DDilatancy stabilises shear failure in rock
Franciscus M. Aben, Nicolas BrantutDepartment of Earth SciencesUniversity College London, UK
Abstract
Failure and fault slip in crystalline rocks is associated with dilation. When pore fluids are presentand drainage is insufficient, dilation leads to pore pressure drops, which in turn lead to strengtheningof the material. We conducted laboratory rock fracture experiments with direct in-situ fluid pressuremeasurements which demonstrate that dynamic rupture propagation and fault slip can be stabilised (i.e.,become quasi-static) by such a dilatancy strengthening effect. We also observe that, for the same effectivepressures but lower pore fluid pressures, the stabilisation process may be arrested when the pore fluidpressure approaches zero and vaporises, resulting in dynamic shear failure. In case of a stable rupture, wewitness continued prolonged slip after the main failure event that is the result of pore pressure rechargeof the fault zone. All our observations are quantitatively explained by a spring-slider model combiningslip-weakening behaviour, slip-induced dilation, and pore fluid diffusion. Using our data in an inverseproblem, we estimate the key parameters controlling rupture stabilisation, fault dilation rate and faultzone storage. These estimates are used to make predictions for the pore pressure drop associated withfaulting, and where in the crust we may expect dilatancy stabilisation or vaporisation during earthquakes.For intact rock and well consolidated faults, we expect strong dilatancy strengthening between 4 and 6km depth regardless of ambient pore pressure, and at greater depths when the ambient pore pressureapproaches lithostatic pressure. In the uppermost part of the crust ( < Pore fluids are ubiquitous throughout the earth’s crust, and may govern or affect fault dynamics in settingsranging from subduction zones, crustal scale strike-slip faults to geothermal reservoirs. Pore fluid pressureis controlled by changes in pore volume, which can be induced by deformation and fault slip. One keypore volume change phenomenon is dilatancy, which is the increase of the pore volume in rocks by small-scale brittle deformation (e.g., microfractures, opening of pore space between grains), and is expected to besignificant in rocks with a low porosity and well consolidated fault gouges. The added pore volume acts as apore pressure sink when the fault zone is partially or entirely undrained – a scenario likely to occur duringfaulting, when new pore volume is created at a rate larger than the rate of fluid recharge from outside thefault zone. A drop in fault zone pore pressure increases the normal stress on the fault, thereby increasingthe shear resistance. Dilatancy strengthening is a transient phenomenon, as the extraneous shear resistancevanishes when pore pressure recovers to its initial value over time. Theoretical work over the past fourdecades (e.g.,
Rice and Rudnicki , 1979;
Rudnicki and Chen , 1988;
Segall and Rice , 1995;
Segall et al. , 2010;
Ciardo and Lecampion , 2019) has shown that dilatancy strengthening can potentially stabilise an otherwise(i.e., under dry or drained conditions) unstable fault by suppressing or delaying acceleration of slip, whichhas primary consequences for earthquake nucleation, rupture dynamics, and our ability to observe potentialearthquake precursory signals (e.g.
Nur , 1972;
Scholz et al. , 1973;
Bouchon et al. , 2013;
Shreedharan et al. ,2021). Despite the central role played by dilatancy in the mechanics of faulting, theoretical predictionshave remained largely untested and only indirect, qualitative experimental evidence of rupture and slip1 a r X i v : . [ phy s i c s . g e o - ph ] J a n tabilisation in granite ( Martin III , 1980) and fault gouge (
Lockner and Byerlee , 1994;
Proctor et al. , 2020)have been brought forward.Two main obstacles remain to assess quantitatively the impact of dilatancy on earthquake nucleationand slip. Firstly, direct experimental evidence of dilatancy-induced rupture stabilisation is still lacking, andit remains unclear whether the common model assumptions are a correct representation of the reality ofhydro-mechanical coupling during rock failure and fault slip. Secondly, quantitative measurements of keymodel parameters under in-situ conditions are still not available. Indeed, most theoretical studies emphasisethe lack of experimental measurements on the effectiveness of a shear fault zone to act as a pore pressuresink, which severally impedes application of their theoretical understanding to make meaningful predictionsof dilatancy stabilisation. Two parameters are required to describe a fault’s effectiveness as a pore pressuresink: The dilation rate, i.e., how porosity increases with fault slip and other evolving quantities, and theso-called storage capacity of the fault zone material, which is a compressibility that relates the pore volumechange to the change in pore pressure.Dilation rates have been measured during slip on natural and simulated fault gouges (e.g.,
Marone et al. ,1990;
Lockner and Byerlee , 1994;
Samuelson et al. , 2009) and after shear failure of intact rock (e.g.,
Rummelet al. , 1978). By contrast, fault zone storage is very difficult to measure. Fault zone storage describes howchanges in pore pressure are buffered by compression/extension of the pore fluid and the pore space. It canbe estimated from the storage capacity S f of the dilated fault zone material (given by the compressibilitiesof the pore space and pore fluid times the porosity (e.g., Jaeger et al. , 2007)) multiplied by the width ofthe fault zone disturbed by dilatancy. Measuring S f and the disturbed fault zone width is challenging: Forfault gouges, storage capacities have been measured only in an undisturbed state (e.g., Wibberley , 2002),and the width of the disturbed gouge layer in natural fault zones may vary over some orders of magnitudeeven during a slip event (see discussion in
Rice , 2006). Measurements of the local pore pressure in the faultzone during fault slip are required to estimate the effective fault zone storage. Previous experimental studiesthat report on dilatancy stabilisation in intact rock (
Martin III , 1980) and stick-slip events in consolidatedgouge (
Lockner and Byerlee , 1994) lacked the on-fault pore pressure measurements to calculate the faultzone storage. Recent advances in laboratory instrumentation (e.g.,
Brantut and Aben , 2020) allow for directmeasurements of the pore pressure drop caused by dilatancy during shear failure of intact rock (
Brantut ,2020) and during slip on a saw-cut granite surface and in a gouge layer (
Proctor et al. , 2020). Such newtechniques open the way for systematic studies of dilatancy strengthening phenomena, including direct testsof model predictions and quantitative parameter estimation.Here, we leverage our newly developed in-situ pore pressure measurement technique (
Brantut , 2020;
Brantut and Aben , 2020) to show direct evidence of rupture stabilisation by dilatancy strengthening, estimatethe key parameters necessary to quantify dilatancy strengthening in faults, and to predict at which depthsand pore pressure conditions dilatancy strengthening can indeed cause stabilisation of shear failure.
The experimental setup used in this study allows us to measure local pore pressure changes at several locationson the surface of a cylindrical rock sample during deformation in a conventional triaxial loading apparatus.The local pore pressure changes were measured with in-house developed miniature pore pressure sensors(
Brantut and Aben , 2020). We used Westerly granite cylindrical samples with two opposite facing notchescut at a 30 ◦ angle with respect to the sample axis, in order to have prior knowledge of the prospective failureplane during axial loading (Figure 1a). The samples were thermally cracked at 600 ◦ C, and intact hydraulicproperties of the samples were characterised prior to deformation (see Methods and Figure S3). The sampleswere deformed at nominal effective pressures of 40 MPa and 80 MPa, with a varying combination of confiningpressure P c and imposed pore pressure p (Table 1). The imposed pore pressure during deformation waskept constant at both sample ends. Axial load was increased by imposing a constant overall deformation rateof 10 − s − . The samples were loaded until shear failure, after which the piston displacement was arresteduntil complete reequilibration of the pore pressure, so that the total pore volume change due to failure inthe sample could be measured. Deformation was subsequently resumed, and slip continued on the newly2able 1: Summary of shear failure experiments. Nominal strain rate was 10 − s − for all experiments.Effective Confining Pore Min. pore Peak slip Dilation Storage Slippressure pressure pressure Failure pressure rate rate capacity weakeningRock P eff P c p mode p min v max dw/dδ S f w δ c (MPa) (MPa) (MPa) (MPa) (mm s − ) (-) (mm GPa − ) (mm)WG6 40 40 0 (dry) stable † - - - - -WG3 a
40 60 20 dynamic 0 b > c .
