviatoric loading (Figs 5, 8 and 10). These deviations reflect relative stiffening by deviatoric loading, most likely caused by closure of cracks perpendicular to the cylinder axis, the direction of travel of the P-waves and the flow direction for the pore fluid, and thus suggesting the development of anisotropy in the crack distribution (see for example Schubnel & Guéguen 2003; Schubnel et al. 2006). Stiffening in axial direction is also documented by the initial increase in tangent modulus with axial strain (Fig. 11a). At pressures above the crack-closure pressure, differences between hydrostatic and directional loading diminish probably because the potential for closure of cracks of any orientation is almost exhausted. Radial strain is a potential indicator of opening and closing of cracks oriented parallel to the sample axis. We approximate radial strain by using the pore-volumetric strain recorded during deviatoric loading as a measure of bulk volumetric strain, θ, that is, εrad = (θ − εax)/2 ≈ (ψ − εax)/2. For an isotropic, linear poroelastic medium, the two volumetric strains obey ψ = (α − ϕcon)θ where α = 1 − Kd/Kr denotes the Biot-Willis coefficient. This relation follows for hydrostatic and general stress states when evaluating eq. (35) in Zimmerman et al. (1986) or eq. (69) in Renner & Steeb (2014) for constant pore-fluid pressure and dpc = Kdθ or $$\mathrm{d}\sigma ^\mathrm{^m} = K_\mathrm{d} \theta$$, respectively. Here, we find α = 0.70 ± 0.04 (Table 3), that is, the change in porosity accounts for more than 60 per cent of the total elastic volumetric strain. According to amount, our approximation overestimates elastic radial strain by less than 30 per cent for a Poisson’s ratio of 0.3 and gets worse the lower the Poisson’s ratio is. In contrast, the inelastic contribution to radial strain is fully reflected by the changes in porosity since inelastic deformation is here accommodated by void creation and elimination. During initial deviatoric loading even the approximated and thus -according to amount- overestimated radial strains are subordinate but P-wave velocity increases significantly. Presumably, the closure of cracks preferentially oriented perpendicular to the maximum principal stress accommodates axial strain with limited Poisson effect. The subsequent reduction in velocity increase correlates with the progressing widening of the sample recorded by the radial strain accompanying the onset of dilation. When normalizing by their maximum in velocities, the four tests at different effective pressures exhibit a ‘universal’ relation between P-wave velocity and radial strain (Fig. 5b). Likewise, normalized permeability exhibits a fairly uniform increase with radial strain when attributing for the contribution to radial strain from hydrostatic loading εrad,hyd, before passing into individual plateaus for the four tests after failure when macroscopic localization has advanced to a fully developed fault or cataclastic flow occurs (Fig. 9b). While all three hydraulic properties seem to correlate better with effective stress ratio λ = (Δσax + peff)/peff than effective mean stress this observation is particularly true for specific storage capacity (Fig. 8) that in contrast to permeability exhibits limited correlation with radial strain (Fig. 9c). Figure 9. View largeDownload slide (a) Normalized P-wave velocity vP, (b) permeability k, (c) specific storage capacity s, and (d) hydraulic diffusivity Dhyd as a function of total radial strain εrad,tot = εrad,hyd + (ψ − εax)/2 at indicated effective confining pressures. P-wave velocities and hydraulic properties were normalized by their maximal and by their hydrostatic reference values, respectively. Diamonds and triangles indicate onset of dilation as deduced from pore-volumetric strain and peak stress, respectively. Figure 9. View largeDownload slide (a) Normalized P-wave velocity vP, (b) permeability k, (c) specific storage capacity s, and (d) hydraulic diffusivity Dhyd as a function of total radial strain εrad,tot = εrad,hyd + (ψ − εax)/2 at indicated effective confining pressures. P-wave velocities and hydraulic properties were normalized by their maximal and by their hydrostatic reference values, respectively. Diamonds and triangles indicate onset of dilation as deduced from pore-volumetric strain and peak stress, respectively. Our observations regarding the relation between permeability and stress ratio qualitatively and quantitatively match with the modelling by Simpson et al. (2001) which predicts changes in normalized permeability of at most a factor of 2 as long as the effective stress ratio remains below about 15, a bound not exceeded in our experiments (Fig. 8). The model lends its support to our interpretation of the relation between properties and radial strain since it finds the permeability evolution in this dilative regime to be controlled by axially oriented cracks widening and/or forming. The modest degree of transversal anisotropy during inelastic deformation predicted by the model suggests that our effective hydraulic properties probably still represent the samples and their evolutions well. More experimental and numerical work is needed to constrain the effect of anisotropy on hydraulic measurements. 4.2.2 Onset of microfracturing Two prominent approaches for the identification of the onset of microfracturing have been followed in the past, analyses of stress-strain relations and monitoring of physical properties presumably sensitive to microfracturing. For dense crystalline rocks, Brace & Martin (1968) introduced the concept of dilatancy, that is, the deviation of volumetric strain induced by deviatoric loading from the initially linear compactive trend. For porous rocks, a separately determined, typically nonlinear hydrostat is used as reference instead (e.g. Zhu & Wong 1997; Baud et al. 2012). Our experiments exemplify the problems encountered for porous rocks. The pore-volumetric strain of the sample deformed at an effective pressure of 100 MPa exhibits a linear relation with (mean) stress almost all the way up to peak stress (Fig. 4b), likely as a consequence of the fortuitous balance between concurrent dilative (extension of tensile cracks) and compactive (pore collapse and crack closure) processes. The comparison to the hydrostatic reference curve is hampered by the restriction of the latter regarding the covered range in pressure (Fig. 10); for the samples deformed at low effective pressures, it is rather difficult to pinpoint the range of stresses where the two curves coincide and where not. In fact, it appears as if they do not show any coincidence for the experiments at the lowest effective pressures possibly due to the development of anisotropy associated with closure of different suites of the pre-existing microcracks during hydrostatic and deviatoric loading, as argued above. For the tests performed at or above crack-closure pressure, pore-volumetric strains observed during hydrostatic and deviatoric loading closely agree to the point when dilation occurs consistent with a diminished role of cracks (and consequently anisotropy). Figure 10. View largeDownload slide Pore-volumetric strain ψ as a function of effective mean stress σm,eff at indicated effective confining pressures. The curves representing pore-volumetric strain recorded during triaxial compression are displayed starting at the respective strains determined during hydrostatic deformation of the reference sample (circles). Figure 10. View largeDownload slide Pore-volumetric strain ψ as a function of effective mean stress σm,eff at indicated effective confining pressures. The curves representing pore-volumetric strain recorded during triaxial compression are displayed starting at the respective strains determined during hydrostatic deformation of the reference sample (circles). Given the relative technical ease with which elastic wave velocities can be ‘continuously’ monitored during deformation experiments, it is not surprising that characteristics of velocity changes were employed to define deformation regimes. For example, Schubnel et al. (2005, 2006) identified passing the maximum in P- and S-wave velocity during triaxial compression with the onset of crack propagation. Different from our study, the velocity measurements were, however, performed perpendicular to the compression axes. Previous studies found significant differences in the sensitivities of P- and S-waves travelling in axial and radial direction to the transition from elastic to inelastic behaviour (e.g. Lockner et al. 1977; Guéguen & Schubnel 2003) in accord with the intuitive notion that crack closure during ‘elastic’ triaxial deformation and inelasticity due to crack extension or formation cause anisotropic crack distributions (e.g. Kachanov 1992). The present experiments yield a significant sensitivity of axial P-wave velocities to all deformation phases. Velocity increases during initial compaction, exhibits a maximum well before peak strength, and then decreases until localized failure occurs (Fig. 9a). The initial increase in velocity can be attributed to either the closure of pre-existing microcracks oriented perpendicular to the direction of the maximum principal stress (e.g. Lockner et al. 1977; Jones 1995; Eslami et al. 2010) or to the stiffening of Hertzian contacts at grain boundaries (e.g. Pyrak-Nolte & Nolte 1995; Toomey & Nakagawa 2003), or to both. The occurrence of the maxima in P-wave velocity coincides with the onset of dilation below crack-closure pressure but the two are separated for higher pressures. The correlation of velocity with radial strain (Fig. 5b) suggests that its decrease is related to opening of cracks oriented subparallel to the direction of maximum principal stress. The extent to which the aspect ratio of cracks affects elastic properties critically depends on the fluid-coupling parameter δ ∝ Kr/Kfξ (Kachanov 1992); that is, the ratio of bulk moduli of solid and fluid (see also Schubnel et al. 2006). This parameter assumes critical limits of δ → ∞ for compressible gases (dry cracks) and δ → 0 for incompressible fluids (water-saturated cracks) leading to insensitivity and sensitivity of velocity to aspect ratio, respectively. Consideration of pore fluid properties and saturation state may contribute to the partly conflicting observations regarding the sensitivity of velocity to cracks aligned with the direction of propagation (e.g. Read et al. 1995; Wulff et al. 1999; Guéguen & Schubnel 2003; Eslami et al. 2010; Blake 2011; Stanchits et al. 2011). It will be important to decipher the underlying reason for the differences in sensitivity in future work owing to its relevance for the analyses of preseismic velocity changes in the field (e.g. Schaff 2012). Obviously, the peculiarities of the inventory of voids (pores versus fractures) are of critical importance because closure of cracks and stiffening of grain contacts oriented perpendicular to the maximum stress may occur concomitantly with opening of cracks parallel to the maximum stress over an extended range of strain. The conceptual focus on cracks parallel and perpendicular to the maximum principal stress may be too gross a simplification, for example, in the light of fracture processes such as wing cracking (e.g. Brace & Bombolakis 1963; Paterson & Wong 2005) or grain comminution in granular media. In the following, we consider the maxima in axial P-wave velocity the latest indicators for crack initiation and tentatively use them to separate ‘elastic’ and ‘inelastic’ deformation where the terms are used in the sense discussed above. This approach is supported by the prominent inflection points in trends of hydraulic properties with axial strain observed at the maxima in velocity (Fig. 7). 4.3 Correlations among physical properties The richness and density of information gained from our tests (continuous resolution of hydraulic properties for strain increments of as small as 1 × 10−4 ) allows us to investigate previously unexplored correlations among determined parameters whose full meaning and potential can only be grasped in the light of yet to be performed hydro-mechanical modelling of heterogeneous media. Correlating elastic properties determined from different stress or strain amplitudes is possibly complicated by nonlinearity (e.g. Fjaer 2009). The oscillation amplitudes applied at the upstream end (<3 MPa) and observed at the downstream end (>0.03 MPa) provide upper and lower bounds, respectively, for the stress perturbations associated with the determination of specific storage capacity. These values are intermediate between the stress intervals of about 2 MPa employed in the calculation of tangent moduli and the stress perturbations associated with ultrasonic waves, for our configuration estimated to be below about 0.01 MPa. Furthermore, the various measures of poroelasticity constrained here likely reflect different parts of the pore-space. Static elastic parameters, that is, derivatives of stress-strain curves, represent bulk properties of the entire sample volume. Ultrasonic velocities reflect the fastest wave paths which in turn depend on wavelength (e.g. Mukerji et al. 1995; Tworzydło & Beenakker 2000). Here, the dominant wavelength is an order of magnitude smaller than sample size and thus dynamic elastic parameters are expected to exceed the static ones. The two directionally dependent mechanical properties that constrain momentary elastic behaviour, tangent modulus and ultrasonic velocity, initially show a positive correlation in the elastic regime. The tangent modulus, however, decreases before the velocity reaches its maximum (Fig. 11a). The occurrence of the two maxima seems to converge with increasing pressure, though. Tangent modulus and velocity show a uniform positive correlation after the maximum in velocity that continues even after peak stress possibly due to uniform damage accumulation before localization (Fig. 11b). Figure 11. View largeDownload slide (a,b) Permeability k, (c,d) specific storage capacity s, (e,f) hydraulic diffusivity Dhyd, and tangent modulus Eapp during ‘elastic’ (left) and ‘inelastic’ (right) triaxial deformation of Wilkeson sst at indicated effective confining pressures. The occurrence of maxima in P-wave velocity was used to distinguish ‘elastic’ from ‘inelastic’ (see the text). Hydraulic data were down-sampled by a factor of 60 to match with the velocity data. To remove distorting scattering from velocity data (cf. Fig. 5a) a polynomial fit was applied to match the relation of velocity to axial strain for each test. Figure 11. View largeDownload slide (a,b) Permeability k, (c,d) specific storage capacity s, (e,f) hydraulic diffusivity Dhyd, and tangent modulus Eapp during ‘elastic’ (left) and ‘inelastic’ (right) triaxial deformation of Wilkeson sst at indicated effective confining pressures. The occurrence of maxima in P-wave velocity was used to distinguish ‘elastic’ from ‘inelastic’ (see the text). Hydraulic data were down-sampled by a factor of 60 to match with the velocity data. To remove distorting scattering from velocity data (cf. Fig. 5a) a polynomial fit was applied to match the relation of velocity to axial strain for each test. The scalar-valued storage capacity is also selective regarding the reflected elements of the crack-distribution. Storage capacity is controlled by the momentary elastic deformation of the hydraulic conduits involved in pressure diffusion, that is, cracks and pores with a favourable combination of width (transport) and stiffness (storage), that is, not necessarily the ones controlling wave velocity. Still, specific storage