On the effective stress law for rock-on-rock frictional sliding, and fault slip triggered by means of fluid injection

Fluid injection into rocks is increasingly used for energy extraction and for fluid wastes disposal, and can trigger/induce small- to medium-scale seismicity. Fluctuations in pore fluid pressure may also be associated with natural seismicity. The energy release in anthropogenically induced seismicity is sensitive to amount and pressure of fluid injected, through the way that seismic moment release is related to slipped area, and is strongly affected by the hydraulic conductance of the faulted rock mass. Bearing in mind the scaling issues that apply, fluid injection-driven fault motion can be studied on laboratory-sized samples. Here, we investigate both stable and unstable induced fault slip on pre-cut planar surfaces in Darley Dale and Pennant sandstones, with or without granular gouge. They display contrasting permeabilities, differing by a factor of 105, but mineralogies are broadly comparable. In permeable Darley Dale sandstone, fluid can access the fault plane through the rock matrix and the effective stress law is followed closely. Pore pressure change shifts the whole Mohr circle laterally. In tight Pennant sandstone, fluid only injects into the fault plane itself; stress state in the rock matrix is unaffected. Sudden access by overpressured fluid to the fault plane via hydrofracture causes seismogenic fault slips. This article is part of the themed issue ‘Faulting, friction and weakening: from slow to fast motion’.


Introduction
Fluid injection into deep rocks is increasingly used in connection with energy extraction [1][2][3][4][5]. Examples include: re-injection for maintenance of reservoir stability, hydraulic fracture for well stimulation and unconventional hydrocarbon production, and deep geothermal energy production whether or not involving hydraulic fracture. Deep injection is also widely used for disposal of fluid wastes from manufacturing or of well flowback fluids, and includes injection for underground gas disposal (e.g. CO 2 ) and storage (e.g. natural gas).
Deep fluid injection can raise the pore pressure in hydraulically conductive rocks, or injection into impermeable rocks can raise the fluid pressure in natural faults and fissures [6]. In these cases, there is potential to promote fault slip. Since the anthropogenically induced tremors at Denver in the 1960s [7], it has been clear that this can lead to induced or triggered seismicity [1][2][3]5,6,[8][9][10][11][12][13][14][15]. However, it must not be assumed that fault slip induced by fluid injection is necessarily seismogenic; it can be stable. The energy release and hence magnitude of such seismicity is strongly sensitive to the amount, rate and pressure of fluid injected, as a result of the way that seismic moment release is related to slipped area [6,10]. McGarr et al. [11] demonstrated that the size of induced earthquakes scales with the amount of fluid injected and that management of fluid injection was the key to minimizing the hazard of induced earthquakes.
The hazards of seismicity induced through fluid injection have been widely studied through in situ monitoring [3,9,16,17] and numerical modelling, but the effects can also be demonstrated on a small scale through laboratory experiments at high pressures, and through mesoscale injection experiments into a natural fault in an underground laboratory [18]. Here, we investigate both stable and unstable induced fault slip on pre-cut planar surfaces in two sandstones, Darley Dale sandstone and Pennant sandstone. The rocks were chosen for their very different permeabilities to fluid flow, while their mineralogies are broadly comparable. After establishing the basic frictional properties of these rocks, we investigate their response to fluid injection while under shear stress and normal stress combinations below those normally required to instigate frictional sliding.

Sample characterization and experiments performed
Two quartz sandstones of different porosities and permeabilities were used: (a) Pennant sandstone, a grey, durable quartz sandstone of upper Carboniferous age from South Wales [19]. Modal composition was obtained by chemical mapping on the scanning electron microscope (SEM), and porosity was determined by gravimetry and by helium porosimetry (table 1). (b) Darley Dale sandstone, an upper Carboniferous quartz sandstone from Derbyshire, England. This decorative stone has previously been widely used in rock mechanics investigations [20][21][22][23].
Both rocks display little evidence of heterogeneity and anisotropy. Their typical microstructures are shown as SEM backscattered electron images in figure 1. Both rocks display a weak grain shape fabric parallel to bedding orientation. Their different permeabilities k (m 2 ) are central to this investigation, and they differ by a factor of approximately 10 5 . Permeability of Darley Dale sandstone to water, normal to bedding, was reported by Zhu & Wong [20] and is sensitive to Terzaghi effective confining pressure Pe (MPa; defined as confining pressure − pore pressure), given by log(k) = −0.878 ± 0.017 log(Pe) − 12.8 ± 0.03.