094 - -WG2 a
40 70 30 dynamic 0 b > c - - -WG5 40 100 60 stable 26 0 .
74 0 .
146 4 .
55 1 . .
27 0 .
171 6 .
14 1 . .
23 0 .
082 1 .
88 1 . b > c .
054 - -WG7 80 160 80 dynamic < d > c .
069 - -WG8 80 160 80 stable 16 0 .
65 0 .
077 1 .
37 1 . † Stable rupture was achieved by controlling acoustic emission rate as in
Aben et al. (2019). a From
Brantut (2020). b Pore water at partially vaporised or degassed. c
10 mm s − is the maximum detectable value in our system. d Pore pressure transducers out of range.formed fault. Periods of stable sliding or stick-slip events occurred. After each slip increment, we pauseddeformation to remeasure pore volume changes. More details are provided in the Methods section.
Under the pressure and temperature conditions of our tests, thermally cracked Westerly granite is brittleand when conducted under dry conditions deformation is always characterised by dynamic failure along ashear fault. Here, in our water saturated experiments, two types of behaviour were observed dependingon the initial pore pressure p (Table 1). At P eff = 40 MPa, tests conducted at p = 20 and 30 MPaunderwent dynamic shear failure. At P eff = 80 MPa, the test conducted at p = 40 MPa and one test where p = 80 MPa also failed dynamically. Dynamic failures are characterised by the well-known ‘bang’. We firstobserved a slow onset of the reduction in stress beyond the peak fracture strength that accelerated strongly.The majority of the stress reduction during dynamic failure was accommodated within less than a second(Figure 1b, dashed segment). No clear deceleration in the stress reduction rate was observed.By contrast, at initially high pore pressure ( p = 60 to 80 MPa and P eff = 40 MPa, and one test at p = 80 MPa and P eff = 80 MPa), sample failure occurred silently, which indicates that failure remainedquasistatic. The entire stress drop took several minutes, and the largest reduction in stress occurred in atimespan of the order of tens of seconds. The peak slip rate was typically less than 1 mm s − . We termthese ruptures “stable”.A clear shear fault formed between the notches for most experiments (e.g., Figure 1a), except for twosamples (deformed at P c = 100 MPa, p = 60 MPa, and P c = 110 MPa, p = 70 MPa) where the fault planedeflected from its prospective path towards the sample’s end.Pore pressure was recorded on the fault by two local pressure sensors, and recorded outside of the failurezone by two additional sensors (Figure 2). In all samples, the pore pressure remained uniform in the sampleuntil the peak differential stress, at about one MPa lower than the imposed pore pressure (Figure 2). Beyondthe peak stress, pore pressure decreased homogeneously throughout the sample at first, and then droppedmore rapidly on-fault than off-fault. In both the dynamic and stable cases, the stress drop during failure wasassociated with a strong pore pressure drop on the fault, of the order of several tens of MPa. The off-faultpore pressure also decreased but in a more gradual manner, and with a delay.The pore pressure evolution during and after failure was different between dynamic and stable ruptures.3igure 1: (a) : Notched Westerly granite samples that are intact (left), thermally cracked (center) andthermally cracked and failed under triaxial loading conditions (right). (b) : Shear stress versus slip duringstable and dynamic shear failure experiments performed at 40 MPa effective pressure. Dynamic failureis indicated by the dashed portion of the curve. Dry quasi-static failure at 40 MPa effective pressure wasachieved by acoustic emission rate controlled loading feedback, and provides a direct estimate of the minimumbreakdown work necessary to reach the residual frictional strength of the fault. Gray dashed line: Cohesion-weakening function f ( δ ) used in spring-slider simulations. Inset: Some stress versus strain curves for stable,dynamic, and controlled shear failure at 40 MPa (black curves) and at 80 MPa (purple curves) effectivepressure. 4n all samples that failed dynamically, the on-fault pore pressure dropped to zero concurrently with themain shear stress drop, and remained constant for the subsequent 5 to 10 minutes, while deformation wasstopped. The off-fault pore pressure did not drop to zero but also remained constant at very small values (afew MPa) for the same time period. The shear stress dropped immediately towards zero, overshooting theresidual frictional strength of the rock (Figure 1b) – a typical observation for dynamic laboratory failures.The lack of an immediate pore pressure recovery in the fault zone can be interpreted as an indication forlocal vaporisation or degassing ( Brantut , 2020).By contrast, during stable failure, the on-fault pore pressure dropped to a minimum significantly abovezero (typically tens of MPa) as slip rate reached it maximum and shear stress experienced a relatively rapidfirst drop. After this first stress drop, we stopped deformation by locking the axial piston in place. The porepressure on the fault immediately increased, while shear stress decreased further and more fault slip wasaccumulated. The on-fault pore pressure then asymptotically recovered to the imposed p at the boundary ofthe sample. The off-fault pore pressure experienced a gradual decrease during failure, followed by a recovery.The pore pressure recovery was slower off than on the fault, to the extent that the off-fault pore pressurewas transiently lower than that measured on the fault, indicating more rapid recharge inside the fault zonethan outside it.The total pore volume in the samples, as measured by the difference in intensifier volume before ruptureand after complete pressure reequilibration, increased strongly during the failure of the intact rock (Figure3). On average, additional pore volume created during failure at 80 MPa initial effective pressure is lowerthan at 40 MPa effective pressure. We express dilation as an increase in fault width w as a function of faultslip δ , which is computed from the measured pore volume change (see Method section). We find dilationrates dw/dδ between 0 .
055 and 0 .
171 for shear failure from an intact state (Table 1) where the two highestdilation rates were observed for samples with a different fault geometry (samples WG5 and WG12). For thetwin experiments performed at the same imposed experimental conditions ( P c = 160 MPa, p = 80 MPa),we observe a marginally higher dilation rate for the stable failure (sample WG8) relative to the dynamicfailure (sample WG7).The large drop in pore pressure localised in the fault zone together with the increase in pore volumeprovide direct evidence for strong fault zone dilatancy during shear failure. We can use the pore pressurerecorded in the fault zone during shear failure to decouple mechanical weakening and pore pressure effects.We compute the effective normal stress on the fault, obtained as σ n − p , where p is the measured on-fault porepressure, and analyse the stress paths in a Mohr diagram (Figure 4). The unloading paths deviate from theexpected unloading path for shear failure under drained conditions: We see that the effective normal stress inall experiments increases strongly at high shear stress (Figure 4). Dynamic shear failure occurs at this stage,where the dynamic unloading path runs parallel to the drained unloading path but at higher normal stress(dashed lines, Figure 4). For stable shear failure, the normal stress increases until the stress state reachesthe frictional strength envelope for Westerly granite ( Byerlee , 1967;
Lockner , 1998). From this point, thefault normal and shear stresses reduce, closely following the frictional envelope, and slip decelerates. Faultslip accumulated during this part of the shear stress drop increases nearly linearly with the recovery of thefault zone pore pressure towards the imposed pore pressure (Figure 6a) and both the rate of pore pressurerecharge and slip decrease with time (Figure 6b, c).