capacity exhibits fairly uniform but non-monotonous correlations with P-wave velocity during inelastic deformation (Fig. 11f); it first increases and then decreases during the continuous velocity decrease. Relations between the two directionally dependent transport properties, P-wave velocity and permeability, are a subject of considerable interest and debate. Obviously, the potential for correspondence between these two properties depends on the specifics of the microstructure, in particular the interconnectivity among the microfractures. Yet, the finite permeability of Wilkeson sandstone and in fact the majority of real rocks suggests that they do not represent the peculiar situation of isolated voids, but interconnected crack networks are maintained even at elevated pressures. Furthermore, the properties of the pore fluid are important since the aspect ratio of cracks, of paramount importance for permeability owing to its dependence on crack width, affects elastic wave velocity only when the pore fluid is rather incompressible (as applicable here, see discussion of pore-fluid coupling parameter above). Thus, it is not surprising that previous studies did not reveal simple or universal relations between P-wave velocity and permeability (e.g. Pyrak-Nolte & Morris 2000; Fortin et al. 2005; Baud et al. 2012; Brantut 2015). Here, however, a systematic inverse correlation between them emerges from the occurrence of the maximum in velocity to peak stress, that is, from the onset of microfracturing to the onset of localization (Fig. 11d). This section of the deformation is characterized by the most pronounced radial straining. During initial elastic compaction, permeability modestly increases or decreases below and above the closure pressure, respectively, and thus does not show a systematic relation to velocity (Fig. 11c). Hydraulic diffusivity expresses per se the correlation between a transport (permeability) and a volumetric (specific storage capacity) property. The observed diffusivity values appear to exhibit less sample-to-sample variability than the two properties from which they calculate, that is, permeability and storage capacity (Figs 11g and h). Since diffusivity matches more closely the evolution of permeability than that of specific storage capacity (Fig. 8) it seems that the gain in connectivity during inelastic deformation outmatches the loss in rigidity. Furthermore, the overall tendency of diffusivity (and permeability) to increase with damage accumulation observed in our experiments suggests that the gain in porosity by dilative processes dominates the evolution of hydraulic properties (Fig. 9) rather than the potentially permeability-decreasing effect of comminution. Hydraulic diffusivity exhibits a systematic inverse correlation with P-wave velocity during inelastic deformation (Fig. 11h). 4.4 Implications Hydraulic diffusivity constitutes the key parameter for fluid-pressure transients and thus also the temporal variations in effective stress, as an expression of hydro-mechanical coupling. In the spirit of the quadrant diagrams of Bernabé et al. (2003) for permeability -porosity relations, we present our results as diffusivity-pore volumetric strain diagrams distinguishing between elastic and inelastic deformation (Fig. 12). The modest changes in diffusivity during elastic compaction fall in quadrant I for hydrostatic and deviatoric deformation. Inelastic deformation is characterized by an inverse relation between diffusivity and pore-volumetric strain, that is, increasing compaction correlates with decreasing diffusivity and vice versa, and thus the curves occupy almost exclusively the quadrants II and IV. Soon after the onset of microfracturing as indicated by the maximum in axial P-wave velocity also bulk dilation commences and diffusivity increases up to more than an order of magnitude during the progressing localization. Only for the sample deforming by non-localized cataclastic flow at the highest explored effective pressure of 100 MPa, we find a range for which the ongoing compaction prevents an increase in diffusivity (Fig. 12). The increase in diffusivity with progressing inelastic deformation should reduce the pore-fluid pressure or its build up and thus, in situ faulting of fluid-bearing rocks should exhibit self-stabilization. Furthermore, an increase in diffusivity increases the spatial scale over which drainage can be effectively reached. Thus, the observed diffusivity evolution is expected to affect hydro-mechanical coupling in a stabilizing and delocalizing way, aspects yet to be fully accounted for by modellers. Figure 12. View largeDownload slide Relation between hydraulic diffusivity and change in connected porosity as documented by pore-volumetric strain ψ during (a) elastic, and (b) inelastic deformation as separated by and identified from the maxima in axial P-wave velocity (Fig. 4) at indicated effective confining pressures. Diffusivity values are normalized by their hydrostatic reference values Dhyd,0 and their values found at the maximum in P-wave velocity Dhyd(vP,max) for elastic and inelastic deformation, respectively. The dashed lines are given to emphasize the quadrant concept originally introduced by Bernabé et al. (2003) for the discussion of permeability-porosity relations. The dotted box in (b) indicates the range in values covered by (a). To ease the comparison in (a), the records of the triaxial deformation tests were tied to the respective points of the hydrostatic relation. Figure 12. View largeDownload slide Relation between hydraulic diffusivity and change in connected porosity as documented by pore-volumetric strain ψ during (a) elastic, and (b) inelastic deformation as separated by and identified from the maxima in axial P-wave velocity (Fig. 4) at indicated effective confining pressures. Diffusivity values are normalized by their hydrostatic reference values Dhyd,0 and their values found at the maximum in P-wave velocity Dhyd(vP,max) for elastic and inelastic deformation, respectively. The dashed lines are given to emphasize the quadrant concept originally introduced by Bernabé et al. (2003) for the discussion of permeability-porosity relations. The dotted box in (b) indicates the range in values covered by (a). To ease the comparison in (a), the records of the triaxial deformation tests were tied to the respective points of the hydrostatic relation. Our experimental results show that -at least in certain yet to be better constrained cases-significant P-wave velocity variations occur in the nominally least sensitive direction, that is, that of the maximum principal stress. On the one hand, this result implies potentially less limitations regarding ray coverage relative to a fault structure necessary for preseismic observations (e.g. Whitcomb et al. 1973; Schaff 2012). On the other hand, it poses a warning regarding the interpretation of damage geometry from velocity variations observed in situ. Furthermore, the results of our study suggest that changes in velocities constitute a qualitative surrogate for changes in hydraulic properties accompanying pre-failure deformation and thus bear constraints on the prediction of fluid flow within seismically active rupture zones. At the advent of comprehensive in situ monitoring of velocity changes associated with large-scale faulting (e.g. Wegler & Sens-Schönfelder 2007; Wegler et al. 2009; Sens-Schönfelder & Wegler 2011), tides (e.g. Takano et al. 2014; Hillers et al. 2015a), or engineering applications (e.g. Olivier et al. 2015), improved constraints on the relation between velocity, stress state, and microstructure become ever more relevant for a significant interpretation of observations. We investigated the correlation between P-wave velocity and various strain (axial, radial, pore-volumetric) and stress (deviatoric, mean, stress ratio) measures to find that radial strain unifies the observed velocity changes during triaxial deformation at a range of effective pressures best. The inelastic radial strain reflects the opening of cracks oriented in the direction of wave propagation. The less significant correlations with stress indicate that the correlation between stress state and microstructure is not unique even for the performed set of experiments on a single rock type. In situ observations have been interpreted in terms of relative velocity changes with strain or stress variations (e.g. Yamamura et al. 2003), the significance of neither being fully understood. Absolute stress-sensitivity values estimated for in situ velocities are at least one order of magnitude larger than the ones typically observed in the laboratory, including this study. Here, sensitivities of relative velocity changes decrease from about 5 × 10−3 MPa−1 to 2 × 10−4 MPa−1 with increasing hydrostatic or mean stress before inelasticity sets in (Figs 3b and 5c). The quantitative discrepancy may be related to a strong nonlinearity in velocity changes with stress at low deviatoric loads and mean stresses far away from crack-closure pressures that exceeds the nonlinearity for hydrostatic loading observed here (Fig. 5c). 5 CONCLUSIONS Our laboratory experiments on samples of Wilkeson sandstone elucidated the structural evolution associated with brittle to semi-brittle faulting as constrained by changes in hydraulic and dynamic elastic properties as well as the correlation among these properties. The chosen experimental procedures and analysis methods provide the complete set of hydraulic properties with unprecedented resolution. In agreement with previous work, we find a general trend of increasing permeability with proceeding brittle deformation. Yet, a transient, rather counterintuitive decrease in permeability and specific storage capacity occurs during localization of faulting. Hydraulic diffusivity, the key parameter for transient flow processes, exhibits larger or smaller increases than permeability depending on the direction and magnitude of changes in specific storage capacity that seem to alter with increasing effective pressure. A uniform relation appears to hold between P-wave velocity and deviatoric radial strain, that is, the strain perpendicular to the direction of propagation. For none of the tested stress measures (maximum principal, deviatoric, or mean stress, stress ratio) a similarly good correlation is found for P-wave velocity. The correlations of hydraulic properties with either total radial strain or stress ratio appear equally systematic. These correlations with radial strain strongly suggest that cracks aligned with the maximum principal stress are responsible for the velocity decrease and the permeability increase. The uniformity of observations for tests performed at different pressures implies a uniformity in the generation and accumulation of damage by microfracturing before localization and suggests that the maximum in velocity serves as a robust indication of inelastic deformation in our experiments. The significant sensitivity of P-wave velocity to characteristics of cracks aligned with the propagation direction differs from results of several previous studies and it thus remains to be resolved what actually determines the degree of sensitivity. Such future work is relevant in the context of in situ velocity monitoring as a tool for detecting changes in deformation state, for which either the propagation direction of the waves cannot be controlled (natural sources) or the orientation of principal stresses is not constrained beforehand. Also, more work is needed to explore the diagnostic potential of velocity variations for the evolution of hydraulic properties, in particular hydraulic diffusivity, indicated by our results and to exploit it in, for example, monitoring of seismically active regions or evolving reservoirs. More sophisticated models than one-dimensional pressure diffusion in an isotropic, homogeneous material have to be elaborated to analyse the observations outside the ‘conventional’ solution space and the complex variations in oscillation characteristics observed right at failure. The understanding of the structural changes occurring during localization and their consequences for fault properties that in turn affect the dynamics of the failure process will largely benefit from such modelling of experimental data. A major challenge for this type of modelling will be to account for anisotropy evolving during deviatoric loading, that as of yet also lacks experimental quantification for hydraulic properties. ACKNOWLEDGEMENTS Generous funding by the Federal Ministry for Economic Affairs and Energy for the project SHynergie is greatly appreciated. We thank all people involved in preparing samples, and setting up and performing experiments, in particular Frank Bettenstedt. REFERENCES Baud P., Meredith P., Townend E., 2012. Permeability evolution during triaxial compaction of an anisotropic porous sandstone, J. geophys. Res. , 117, B05203, doi:10.1029/2002JB002005. Bennion D., Goss M., 1977. A sinusoidal pressure response method for determining the properties of a porous medium and its in-situ fluid, Can. J. Chem. Eng. , 55, 113– 117. Google Scholar CrossRef Search ADS   Benson P., Meredith P., Platzman E., White R., 2005. 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This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Geophysical Journal International Oxford University Press

Relations between hydraulic properties and ultrasonic velocities during brittle failure of a low-porosity sandstone in laboratory experiments

Relations between hydraulic properties and ultrasonic velocities during brittle failure of a low-porosity sandstone in laboratory experiments

Geophysical Journal International Geophys. J. Int. (2018) 212, 627–645 doi: 10.1093/gji/ggx419 Advance Access publication 2017 September 27 GJI Rock and mineral physics, rheology Relations between hydraulic properties and ultrasonic velocities during brittle failure of a low-porosity sandstone in laboratory experiments B. Ahrens, M. Duda and J. Renner Institute for Geology, Mineralogy, and Geophysics, Ruhr-Universitat ¨ Bochum, 44780 Bochum, Germany. E-mail: benedikt.ahrens@rub.de Accepted 2017 September 26. Received 2017 September 22; in original form 2017 March 22 SUMMARY We continuously monitored hydraulic properties and ultrasonic P-wave velocity during triaxial compression of Wilkeson sandstone samples with an initial total porosity of about 9.5 per cent to elucidate the microstructural evolution and its consequences for the relation between these properties during elastic and inelastic deformation. Samples were deformed at drained con- ditions at a range of effective pressures from below to above the determined crack-closure pressure of about 50 MPa and hydraulic properties were determined using the oscillatory pore- pressure method. As a result of the employed period the evolution of hydraulic properties was −4 resolved for strain increments of 10 and thus constrained during imminent localized failure. Except during the development of a localized fault at the lowest imposed effective confining pressures, we found permeability and hydraulic diffusivity to generally increase during pro- gressing brittle deformation associated with dilation. Thus, in situ faulting of fluid-bearing rocks should in general exhibit self-stabilization. Axial P-wave velocity shows a rather uni- form correlation with radial strain during inelastic deformation suggesting a prominent role of cracks aligned with the direction of wave propagation, an inference that needs further analyses regarding the varying sensitivity of longitudinal waves for (micro)structural...