Application of 70 MPa effective pressure decreases permeability from 10 −13.2 m 2 at 3 MPa effective pressure to 10 −14.6 m 2 (figure 2b). The permeability of Pennant sandstone to argon gas was measured normal to bedding using the oscillating pore pressure method [24] over the effective pressure range 1-70 MPa and is much lower than that of Darley Dale sandstone. It may    are not shown, but they would lie at slightly higher stresses than the best fits to the resolved stresses on the fault planes that formed. Except at low pressure, fault angles are systematically larger than would be predicted by the simple Mohr-Coulomb criterion. Uncertainties in fits are shown in table 1. Errors of measurement are generally smaller than plotted point size. These data were previously reported in Hackston & Rutter [23]. (Online version in colour.) be described by log k = −0.0113 ± 0.001Pe − 18.2 ± 0.05, thus permeability decreases from about 10 −18.2 to 10 −19 m 2 over the same effective pressure range (figure 2a). Effective pressure change produced by varying pore pressure at constant confining pressure has the same effect on permeability as changing confining pressure at constant pore pressure. The permeability of this rock is comparable to that of many shales [25]. It is a tight sandstone. This is an important consideration when pore fluid pressure is applied.
Preliminary measurement [26] of the hydraulic conductivity kt (m 3 ) of the sawcut interface for Pennant sandstone follows the law log kt = −17.05 − 0.009Pe.
Pe is equivalent to normal stress across the interface. To compare, note that matrix permeability is equivalent to the hydraulic conductivity of a layer 1 m thick. Thus, an interface of nominal 10 µm thickness (t) under 1 MPa normal stress behaves as if its permeability (k) were 10 6 × greater than the bulk permeability, with the contrast becoming greater at higher hydrostatic pressure. Owing to its greater bulk permeability we have no data for the hydraulic conductivity of a planar interface in Darley Dale sandstone, but we can expect it to be comparable to Pennant sandstone. In this case, the 'permeability' of a planar crack will be comparable with that for flow through the rock matrix, i.e. a crack conveys no conductive advantage in Darley Dale sandstone.
The influence of effective confining pressure on the strength of these rocks was reported by Hackston & Rutter [23]. Samples were tested oven (60°C) dry under axisymmetric shortening (conventional triaxial) loading conditions (figure 3). The porosity differences have a major impact on the relative unconfined and cohesive strengths of these rocks (table 1). The coefficient of frictional sliding was measured both on freshly produced fault surfaces and on planar sawcut surfaces (figure 4).
They are almost identical (0.66 and 0.65, respectively, for Pennant and Darley Dale sandstones) and correspond with the generalization of Byerlee [27,28] that friction in rocks is, to a useful approximation, almost independent of rock type. In Hackston & Rutter's [23] experiments, these rocks displayed stable sliding on fault and sawcut surfaces under the full range of conditions imposed, both with and without static pore fluid pressures applied.