After shear failure, we imposed between 8 and 10 intervals of slip on the freshly formed fault at a slip rateof 1.15 µ m s − or 11.5 µ m s − . We arrested the slip rate between each interval to allow pore pressure toreequilibrate and measure total pore volume change. For each slip interval, fault slip was accommodatedeither by stable sliding, by stick slip, or by a combination of the two. We measured 6 to 8 stick slip events persample. The pore pressure initially increased at the onset of each interval, concurrently with an increase inshear stress (Figure 5). This was followed by a decrease in pore pressure, whilst shear stress approached eithera new steady state value and fault slip remained stable, or the shear stress approached a peak value beforedropping dynamically during a stick slip event. During the stick slip events, the pore pressure dropped byaround 1 to 10 MPa in the fault zone (Figure 5). After a stick slip event, we observed a similar behaviour of5 s h ea r s t r e ss [ M P a ] (a): stable failure P c = , p = po r e p r e ss u r e [ M P a ] off-faulton-fault s li p [ mm ] (b): dynamic failure P c = , p = s h ea r s t r e ss [ M P a ] (c): stable failure P c = , p = po r e p r e ss u r e [ M P a ]
100 0 100 200time since main failure [s]012 s li p [ mm ] (d): dynamic failure P c = , p = Figure 2: Records of shear stress (top), pore pressure (middle), and fault slip (bottom) for stable failure at (a) : P eff = 40 MPa (sample WG4) and at (b) : P eff = 80 MPa (sample WG8), and for dynamic failure at (c) : P eff = 40 MPa (sample WG3) and at (d) : P eff = 80 MPa (sample WG10). Interval for dynamic intervalshown as a dashed curve. Middle panel: On-fault pore pressure records shown as black curves, off-fault porepressure records shown as gray curves. Insets show pore pressures around the main failure event. Data fromother samples are shown in Figure S2. 6 fault slip [mm] po r e vo l u m ec h a ng e [ c m ] P e ff = M P a P e ff = M P a Figure 3: Cumulative pore volume change versus fault slip for shear failure (large markers) and subsequentepisodes of fault slip (small markers) in samples deformed at an initial effective pressure of 40 MPa (blackcurves) and 80 MPa (gray curves). Left-pointing triangle = WG3, diamond = WG4, circle = WG5, square= WG7, inverted triangle = WG8, hexagon = WG10, right-pointing triangle = WG12.
50 100 150 200 250 300 normal stress [MPa] s h ea r s t r e ss [ M P a ] dynamic d r a i n e d d r a i n e d stable s t a b l e σ n τ fr i c ti o n a l s t r e n g t h fr a c t u r e s t r e n g t h log slip rate[mm s − ] Figure 4: (a) : Shear failure stress paths from the fracture strength (peak shear stress) down to the residualfrictional strength for dry or drained shear failure (gray curves), and for partially drained stable and dynamicshear failure (black curves, unstable experiments shown up to the onset of dynamic failure). Fault normalstress is corrected for the fault zone pore pressure measured during failure. Frictional strength for Westerlygranite from
Byerlee (1967) (dashed line). The inset shows the overall trend of fracture and frictionalstrength (
Byerlee , 1967) and highlights the approximate range of our experiments in the main panel.7 s h ea r s t r e ss [ M P a ] P c = p = po r e p r e ss u r e [ M P a ] off-faulton-fault time since shear failure [s] s li p [ mm ] (cid:219) δ = (cid:219) δ = (cid:219) δ = . µ ms − (cid:219) δ = . µ ms − Figure 5: Records of the shear stress (top), pore pressure (middle), and fault slip (bottom) during progressivesliding on a fresh fault in granite. The records contains stick slip events at imposed slip rates of 1.15 µ m s − and 11.5 µ m s − (dynamic intervals shown by dashed curve). On-fault pore pressure records shown as blackcurves, off-fault pore pressure records shown as gray curves. Inset shows shear stress for the entire series ofstick slip events recorded for this sample (sample WG8).pore pressure, fault slip, and shear stress to that observed after stable shear failure: Pore pressure recoveredtowards its imposed value, while shear stress decreases further and the fault continued to accumulate slip.The pore volume in the samples continued to increase during prolonged slip along the fault at a lesser ratethan during shear failure (Figure 3). Dilation rates computed for sliding along the freshly created fault varybetween 0 .
011 and 0 . Proctor et al. (2020). 8 slip since failure [mm] po r e p r e ss u r e r ec ov e r y [ M P a ] (a) WG4WG8 W G WG5 time since failure [s] po r e p r e ss u r e r ec ov e r y [ M P a ] (b) W G W G W G W G time since failure [s] s li p s i n ce f a il u r e [ mm ] (c) W G W G WG12 W G Figure 6: (a) : Pore pressure recovery p in the fault zone after shear failure as a function of slip δ , with δ = 0 at p = p min . p min is the lowest value for pore pressure reached during the failure process. Purple linesgives the relation between pore pressure and slip based on the unloading stiffness of the triaxial apparatus(equation 37). b) , (c) : Pore pressure recharge and fault slip versus time, shown from the lowest pore pressurereached during failure. Diffusion driven pore pressure recharge and slip are expected to scale with time asequation (2) (shown by purple curves). Our experiments demonstrate that dilatancy strengthening leads to stabilisation of the shear failure processof intact rock by increasing the normal stress on the fault, whereas shear failure of Westerly granite under dryor drained conditions is typically unstable. Our data also shows a transition from stable to dynamic shearfailure at the same effective pressure, which we propose results from insufficient initial pore pressure leadingto pore fluid vaporisation or degassing, which in turns caps the efficiency of dilatancy strengthening andleads to dynamic failure. These observations are similar to those of
Martin III (1980), but we have uniquemeasurements of the pore pressure evolution during stable and dynamic shear failure. These measurementindicate that stable shear failure consists of two consecutive stages: A stage dominated by a strong increasein fault strength by the dilatancy-induced pore pressure drop, followed by a stage of continued sliding duringpore pressure recharge. As the latter stage follows the frictional strength envelope of Westerly granite,we conclude that the fault itself must have formed during the first stage. Stable failure thus effectivelyseparates the loss of cohesion of intact rock from frictional sliding on a fresh fault. These findings ondilatancy strengthening during faulting have implications for earthquake nucleation and rupture dynamicsthat we shall discuss in this section.Ultimately, stability of rock failure in the laboratory is determined by the rock and machine stiffness andby the weakening of the fault zone material, the latter being impacted by pore pressure. In the following,we present a simple fault model which includes those key ingredients, and use it to fit our data and retrievequantitative estimates of hydro-mechanical parameters, allowing us to make predictions for stable shearfailure in the crust.