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Abstract

SUMMARY We continuously monitored hydraulic properties and ultrasonic P-wave velocity during triaxial compression of Wilkeson sandstone samples with an initial total porosity of about 9.5 per cent to elucidate the microstructural evolution and its consequences for the relation between these properties during elastic and inelastic deformation. Samples were deformed at drained conditions at a range of effective pressures from below to above the determined crack-closure pressure of about 50 MPa and hydraulic properties were determined using the oscillatory pore-pressure method. As a result of the employed period the evolution of hydraulic properties was resolved for strain increments of 10−4 and thus constrained during imminent localized failure. Except during the development of a localized fault at the lowest imposed effective confining pressures, we found permeability and hydraulic diffusivity to generally increase during progressing brittle deformation associated with dilation. Thus, in situ faulting of fluid-bearing rocks should in general exhibit self-stabilization. Axial P-wave velocity shows a rather uniform correlation with radial strain during inelastic deformation suggesting a prominent role of cracks aligned with the direction of wave propagation, an inference that needs further analyses regarding the varying sensitivity of longitudinal waves for (micro)structural changes in their direction of travel. Hydraulic properties exhibit systematic correlations with radial strain during inelastic deformation, too, a rather expected result for permeability. During nominally elastic deformation, however, stress ratio and mean stress seem to control hydraulic properties. Differences between elastic properties derived from mechanical and hydraulic excitations probably indicate heterogeneity of the microstructures. An additional contribution of anisotropy in hydraulic properties cannot be ruled out that up to now evades experimental quantification. The observed fairly uniform correlation between hydraulic diffusivity and P-wave velocity in the direction of fluid flow suggests that monitoring changes in elastic wave velocities bears some potential to constrain changes in conditions for transient fluid flow. Fracture and flow, Microstructure, Permeability and porosity, Creep and deformation, Elasticity and anelasticity, Acoustic properties 1 INTRODUCTION The derivation of structure and state of rocks from constraints on the spatial distribution of their physical properties represents one of the key objectives in geophysics. Constraints on structure are gained from borehole logging and surface surveys covering spatial scales from centimetres to kilometres. Laboratory experiments with the ability to control the state of rocks are crucial for amending results for structure with information on state. Using information from the propagation of elastic waves in rock volumes of interest probably constitutes the most common approach across all scales. In particular, the sensitivity of wave velocities to structure and state permits monitoring natural or anthropogenically induced subsurface processes using time-dependent variations in their characteristics, for example related to precursory phenomena, earthquake-source mechanisms, and aftershock sequences (e.g. Rice & Rudnicki 1979; Miller et al. 2004; Zaliapin & Ben Zion 2011; Froment et al. 2014; Hillers et al. 2015b), faulting in general (e.g. Faulkner et al. 2010), gas and oil production (e.g. Landrø 2001), or stimulation of geothermal reservoirs (e.g. Charléty et al. 2006). The determination of hydraulic properties of rocks or, more general, the subsurface serves the dual purpose of constraining structure and of providing effective properties for a specific system of interest. Such effective properties permit modelling subsurface fluid flow associated with natural processes and predicting the outcome of pumping operations as associated with, for example, freshwater supply, exploitation of hydrocarbon reservoirs, operation of subsurface liquid-waste repositories, or geothermal energy provision. Furthermore, hydraulic properties are pivotal for the evolution of mechanical states when a system containing fluids is perturbed by either changes in bulk stresses or fluid pressures. The related concept of effective pressure and stress goes back to at least Terzaghi (1936). Among other phenomena, hydro-mechanical coupling, that is, the mutual influencing of bulk stresses and fluid pressures (Rice & Cleary 1976), has been held responsible for aftershock activity (e.g. Miller et al. 2004) and induced seismicity (e.g. Majer et al. 2007; Zang et al. 2014). In situ, wave observations potentially provide better coverage of rock volumes in space and time than hydraulic investigations and thus constraints on correlations between elastic and hydraulic properties may provide the basis for improved subsurface characterization with respect to hydro-mechanical aspects. Clearly, wave velocities and hydraulic properties share their general sensitivity to the structure of pores and fractures. Yet, the specific aspects of the structure that affect these two properties might differ considerably. Consequently, their information content may be complementary rather than redundant. Laboratory experiments permit to elucidate the structural evolution associated with brittle faulting and determine the sensitivity of hydraulic and dynamic elastic properties to bulk elastic and inelastic deformation and the relation among them. In this laboratory study we continuously monitored hydraulic properties and elastic wave velocities of sandstone samples during conventional triaxial compression. We suppressed a potentially biasing contribution of hydro-mechanical coupling by maintaining the samples at effectively drained conditions, that is, the pore-fluid pressure remained constant in the samples throughout the deformation experiments. This approach serves two linked objectives, (1) constraining the interrelation between the evolution in structure and hydraulic properties and (2) investigating the correlation between wave velocities and hydraulic properties during brittle faulting. These objectives address the hold-up of substantial modelling of, for example, aftershock activity and induced seismicity, imposed by the lack of fundamental observations regarding the deformation-related evolution in hydraulic properties and how it affects the space-time characteristics of the pore-pressure field surrounding a failure event and thus the potential spread of failure. Our approach intentionally contrasts and extends previous studies in several respects. In the suite of compression experiments, applied confining pressures cover the range from below to above a pressure level identified as critical for crack closure to control the state of pre-existing cracks. Of the main three hydraulic methods, steady-state flow tests, pulse tests, and oscillatory tests, we selected the latter for their advantages regarding continuous high-resolution monitoring of hydraulic properties during deformation (Fischer 1992; Zhang et al. 1994; Mitchell & Faulkner 2008). Studies on correlations between hydraulic properties and velocities of elastic waves that were restricted to hydrostatic loading (Benson et al. 2005; Fortin et al. 2005) are extended to conventional triaxial compression. 2 MATERIAL AND METHODS The experiments were performed on samples of Wilkeson sandstone (WS) to exploit the extensive mechanical characterization performed by Duda & Renner (2013). In particular, the previously elaborated requirements for drained experiments and the constraints on deformation regimes build the foundation for the current study. We report here the relevant specifics of the used apparatus but also refer to its extensive description in Duda & Renner (2013). 2.1 Experimental apparatus and procedure 2.1.1 Setup for triaxial deformation experiments Conventional triaxial compression tests were performed at constant confining pressure pc and nominally constant pore-fluid pressure pf. Here, we refer to their difference as effective pressure, that is, peff = pc − pf, and more generally to any difference between a normal stress and fluid pressure as effective stress. We refrain from introducing a weighting parameter differing from 1 in these differences for two reasons: (1) The entire suite of experiments was performed at the same nominal fluid pressure and thus all pressure and stress levels would simply be shifted by the same constant value; (2) brittle strength seems to obey a scaling with these unweighted differences (see review by Paterson & Wong 2005). The pore-pressure and confining-pressure systems were filled with distilled water and hydraulic oil, respectively. Samples were jacketed by rubber tubes to prevent a connection between the two fluid systems. Porous spacers at the end faces of the samples guaranteed a spatially uniform fluid pressure (Fig. 1). The upper axial loading piston had a central bore to which an external pressure transducer (strain-gauge based, range 100 MPa, accuracy 0.1 per cent) was connected. A flexible high-pressure capillary tube connected the bores of the plug at the sample bottom and of the bottom closure of the vessel enabling connection of a second, identical pressure transducer outside of the vessel that continuously recorded the pore pressure at the lower sample end at hydrostatic conditions (i.e. separated bottom plug and bottom closure) and during deviatoric loading. Figure 1. View largeDownload slide Experimental setup for continuous determination of hydraulic properties and ultrasound measurements during triaxial compression experiments (modified after Duda & Renner, 2013). Distilled water and hydraulic oil were used as pore fluid (blue) and as confining medium (green), respectively. Figure 1. View largeDownload slide Experimental setup for continuous determination of hydraulic properties and ultrasound measurements during triaxial compression experiments (modified after Duda & Renner, 2013). Distilled water and hydraulic oil were used as pore fluid (blue) and as confining medium (green), respectively. Axial piston movement was controlled using a couple of external inductive displacement transducers (range 20 mm, accuracy 0.1 per cent). Axial load was measured with an external load cell (strain gauge based, range 5000 kN, accuracy 0.01 per cent). Confining pressure and pore pressure were applied by servohydraulically controlled pressure intensifiers equipped with inductive displacement transducers (range 300 mm, accuracy 0.1 per cent). The pore-pressure intensifier served as pore-volumometer allowing us to determine pore-volumetric strain ψ = −Δϕcon, that is, the change in connected porosity. Axial strain εax and pore-volumetric strain ψ were determined correcting the corresponding piston displacements for system characteristics as derived from tests on a steel dummy replacing the actual sample. Stress difference Δσ was calculated from loads in excess of the load at hit-point divided by the initial cross-section of the samples, an appropriate approximation for the moderate axial strains achieved. The tangent modulus or apparent Young’s modulus Eapp = dΔσ/dεax representing the local derivative of axial stress with respect to axial strain was calculated for strain increments of 1/100 of axial strain at peak stress. Following the geoscientific convention, compressive stresses and strains are considered positive. 2.1.2 Ultrasonic measurements Two identical, piezo-ceramic ultrasound P-wave transducers (1 MHz centre frequency, 38.1 mm diameter) acting as source and receiver were placed above and below the vessel, respectively, in line with the axial loading pistons (Fig. 1). The transducers were embedded in hollow steel cylinders to prevent them from carrying load. A spring and a coupling medium (graphite lubricant) were used to provide reproducible and constant coupling conditions between transducers and assembly parts. A rectangular signal with an amplitude of 400 V produced by an external waveform generator (Panametrics EPOCH 4B) activated the source to emit 60 mechanical pulses per second into the rock sample. The receiver converted the transmitted elastic wave to an electrical signal. Every 600 s, stacks of 1000 records each covering 400 μs following the excitation by the waveform generator were stored within 17 s with a sampling interval of 8 × 10−9 s using a digital oscilloscope (Tektronix TDS5034B). The first arrival of each signal corresponds to the direct P-wave propagating in axial direction through sample and assembly. Arrival times were determined using a semi-automatic method based on the similarity of the recorded signals. A reference-arrival time was manually picked using the waveform with the highest signal-to-noise ratio of a specific experiment. Arrival times of all other recordings were then determined accounting for the time shift gained from two successive cross correlations performed between consecutively stored signals. The first cross correlation of full time series provided an estimate for the group travel time of the entire coda and allowed for automatically constraining the relevant time interval for the second cross correlation restricted to the first-break wavelet, which provided the finally used time shift. This procedure mitigates some of the ambiguity problems associated with manual picking of signals with variable frequency content and detects subtle changes in arrival time from one transmission to the next. Arrival times were corrected for the travel time in the assembly parts to calculate effective travel times for the samples. The correction values were determined from triaxial tests on a steel sample at the same conditions as the tests on the rock samples. The analysis of the calibration accounted for the contribution of changes in length and P-wave velocity of the steel cylinder with increasing pressure and axial stress (Gerlich & Hart 1984). Velocities were calculated by dividing the current sample length by the determined travel times. Uncertainties in sample geometry, displacement and force measurements, as well as accuracies of Young’s modulus of steel (accuracy 5 per cent), travel and dead time (accuracy 0.3 per cent) lead to absolute errors in P-wave velocity values of less than 110 m s−1. Differences between two determined velocity values (be it at ambient pressure or under triaxial stress) can be resolved to 10 m s−1. 