Experimental methods employed
All experiments in this study were carried out on 20 mm diameter × 45 mm nominal length cylindrical samples cored normal to bedding orientation. Cylinders were sawcut at an angle of either 35°, 45°or 55°to the cylinder axis and the cut surfaces were ground using a 60 µm grinding wheel and lightly finished using 16 µm abrasive paper. Prepared samples were stored in an oven at 60°C until used. Samples were jacketed with an inner sleeve 2 mm thick of silicone rubber and an outer jacket of heat-shrink rubber. The purpose of the inner jacket was to inhibit perforation of the outer, sealing jacket by the sharp edge of the sawcut sample half during slip events. The jackets were established not to contribute any significant load-bearing capacity to the sample assembly (0.1 MPa or less). All samples were shortened parallel to the cylinder axis. Loading stress paths included deformation both at constant confining pressure (smallest principal stress σ 3 = σ 2 ) and at constant resolved normal stress across the slip plane, maintained by servocontrol of the confining pressure (table 2). The confining pressure fluid was a synthetic ester, dioctyl sebacate (Reolube DOS ® ), a linear viscous fluid whose viscosity at atmospheric pressure is 0.024 Pa s, about 24 times that of water. Compared with common mineral oils, it displays only a small viscosity increase over the 100 MPa pressure range used in this study. Compressibility was measured to be 4.71 × 10 −10 Pa −1 , insensitive to pressure above 50 MPa. The same fluid was used for pore fluid pressure when a relatively viscous fluid was required. In other tests, argon gas was used as pore fluid. It has a viscosity of approximately 6 × 10 −4 Pa s at 50 MPa pressure and 20°C [29]. Compressibility is nearly insensitive to pressure up to about 150 MPa, beyond which it decreases by 60% by 100 MPa pressure. Water was avoided as a pore fluid to eliminate possible water weakening effects relative to tests on the dry rock, such as have been previously reported for some sandstones [30][31][32]. Axial load was measured using an internal load cell that permitted stress measurements to an accuracy of better than 0.5 MPa. Axial loading and confining pressure regulation was achieved by computer-controlled electromechanical servo-systems. Confining and pore fluid pressures were measured to an accuracy of better than 0.3 MPa. Axial displacement is measured outside the pressure vessel, and specimen displacements were determined by subtracting the calculated elastic machine displacement from the total measured displacement using an apparatus stiffness of 82 kN mm −1 . All loading increments were made at a total axial displacement rate of 8.0 × 10 −4 mm s −1 The aim of the experiments was to inject pressurized fluid into rock samples containing prestressed fault planes and hence to induce fault slip. Both high-and low-viscosity fluids were used (oil and argon gas) and fault planes were made less or more hydraulically conductive either by leaving them clear and close-fitting, or by introducing a 0.5 mm thick layer of fault 'gouge' in the form of granulated quartz with 75% of grains in the size range 38-63 µm. For most experiments    In the case of Pennant sandstone, experiments were also carried out in which the connection from the pore fluid pipe in the loading piston was made through the upper forcing block to the fault plane by means of a hydraulic fracture. This required a substantial fluid overpressure to develop in the pore pressure system that was suddenly released into the fault plane when the hydrofracture eventually formed and penetrated to the fault plane. To initiate the hydrofracture, a shallow hole, 4 mm long, was drilled into the upper forcing block (figure 5c). The end of the hole could be pressurized well in excess of the applied confining pressure, until leak-off (hydraulic fracture initiation) and then shear failure (differential stress drop) occurred. Slip on an inclined sawcut in the 'triaxial' testing configuration is not perfect. Substantial shear offsets can change the cross-sectional area of contact between the specimen halves, and the lateral displacement of the axial column tends to modify the stress state in the sample as a result of induced bending forces [33]. These effects can be minimized by keeping inelastic shear offsets small, less than 3 mm. Further, the apparatus used is able to deform samples in axial extension as well as in axial shortening. This allows a sheared sample to be reverse-sheared to restore the alignment of the forcing blocks. It was found that re-shearing the same sample did not significantly affect the frictional behaviour [23].
Faults were pre-stressed to combinations of resolved shear and normal stress corresponding to the onset of slip, over a time period of 30 min but without any pore pressure. Then pore pressure was increased at a uniform rate until slip began (e.g. after about 3 min). Slip on the sawcut surfaces, whether or not gouge was present, was indicated after the test by the presence of slip striations on the faults. The progression of slip as pore pressure was increased caused the differential stress to decrease, i.e. the diameter of the Mohr circle representing the stress state decreased, while the total confining pressure was maintained constant. This caused the stress condition (combination of resolved shear stress and normal stress) on the fault to decrease below the sliding condition, but sliding was maintained by continually increasing the pore pressure, by pumping pore fluid into the specimen.
To summarize, three different test configurations, (a), (b) and (c), were investigated, and these are illustrated in figure 5.

Experimental results
For each of the test configurations employed, we present an anticipated result followed by actual experimental observations. Details of experiments performed are shown in table 2. In each case the expected friction sliding criterion is as shown in figure 4, and confining pressure used was held constant at 69 MPa as pore fluid was injected, rising at a constant rate of change of pressure of nominally 5 × 10 −2 MPa s −1 , until the sample was fully offloaded after a period of about 30 min. Volume rate of injection thus varies according to pressure and fluid compressibility, and the storage capacity of the rock sample and pipework. The relative magnitudes of permeability of the rock matrix, the hydraulic conductivity of the fault plane and the viscosity of the infiltrating fluid are expected to determine mechanical behaviour of the rock/fluid system. Depending on their values, different things (cases (a) through (d) below) can be expected to happen.