Prediction of dilatancy strengthening requires knowledge on the pore pressure drop in the part of the faultzone disturbed by dilatancy. For undrained conditions, where fluids within the fault zone are hydrologicallyisolated from their surroundings, the undrained pore pressure drop ∆ p undrained is simply expressed as thedilation rate (here expressed as an increase in fault width, Table 1 and Figure 3) times slip on the fault,divided by the effective fault zone storage. The fault storage consists of the storage capacity of the fault9one S f and the width of the fault, so that the undrained isothermal pore pressure drop is∆ p undrained = ( dw/dδ ) δ/S f w. (1)Undrained conditions may have been approached during dynamic shear failure of intact rock; however, thisdata cannot be used to estimate fault zone storage since the pore pressure drop was capped by the imposedpore pressure. We can estimate fault zone storage directly from the pore pressure drop measured during thedynamic interval of the stick slip events, assuming undrained conditions. To do so, we quantified from thedata the pore pressure drop ∆ p undrained and concurrent fault slip δ during the dynamic slip events (typicallyoccurring between 1 or 2 data points, i.e., <
200 ms). We used the dilation rate measured during progressiveslip on the fault, which we approximate as constant regardless of stable sliding or episodes of stick slip(Figure 3). For a total of 25 analysed stick slip events measured on 4 samples, most values for S f w liebetween 1 × − Pa − m and 6 × − Pa − m (3 stick slip events fall below this range, and 3 above). Notethat Brantut (2020) followed a similar approach to assess fault zone storage, but a larger time interval wasused that includes a partially drained pressure drop and slip preceding dynamic slip. His values thereforemust be treated as an upper bound as they may overestimate fault zone storage.Stable shear failure of intact rock occurs over the timespan of tens of seconds, and so the fault zone ispartially drained and we cannot obtain fault zone storage from equation (1). Instead, we use a 1D spring-slider model to simulate shear failure of intact rock under partially drained conditions (Figure 7a), adaptedfrom
Rudnicki and Chen (1988). The spring-slider model uses a cohesive type constitutive law for thestrength of the fault, with a residual fault friction that depends on the pore pressure, and allows for fault-normal fluid flow and dilation as a function of slip (see Methods section). We use the spring-slider modeland our experimental data in an inverse problem to estimate fault zone storage S f w and cohesion-weakeningdistance δ c for each experiment that exhibited stable shear failure (see Methods section). δ c in the model isakin to the slip-weakening distance measured in triaxial (controlled) rupture experiments on intact granite(Figure 1, sample WG6 in Table 1) (e.g., Wong , 1982;
Lockner et al. , 1991;
Aben et al. , 2020). However, δ c in the model depends on the definition of cohesion-weakening function and may vary from the measuredslip-weakening distances. We therefore treat δ c as an unknown parameter.Spring-slider simulations with the best fitting pair of values for S f w and δ c simulates fairly well theaccelerating slip rate and pore pressure reduction leading up to stable failure (Figure 7c, d). The simulatedon-fault pore fluid pressure recharge after failure does not follow the experimental record (Figure 7d), likelythe result of assuming fault-normal flow only in simulations. However, significant fault-parallel flow existsafter failure, aiding in faster recovery of the pore pressure, as directly evidenced by the transient time intervalwhere pore pressure is lower off- than on the fault (Figure 2 – a situation that is impossible if fluid flowwas only one-dimensional. In addition, the host rock was damaged during failure, resulting in faster porepressure recharge. This is best illustrated in the simulation for the experiment at P c = 110 and p = 70 MPa(Figure S6, sample WG12), where the fault deflected from its prospective trajectory towards the sample end,opening a direct flow path to the fluid inlet.The four experiments yield best fits for δ c that vary between 1.1 mm to 1.4 mm (Table 1), similar tovalues measured for the slip-weakening distance of granite ( Aben et al. , 2020) (Figure 1). We use this toquantify the transition from stable to dynamic failure in our experiments, which occurs when the loadingstiffness of the machine k (see Method section for computation of k ) is lower than the critical stiffness ofthe fault k cr . k cr depends on the sharpest decrease of the cohesion weakening function ( f ( δ/δ c ) in thespring-slider model, Methods section). For our expression for f , k drainedcr = 1 . τ p /δ c , where τ p is the cohesionshear stress drop at constant normal stress (Methods section). The undrained critical stiffness is given by k undrainedcr = (1 . τ p − µ ∆ p undrained ) /δ c . Using representative values for failure at 40 MPa effective pressure( τ p ≈
60 MPa, δ c ≈ µ = 0 . k drainedcr /k = 1 .
23 and drained conditions always leadto failure. However, k undrainedcr /k < P c = 110 , p = 70, Table 1). Note that, although the measured partiallydrained pore pressure drop is not the undrained pore pressure drop, it has the same effect on the critical faultstiffness. As proposed, the dilatancy strengthening effect cannot reach a sufficient magnitude to stabilise10 y = 0, p = p )(y = L)S, κ w k fault zone σ n fluid inlet S f , dw/dδ τ host rock spring slider model schematic(a) time since failure [s] f a u lt z on e po r e p r e ss u r e [ M P a ] (d)cohesion weakening distance [mm] f a u lt z on e s t o r a g e [ P a − m ] (b) sample WG4time since failure [s] s li p r a t e [ mm / s ] ob s e r v e d simulated(c) Figure 7: (a) : Sketch of the spring-slider model that allows for dilation within the fault zone. (b) : Prob-ability density resulting from exploring ( δ c , S f w )-space, computed using a least absolute value criterion forthe misfit between observed and simulated data for experiment WG4. Best fitting simulation indicated byblack marker. (c) : Observed (gray curve) and simulated (black) slip rate over time. (d) : Observed (graycurve) and simulated (black) fault zone pore pressure.11hear failure if the potential pore pressure drop is limited by the magnitude of the imposed pore pressure.Thus, according to our estimate, we expect vaporisation and dynamic failure at p <
28 MPa. Indeed, weobserve dynamic failure at the experiment performed at P c = 70 , p = 30. From this, we conclude that in ourexperimental setup we may always expect stable failure, provided that the initial pore pressure is sufficientlyhigh to sustain the on-fault pore pressure drop during failure.The best fitting values for effective fault zone storage S f w for the two experiments with the smallestdilation rates are around 1 . × − , the two experiments with a larger dilation rate yield values thatare approximately 3 to 4 times as large (Table 1). We then simulated the dynamic failure experimentswith values for S f w between 1 × − and 6 × − . Using this range, we always predict dynamic shearfailure associated with fluid vaporisation (i.e., on-fault pore pressure reaches zero and is capped there in thesimulation) for the samples with the lowest imposed pore pressures ( p = 20 MPa and p = 30 MPa, samplesWG2 and WG3). For dynamic failure at ( P c = 120 MPa, p = 40 MPa) and ( P c = 160 MPa, p = 80 MPa),dynamic failure was predicted only when S f w was smaller than 1 . × − and 1 . × − , respectively.From this simple analysis, we may conclude that S f w ≈ . × − are representative for more “planar”fault zones, whereas higher values up to S f w ≈ × − represent “non-planar” fault zone geometries witha higher dilation rate.Using equation (1), we get ∆ p undrained = 105 MPa and 79 MPa for failure with a lower dilation rate(samples WG4 and WG8, respectively) and ∆ p undrained = 50 and 39 MPa for non-planar faults (samplesWG5 and WG12, respectively). The measured partially drained and computed fully drained pore pressuredrops during shear failure of intact rock is larger for fault zones with a lower dilation rate compared withstrongly dilatant faults (Table 1). This may seem counterintuitive at a first glance, but ∆ p undrained iseffectively the ratio of dilation rate over fault zone storage: The dilation rate doubles from planar to non-planar fault zones, whereas the fault zone storage increases by three or fourfold. Of the twin experimentsperformed at P c = 160 MPa and p = 80 MPa, dynamic failure occurred for a lower dilation rate (and so alarger pore fluid pressure drop leading to vaporisation) compared to the stable failure.Effective fault zone storage for stick slip events is about one order of magnitude smaller than storageestimated for shear failure of intact rock. The difference may be ascribed to i) a change in storage capacityof the fault zone material, which evolves from a micro-fracture dominated zone at the onset of shear failureto a gouge and cataclasite bearing zone of deformation as fault slip progresses, and ii) a difference in thewidth of the fault zone that is disturbed by dilation. Visual inspection of the samples, and microstructurespublished by Brantut (2020), suggest a fault zone width of around 1 to 3 mm for shear failure of intactrock. Microstructures in simulated gouge show strong localisation during stick slip events, with shear zonesof the order of 10 µ m wide ( Scuderi et al. , 2017). Dilation outside these principal shear zones is likely, andlocalisation may be less in rough faults presented here than in simulated gouge. Nonetheless, an order ofmagnitude decrease in disturbed fault zone width for stick slip events compared to shear failure remainsrealistic.
We observed progressive fault slip after stable shear failure that scales linearly with on-fault pore pressurerecovery (Figure 6a). As the pore pressure recovers, the shear resistance to faulting decreases by τ = µ ( σ n − p ).Elastic unloading of the fault’s surrounding medium (i.e., the loading piston and the host rock) providesthe driving force to overcome the shear resistance, and is given by the elastic unloading stiffness − k of thesurrounding medium times fault slip δ . For stable sliding observed in the experiments, shear resistance τ and imposed load − kδ are in balance. From this, it follows that afterslip is linearly proportional to porepressure recovery, with a slope proportional to µ/k . Indeed, the slope of the data can be fitted with thisratio, adjusted for the triaxial conditions of the experiment (Figure 6) (see Methods section for details).Post-failure slip is primarily driven by diffusive pore pressure recharge of the fault: The fault zone afterfailure is at a lower pore pressure than its surroundings, and pore pressure reequilibration subsequentlyoccurs at a rate controlled by the hydraulic diffusivity of the fault walls. Specifically, if we assume that thefault is embedded in an infinite medium (i.e., the distance to any boundary where pore pressure is maintainedconstant is very large compared to hydraulic diffusion length), the on-fault pore pressure evolution is given12y ( Carslaw and Jaeger , 1959, , Chap. 12, p. 306) p ( t ) − p ≈ ∆ p undrained (cid:0) − e t/t recharge erfc( (cid:113) t/t recharge ) (cid:1) , (2)where t recharge = ηS w / (4 Sκ ) is the characteristic recharge time. Using independently measured repre-sentative values for each parameter, we obtain t recharge ≈
16 s, and the time evolution of post-failure porepressure is well predicted by Equation 2 (Figure 6b). As stated above and directly observed in the data(Figure 6a), the slip evolution is proportional to the pore pressure evolution, and is also well predicted bythe recharge equation 2 (Figure 6c).The details of the pore pressure and slip evolution deviate from the simple semi-infinite model because (1)further slip is likely to generate some dilation, limiting the recharge, (2) the imposed constant pore pressureat the ends of the sample accelerates the recharge, and (3) the fault geometry likely leads to along-fault porepressure diffusion, which also accelerates the recharge. Despite these caveats, the simple recharge modelcoupled to elastic relaxation explains remarkably well the phenomenon of short-term post-failure slip.