2.1.3 Determination of hydraulic properties Hydraulic properties of the samples were determined using the oscillatory pore-pressure method (e.g. Turner 1958; Stewart et al. 1961; Bennion & Goss 1977). The ends of the saturated porous samples were separately connected to an upstream (composed of upper axial piston, pressure intensifier, and pressure transducer) and a downstream reservoir (composed of bottom plug, capillary tube, bottom closure, and pressure transducer, Fig. 1). Sinusoidal pore-pressure variations were initiated in the upstream reservoir. Amplitude ratio and phase shift between the pressure oscillations in the two reservoirs were derived by Fourier analysis applied to time windows with a duration of four oscillation periods that were successively shifted by 1 s. The standard deviation of amplitude ratio and phase shift gained from 10 successive analyses serves as an uncertainty measure. Response characteristics were inverted to effective hydraulic parameters, that is, permeability k, specific storage capacity s, and hydraulic diffusivity Dhyd = k/(μs), where μ denotes pore-fluid viscosity. Our inversion relies on the analytical solution of the one-dimensional diffusion equation for harmonic flow through an isotropic, homogeneous medium (e.g. Kranz et al. 1990; Fischer 1992; Bernabé et al. 2006), as is true for all currently practiced transient methods for the determination of hydraulic properties (e.g. Brace et al. 1968; Neuzil et al. 1981; Song et al. 2004; Song & Renner 2006). Specific storage capacity has two contributions, the compressibility of the pore-fluid cf and the pore compressibility in response to pore-pressure variations, cpp, following the notation of Zimmerman et al. (1986) for compressibilities:   \begin{equation} s = \phi _\mathrm{con} c_\mathrm{f} + c_\mathrm{pp} . \end{equation} (1)We used cf for water (Wagner 2009) to quantitatively compare our results to the contribution of the pore-fluid compressibility. Apart from the fundamental contemplation of the validity of the simplified model underlying the employed one-dimensional diffusion equation, the uncertainty of hydraulic parameters also strongly depends on the peculiarities of the analytical solution space. Oscillation period and the size of the downstream reservoir are the dominant factors controlling where test results plot in the solution space (e.g. Faulkner & Rutter 2000; Bernabé et al. 2006; Song & Renner 2006). Our tests were carried out with a downstream-reservoir whose storage capacity is only (1.2 ± 0.1) × 10−15 m3 Pa−1 corresponding to about half the samples’ storage capacity and thus providing significant sensitivity for storage-capacity determination. Before commencing triaxial compression we performed oscillatory tests at a range of periods and amplitudes to optimize the parameters of the harmonic excitation subsequently applied during the axial compression regarding avoidance of the peculiar tail of the solution space with poor resolution of hydraulic parameters. The absolute uncertainty of a single value for each of the hydraulic properties amounts to 10 per cent, yet relative variations in permeability, specific storage capacity, and hydraulic diffusivity as low as 4 per cent, 2 per cent and 2 per cent, respectively, can be resolved in a single experiment. 2.2 Sample material WS forms part of the Pudged Group of the Mid Miocene (Gard 1968); our blocks originate from an outcrop in Pierce County, Washington (USA). The sandstone is composed of approximately 55  per cent quartz, 15  per cent plagioclase, 15  per cent sericite, 10  per cent muscovite, 4  per cent opaque minerals, and 1  per cent microcline. The poorly sorted grains exhibit an average size of about 200 μm. The grain-supported fabric features convex-concave grain and phase boundaries (Fig. 2), which indicate an advanced diagenesis of the formation. Plagioclase grains are partly altered to sericite and saussurite. Some quartz grains contain fluid inclusions and show undulatory extinction as well as recrystallization features. Muscovite grains are partly kinked. Figure 2. View largeDownload slide (a) Optical micrograph (crossed polarizers) and (b) scanning-electron microscope image of a polished thin section of intact Wilkeson sst. Long edges of images are parallel to the symmetry axis of the cylindrical test samples. Figure 2. View largeDownload slide (a) Optical micrograph (crossed polarizers) and (b) scanning-electron microscope image of a polished thin section of intact Wilkeson sst. Long edges of images are parallel to the symmetry axis of the cylindrical test samples. Measurements on the block from which samples were cored indicate an anisotropy of 10 to 15 per cent in ultrasonic P-wave velocities at ambient conditions despite the lack of visible layering. Five cylindrical samples with a diameter of 30 mm were diamond-drilled parallel to the direction of the highest velocity and their end faces were ground square to a final length of 75.00 ± 0.02 mm, that is, the sample geometry complied with a length-to-diameter ratio of 2.5:1 that limits the effect of friction at the interface with the steel pistons on the state of stress in the sample centre (Paterson & Wong 2005). Preparation was conducted with water as coolant. Fundamental physical properties were determined on all samples at ambient conditions to evaluate the homogeneity of the suite considered for triaxial testing. Bulk density ρ of samples was calculated from their masses after drying at 60 °C and geometrical volume determination. The average mineral density $$\bar{\rho }_{{\rm min}} = 2623 \pm 27\,{\rm kg \, m}^{-3}$$ was gained from pycnometer measurements on powder produced by crushing and grinding pieces of the sandstone permitting to derive total porosity according to $$\phi _{{\rm tot}}= 1 - (\rho /\bar{\rho }_{{\rm min}})$$. Connected porosity was calculated from the increase in masses after evacuation and subsequent saturation of samples with distilled water (for more details see Duda & Renner 2013). 2.2.1 Physical properties at ambient conditions The measurements of the basic physical properties of the cylindrical samples confirms the general homogeneity of the sandstone block but the resolution of the individual properties indicates some variability (Table 1). Total and connected porosity differ by about 0.5  per cent, yet, this difference is barely significant considering absolute uncertainties. Dry samples exhibit ratios of P- to S-wave velocities close to 1.5. Upon saturation, P-wave velocities increase by almost 30  per cent. The properties of the samples used in this study deviate only slightly from the ones used by Duda & Renner (2013) that were prepared from a different block. The previously used samples were slightly less porous (8.4  per cent connected porosity compared to 8.9  per cent), but this difference will not affect the validity of results from Duda & Renner (2013) relevant to and for this study. Table 1. Basic physical properties of tested samples as determined at ambient conditions. Sample  ρ (kg m−3)  ϕtot (per cent)  ϕcon (per cent)  vP,dry (m s−1)  vS,dry (m s−1)  vP,sat (m s−1)  WS-4 K  2375 ± 8  9.5 ± 0.7  8.9 ± 1.0  2797 ± 83  1835 ± 219  3974 ± 153  WS-4 L  2364 ± 8  9.9 ± 0.7  9.4 ± 1.0  2687 ± 80  1827 ± 229  3949 ± 150  WS-4 N  2381 ± 8  9.2 ± 0.7  8.9 ± 1.0  2705 ± 80  1860 ± 222  4034 ± 156  WS-4 O  2383 ± 8  9.2 ± 0.7  8.9 ± 1.0  2776 ± 83  1903 ± 222  3937 ± 148  WS-4 R  2370 ± 8  9.7 ± 0.7  9.0 ± 1.0  2738 ± 82  1871 ± 224  3938 ± 150  mean ±SD  2378 ± 8  9.5 ± 0.3  8.9 ± 0.2  2738 ± 46  1860 ± 30  3949 ± 41  Sample  ρ (kg m−3)  ϕtot (per cent)  ϕcon (per cent)  vP,dry (m s−1)  vS,dry (m s−1)  vP,sat (m s−1)  WS-4 K  2375 ± 8  9.5 ± 0.7  8.9 ± 1.0  2797 ± 83  1835 ± 219  3974 ± 153  WS-4 L  2364 ± 8  9.9 ± 0.7  9.4 ± 1.0  2687 ± 80  1827 ± 229  3949 ± 150  WS-4 N  2381 ± 8  9.2 ± 0.7  8.9 ± 1.0  2705 ± 80  1860 ± 222  4034 ± 156  WS-4 O  2383 ± 8  9.2 ± 0.7  8.9 ± 1.0  2776 ± 83  1903 ± 222  3937 ± 148  WS-4 R  2370 ± 8  9.7 ± 0.7  9.0 ± 1.0  2738 ± 82  1871 ± 224  3938 ± 150  mean ±SD  2378 ± 8  9.5 ± 0.3  8.9 ± 0.2  2738 ± 46  1860 ± 30  3949 ± 41  ρ: bulk density; ϕtot: total porosity; ϕcon: connected porosity; vP,dry, vP,sat: P-wave velocity of dry and saturated sample; vS,dry: S-wave velocity of dry sample. Quoted uncertainties reflect accuracies of the measurements. The resolution of measurements relevant for assessing the significance of differences between two bulk densities, total porosities, connected porosites, P- and S-wave velocities of dry sample, and P-wave velocities of saturated samples amount to 0.8 kg m−3, 0.3  per cent, 0.1  per cent, 10 m s−1, 46 m s−1, and 21 m s−1, respectively. View Large 2.2.2 Physical properties at effective hydrostatic pressures up to 160 MPa Axial, pore- and bulk-volumetric strain, ultrasonic P-wave velocity, and hydraulic properties were determined for one sample during hydrostatic loading at effective pressures up to 160 MPa to constrain the crack-closure pressure, basic elastic parameters, and the pressure sensitivity of hydraulic properties (Fig. 3). Confining pressure was increased in steps of 10 MPa starting from 40 MPa while the pore pressure had a nominally constant value of 30 MPa but was continuously oscillated at the upstream reservoir with an amplitude of 1 MPa and a period of 20 s. After each increase in confining pressure we waited until pore pressure was re-equilibrated within the sample as indicated by a ceasing movement of the volumometer piston to ensure effectively drained conditions. Changes in sample length and corresponding axial strain were derived from monitoring hit-points. Bulk-volumetric strain was estimated from the axial strain assuming isotropy, that is, θ = 3εax. Three pressure settings were revisited on the way back to initial conditions to test for permanent changes induced by the pressurization. Figure 3. View largeDownload slide (a) Bulk-volumetric strain θ = 3εax (black circles) and pore-volumetric strain $$\psi = \Delta \psi + \theta (p_\mathrm{eff}=10\,{\rm MPa})$$ (red triangles), (b) P-wave velocity vP,sat, (c) permeability k, (d) specific storage capacity s, and (e) hydraulic diffusivity Dhyd as a function of effective confining pressure peff during hydrostatic deformation of Wilkeson sst up to 160 MPa. In (a), pore-volumetric strains that were determined starting from $$p_\mathrm{eff}=10\,{\rm MPa}$$ were shifted by the bulk-volumetric strain at 10 MPa assuming isotropic deformation. The pressure range highlighted in grey indicates the crack-closure pressure (see the text for discussion). The dashed black line in (d) represents the contribution from the storage capacity of water alone (eq. 1). Figure 3. View largeDownload slide (a) Bulk-volumetric strain θ = 3εax (black circles) and pore-volumetric strain $$\psi = \Delta \psi + \theta (p_\mathrm{eff}=10\,{\rm MPa})$$ (red triangles), (b) P-wave velocity vP,sat, (c) permeability k, (d) specific storage capacity s, and (e) hydraulic diffusivity Dhyd as a function of effective confining pressure peff during hydrostatic deformation of Wilkeson sst up to 160 MPa. In (a), pore-volumetric strains that were determined starting from $$p_\mathrm{eff}=10\,{\rm MPa}$$ were shifted by the bulk-volumetric strain at 10 MPa assuming isotropic deformation. The pressure range highlighted in grey indicates the crack-closure pressure (see the text for discussion). The dashed black line in (d) represents the contribution from the storage capacity of water alone (eq. 1). Bulk- and pore-volumetric strain, and P-wave velocity exhibit similar relations to increasing pressure (Fig. 3). Strains and velocity exhibit steep nonlinear increases with increasing effective pressure that give way to modest linear increases above about 40 to 50 MPa. The transitional behaviour is commonly attributed to the successive closure of microcracks (Walsh 1966; Mavko & Nur 1978) - the higher the pressure the lower the aspect ratio ξ (ratio between width and length) of closing cracks. The pressure at the beginning of a linear relation between bulk-volumetric strain and confining pressure is commonly referred to as the critical crack-closure pressure pcc. Non-porous, but cracked rocks exhibit a velocity plateau once this critical pressure is exceeded (e.g. Mavko & Nur 1978). The continuous velocity increase exhibited by the tested porous sandstone demonstrates that microstructural changes, for example, modifications of grain contacts, continue to occur up to the highest explored pressure. David & Zimmerman (2012) associated such behaviour with the distribution function of aspect ratios. By somewhat generalizing its original meaning, we will use the term crack-closure pressure to address the observed change in sensitivity to pressure probably associated with the presence of a dominant aspect ratio of microcracks perceptible to closure. During hydrostatic loading permeability and specific storage capacity vary from 5 to 12 × 10 − 18 m2 and 2 to 6 × 10 − 11 Pa − 1, respectively, and both exhibit a peculiar difference between the values determined at truly hydrostatic conditions and at a stress difference as little as 1 MPa applied to determine current sample length from the hit-point (Figs 3c and d). At axial load, values of k and s are smaller than at hydrostatic conditions. These differences are the largest below the closure pressure ( ∼ 50 MPa) up to which k and s at axial load tend to increase whereas values determined at hydrostatic conditions more systematically decrease with increasing effective pressure throughout. Almost all of the determined storage capacity values correspond to apparent porosities exceeding, albeit only slightly, the nominal connected porosity determined at ambient conditions (Fig. 3d) underlining that the pore space dominating storage is rather undeformable as also indicated by the modest pressure dependence of s (Fig. 3d). Hydraulic diffusivity ranges between 1.1 and 1.8 × 10 − 4 m2 s − 1 and exhibits a relation between values determined at hydrostatic conditions and at small load that is opposite to the one for k and s (Fig. 3e). The reproducibility during pressure decrease differs for the various properties (Fig. 3). The match between volumetric strains during pressurization and depressurization excludes significant inelastic deformation during the performed cycle of hydrostatic loading but a slight permanent compaction (reduction in bulk volume by loss of porosity of a few tenths of a percent). Permeability decreases by almost 20  per cent in accord with the indicated loss in porosity. Specific storage capacity is however reproducible. These observations indicate permanent closure of voids to which permeability is sensitive but storage capacity is not. The slope of the linear increase of the approximated bulk-volumetric strain with effective pressure (Fig. 3a) corresponds to a lower bound for the quasi-static drained bulk modulus of $$K_\mathrm{d}=c_\mathrm{bc}^{-1}=15.6 \pm 0.6\,{\rm GPa}$$ (Table 3) assuming isotropy. The pronounced pressure dependence of strains and velocity demonstrates that the deformation is dominated by crack closure. Previous work on cracked samples demonstrated that anisotropy diminishes with crack closure (e.g. Zang et al. 1989; Wang 2002) and thus the degree of anisotropy indicated by the velocity measurements at ambient conditions likely does not hold for the samples at test conditions. This estimate constitutes an upper bound for the true value if the anisotropy in P-wave velocity observed at ambient conditions holds also for static elastic parameters at elevated pressure. Even this upper bound is fairly low compared to typical bulk moduli of rock-forming minerals likely due to the still significant total porosity of about 7–8  per cent at the explored effective pressures, estimated by reducing the total porosity determined at ambient pressure by the change in porosity associated with crack closure gained from the intercept of pore-volumetric strain versus effective pressure, that is, $$\psi _\mathrm{crack} \approx 0.9\,{\rm \hbox{\,per\,cent}}$$ (Fig. 3a). 