(a) Case of injection into a rock of high permeability
If the rock permeability is high, even for a range of fluid viscosities, the fluid pressure is expected to be transmitted throughout all the rock pores and onto the fault plane. This is the 'classic' case of the effect of pore pressure on rock failure, in which the whole Mohr circle is shifted to the left (lower resolved normal stress, σ n , values) as a result of increasing pore pressure. Anticipated behaviour for a 45°sawcut is shown in figure 6. A loading path at constant confining pressure (50 MPa) and zero pore pressure is shown as an expanding series of Mohr circles. The locus of points representing shear stress/normal stress on the fault plane joins the points of maximum  shear stress and is steeper than the frictional sliding criterion. At the point of initiation of sliding, pore pressure can then be progressively raised, reducing effective normal stress components and moving the stress circle to the left. In this way the stress conditions on the fault could be made to track the entire frictional sliding line on a Mohr diagram. In figure 6, the rock would be sufficiently strong to resist the formation of a new fault plane. Experimental results exhibiting this behaviour are shown in figure 7 for the permeable Darley Dale sandstone (oil injection), for sawcuts at 35°and 45°to maximum compressive stress direction. The leftward-declining values of shear stress and effective normal stress (the latter calculated as resolved normal stress minus applied pore pressure) on the fault plane are shown as pore pressure was increased. They follow tracks close to but apparently slightly above the friction sliding criterion established from tests at zero pore pressure, which may mean that the fluid pressure was not completely uniform over the fault plane.   Pre-cut orientations of 35°and 45°to the cylinder axis were sufficiently favourable for slip that fresh faulting was not initiated by displacing the Mohr circle to the left, despite the relatively low cohesive strength of Darley Dale sandstone. Figure 8 shows the expected behaviour when fluid pressure is increased when the pre-cut is unfavourably oriented (55°). The displaced Mohr circle intersects the fresh faulting criterion before the slip condition on the sawcut is reached. If this happens, it tells us that the whole Mohr circle has been displaced. Continued increase in fluid pressure should offload the system via slip on the new fault surface, rather than the pre-existing one.
Darley Dale sandstone was observed to exhibit this behaviour (sample DDN3) with a 55°s awcut (oil injection, figure 9). A fresh fault formed at 34°to maximum principal stress and the specimen offloaded by slip along this plane as pore pressure continued to be increased. There was no slip along the pre-cut plane. Figure 10 shows the production of a fresh fault in this way.

(b) Case of rock of low matrix permeability, but conductive fault plane
If the matrix conductance is low but that of the fault plane remains high, pressurized pore fluid can be injected only along the fault plane via the axial hole in the specimen, reducing the effective normal stress σ n on the fault plane below the total σ n applied from the far field.   Figure 11. Pennant sandstone (impermeable) showing progressive offloading by slip along 45°sawcuts as a result of gas injection into the fault plane for the specimen numbers indicated. Total confining pressure = 69 MPa. The gas cannot permeate the whole sample but it can penetrate into the sawcut. Large symbols show tests in which a static pore pressure was applied before loading. In all cases the law of effective stress is followed for the fault plane, but the Mohr circle is not shifted leftward. (Online version in colour.) Figure 11 shows offloading of this rock by gas injection-induced slip on a 45°sawcut both with and without fault gouge in the slip plane. Gas injection was used in the experiments to ensure flow into the fault plane while flow into the rock matrix was still inhibited, although the hydraulic conductivity of a gouge-bearing fault plane would still be sufficiently high even if oil injection were to be used (see below). Friction measured by offloading during gas injection corresponds closely to the friction coefficient measured in tests without pore pressure over a range of normal stress values, hence the effective stress law applies.