Dilatancy-induced pore pressure changes in the crust
We shall use the experimental constraintson fault zone storage and dilation rate to predict dilatancy stabilisation of shear rupture in the crust formaterials with large cohesion, such as intact rock and consolidated faults. We make predictions for two end-member cases of fully drained and fully undrained conditions. The dilation rate during shear failure of intactrock decreases with increasing effective pressure, as we can see from our data and those of
Rummel et al. (1978, Figure 6), which can be described by an exponential function (Figure S5). We used this function fora mean dilation rate, an upper bound for non-planar faults, and a lower bound for planar faults (Figure S5),as observed in our experiments. With these values, the undrained isothermal pore pressure drop (equation(1)) is between 50 and 100 MPa at effective pressures below 50 MPa, and becomes negligible at effectivepressures above 200 MPa (Figure S7a).Effective pressure in the crust depends on the depth and the ambient pore fluid pressure, which typicallyvaries between hydrostatic and lithostatic pore pressure. Using the pressure-dependent undrained porepressure drop, we predict that vaporisation during shear failure is likely to occur down to a depth of 4 km(for average dilation) regardless of the initial pore pressure (Figure 8). At very low pore pressure, vaporisationmay occur down to 8 km depth. We thus do not expect significant failure stabilisation in the uppermost partof the crust. We do however expect a strong dilatancy strengthening effect in the 2 kilometres directly belowthe vaporisation zone, where the undrained pore pressure drop is between 20% and 100% of the ambientpore pressure. Dilatancy strengthening becomes negligible at high effective pressures, below around 6 kmdepth, but remains significant in regions in the crust with a low effective pressure – i.e., regions where thepore pressure approaches lithostatic pressure.In the above analyses, the fault zone storage S f w remained constant with effective pressure. Whether thedisturbed fault zone width w changes with pressure remains speculative. Storage capacities of rocks typicallydecrease with increasing pressure (see measurements for undeformed thermally cracked Westerly, Figure S3),and we expect the same for the fault zone storage capacity S f . Qualitatively, a pressure-dependent fault zonestorage capacity would yield a larger undrained pore pressure drop at higher effective pressure, i.e., greaterdepth. Implications for fault slip
We predict strong dilatancy strengthening in the upper crust (between 4and 6 km) and at increased depth if the ambient pore pressure approaches lithostatic pressure. This hasimplications for the behaviour of fault slip and rupture dynamics across the entire ”spectrum of fault slip”from slow slip rates to earthquakes.In the last two decades, a range of fault slip rates in different crustal settings have been measured thatare slower than dynamic slip during earthquakes (
Peng and Gomberg , 2010;
B¨urgmann , 2018), such as slowslip events (SSEs) at the base of the seismogenic crust in subduction zones and in crustal-scale strike slip13
100 200 300 400 ambient pore pressure p [MPa] d e p t h [ k m ] v a po r i s a ti on ∆ p u n d r . = p , a v e r a g e d il a ti o n l a r g e d il a ti o n s m a ll d il a ti o n lit ho s t a ti c po r e p r e ss u r e hyd r o s t a ti c po r e p r e ss u r e ∆ p undrained / p critical fault length [m] d r a i n e d u n d r a i n e d , h y d r o s t a ti c und r a i n e d , n ea r lit ho s t a ti c Figure 8: Undrained isothermal pore pressure drop during shear failure of intact material in a crustal depthprofile with a range of ambient pore pressures (hydrostatic and lithostatic pore pressure profiles shown asdashed lines). The undrained pore pressure drop is normalised by the ambient pore pressure, shown here foraverage fault zone dilation. Black curves show where the undrained pore pressure drop equals the ambientpore pressure (i.e., net zero pore pressure), for average, small, and large dilation. At depths smaller thanthese curves, vaporisation is expected. Left panel: Critical length of the slipping patch of the fault, belowwhich the fault is stable. Gray curve: drained critical fault length, black curves: undrained critical faultlength for a hydrostatic pore pressure profile and a near lithostatic pore pressure profile. Critical fault lengthcalculated with a shear modulus of 25 GPa and poisson’s ratio of 0.22.14aults. Ambient pore pressure in these regions are typically high and may approach the lithostatic porepressure (e.g.,
Thomas et al. , 2009;
Matsubara et al. , 2009). Many SSEs are episodic, and may occur onparts of the fault that are also subject to fast slip (i.e., earthquakes) (
B¨urgmann , 2018). Slow slip eventsrequire initial weakening of the fault zone material to achieve a notable increase of fault slip, followed bysome strengthening mechanism to limit the slip rate and rupture velocity. Dilatancy strengthening of avelocity-weakening material has been proposed as one of these strengthening mechanisms (e.g.,
Segall et al. ,2010), motivated by the abundance of pore fluids within most SSE settings. Our experiments support thisidea directly, observing that dilatancy strengthening leads to the formation of an 8 cm long fault in around3 s, giving a propagation velocity of around 2.3 km per day that matches with propagation velocities forSSEs of around 10 km per day (
Gomberg et al. , 2016).We observed that dilatancy generates two separate phases of fault slip: The initial weakening process (heremodelled as the slip-dependent drop in cohesion), followed by prolonged fault slip and stress reduction drivenby pore pressure recharge of the fault. The timescale of the first phase is an elastodynamic one, governedby the slip-weakening distance and, in our tests, machine stiffness, and in nature by elastic properties of thesurrounding rock. The rate of the second phase is determined by the characteristic timescale t recharge givenin equation 2, where w is the width where dilation occurred (not necessarily the same as the width of theslipping zone). If the recharge timescale is much larger than the initial elastodynamic weakening timescale,pore pressure reequilibration can lead to afterslip, or at least to prolonged slip far from the rupture tip withina single rupture event. The characteristic timescale for the experiments is around 16 s. However, in nature,the dilatant region w could be orders of magnitude larger due to fault roughness (e.g., at dilatant jogs), sothat simple estimates from laboratory data might severely underestimate the diffusion timescale. In thiscase, afterslip may be observed as transient post-seismic slip, with post-seismic fluid migration potentiallydriving aftershock sequences (e.g., Nur and Booker , 1972;
Miller , 2020).