2.3 Experimental conditions We chose effective pressures of 10, 20, 50 and 100 MPa for the triaxial compression experiments to cover the full range from below to above the identified crack-closure pressure (Fig. 3) and from brittle faulting to cataclastic flow as deduced from our previous experiments (Duda & Renner 2013). The hydraulic diffusivity of the samples determined at hydrostatic conditions just before the triaxial deformation confirmed the validity of our previously determined requirements for drained conditions (Duda & Renner 2013). Strain rates between 1 × 10−7 and 3 × 10−7 s−1 (Table 2) were thus imposed that fall significantly below the estimated critical strain rates of 4 × 10−6 to 1 × 10−5 s−1. The nominal pore-fluid pressure of 30 MPa was continuously oscillated at the upstream end with an amplitude of 1 MPa, that is, a modest perturbation of about 3 per cent, and a period of 20 s. Oscillation characteristics were inverted to hydraulic properties every 10 s, corresponding to axial strain increments of 1 × 10−6 to 3 × 10−6 for the imposed strain rates. Stacked seismograms were stored for axial strain increments of about 6 × 10−5 to 1.8 × 10−4 . Table 2. Conditions of performed deformation experiments and hydraulic properties as determined at hydrostatic conditions before deviatoric loading was commenced with the indicated strain rates $$\dot{\epsilon }$$ in comparison to properties of sample WS-4 R exclusively tested at hydrostatic conditions between 10 and 160 MPa effective pressure (see Fig. 3). peff (MPa)  Sample  k (m2)  s (Pa−1)  Dhyd (m2 s−1)  vP (m s−1)  $$\dot{\epsilon }$$ (s−1)  10  WS-4 K  (5.0 ± 0.3) × 10 − 18   (4.3 ± 0.1) × 10 − 11   (9.1 ± 0.4) × 10 − 5   4283 ± 11   1 × 10−7     WS-4 R  (1.0 ± 0.1) × 10 − 17   (5.6 ± 0.1) × 10 − 11   (1.4 ± 0.1) × 10 − 4   4240 ± 11   –  20  WS-4 O  (3.8 ± 0.3) × 10 − 18   (4.1 ± 0.2) × 10 − 11   (7.2 ± 0.2) × 10 − 5   4398 ± 11   1 × 10−7     WS-4 R  (1.1 ± 0.1) × 10 − 17   (5.3 ± 0.1) × 10 − 11   (1.5 ± 0.1) × 10 − 4   4400 ± 11   –  50  WS-4 L  (1.2 ± 0.1) × 10 − 17   (5.1 ± 0.1) × 10 − 11   (1.8 ± 0.1) × 10 − 4   4562 ± 11   3 × 10−7     WS-4 R  (1.1 ± 0.1) × 10 − 17   (5.6 ± 0.1) × 10 − 11   (1.5 ± 0.1) × 10 − 4   4620 ± 11   –  100  WS-4 N  (4.1 ± 0.3) × 10 − 18   (4.3 ± 0.2) × 10 − 11   (7.3 ± 0.6) × 10 − 5   4696 ± 11   3 × 10−7     WS-4 R  (8.4 ± 0.1) × 10 − 18   (4.6 ± 0.1) × 10 − 11   (1.4 ± 0.6) × 10 − 4   4710 ± 11   –  peff (MPa)  Sample  k (m2)  s (Pa−1)  Dhyd (m2 s−1)  vP (m s−1)  $$\dot{\epsilon }$$ (s−1)  10  WS-4 K  (5.0 ± 0.3) × 10 − 18   (4.3 ± 0.1) × 10 − 11   (9.1 ± 0.4) × 10 − 5   4283 ± 11   1 × 10−7     WS-4 R  (1.0 ± 0.1) × 10 − 17   (5.6 ± 0.1) × 10 − 11   (1.4 ± 0.1) × 10 − 4   4240 ± 11   –  20  WS-4 O  (3.8 ± 0.3) × 10 − 18   (4.1 ± 0.2) × 10 − 11   (7.2 ± 0.2) × 10 − 5   4398 ± 11   1 × 10−7     WS-4 R  (1.1 ± 0.1) × 10 − 17   (5.3 ± 0.1) × 10 − 11   (1.5 ± 0.1) × 10 − 4   4400 ± 11   –  50  WS-4 L  (1.2 ± 0.1) × 10 − 17   (5.1 ± 0.1) × 10 − 11   (1.8 ± 0.1) × 10 − 4   4562 ± 11   3 × 10−7     WS-4 R  (1.1 ± 0.1) × 10 − 17   (5.6 ± 0.1) × 10 − 11   (1.5 ± 0.1) × 10 − 4   4620 ± 11   –  100  WS-4 N  (4.1 ± 0.3) × 10 − 18   (4.3 ± 0.2) × 10 − 11   (7.3 ± 0.6) × 10 − 5   4696 ± 11   3 × 10−7     WS-4 R  (8.4 ± 0.1) × 10 − 18   (4.6 ± 0.1) × 10 − 11   (1.4 ± 0.6) × 10 − 4   4710 ± 11   –  peff: effective confining pressure; k: permeability; s: specific storage capacity; Dhyd: hydraulic diffusivity; vP: P-wave velocity. Individual values are amended by the resolution relevant for comparison of the results for the tested samples. The absolute accuracies of the measurements are quoted in the text. View Large 3 RESULTS The homogeneity of the used block, already indicated by the close match between the physical properties of the prepared samples determined at ambient pressure (Table 1), was further confirmed by the measurements performed after pressurization of the samples and before commencing axial deformation. In detail, we find, however, the properties of the two samples with the highest total porosity, WS-4 R tested at hydrostatic conditions and WS-4 L axially deformed at 50 MPa effective pressure, to deviate slightly from those of the other three samples (Tables 1 and 2). Their permeability and diffusivity are initially about twice as high as those of the other samples and their ultrasonic P-wave velocities are the lowest of the sample suite. Yet, these differences are small enough to safely attribute the differences in results of the four deformation tests to the differences in imposed effective pressure rather than to sample variability. 3.1 Deformation characteristics The stress-strain curves are characteristic for the pressure-induced transition from dilative brittle failure to compactive cataclastic flow (Fig. 4) as confirmed by the observed macroscopic failure modes. Here, we use ‘compaction’ and ‘compactive’ or ‘dilation’ and ‘dilative’ to indicate a relative decrease or increase in current sample volume, respectively. Furthermore, ‘low’ and ‘high’ pressure will be meant relative to the determined crack-closure pressure of 40 to 50 MPa. Up to effective pressures of 50 MPa, initial compaction reverses into significant dilation accompanying a drop in axial stress to constant, pressure-dependent residual-stress levels (Fig. 4a). The magnitude of the drop after peak stress is the lower the higher the effective pressure. Likewise, the degree of dilation systematically decreases with increasing effective pressure; the sample tested at 100 MPa deforms at almost constant pore volume beyond the modest peak in stress (Fig. 4b). The stress sensitivity of the initial pore compaction seems unaffected by pressure whereas the slope of axial stress-axial strain curves systematically increase with pressure. The latter are never really linear but exhibit an S-shape as commonly observed for porous rocks (e.g. Duda & Renner 2013). Figure 4. View largeDownload slide Axial stress difference Δσ as a function of (a) axial strain εax and (b) pore-volumetric strain ψ during triaxial compression experiments on saturated Wilkeson sst at indicated effective confining pressures. Diamonds and squares give maxima in P-wave velocity and onset of dilation as deduced from pore-volumetric strain, respectively. Figure 4. View largeDownload slide Axial stress difference Δσ as a function of (a) axial strain εax and (b) pore-volumetric strain ψ during triaxial compression experiments on saturated Wilkeson sst at indicated effective confining pressures. Diamonds and squares give maxima in P-wave velocity and onset of dilation as deduced from pore-volumetric strain, respectively. The morphological features of deformed samples retrieved from the vessel almost fully cover the transition from a single shear fault to pervasive cataclasis barely localized on macroscopic scale (see for example Hadizadeh & Rutter 1983). The angle of the fault to the cylinder axis decreases from about 70° to about 50° and the fault zone widens from about 2 to almost 10 mm with increasing effective pressure up to 50 MPa; the sample tested at peff = 100 MPa exhibits only a hint of a localized fault in a pervasively deformed zone that covers half of the sample. These observations on fault morphology match well with the characteristics of the stress-strain curves exhibiting a transition from significant stress drop to modest softening after peak stress with increasing effective pressure (Fig. 4). 3.2 Evolution of physical properties during triaxial deformation 3.2.1 Ultrasound P-wave velocities During triaxial compression, P-wave velocity evolves qualitatively similar for all investigated effective pressures but quantitative differences are significant (Fig. 5a). The evolution with increasing strain resembles inverted parabolas: P-wave velocity increases significantly right upon the onset of deviatoric loading and subsequently exhibits a maximum well before peak stress. The higher the pressure is the higher the velocities are but the less relative changes they exhibit during the course of an experiment. The axial strains at the maxima in velocity differ only slightly between the four experiments and thus fall increasingly short of the onset of dilation as determined from pore-volumetric strains with increasing effective pressure. At an effective pressure of 100 MPa, the strain at the onset of dilation is more than twice the one at the velocity maximum. The stress drop likely associated with localized brittle failure is accompanied by a distinct drop in velocity whose magnitude decreases with increasing effective pressure. At residual stresses, P-wave velocity falls below its initial value determined on intact samples at hydrostatic conditions for all effective pressures but 10 MPa. Figure 5. View largeDownload slide P-wave velocity vP as a function of (a) axial strain εax, (b) deviatoric radial strain with respect to its value at maximum P-wave velocity, εrad − εrad(vP,max), (c) effective mean stress σm,eff = Δσax/3 + peff, and (d) effective stress ratio λ = (Δσax + peff)/peff during (c) hydrostatic (circles) and deviatoric (lines) loading at indicated effective confining pressures. In (b,d) P-wave velocities were normalized by their maxima vP, max and by the values measured at hydrostatic conditions before triaxial loading was commenced (vP,0), respectively. Squares and diamonds indicate maxima in P-wave velocity (amended by uncertainty range) and onset of dilation as deduced from pore-volumetric strain, respectively. The arrow in (b) indicates the progression of deformation. Figure 5. View largeDownload slide P-wave velocity vP as a function of (a) axial strain εax, (b) deviatoric radial strain with respect to its value at maximum P-wave velocity, εrad − εrad(vP,max), (c) effective mean stress σm,eff = Δσax/3 + peff, and (d) effective stress ratio λ = (Δσax + peff)/peff during (c) hydrostatic (circles) and deviatoric (lines) loading at indicated effective confining pressures. In (b,d) P-wave velocities were normalized by their maxima vP, max and by the values measured at hydrostatic conditions before triaxial loading was commenced (vP,0), respectively. Squares and diamonds indicate maxima in P-wave velocity (amended by uncertainty range) and onset of dilation as deduced from pore-volumetric strain, respectively. The arrow in (b) indicates the progression of deformation. The relation between the P-wave velocities recorded for the four triaxial compression experiments and the ones observed for hydrostatic conditions exhibit a peculiar change with increasing effective mean stress, σm,eff = σm − pf = Δσax/3 + peff (Fig. 5c). Below the crack-closure pressure (i.e. at peff = 10 and 20 MPa) velocities recorded in the triaxial experiments immediately exceed their hydrostatic counterparts until the maximum is reached that closely matches the hydrostatic reference value. Above the crack-closure pressure, velocities at triaxial conditions fall persistently short of the ones found for hydrostatic conditions. 3.2.2 Hydraulic properties The reported hydraulic properties refer to hypothetical, equivalent, homogeneous and isotropic media that were to give the same phase shift and amplitude ratio as the observed ones. Although samples obviously become heterogeneous during faulting (and likely develop anisotropy that even changes during the course of deformation), such an equivalent medium is found, that is, observed phase shift and amplitude ratio actually fall into the solution space of the diffusion equation. We can determine effective hydraulic properties, but for two strain sequences. The oscillation characteristics observed for the samples tested at effective pressures of 10 MPa and 50 MPa temporarily fall outside of the analytical solution space during the stress drop and during the softening phase, respectively (Fig. 6). Figure 6. View largeDownload slide Graphical representation of the solution space for the one-dimensional pressure-diffusion equation in the domain of phase shift φ and amplitude ratio δpp. The grid is spanned by isolines of dimensionless storage capacity ξ and dimensionless permeability η (increasing in the indicated directions). The coloured lines represent phase shift-amplitude ratio pairs determined during triaxial deformation of Wilkeson sst at indicated effective confining pressures (the black arrow indicates deformation progress). Figure 6. View largeDownload slide Graphical representation of the solution space for the one-dimensional pressure-diffusion equation in the domain of phase shift φ and amplitude ratio δpp. The grid is spanned by isolines of dimensionless storage capacity ξ and dimensionless permeability η (increasing in the indicated directions). The coloured lines represent phase shift-amplitude ratio pairs determined during triaxial deformation of Wilkeson sst at indicated effective confining pressures (the black arrow indicates deformation progress). Our a priori analysis for the drainage conditions of the experiments remains valid in the light of the a posteriori knowledge of the range in diffusivity determined during triaxial compression. We estimate a penetration depth of $$\,r_\mathrm{p}\sim \sqrt{D_\mathrm{hyd}T}$$ of about half the sample length (the longest distance to a reservoir), even for the lowest observed diffusivity values of 5 × 10−5 m2 s−1 on the timescale of the imposed oscillation period of T = 20 s. Also, we did not observe any of the characteristic features pointing to a potential lack in drainage as described in Duda & Renner (2013). We initially restrict to comparing the evolution of physical properties of the experiments performed at 20 and 100 MPa effective pressure (Fig. 7) exemplifying the two distinct failure regimes covered by our experiments, localized brittle failure and incipient cataclastic flow. The relative changes in hydraulic properties during the course of the deformation are larger for lower effective pressure, for example, one and a half orders of magnitude increase in permeability at 20 MPa compared to one order of magnitude increase at 100 MPa. The variations in specific storage capacity and hydraulic diffusivity cover smaller ranges than the ones in permeability. The moderate variability of diffusivity is an expression of the general tendency for permeability and specific storage capacity to covary. Figure 7. View largeDownload slide (a,b) Permeability k, (c,d) specific storage capacity s (in comparison to the contribution of the compressible pore fluid alone as indicated by the horizontal dashed lines calculated employing the connected porosity ϕcon at ambient conditions), and (e,f) hydraulic diffusivity Dhyd during triaxial deformation of Wilkeson sst at 20 and 100 MPa effective confining pressure as a function of axial strain εax (left) and pore-volumetric strain ψ (right). The grey curves indicate the ranges of absolute uncertainty. Where these are not visible, uncertainties do not exceed the thickness of the coloured curves. Squares, diamonds, and triangles indicate maxima in P-wave velocity, onset of dilation as deduced from pore-volumetric strain, and peak stress, respectively. Figure 7. View largeDownload slide (a,b) Permeability k, (c,d) specific storage capacity s (in comparison to the contribution of the compressible pore fluid alone as indicated by the horizontal dashed lines calculated employing the connected porosity ϕcon at ambient conditions), and (e,f) hydraulic diffusivity Dhyd during triaxial