In case (b) raising the pore pressure is not expected to shift the whole Mohr circle to the left, only the shear stress/normal stress combination that applies on the fault plane. In this way it should be possible to bring about fault slip on an unfavourably oriented fault plane where, for example, maximum principal stress is at a high angle to the fault plane. This happens without shifting the left side of the Mohr circle into the tensile failure regime, where hydraulic fracture might occur, and without precipitating the formation of a fresh, optimally oriented fault surface. This is the crux of the widely discussed point concerning the role of pore pressure on promotion of slip on unfavourably oriented fault planes in nature, e.g. low-angle extensional detachments and faults such as the San Andreas Fault [34][35][36][37].
As the effective stress state on the fault plane is displaced leftward, we expect it to intersect the frictional sliding criterion, causing slip and hence reduction of the shear stress. As previously, continued raising of the fluid pressure should cause the effective stress state to track the sliding criterion down to progressively lower resolved shear and normal stresses ( figure 12). If fluid pressure is less than fully effective, for example, if there are poorly conductive patches on the fault plane, this is expected to cause an apparent deviation of the stress state to the left of the friction sliding line. Figure 13 demonstrates the above predictions for oil injection into an unfavourably oriented (55°) fault plane in Pennant sandstone (e.g. test Pa8), with quartz gouge to enhance the conductivity of the fault plane itself. The initial loading path (combination of normal and shear stress resolved on the weak plane) is subparallel to the sliding criterion, hence sliding would not normally occur on this weak plane before a fresh fault forms due to progressive expansion of the Mohr circle. However, by raising fluid pressure, sliding was initiated on the weak plane, and the system was offloaded by slip on the weak plane as evidenced by frictional wear grooves on the slip plane. No fresh fault was produced, as would happen if the whole Mohr circle were displaced, and no hydraulic fracturing was produced. In contrast with the case of a granular gouge-bearing fault, Pennant sandstone with no gouge in the fault plane is expected to be poorly conductive for oil injection. In the time scale of an experiment, higher fluid pressures should, therefore, be required to inject the oil, and this would be expected to cause apparent deviations from the effective stress law, as illustrated in the hypothesis shown in figure 14. The measured injection pressure would exceed the true effective pressure inside the fault plane. The latter would be expected to accord with the effective stress law. The difference between the injection fluid pressure and the fluid pressure actually acting in the fault plane can be termed the injection overpressure. Figure 15 shows experimental results for samples of Pennant sandstone with 45°sawcuts and no quartz gouge with oil injection. Overpressures were required to initiate sliding in experimental sawcut faults (tests Pa4a and Pa4b). As the faults are progressively offloaded, the amount of required fluid overpressure decreases, probably because the decreasing effective normal and shear stresses on the fault, coupled with possible slip dilatancy, increase its hydraulic conductivity. This is illustrated in figure 16, in which the pore pressure excess factor Px, defined as the ratio of the applied pore pressure to the pore pressure in the fault plane that corresponds to the effective stress law being followed (Px = 1), is plotted versus shear stress. Px = 1 for gas  figure 14, for Pennant sandstone with a 45°sawcut, oil pore fluid and no gouge, to ensure poor initial hydraulic conductivity of the fault (tests Pa4a and Pa4b). Overpressure of the injected oil (small symbols, leftward deviation from the friction line) is required to make it inject into the fault plane in the time scale of the experiment. Each test is terminated when the overpressured oil breaks through to the outer surface of the specimen and starts to inflate the jacket. The large symbols show tests in which a fixed pore pressure was applied before loading. These follow the law of effective stress. (Online version in colour.) injection, when the effective stress law is demonstrably followed. In these tests, pore pressure injection is terminated when the fluid breaks through to the outside cylindrical surface of the specimen, when it becomes buffered by the confining pressure.
When tests with constant pore pressures are carried out, in which fixed pore pressure is applied before loading, the rock follows the effective stress law rather precisely ( figure 15). These results demonstrate that it is not easy to inject fluids into shear-oriented faults and cracks when under load, but becomes easier when such planes are filled with granular gouge, or as normal and shear stresses are decreased, or if slippage induces local dilatancy.
The requirement for pore pressure excess (overpressure) in these injection tests implies either that in the time scale of the experiments excess fluid pressure is required to force fluid into the fault plane where it intersects the injection hole, and/or that the fluid 'fingers' its way into the slip plane, only partially supporting the applied normal load, unless sufficient excess pressure is applied in the fingers.