Implications for earthquake nucleation
Strong dilatancy strengthening leading to slow slip in ourexperiments may not always prevent earthquake nucleation. An earthquake nucleates when the loadingstiffness of the material surrounding the fault is lower than the critical stiffness of the fault ( k cr ). Theloading stiffness decreases with growth of the slipping fault section, whilst dilatancy stabilisation decreasesthe faults’ critical stiffness. We use the same formulation for drained and undrained k cr as in section 3.1, withpressure dependent values for ∆ p undrained and τ p . The pressure dependence of τ p for Westerly granite wasobtained from Byerlee (1967, Figure 5 and 9). The undrained critical stiffness at effective pressures below200 MPa is greatly reduced compared with the critical drained stiffness (Figure S7b). At higher effectivepressures, the dilatancy strengthening effect on the critical stiffness vanishes. The effective stiffness of thesurrounding material for faults embedded in an elastic continuum is approximated by k = G/W , where G is the shear modulus and W is the length of the slipping patch on the fault. Therefore, the critical stiffnessis inversely proportional to a critical length of the slipping region of the fault zone, and the fault becomesunstable when the dimension of the slipping patch exceeds a critical nucleation length. Using our data onWesterly Granite, the drained nucleation length remains more or less stable throughout the brittle part ofthe crust at around 0.3 to 0.4 m (Figure 8). For ambient hydrostatic pore pressure in an undrained faultzone, the nucleation length in the top two km of the crust increase fourfold, and reverts to the drainednucleation length at increased depth. We do not expect the increased nucleation length to have a large effectat such shallow depths: We predict vaporisation of pore fluids that reverts the fault to drained conditions.For ambient pore pressure profiles that approach lithostatic pore pressure, we predict that undrained failureincreases the nucleation length by a factor of 4 below the vaporisation zone and by a factor of 3 at 15 kmdepth (Figure 8).The near-lithostatic pore pressure zones correspond to the zones where SSEs occur, and although thenucleation length of Westerly granite may not be representative for the material that host SSEs, we stillexpect a strong increase in nucleation length for these materials. The critical nucleation length for Westerlygranite is directly applicable in regions in the crust with human subsurface activities, such as geothermalenergy reservoirs. These are generally located between 2 and 5 km depth (e.g., Tomac and Sauter , 2018),which overlaps with the depth range for pore fluid vaporisation and for strong dilatancy hardening (Figure15). Dilatancy strengthening leads to slow slip over increased distances, but does not necessarily change theinherent seismogenic character of the material and also allows for seismicity. This is evident from the stickslip events during continued fault slip in our experiments, where dilatancy strengthening was insufficientto suppress dynamic rupture. The approach towards dynamic rupture is however extended by dilatancystrengthening. Such a longer precursory phase may allow for better identification of active earthquakeprecursory phenomena such as foreshocks (e.g.,
Ohnaka , 1992;
Bouchon et al. , 2013) or precursory creep(e.g.,
Roeloffs , 2006), and passive phenomena such as changes in v p /v s ratios (e.g., Nur , 1972) and changesin seismic wave amplitudes (e.g.,
Shreedharan et al. , 2021). These two latter phenomena may be particularlysensitive to local pressure changes and fracture damage induced by dilatancy (
Shreedharan et al. , 2021).Hence, crustal regions with strong dilatancy strengthening may be best suitable to find reliable precursoryphenomena.
Implications for rupture energy budget
Dilatancy strengthening affects the dynamics of rupture.Rupture propagation is governed by the partitioning of stored elastic strain energy: A small part of it maybe released as radiated energy when failure is dynamic, whereas most is dissipated to overcome residualfault friction during sliding (frictional work) and as breakdown work done in excess of the frictional work.Breakdown work W b is a collective term for energy spent to reduce the intact strength of the fault zonetowards its residual steady-state frictional strength. The work done to overcome the dilatancy-inducedextraneous shear resistance that we observe in our experiments is in excess of the (drained) residual friction,and is thus part of the breakdown work. Breakdown work can be estimated simply as the area underthe shear stress versus slip curve in excess of the residual shear stress ( Wong , 1982) (Figure 1b). Doingso for the experimental data provides values for W b at stable shear failure of 83 kJm − (sample WG12),84 kJm − (sample WG5), and 100 kJm − (sample WG4) at 40 MPa effective pressure, and 118 kJm − at80 MPa effective pressure (sample WG8). The minimum breakdown work necessary to form a fault zoneand reach the residual frictional strength of a rock (also known as shear fracture energy) may be measuredfrom a quasi-static or “controlled” shear failure experiment (e.g., Lockner et al. , 1991;
Aben et al. , 2020). Weperformed such experiments on a notched, thermally cracked Westerly granite samples at 40 MPa effectivepressure, and obtain W b = 60 kJm − . For slow shear rupture, we thus see a 37-67% increase in breakdownwork attributed to dilatancy strengthening at 40 MPa effective pressure. The peak shear stress for dry andsaturated conditions at a given effective pressure is the same (Figure 1b), so that the strain energy storedin the loading apparatus does not vary. An increase in breakdown work thus implies that less strain energymay be used to accelerate rupture and slip – the failure process remains stable.We expect a further increase in breakdown work as rupture accelerates. At low rupture velocity, wehave shown that the dilatancy-induced pore pressure drop near the rupture tip is partially drained, as themeasured pore pressure drop is less than ∆ p undrained . At higher velocity, the fault zone dilates in a shortertime interval and so the pore pressure drop approaches the undrained pore pressure drop. The resistanceto slip thus increases more for a fast rupture than for a slow rupture. The strengthening effect remainstransient and vanishes with pore pressure recharge, but the time delay, or distance along the fault, betweenthe initial stress drop at the rupture tip and the second stress drop from pore pressure recovery is controlledby how fast the rupture propagates and how fast pore pressure diffusion can compensate the undrainedpressure drop near the tip. Thus, the breakdown work increases with rupture velocity 1) because of a largerdilatancy strengthening effect, and 2) because a larger amount of slip is accumulated before the fault zonepore pressure reaches its ambient value.Weakening processes acting behind the rupture tip at faster fault slip may be impacted by dilatancystrengthening. For example, we expect that the onset of weakening by thermal pressurisation of pore fluids(e.g., Lachenbruch , 1980;
Rice , 2006) will be delayed by the dilatancy-induced pore pressure drop at therupture tip compared to a fully drained case, as the thermal pressurisation process needs to overcome thedeficit in pore pressure first. This may increase the temperature in the fault zone due to frictional heating,which is otherwise buffered by thermal pressurisation weakening (
Garagash and Rudnicki , 2003).16
Conclusions
Our laboratory experiments demonstrate that dilatant strengthening stabilises rock failure and fault slip.The effect of dilatancy is capped by the zero lower bound for pore pressure, where fluid vaporises. All ourtests where failure was dynamic experienced transient fluid vaporisation. In the presence of pressurised fluids,rupture occurs in two stages: An initial stage driven by intrinsic weakening and elastic energy release form thesurrounding medium, and a second stage where post-failure pore pressure reequilibration leads to prolongedslip and stress drop, purely controlled by pore pressure changes. Our laboratory data are quantitativelyexplained by a simple spring-slider model, which we use to constrain a key previously unknown quantity,the fault zone storage capacity. The consequences of dilatant stabilisation of rupture are manifold, includingan increase in nucleation size, slowing of rupture propagation and increases in breakdown work and/orfracture energy. Our laboratory techniques opens the way to systematic quantification of hydro-mechanicalparameters under in-situ conditions, so that the wealth of theoretical knowledge on diltancy (e.g.,
Rice andRudnicki , 1979;
Segall and Rice , 1995;
Segall et al. , 2010;
Ciardo and Lecampion , 2019) can be used andtestable predictions can be made.