deformation of Wilkeson sst at 20 and 100 MPa effective confining pressure as a function of axial strain εax (left) and pore-volumetric strain ψ (right). The grey curves indicate the ranges of absolute uncertainty. Where these are not visible, uncertainties do not exceed the thickness of the coloured curves. Squares, diamonds, and triangles indicate maxima in P-wave velocity, onset of dilation as deduced from pore-volumetric strain, and peak stress, respectively. The evolution of the hydraulic properties changes notably when the maximum in P-wave velocity occurs and when dilation sets in (Fig. 7). At low effective pressure, the two conditions are simultaneously reached and mark the onset of a significant increase in all hydraulic properties; at high effective pressure, the maximum in P-wave velocity occurs first marking the minimum in hydraulic properties while the onset of dilation coincides with an inflection point of their subsequent increase with axial strain. The stress drop associated with macroscopic brittle failure at low effective pressure is accompanied by a sudden, counterintuitive, transient decrease in permeability and specific storage capacity, yet in combination still resulting in an increase in diffusivity. Hydraulic properties seem to approach fairly constant values when frictional sliding on the developed fault prevails at residual stresses. Permeability and specific storage capacity reach their highest levels at residual stress that however correspond to a diffusivity that falls below values reached earlier during triaxial deformation. During the slight post-peak softening associated with a continuous increase in pore volume observed for cataclastic flow at high effective pressure, permeability stays constant and specific storage capacity exhibits a modest decrease resulting in an increase in diffusivity. Over the course of a deformation experiment, the variations of the three hydraulic properties, permeability, specific storage capacity, and hydraulic diffusivity, significantly exceed the changes due to hydrostatic loading at comparable effective mean stresses (Figs 8a, c and e). Notably, the initial trends with mean stress do not match the ones with pressure in particular for the two experiments performed at 10 and 20 MPa, that is, below crack-closure pressure. The immediate decrease-contrasting the counterintuitive increase of the property values with increasing hydrostatic pressure below the crack-closure pressure- is followed by an increase exceeding the ones observed for hydrostatic conditions. Compared to k and s, diffusivity exhibits the least deviation from the hydrostatic reference curve before dilation sets in. Figure 8. View largeDownload slide Permeability k, specific storage capacity s, and hydraulic diffusivity Dhyd as a function of effective mean stress σm,eff = Δσax/3 + peff (a,c,e) and effective stress ratio λ = (Δσax + peff)/peff (b,d,f) during hydrostatic (circles) and deviatoric (lines) loading at indicated effective confining pressures. For (a,c,e) we shifted the results for the onset of deviatoric loading to match with the reference values gained from hydrostatic loading to remove the slight sample-to-sample variability (Table 1) and thus facilitate comparison of the mean-stress dependence. For (b,d,f) we normalized the hydraulic properties by their respective magnitude at the onset of deviatoric loading (subscript 0) to eliminate the effect of confining pressure on their absolute level. Squares and diamonds indicate maxima in P-wave velocity and onset of dilation as deduced from pore-volumetric strain, respectively. Figure 8. View largeDownload slide Permeability k, specific storage capacity s, and hydraulic diffusivity Dhyd as a function of effective mean stress σm,eff = Δσax/3 + peff (a,c,e) and effective stress ratio λ = (Δσax + peff)/peff (b,d,f) during hydrostatic (circles) and deviatoric (lines) loading at indicated effective confining pressures. For (a,c,e) we shifted the results for the onset of deviatoric loading to match with the reference values gained from hydrostatic loading to remove the slight sample-to-sample variability (Table 1) and thus facilitate comparison of the mean-stress dependence. For (b,d,f) we normalized the hydraulic properties by their respective magnitude at the onset of deviatoric loading (subscript 0) to eliminate the effect of confining pressure on their absolute level. Squares and diamonds indicate maxima in P-wave velocity and onset of dilation as deduced from pore-volumetric strain, respectively. 4 DISCUSSION Our study exceeds previous ones in terms of comprehensiveness of constraints on effective hydraulic properties. Compared to studies that also used the oscillatory method we achieve a higher resolution of permeability and storage capacity during deformation (Fischer 1992; Zhang et al. 1994; Mitchell & Faulkner 2008). Studies employing pulse techniques (e.g. Zoback & Byerlee 1975; Zhu & Wong 1997; Renner et al. 2000) yield only permeability and typically suffer from relaxation effects when deformation is paused for the hydraulic measurement. These effects could potentially be mitigated by adjusting the confining pressure or the axial load (Keaney et al. 1998), but only at the expense of an alteration of the deformation process. The instances during which the oscillation characteristics do not fall into the solution space (Fig. 6) coincide with localization of deformation and thus bear the potential to characterize the development of heterogeneity. Exploitation of this potential requires derivation of solution spaces for more advanced models, a subject that goes beyond the scope of the current presentation. The chosen strain rate ensured internal drainage. One should, however, not confuse the timescale resulting from the analysis of the critical strain rate with a requirement for the oscillation period to be employed in an oscillatory test. The scaling relation between timescale and penetration depth corresponds to a significant pressure perturbation, typically on the order of 10  per cent or more depending on flow geometry (e.g. Weir 1999). The pore-pressure method, in contrast, yields information regarding the entire sample length independent of chosen oscillation period because Fourier analysis allows for the detection of rather weak signals. Here, the lowest amplitude ratios analysed are on the order of 10−2 (Fig. 6). The gained effective hydraulic properties represent the path(s) connecting the sample ends along which pressure diffusion is most effective. Oscillation period affects to what extent subsidiary branches of the pore and crack network (including dead ends) contribute to transport and storage in a heterogeneous sample. In the following, we first discuss the microstructural state of the Wilkeson sandstone samples before and then indicators for changes in microstructure during deviatoric deformation, such as anisotropy and onset of inelastic deformation. We proceed by analysing the interrelation between the monitored properties before concluding with implications for contributions of hydro-mechanical coupling to in situ processes. 4.1 Microstructural state of samples before deformation as derived from their physical properties Wilkeson sandstone exhibits a peculiar combination of the characteristics of cracked and porous media. The pronounced changes in the pressure dependences of axial and pore-volumetric strain, and P-wave velocity under hydrostatic conditions (Figs 3a and b) demonstrate the presence of microfractures that close in the explored range of effective pressures. Yet, the modest pressure dependence of the hydraulic properties and the absolute value of specific storage capacity, which deviates little from what is expected from the fluid compressibility alone (Fig. 3d), suggest a well connected rather stiff pore space. Thus, the microfractures closing under pressure have apparently restricted relevance for the interconnectivity of the conduit network and storage dominantly occurs in pores that are not sensitive to pressure. The absolute value of specific storage capacity and its insensitivity to pressure suggest about 8  per cent stiff pores that are complemented by deformable microfractures with ϕcrack ∼ 1  per cent as indicated by the intercepts of the linear relations of bulk and pore-volumetric strain to pressure (Fig. 3a). These fractions appear in good agreement with the visual impression gained from SEM imaging (Fig. 2b). The conceptual notion of a combination of pores and cracks can also explain the contrasting observations of hysteresis in the examined physical properties during the exerted cycle of hydrostatic pressurization and depressurization (Fig. 3). Likely, cracks oriented parallel to the cylinder axis contribute significantly to bulk permeability and remain partly closed during unloading with no major impact on axial strain or axial P-wave velocity, and only subordinately affect specific storage capacity. Obviously, the small increase in permeability with increasing pressure and the curious difference between hydraulic properties at hydrostatic conditions and at small deviatoric loads are also related to the ‘closable’ fractures since this phenomenon essentially vanishes above the identified crack-closure pressure (Figs 3c–e). Intrinsic anisotropy and its development during hydrostatic loading -not accounted for in the analyses- may cause artefacts in the followed determination of hydraulic properties that result in these peculiar observations. This behaviour may, however, also be a result of the shift in density and thus relevance of the two void features involved, fractures with large aspect ratios and fairly equant pores as representative of a granular medium. The closing of the fractures may lead to a subtle change towards a less fracture-affected pore network for which the averaging involved in transient hydraulic tests actually yields an increase in permeability. The mechanical response of an isotropic, homogeneous, linear poroelastic medium to hydrostatic loading is fully described by three bulk moduli (compressibilities), for example, the drained (also addressed as dry or skeleton) modulus $$K_\mathrm{d}(=c_\mathrm{bc}^{-1})$$, the average mineral modulus $$K_\mathrm{r}(=c_\mathrm{r}^{-1})$$, and the fluid modulus $$K_\mathrm{f}(=c_\mathrm{f}^{-1})$$. Our experiments allow us to derive three independent elastic parameters (Kd and cpc from the linear sections of the corresponding strain records, Fig. 3a, and s from the hydraulic testing, Fig. 3d) and thus -considering fluid compressibility to be known- we can evaluate their consistency with linear poroelasticity for a homogeneous and isotropic medium. Specific storage capacity derived from compressibility values is twice as large as the value actually observed above the closure pressure (Table 3, Fig. 3). This apparent discrepancy may in principle point to violated assumptions regarding isotropy and linearity. However, we suppose that the quantitative mismatch reflects structural heterogeneity as also indicated by a storage capacity below the contribution of the pore fluid alone during parts of the deformation suggesting hydraulically preferred pore features (Figs 7c and d). Observed storage capacity is very much expected to be smaller than the one predicted from static compressibility (moduli) measurements since only the conducting fractures contribute to it while the static moduli are sensitive to all compliant pore features. Table 3. Measured and derived dynamic and quasi-static mechanical parameters for Wilkeson sst.   Parameter  Relation  Value    Results from ultrasonic wave transmission at ambient conditions    Drained bulk modulus  $$\tilde{K}_{{\rm d}} = \rho _{{\rm dry}}(v_{{\rm P,dry}}^2-\frac{4}{3}v_{{\rm S,dry}}^2)$$  (6.9 ± 0.7) GPa    Undrained bulk modulus  $$\tilde{K}_{{\rm ud}} = \rho _{{\rm sat}}(v_{{\rm P,sat}}^2-\frac{4}{3}v_{{\rm S,sat}}^2)$$  (27.5 ± 0.3) GPa  dynamic  Shear modulus  $$\tilde{G} = \rho _{{\rm dry}}v_{{\rm S,dry}}^2$$  (8.2 ± 0.3) GPa    Measured parameters at hydrostatic conditions (peff > 40 MPa)    Drained bulk compressibility  $$c_{{\rm bc}}= 3\mathrm{d}\epsilon _\mathrm{ax}/\mathrm{d} \Delta \sigma |_{p_\mathrm{f}}$$  (6.5 ± 0.1) × 10 − 11 Pa − 1    Drained bulk modulus  $$K_{{\rm d}}=c_{{\rm bc}}^{-1}$$  (15.6 ± 0.6) GPa    Pore compressibility  $$c_{\mathrm{pc}}=\mathrm{d}\psi /\mathrm{d} \Delta \sigma |_{p_\mathrm{f}}/\phi _\mathrm{con}$$  (5.1 ± 0.1) × 10 − 10 Pa − 1    Specific storage capacity  s  (4.1 ± 0.3) × 10 − 11 Pa − 1  quasi-static  Derived parameters at hydrostatic conditions (peff > 40 MPa)    Average bulk modulus of minerals  $$K_{\mathrm{r}}(c_{\mathrm{bc}},c_{\mathrm{pc}}) = c_{\mathrm{r}}^{-1} = (c_{\mathrm{bc}}-\phi _\mathrm{con}c_{\mathrm{pc}})^{-1}$$  (51.7 ± 4.1) GPa    Biot–Willis parameter  α(cbc, cpc) = ϕconcpc/cbc  (0.70 ± 0.04)    Specific storage capacity  s(cbc, cpc) = ϕcon[cf − cbc + (1 + ϕcon)cpc]  (8.2 ± 0.4) × 10 − 11 Pa − 1    Parameter  Relation  Value    Results from ultrasonic wave transmission at ambient conditions    Drained bulk modulus  $$\tilde{K}_{{\rm d}} = \rho _{{\rm dry}}(v_{{\rm P,dry}}^2-\frac{4}{3}v_{{\rm S,dry}}^2)$$  (6.9 ± 0.7) GPa    Undrained bulk modulus  $$\tilde{K}_{{\rm ud}} = \rho _{{\rm sat}}(v_{{\rm P,sat}}^2-\frac{4}{3}v_{{\rm S,sat}}^2)$$  (27.5 ± 0.3) GPa  dynamic  Shear modulus  $$\tilde{G} = \rho _{{\rm dry}}v_{{\rm S,dry}}^2$$  (8.2 ± 0.3) GPa    Measured parameters at hydrostatic conditions (peff > 40 MPa)    Drained bulk compressibility  $$c_{{\rm bc}}= 3\mathrm{d}\epsilon _\mathrm{ax}/\mathrm{d} \Delta \sigma |_{p_\mathrm{f}}$$  (6.5 ± 0.1) × 10 − 11 Pa − 1    Drained bulk modulus  $$K_{{\rm d}}=c_{{\rm bc}}^{-1}$$  (15.6 ± 0.6) GPa    Pore compressibility  $$c_{\mathrm{pc}}=\mathrm{d}\psi /\mathrm{d} \Delta \sigma |_{p_\mathrm{f}}/\phi _\mathrm{con}$$  (5.1 ± 0.1) × 10 − 10 Pa − 1    Specific storage capacity  s  (4.1 ± 0.3) × 10 − 11 Pa − 1  quasi-static  Derived parameters at hydrostatic conditions (peff > 40 MPa)    Average bulk modulus of minerals  $$K_{\mathrm{r}}(c_{\mathrm{bc}},c_{\mathrm{pc}}) = c_{\mathrm{r}}^{-1} = (c_{\mathrm{bc}}-\phi _\mathrm{con}c_{\mathrm{pc}})^{-1}$$  (51.7 ± 4.1) GPa    Biot–Willis parameter  α(cbc, cpc) = ϕconcpc/cbc  (0.70 ± 0.04)    Specific storage capacity  s(cbc, cpc) = ϕcon[cf − cbc + (1 + ϕcon)cpc]  (8.2 ± 0.4) × 10 − 11 Pa − 1  View Large The decrease in permeability with increasing effective pressure above the closure pressure corresponds to a permeability modulus $$K_k=\mathrm{d}p_\mathrm{eff}/\mathrm{d}\ln k = 221 \pm 2\,{\rm MPa}$$. An average aspect ratio on the order of $$\bar{\xi }_k \sim K_\mathrm{r}/3 K_k\sim 10^2$$ results when employing the permeability model presented by Gavrilenko & Gueguen (1989). This average aspect ratio is about an order of magnitude smaller than the estimate (see Walsh 1966; Mavko & Nur 1978) of ξcc ∼ Kr/pcc ∼ 103 corresponding to the observed crack-closure pressure. The relation between these two estimates of aspect ratios is, however, consistent. The cracks controlling permeability above the identified ‘closure pressure’ have to exhibit a smaller aspect ratio than those that are closed when this pressure level is reached. The observed continuous velocity increase with increasing pressure (Fig. 3b) is also suggestive of an ensemble of fractures with a range of geometrical properties (see David & Zimmerman 2012). 