Thus the effective normal stress σ n is given by  Here, α represents the fraction of the fault area that is flooded with fluid injected at pressure P at any instant, and it cannot be presumed always to be unity. There is reason to believe [38,39] that fingering of fluid pathways into a stressed plane commonly occurs, so that not all the externally applied normal stress may be reduced by the amount of the fluid pressure, and an excess pressure in the fluid-wetted areas is required to overcome the normal load sufficiently to cause slip. Figure 17 shows fluid fingering of a fluid injected into a transparent polymethyl methacrylate (PMMA) block used to demonstrate hydraulic fracture. In none of the experiments described thus far were seismogenic stress drops observed during offloading, although episodes of relatively rapid but stable sliding were seen.

(d) Case of fluid injection into a stressed fault via hydraulic fracture
Having to produce a hydraulic fracture before a fluid can be injected into a fault potentially allows a large fluid overpressure to be built that is then suddenly discharged into a fault plane via the hydraulic fracture. Figure 18 shows predicted behaviour of the externally measured fluid pressure evolution in this case, and the interpreted fluid pressure pathway in the fault plane.  (table 2 and figure 19, 45°and 55°sawcuts, quartz gouge present) demonstrate these effects through experiments. Successful underground hydraulic fracturing from a borehole with a minimum of overpressure is likely to depend on the wallrock being damaged either by explosions from casing perforations or by taking advantage of preexisting flaws (cracks or joints). The critical crack length for initiation of hydraulic fracture at approximately 100 MPa fluid pressure in brittle rocks is expected to be of the order of 0.5 mm [40,41] and this is the order of expected size of flaws in our experiments. In these experiments the starter flaw was the bottom of a short (4 mm) hole at the top of the specimen (figure 5). In common with most experiments, the total confining pressure was made to be constant at 69 MPa and a shear stress of approximately 100 MPa was applied, resulting in an initial resolved normal stress of approximately 180 MPa on a 45°sawcut and a differential stress of similar magnitude. This shear stress is not sufficient to cause sliding on the pre-cut weak plane, and the Mohr circle is still well below the fresh failure criterion for Pennant sandstone (figure 3). Fluid pressure was applied to the notch at the top of the specimen. About 140 MPa total oil pressure was required to cause hydraulic fracture in each case, about twice the confining pressure on the outer surface of the rock cylinder. The pressure seal around the entry hole at the top of the specimen prevented lateral leakage of injection fluid away from the injection point. The eventual intersection of the hydraulic fractures with the top of the fault plane rapidly injected excess pressure stepwise into the fault plane, causing a time sequence of five or six seismogenic slip events ( figure 19). Typically three or four hydraulic fractures were produced, radiating from the fluid injection point, and they could be recognized on the outer surface of the specimen after the test because they continued to seep oil (figure 20, sample Pa9).
The effective pressures plotted on figure 19 are calculated normal stresses on the fault plane minus the fluid pressure applied at the top of the specimen. Thus, they are not the true effective normal stresses on the fault plane, except towards the end of the experiment, when connected pathways between the fluid injection point and the fault plane have become established. At this point the fluid pressure has dropped to become equal to the confining pressure. Any excess fluid volume is able to bulge the jacket away from the specimen and the shear stress on the fault plane falls to zero.

Discussion
These experiments were carried out at a radically different length scale compared with fluid injection into deep boreholes. Nevertheless, the results provide useful insight into the behaviour of rocks in real fluid injection scenarios. The results show that the relative permeability of the rock matrix compared with the hydraulic conductance of the network of cracks and faults that may exist in the subsurface determines the behaviour of the rock mass for a given fluid viscosity.
When the rock matrix is sufficiently permeable, combined with a sufficiently low injection fluid viscosity, the effective pressure on fault and joint planes is the same as on similarly oriented imaginary planes in the rock mass, and the effective stress law can be simply applied. Increasing pore pressure by injection shifts the whole Mohr circle to the left by an amount equal to the pore pressure. If there is a pre-existing weak plane unfavourably oriented with respect to the principal stresses, for example, at a high angle to σ 1 , then the leftward-shifting Mohr circle may intersect the fresh faulting criterion before conditions for slip on the weak plane are attained. If the differential stress is sufficiently low to prevent fresh shear faulting, then hydraulic fracturing may occur as