Acknowledgements
This study was funded by the UK Natural Environmental Research Council grant NE/K009656/1 to N.B.,and the European Research Council under the European Union’s Horizon 2020 research and innovation pro-gramme (project RockDEaF, grant agreement
References
Aben, F., N. Brantut, T. Mitchell, and E. David (2019), Rupture Energetics in Crustal RockFrom Laboratory-Scale Seismic Tomography,
Geophysical Research Letters , , 7337–7344, doi:10.1029/2019GL083040.Aben, F., N. Brantut, and T. Mitchell (2020), Off-fault damage characterisation during and after experi-mental quasi-static and dynamic rupture in crustal rock from laboratory, Journal of Geophysical Research:Solid Earth , doi:10.1029/2020JB019860.Bouchon, M., V. Durand, D. Marsan, H. Karabulut, and J. Schmittbuhl (2013), The long precursory phaseof most large interplate earthquakes,
Nature Geoscience , (4), 299–302, doi:10.1038/ngeo1770.Brantut, N. (2020), Dilatancy-induced fluid pressure drop during dynamic rupture: Direct experimentalevidence and consequences for earthquake dynamics, Earth and Planetary Science Letters , , 116,179,doi:10.1016/j.epsl.2020.116179.Brantut, N., and F. Aben (2020), Fluid pressure heterogeneity during fluid flow in rocks: New laboratorymeasurement device and method, doi:arXiv:2006.16699v1.B¨urgmann, R. (2018), The geophysics, geology and mechanics of slow fault slip, Earth and Planetary ScienceLetters , , 112–134, doi:10.1016/j.epsl.2018.04.062.Byerlee, J. (1967), Frictional Characteristics of Granite under High Confining Pressure, Journal of Geophys-ical Research , (14), 3639–3648.Carslaw, H., and J. Jaeger (1959), Conduction of heat in solids , 2nd ed., Clarendon Press, Oxford.17iardo, F., and B. Lecampion (2019), Effect of Dilatancy on the Transition From Aseismic to Seismic SlipDue to Fluid Injection in a Fault,
Journal of Geophysical Research: Solid Earth , , 3724–3743, doi:10.1029/2018JB016636.Garagash, D., and J. Rudnicki (2003), Shear heating of a fluid-saturated slip-weakening dilatant fault zone 1.Limiting regimes, Journal of Geophysical Research: Solid Earth , (B2), 1–19, doi:10.1029/2001jb001653.Gomberg, J., A. Wech, K. Creager, K. Obara, and D. Agnew (2016), Reconsidering earthquake scaling, Geophysical Research Letters , (12), 6243–6251, doi:10.1002/2016GL069967.Jaeger, J., N. Cook, and R. Zimmerman (2007), Fundamentals of Rock Mechanics , fourth edi ed., 488 pp.,Blackwell Publishing.Kennedy, C., and M. Carpenter (2003), Additive Runge–Kutta schemes for convection–diffusion–reactionequations,
Applied Numerical Mathematics , (1-2), 139–181.Lachenbruch, A. (1980), Frictional heating, fluid pressure, and the resistance to fault motion., Journal ofGeophysical Research , (B11), 6097–6112, doi:10.1029/JB085iB11p06097.Lockner, D. (1998), A generalized law for brittle deformation of Westerly granite, Journal of GeophysicalResearch , (B3), 5107–5123.Lockner, D., and J. Byerlee (1994), Dilatancy in hydraulically isolated faults and the suppression of insta-bility, Geophysical Research Letters , (22), 2353–2356.Lockner, D., J. Byerlee, V. Kuksenko, A. Ponomarev, and A. Sidorin (1991), Quasi-static fault growth andshear fracture energy in granite, Nature , (6313), 39–42, doi:10.1038/350039a0.Marone, C., C. Raleigh, and C. Scholz (1990), Frictional behavior and constitutive modeling of simulatedfault gouge, Journal of Geophysical Research , (B5), 7007–7025, doi:10.1029/JB095iB05p07007.Martin III, R. (1980), Pore pressure stabilization of failure in westerly granite, Geophysical Research Letters , (5), 404–406.Matsubara, M., K. Obara, and K. Kasahara (2009), High-VP/VS zone accompanying non-volcanictremors and slow-slip events beneath southwestern Japan, Tectonophysics , (1-4), 6–17, doi:10.1016/j.tecto.2008.06.013.Miller, S. (2020), Aftershocks are fluid-driven and decay rates controlled by permeability dynamics, NatureCommunications , (1), 1–11, doi:10.1038/s41467-020-19590-3.Nur, A. (1972), Dilatancy, pore fluids, and premonitory variations of ts/tp travel times, Bulletin of theSeismological Society of America , (5), 1217–1222.Nur, A., and J. Booker (1972), Aftershocks caused by pore fluid flow?, Science , (4024), 885–887, doi:10.1126/science.175.4024.885.Ohnaka, M. (1992), Earthquake source nucleation: A physical model for short-term precursors, Tectono-physics , (1-4), 149–178, doi:10.1016/0040-1951(92)90057-D.Peng, Z., and J. Gomberg (2010), An integrated perspective of the continuum between earthquakes andslow-slip phenomena, Nature Geoscience , (September), doi:10.1038/ngeo940.Proctor, B., D. Lockner, B. Kilgore, T. Mitchell, and N. Beeler (2020), Direct Evidence for Fluid Pressure,Dilatancy, and Compaction Affecting Slip in Isolated Faults, Geophysical Research Letters , (16), doi:10.1029/2019GL086767. 18ackauckas, C., and Q. Nie (2017), DifferentialEquations.jl – A Performant and Feature-Rich Ecosystem forSolving Differential Equations in Julia, Journal of Open Research Software , (1), 15, doi:10.5334/jors.151.Rice, J. (2006), Heating and weakening of faults during earthquake slip, Journal of Geophysical Research:Solid Earth , (5), 1–29, doi:10.1029/2005JB004006.Rice, J., and J. Rudnicki (1979), Earthquake Precursory Effects Due to Pore Fluid Stabilization of a Weak-ening Fault Zone, Journal of Geophysical Research , (B5), 2177–2193.Roeloffs, E. (2006), Evidence for aseismic deformation rate changes prior to earthquakes, Annual Review ofEarth and Planetary Sciences , , 591–627, doi:10.1146/annurev.earth.34.031405.124947.Rudnicki, J., and C.-H. Chen (1988), Stabilization of Rapid Frictional Slip on a Weakening Fault by DilatantHardening, Journal of Geophysical Research , (B5), 4745–4757.Rummel, F., H. Alheid, and C. Frohn (1978), Dilatancy and fracture induced velocity changes in rockand their relation to frictional sliding, Pure and Applied Geophysics PAGEOPH , (4-5), 743–764, doi:10.1007/BF00876536.Samuelson, J., D. Elsworth, and C. Marone (2009), Shear-induced dilatancy of fluid-saturated faults: Experi-ment and theory, Journal of Geophysical Research: Solid Earth , (12), 1–15, doi:10.1029/2008JB006273.Scholz, C., L. Sykes, and Y. Aggarwal (1973), Earthquake Prediction: A Physical Basis, Science , (4192),803–810.Scuderi, M., C. Collettini, C. Viti, E. Tinti, and C. Marone (2017), Evolution of shear fabric in granularfault gouge from stable sliding to stick slip and implications for fault slip mode, Geology , (8), 731–734,doi:10.1130/G39033.1.Segall, P., and J. Rice (1995), Dilatancy, compaction, and slip instability of a fluid-infiltrated fault, Journalof Geophysical Research , (B11), 22,155–22,171.Segall, P., A. Rubin, A. Bradley, and J. Rice (2010), Dilatant strengthening as a mechanism for slow slipevents, Journal of Geophysical Research , (B12305), 1–37, doi:10.1029/2010JB007449.Shreedharan, S., D. Bolton, J. Rivi`ere, and C. Marone (2021), Competition between preslip and deviatoricstress modulates precursors for laboratory earthquakes, Earth and Planetary Science Letters , , 116,623,doi:10.1016/j.epsl.2020.116623.Tarantola, A. (2005), Inverse problem theory and methods for model parameter estimation , 2nd ed., Societyfor Industrial and Applied Mathematics, Philadelphia.Thomas, A., R. Nadeau, and R. B¨urgmann (2009), Tremor-tide correlations and near-lithostatic pore pressureon the deep San Andreas fault,
Nature , (7276), 1048–1051, doi:10.1038/nature08654.Tomac, I., and M. Sauter (2018), A review on challenges in the assessment of geomechanical rock performancefor deep geothermal reservoir development, Renewable and Sustainable Energy Reviews , (May 2016),3972–3980, doi:10.1016/j.rser.2017.10.076.Wibberley, C. (2002), Hydraulic diffusivity of fault gouge zones and implications for thermal pressurizationduring seismic slip, Earth, Planets and Space , (11), 1153–1171, doi:10.1186/BF03353317.Wong, T.-F. (1982), Micromechanics of faulting in westerly granite, International Journal of Rock Mechanicsand Mining Sciences and , (2), 49–64, doi:10.1016/0148-9062(82)91631-X.19 ethods Sample preparation
Cylindrical wWesterly granite samples of 40 mm diameter and 100 mm height werecored and their surfaces were ground parallel. We cut two 17 mm deep notches into the cylindrical surfaceat a 30 ◦ angle with the cylinder axis (Figure S1a). The notches were aligned opposite each other so thatthe plane in between was most likely to fail during axial loading. The samples were subjected to thermalmicrocracking to increase the hydraulic diffusivity of the rock, achieved by placing the samples in a tubefurnace that was heated at a rate of 3 ◦ C min − to 600 ◦ C. This temperature was maintained for the durationof two hours, followed by cooling over the course of about 12 hours by switching off the furnace. The notcheswere filled with teflon disks prior to insertion in a rubber jacket (Figure S1a). The jacket was equipped withfour miniature pore pressure sensors: Two on the prospective failure plane, and two at the same height onthe intact part of the sample.