4.2 Microstructural evolution during deviatoric loading as deduced from the monitored physical properties Deformation of rocks at temperatures at which crystal plasticity is insignificant, like here, is inherently related to the processes involving pre-existing microcracks, that is, their closure, frictional sliding, and eventually their growth. Frictional processes on pre-existing microfractures are intrinsically irreversible and lead to distinct asymmetry and hysteresis during loading and unloading cycles at stress levels too low to initiate microfracturing (e.g. Walsh 1965; David et al. 2012). Thus, rather than addressing the reversibility of deformation -in accord with the fundamental thermodynamic definition of elastic behaviour- the separation of ‘elastic’ versus ‘inelastic’ deformation of rocks by brittle mechanisms often refers (loosely) to the onset of microfracturing, that is, permanent changes in the defect inventory. We will also use this terminology here. Closure and growth of pre-existing microfractures cause anisotropy in physical properties when rocks are subjected to deviatoric loading (e.g. Johnston et al. 1979; Toksöz et al. 1979; Crampin et al. 1980; Jones 1995; Sharma 2005). Previous experimental and numerical studies provided ample evidence that the initial response of rocks to deviatoric loading is controlled by the preferential closure of pre-existing cracks oriented perpendicular to the maximum principal stress. When stresses exceed the elastic limit, brittle deformation is induced on grain scale in the form of directionally dependent extension of existing and formation of new cracks. Anisotropy -well documented for elastic velocities during elastic as well as inelastic deformation (e.g. Guéguen & Schubnel 2003; Schubnel & Guéguen 2003)- may bias the absolute values of the reported effective hydraulic parameters derived relying on an analytical solution assuming isotropy. Yet, up to now the anisotropy in hydraulic parameters is hardly accessible in experiments (but see Dautriat et al. 2009). Thus, we do not have direct evidence for anisotropy in either elastic nor hydraulic properties (measured only in one direction). However, differences between observations for hydrostatic and deviatoric loading serve as an indicator for anisotropy. 4.2.1 Evolving anisotropy The relation between observations for hydrostatic and deviatoric loading is controlled by the identified crack-closure pressure. For experiments conducted at effective confining pressures below this level, pore-volumetric strain, hydraulic properties, and P-wave velocity tend to deviate from their hydrostatic reference curves immediately at the onset of deviatoric loading (Figs 5, 8 and 10). These deviations reflect relative stiffening by deviatoric loading, most likely caused by closure of cracks perpendicular to the cylinder axis, the direction of travel of the P-waves and the flow direction for the pore fluid, and thus suggesting the development of anisotropy in the crack distribution (see for example Schubnel & Guéguen 2003; Schubnel et al. 2006). Stiffening in axial direction is also documented by the initial increase in tangent modulus with axial strain (Fig. 11a). At pressures above the crack-closure pressure, differences between hydrostatic and directional loading diminish probably because the potential for closure of cracks of any orientation is almost exhausted. Radial strain is a potential indicator of opening and closing of cracks oriented parallel to the sample axis. We approximate radial strain by using the pore-volumetric strain recorded during deviatoric loading as a measure of bulk volumetric strain, θ, that is, εrad = (θ − εax)/2 ≈ (ψ − εax)/2. For an isotropic, linear poroelastic medium, the two volumetric strains obey ψ = (α − ϕcon)θ where α = 1 − Kd/Kr denotes the Biot-Willis coefficient. This relation follows for hydrostatic and general stress states when evaluating eq. (35) in Zimmerman et al. (1986) or eq. (69) in Renner & Steeb (2014) for constant pore-fluid pressure and dpc = Kdθ or $$\mathrm{d}\sigma ^\mathrm{^m} = K_\mathrm{d} \theta$$, respectively. Here, we find α = 0.70 ± 0.04 (Table 3), that is, the change in porosity accounts for more than 60 per cent of the total elastic volumetric strain. According to amount, our approximation overestimates elastic radial strain by less than 30 per cent for a Poisson’s ratio of 0.3 and gets worse the lower the Poisson’s ratio is. In contrast, the inelastic contribution to radial strain is fully reflected by the changes in porosity since inelastic deformation is here accommodated by void creation and elimination. During initial deviatoric loading even the approximated and thus -according to amount- overestimated radial strains are subordinate but P-wave velocity increases significantly. Presumably, the closure of cracks preferentially oriented perpendicular to the maximum principal stress accommodates axial strain with limited Poisson effect. The subsequent reduction in velocity increase correlates with the progressing widening of the sample recorded by the radial strain accompanying the onset of dilation. When normalizing by their maximum in velocities, the four tests at different effective pressures exhibit a ‘universal’ relation between P-wave velocity and radial strain (Fig. 5b). Likewise, normalized permeability exhibits a fairly uniform increase with radial strain when attributing for the contribution to radial strain from hydrostatic loading εrad,hyd, before passing into individual plateaus for the four tests after failure when macroscopic localization has advanced to a fully developed fault or cataclastic flow occurs (Fig. 9b). While all three hydraulic properties seem to correlate better with effective stress ratio λ = (Δσax + peff)/peff than effective mean stress this observation is particularly true for specific storage capacity (Fig. 8) that in contrast to permeability exhibits limited correlation with radial strain (Fig. 9c). Figure 9. View largeDownload slide (a) Normalized P-wave velocity vP, (b) permeability k, (c) specific storage capacity s, and (d) hydraulic diffusivity Dhyd as a function of total radial strain εrad,tot = εrad,hyd + (ψ − εax)/2 at indicated effective confining pressures. P-wave velocities and hydraulic properties were normalized by their maximal and by their hydrostatic reference values, respectively. Diamonds and triangles indicate onset of dilation as deduced from pore-volumetric strain and peak stress, respectively. Figure 9. View largeDownload slide (a) Normalized P-wave velocity vP, (b) permeability k, (c) specific storage capacity s, and (d) hydraulic diffusivity Dhyd as a function of total radial strain εrad,tot = εrad,hyd + (ψ − εax)/2 at indicated effective confining pressures. P-wave velocities and hydraulic properties were normalized by their maximal and by their hydrostatic reference values, respectively. Diamonds and triangles indicate onset of dilation as deduced from pore-volumetric strain and peak stress, respectively. Our observations regarding the relation between permeability and stress ratio qualitatively and quantitatively match with the modelling by Simpson et al. (2001) which predicts changes in normalized permeability of at most a factor of 2 as long as the effective stress ratio remains below about 15, a bound not exceeded in our experiments (Fig. 8). The model lends its support to our interpretation of the relation between properties and radial strain since it finds the permeability evolution in this dilative regime to be controlled by axially oriented cracks widening and/or forming. The modest degree of transversal anisotropy during inelastic deformation predicted by the model suggests that our effective hydraulic properties probably still represent the samples and their evolutions well. More experimental and numerical work is needed to constrain the effect of anisotropy on hydraulic measurements. 4.2.2 Onset of microfracturing Two prominent approaches for the identification of the onset of microfracturing have been followed in the past, analyses of stress-strain relations and monitoring of physical properties presumably sensitive to microfracturing. For dense crystalline rocks, Brace & Martin (1968) introduced the concept of dilatancy, that is, the deviation of volumetric strain induced by deviatoric loading from the initially linear compactive trend. For porous rocks, a separately determined, typically nonlinear hydrostat is used as reference instead (e.g. Zhu & Wong 1997; Baud et al. 2012). Our experiments exemplify the problems encountered for porous rocks. The pore-volumetric strain of the sample deformed at an effective pressure of 100 MPa exhibits a linear relation with (mean) stress almost all the way up to peak stress (Fig. 4b), likely as a consequence of the fortuitous balance between concurrent dilative (extension of tensile cracks) and compactive (pore collapse and crack closure) processes. The comparison to the hydrostatic reference curve is hampered by the restriction of the latter regarding the covered range in pressure (Fig. 10); for the samples deformed at low effective pressures, it is rather difficult to pinpoint the range of stresses where the two curves coincide and where not. In fact, it appears as if they do not show any coincidence for the experiments at the lowest effective pressures possibly due to the development of anisotropy associated with closure of different suites of the pre-existing microcracks during hydrostatic and deviatoric loading, as argued above. For the tests performed at or above crack-closure pressure, pore-volumetric strains observed during hydrostatic and deviatoric loading closely agree to the point when dilation occurs consistent with a diminished role of cracks (and consequently anisotropy). Figure 10. View largeDownload slide Pore-volumetric strain ψ as a function of effective mean stress σm,eff at indicated effective confining pressures. The curves representing pore-volumetric strain recorded during triaxial compression are displayed starting at the respective strains determined during hydrostatic deformation of the reference sample (circles). Figure 10. View largeDownload slide Pore-volumetric strain ψ as a function of effective mean stress σm,eff at indicated effective confining pressures. The curves representing pore-volumetric strain recorded during triaxial compression are displayed starting at the respective strains determined during hydrostatic deformation of the reference sample (circles). Given the relative technical ease with which elastic wave velocities can be ‘continuously’ monitored during deformation experiments, it is not surprising that characteristics of velocity changes were employed to define deformation regimes. For example, Schubnel et al. (2005, 2006) identified passing the maximum in P- and S-wave velocity during triaxial compression with the onset of crack propagation. Different from our study, the velocity measurements were, however, performed perpendicular to the compression axes. Previous studies found significant differences in the sensitivities of P- and S-waves travelling in axial and radial direction to the transition from elastic to inelastic behaviour (e.g. Lockner et al. 1977; Guéguen & Schubnel 2003) in accord with the intuitive notion that crack closure during ‘elastic’ triaxial deformation and inelasticity due to crack extension or formation cause anisotropic crack distributions (e.g. Kachanov 1992). The present experiments yield a significant sensitivity of axial P-wave velocities to all deformation phases. Velocity increases during initial compaction, exhibits a maximum well before peak strength, and then decreases until localized failure occurs (Fig. 9a). The initial increase in velocity can be attributed to either the closure of pre-existing microcracks oriented perpendicular to the direction of the maximum principal stress (e.g. Lockner et al. 1977; Jones 1995; Eslami et al. 2010) or to the stiffening of Hertzian contacts at grain boundaries (e.g. Pyrak-Nolte & Nolte 1995; Toomey & Nakagawa 2003), or to both. The occurrence of the maxima in P-wave velocity coincides with the onset of dilation below crack-closure pressure but the two are separated for higher pressures. The correlation of velocity with radial strain (Fig. 5b) suggests that its decrease is related to opening of cracks oriented subparallel to the direction of maximum principal stress. The extent to which the aspect ratio of cracks affects elastic properties critically depends on the fluid-coupling parameter δ ∝ Kr/Kfξ (Kachanov 1992); that is, the ratio of bulk moduli of solid and fluid (see also Schubnel et al. 2006). This parameter assumes critical limits of δ → ∞ for compressible gases (dry cracks) and δ → 0 for incompressible fluids (water-saturated cracks) leading to insensitivity and sensitivity of velocity to aspect ratio, respectively. Consideration of pore fluid properties and saturation state may contribute to the partly conflicting observations regarding the sensitivity of velocity to cracks aligned with the direction of propagation (e.g. Read et al. 1995; Wulff et al. 1999; Guéguen & Schubnel 2003; Eslami et al. 2010; Blake 2011; Stanchits et al. 2011). It will be important to decipher the underlying reason for the differences in sensitivity in future work owing to its relevance for the analyses of preseismic velocity changes in the field (e.g. Schaff 2012). Obviously, the peculiarities of the inventory of voids (pores versus fractures) are of critical importance because closure of cracks and stiffening of grain contacts oriented perpendicular to the maximum stress may occur concomitantly with opening of cracks parallel to the maximum stress over an extended range of strain. The conceptual focus on cracks parallel and perpendicular to the maximum principal stress may be too gross a simplification, for example, in the light of fracture processes such as wing cracking (e.g. Brace & Bombolakis 1963; Paterson & Wong 2005) or grain comminution in granular media. In the following, we consider the maxima in axial P-wave velocity the latest indicators for crack initiation and tentatively use them to separate ‘elastic’ and ‘inelastic’ deformation where the terms are used in the sense discussed above. This approach is supported by the prominent inflection points in trends of hydraulic properties with axial strain observed at the maxima in velocity (Fig. 7). 4.3 Correlations among physical properties The richness and density of information gained from our tests (continuous resolution of hydraulic properties for strain increments of as small as 1 × 10−4 ) allows us to investigate previously unexplored correlations among determined parameters whose full meaning and potential can only be grasped in the light of yet to be performed hydro-mechanical modelling of heterogeneous media. Correlating elastic properties determined from different stress or strain amplitudes is possibly complicated by nonlinearity (e.g. Fjaer 2009). The oscillation amplitudes applied at the upstream end (<3 MPa) and observed at the downstream end (>0.03 MPa) provide upper and lower bounds, respectively, for the stress perturbations associated with the determination of specific storage capacity. These values are intermediate between the stress intervals of about 2 MPa employed in the calculation of tangent moduli and the stress perturbations associated with ultrasonic waves, for our configuration estimated to be below about 0.01 MPa. Furthermore, the various measures of poroelasticity constrained here likely reflect different parts of the pore-space. Static elastic parameters, that is, derivatives of stress-strain curves, represent bulk properties of the entire sample volume. Ultrasonic velocities reflect the fastest wave paths which in turn depend on wavelength (e.g. Mukerji et al. 1995; Tworzydło & Beenakker 2000). Here, the dominant wavelength is an order of magnitude smaller than sample size and thus dynamic elastic parameters are expected to exceed the static ones. The two directionally dependent mechanical properties that constrain momentary elastic behaviour, tangent modulus and ultrasonic velocity, initially show a positive correlation in the elastic regime. The tangent modulus, however, decreases before the velocity reaches its maximum (Fig. 11a). The occurrence of the two maxima seems to converge with increasing pressure, though. Tangent modulus and velocity show a uniform positive correlation after the maximum in velocity that continues even after peak stress possibly due to uniform damage accumulation before localization (Fig. 11b). Figure 11. View largeDownload slide (a,b) Permeability k, (c,d) specific storage capacity s, (e,f) hydraulic diffusivity Dhyd, and tangent modulus Eapp during ‘elastic’ (left) and ‘inelastic’ (right) triaxial deformation of Wilkeson sst at indicated effective confining pressures. The occurrence of maxima in P-wave velocity was used to distinguish ‘elastic’ from ‘inelastic’ (see the text). Hydraulic data were down-sampled by a factor of 60 to match with the velocity data. To remove distorting scattering from velocity data (cf. Fig. 5a) a polynomial fit was applied to match the relation of velocity to axial strain for each test. Figure 11. View largeDownload slide (a,b) Permeability k, (c,d) specific storage capacity s, (e,f) hydraulic diffusivity Dhyd, and tangent modulus Eapp during ‘elastic’ (left) and ‘inelastic’ (right) triaxial deformation of Wilkeson sst at indicated effective confining pressures. The occurrence of maxima in P-wave velocity was used to distinguish ‘elastic’ from ‘inelastic’ (see the text). Hydraulic data were down-sampled by a factor of 60 to match with the velocity data. To remove distorting scattering from velocity data (cf. Fig. 5a) a polynomial fit was applied to match the relation of velocity to axial strain for each test. The scalar-valued storage capacity is also selective regarding the reflected elements of the crack-distribution. Storage capacity is controlled by the momentary elastic deformation of the hydraulic conduits involved in pressure diffusion, that is, cracks and pores with a favourable combination of width (transport) and stiffness (storage), that is, not necessarily the ones controlling wave velocity. Still, specific storage capacity exhibits fairly uniform but non-monotonous correlations with P-wave velocity during inelastic deformation (Fig. 11f); it first increases and then decreases during the continuous velocity decrease. Relations between the two directionally dependent transport properties, P-wave velocity and permeability, are a subject of considerable interest and debate. Obviously, the potential for correspondence between these two properties depends on the specifics of the microstructure, in particular the interconnectivity among the microfractures. Yet, the finite permeability of Wilkeson sandstone and in fact the majority of real rocks suggests that they do not represent the peculiar situation of isolated voids, but interconnected crack networks are maintained even at elevated pressures. Furthermore, the properties of the pore fluid are important since the aspect ratio of cracks, of paramount importance for permeability owing to its dependence on crack width, affects elastic wave velocity only when the pore fluid is rather incompressible (as applicable here, see discussion of pore-fluid coupling parameter above). Thus, it is not surprising that previous studies did not reveal simple or universal relations between P-wave velocity and permeability (e.g. Pyrak-Nolte & Morris 2000; Fortin et al. 2005; Baud et al. 2012; Brantut 2015). Here, however, a systematic inverse correlation between them emerges from the occurrence of the maximum in velocity to peak stress, that is, from the onset of microfracturing to the onset of localization (Fig. 11d). This section of the deformation is characterized by the most pronounced radial straining. During initial elastic compaction, permeability modestly increases or decreases below and above the closure pressure, respectively, and thus does not show a systematic relation to velocity (Fig. 11c). Hydraulic diffusivity expresses per se the correlation between a transport (permeability) and a volumetric (specific storage capacity) property. The observed diffusivity values appear to exhibit less sample-to-sample variability than the two properties from which they calculate, that is, permeability and storage capacity (Figs 11g and h). Since diffusivity matches more closely the evolution of permeability than that of specific storage capacity (Fig. 8) it seems that the gain in connectivity during inelastic deformation outmatches the loss in rigidity. Furthermore, the overall tendency of diffusivity (and permeability) to increase with damage accumulation observed in our experiments suggests that the gain in porosity by dilative processes dominates the evolution of hydraulic properties (Fig. 9) rather than the potentially permeability-decreasing effect of comminution. Hydraulic diffusivity exhibits a systematic inverse correlation with P-wave velocity during inelastic deformation (Fig. 11h). 4.4 Implications Hydraulic diffusivity constitutes the key parameter for fluid-pressure transients and thus also the temporal variations in effective stress, as an expression of hydro-mechanical coupling. In the spirit of the quadrant diagrams of Bernabé et al. (2003) for permeability -porosity relations, we present our results as diffusivity-pore volumetric strain diagrams distinguishing between elastic and inelastic deformation (Fig. 12). The modest changes in diffusivity during elastic compaction fall in quadrant I for hydrostatic and deviatoric deformation. Inelastic deformation is characterized by an inverse relation between diffusivity and pore-volumetric strain, that is, increasing compaction correlates with decreasing diffusivity and vice versa, and thus the curves occupy almost exclusively the quadrants II and IV. Soon after the onset of microfracturing as indicated by the maximum in axial P-wave velocity also bulk dilation commences and diffusivity increases up to more than an order of magnitude during the progressing localization. Only for the sample deforming by non-localized cataclastic flow at the highest explored effective pressure of 100 MPa, we find a range for which the ongoing compaction prevents an increase in diffusivity (Fig. 12). The increase in diffusivity with progressing inelastic deformation should reduce the pore-fluid pressure or its build up and thus, in situ faulting of fluid-bearing rocks should exhibit self-stabilization. Furthermore, an increase in diffusivity increases the spatial scale over which drainage can be effectively reached. Thus, the observed diffusivity evolution is expected to affect hydro-mechanical coupling in a stabilizing and delocalizing way, aspects yet to be fully accounted for by modellers. Figure 12. View largeDownload slide Relation between hydraulic diffusivity and change in connected porosity as documented by pore-volumetric strain ψ during (a) elastic, and (b) inelastic deformation as separated by and identified from the maxima in axial P-wave velocity (Fig. 4) at indicated effective confining pressures. Diffusivity values are normalized by their hydrostatic reference values Dhyd,0 and their values found at the maximum in P-wave velocity Dhyd(vP,max) for elastic and inelastic deformation, respectively. The dashed lines are given to emphasize the quadrant concept originally introduced by Bernabé et al. (2003) for the discussion of permeability-porosity relations. The dotted box in (b) indicates the range in values covered by (a). To ease the comparison in (a), the records of the triaxial deformation tests were tied to the respective points of the hydrostatic relation. Figure 12. View largeDownload slide Relation between hydraulic diffusivity and change in connected porosity as documented by pore-volumetric strain ψ during (a) elastic, and (b) inelastic deformation as separated by and identified from the maxima in axial P-wave velocity (Fig. 4) at indicated effective confining pressures. Diffusivity values are normalized by their hydrostatic reference values Dhyd,0 and their values found at the maximum in P-wave velocity Dhyd(vP,max) for elastic and inelastic deformation, respectively. The dashed lines are given to emphasize the quadrant concept originally introduced by Bernabé et al. (2003) for the discussion of permeability-porosity relations. The dotted box in (b) indicates the range in values covered by (a). To ease the comparison in (a), the records of the triaxial deformation tests were tied to the respective points of the hydrostatic relation. Our experimental results show that -at least in certain yet to be better constrained cases-significant P-wave velocity variations occur in the nominally least sensitive direction, that is, that of the maximum principal stress. On the one hand, this result implies potentially less limitations regarding ray coverage relative to a fault structure necessary for preseismic observations (e.g. Whitcomb et al. 1973; Schaff 2012). On the other hand, it poses a warning regarding the interpretation of damage geometry from velocity variations observed in situ. Furthermore, the results of our study suggest that changes in velocities constitute a qualitative surrogate for changes in hydraulic properties accompanying pre-failure deformation and thus bear constraints on the prediction of fluid flow within seismically active rupture zones. At the advent of comprehensive in situ monitoring of velocity changes associated with large-scale faulting (e.g. Wegler & Sens-Schönfelder 2007; Wegler et al. 2009; Sens-Schönfelder & Wegler 2011), tides (e.g. Takano et al. 2014; Hillers et al. 2015a), or engineering applications (e.g. Olivier et al. 2015), improved constraints on the relation between velocity, stress state, and microstructure become ever more relevant for a significant interpretation of observations. We investigated the correlation between P-wave velocity and various strain (axial, radial, pore-volumetric) and stress (deviatoric, mean, stress ratio) measures to find that radial strain unifies the observed velocity changes during triaxial deformation at a range of effective pressures best. The inelastic radial strain reflects the opening of cracks oriented in the direction of wave propagation. The less significant correlations with stress indicate that the correlation between stress state and microstructure is not unique even for the performed set of experiments on a single rock type. In situ observations have been interpreted in terms of relative velocity changes with strain or stress variations (e.g. Yamamura et al. 2003), the significance of neither being fully understood. Absolute stress-sensitivity values estimated for in situ velocities are at least one order of magnitude larger than the ones typically observed in the laboratory, including this study. Here, sensitivities of relative velocity changes decrease from about 5 × 10−3 MPa−1 to 2 × 10−4 MPa−1 with increasing hydrostatic or mean stress before inelasticity sets in (Figs 3b and 5c). The quantitative discrepancy may be related to a strong nonlinearity in velocity changes with stress at low deviatoric loads and mean stresses far away from crack-closure pressures that exceeds the nonlinearity for hydrostatic loading observed here (Fig. 5c). 5 CONCLUSIONS Our laboratory experiments on samples of Wilkeson sandstone elucidated the structural evolution associated with brittle to semi-brittle faulting as constrained by changes in hydraulic and dynamic elastic properties as well as the correlation among these properties. The chosen experimental procedures and analysis methods provide the complete set of hydraulic properties with unprecedented resolution. In agreement with previous work, we find a general trend of increasing permeability with proceeding brittle deformation. Yet, a transient, rather counterintuitive decrease in permeability and specific storage capacity occurs during localization of faulting. Hydraulic diffusivity, the key parameter for transient flow processes, exhibits larger or smaller increases than permeability depending on the direction and magnitude of changes in specific storage capacity that seem to alter with increasing effective pressure. A uniform relation appears to hold between P-wave velocity and deviatoric radial strain, that is, the strain perpendicular to the direction of propagation. For none of the tested stress measures (maximum principal, deviatoric, or mean stress, stress ratio) a similarly good correlation is found for P-wave velocity. The correlations of hydraulic properties with either total radial strain or stress ratio appear equally systematic. These correlations with radial strain strongly suggest that cracks aligned with the maximum principal stress are responsible for the velocity decrease and the permeability increase. The uniformity of observations for tests performed at different pressures implies a uniformity in the generation and accumulation of damage by microfracturing before localization and suggests that the maximum in velocity serves as a robust indication of inelastic deformation in our experiments. The significant sensitivity of P-wave velocity to characteristics of cracks aligned with the propagation direction differs from results of several previous studies and it thus remains to be resolved what actually determines the degree of sensitivity. Such future work is relevant in the context of in situ velocity monitoring as a tool for detecting changes in deformation state, for which either the propagation direction of the waves cannot be controlled (natural sources) or the orientation of principal stresses is not constrained beforehand. Also, more work is needed to explore the diagnostic potential of velocity variations for the evolution of hydraulic properties, in particular hydraulic diffusivity, indicated by our results and to exploit it in, for example, monitoring of seismically active regions or evolving reservoirs. More sophisticated models than one-dimensional pressure diffusion in an isotropic, homogeneous material have to be elaborated to analyse the observations outside the ‘conventional’ solution space and the complex variations in oscillation characteristics observed right at failure. The understanding of the structural changes occurring during localization and their consequences for fault properties that in turn affect the dynamics of the failure process will largely benefit from such modelling of experimental data. 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