Pore pressure sensors
The pore pressure sensors consist of a steel 12 mm diameter cap with a thicknessof 2.5 mm (Figure S1b). On the inside edge of the cap, an 0.2 mm high lip creates a small reservoir inwhich pore fluid resides. The cap is placed over a metal stem so that the inside lip rests on the stem’ssurface. The pore fluid within the cap is isolated from the confining medium by an O-ring on the stem.The metal stem is glued into the rubber jacket, and allows pore fluid to pass from the metal cap to thesurface of the sample through an 0.4 mm diameter wide bore in its centre. A diaphragm strain gauge (fourindividual strain gauges arranged in a circular pattern, where two strain gauges measure tangential strainand two measure circumferential strain, wired in a full bridge configuration) is bonded to the outer surfaceof the steel cap. When pressure is applied to the steel cap from the outside (i.e., by confining pressure),the steel cap will elastically deflect inwards. Pore fluid pressure applied from the inside of the steel cap willinduce elastic deflection in the opposite direction. The elastic strain of the steel cap with effective pressure(confining pressure minus pore fluid pressure) is measured as a linear change in resistivity of the diaphragmstrain gauge. The change in resistivity was measured by a Fylde DC Transducer and Amplifier and digitallylogged. The sensor were calibrated prior to the onset of deformation by a number of confining pressure stepsand pore pressure steps. Further details on these sensors and their calibration are found in
Brantut andAben (2020).
Experimental setup
The shear failure experiments were executed in a conventional oil-medium triaxialloading apparatus at University College London. Axial load was measured by an external load cell correctedfor friction at the piston seal. Axial shortening was measured by a pair of Linear Variable DifferentialTransducers (LVDTs) outside the confining pressure vessel, corrected for the elastic shortening of the piston.Up- and downstream pore pressures were measured by pressure transducers located outside the pressurevessel, and pore pressure was controlled by a single pore pressure intensifier equipped with an LVDT tomeasure the volume change in the intensifier reservoir. The shear stress and normal stress on the fault werecalculated from the recorded differential stress and confining pressure, using the 30 ◦ angle between the faultand the direction of axial load.The samples were deformed at nominal effective pressures of 40 MPa or 80 MPa, with a varying combi-nation of confining and imposed pore pressure (Table 1). The imposed pore pressure during deformation waskept constant at both sample ends. Axial load was increased by imposing a constant piston displacementrate of 1 × − mm s − that corresponds to a strain rate of 1 × − s − . The samples were loaded untilshear failure, after which the piston displacement was arrested to measure the pore volume change ∆ V p inthe sample. After the measurements, we continued to accumulate slip on the newly formed fault; either bystable sliding or by stick slip events. After each slip increment, we paused deformation to remeasure porevolume changes. Pore volume change was measured from the volume change in the pore pressure intensifier,which we ascribe entirely to the change in pore volume in the sample. After arresting the movement ofthe loading piston, the pore pressure throughout the sample was allowed to recover towards the imposedvalue. This recovery was achieved when the volume of the pore pressure intensifier reached a stable value.We defined the pore volume change for an interval of fault slip between two such equilibration points. The20rst point in this series is at zero fault slip, defined at the peak differential stress where we did not arrestdeformation, but nonetheless recorded a nearly homogenous pore pressure throughout the sample. Calculation of dilation rate
We compute dilation rate from the pore volume data. We express dilationas the rate of increase in fault zone width w with fault slip δ , and we assume that any change in pore volumeoccurred in the fault zone after surpassing the peak stress. The volume of the fault zone is approximated asan elliptic cylinder with fault zone width w , a long axis radius a = 40 mm and a short axis radius b = 20 mm.The axes remains constant during deformation so that all pore volume change that has been measured isaccommodated by a change in w . This allows us to calculate dw = ∆ V p /πab for the slip interval in whichintact failure occurred, and for the slip intervals after shear failure. Fault slip was calculated from the 30 ◦ angle between the fault and the direction of axial load and the axial strain measurements corrected by theintact Young’s modulus of the sample, thereby assuming that all deformation after the peak stress wasaccommodated by fault slip. Projecting the pore volume change in an elliptic cylinder is a simplification:i) Parts of the elliptic cylinder are occupied by teflon disks that contribute less to the overall pore volumeincrease, leading to an underestimation of the fault zone porosity change. This underestimate is offset byii) an increase in the cylinder surface area as slip is accumulated and iii) some damage observed outside themain failure zone by visual inspection of the post-mortem samples. Hydraulic characterisation of intact rock
Permeability κ and storage capacity S of the intact rockwere obtained prior to deformation by a transient pore pressure front method akin to the pulse-decay method.A 5 MPa pore pressure step was produced at a rate of 5 MPa s − in the upstream pore pressure intensifier.The downstream end of the sample was undrained, with a known downstream storage capacity. The porepressure records and the upstream pore volume change for the pore pressure pulse may be expressed bya closed-form solution with two unknown parameters that are commensurate to permeability and storagecapacity. The solution can be used for any location along the sample height between the upstream anddownstream reservoirs, assuming homogenous hydraulic properties in the sample. This allows us to invertthe pore pressure and pore pressure intensifier volume records to obtain the best-fitting pair of values forthe permeability and storage capacity, with the intermediate pore pressure records measured by the effectivepressure transducers providing some additional constraints to the solution. For more details on the porepressure front method, see Brantut and Aben (2020). The results of the hydraulic characterisation of thesamples are presented in Figure S3.
Spring-slider model: Setup
We consider the sample as a rigid body split by a shear fault (Figure 7a).We impose a fault slip rate v ∞ through an elastic medium with stiffness k in the direction of fault slip δ .For quasistatic fault motion, the imposed load is equal to the shear resistance τ of the fault: k ( v ∞ t − δ ) − τ − χv = 0 , (3)where v is the slip rate, and χ is a viscous damping parameter preventing velocity to become unboundedduring instabilities. Typically χv remains negligible compared to other terms unless v becomes extremelylarge (e.g., Segall and Rice , 1995). We then follow
Rudnicki and Chen (1988) and use a cohesive typeconstitutive law for the strength of the fault, with a residual fault friction that depends on the pore pressure(as seen in our data, Figure 6): τ = τ p (1 − f ( δ/δ c )) (cid:124) (cid:123)(cid:122) (cid:125) cohesion + µ ( σ n − p ) (cid:124) (cid:123)(cid:122) (cid:125) friction , (4)where τ p is the drop in shear stress from fracture strength to frictional strength for a constant normal stress(Figure S2). The function f describes the loss of cohesion with slip, with f (0) = 0 , f (1) = 1. δ c is acharacteristic slip-weakening distance (Figure S2a). Here, we use f = − δ/δ c ) + 3( δ/δ c ) as suggested by Rudnicki and Chen (1988, equation 3) (Figure S2a). The shape of this cohesion-weakening function matcheswell with the overall shape of the quasi-static controlled rupture (Figure 1a).21o simulate changes in p , we first consider that the fault zone itself is of a uniform width w and hasa uniform pore pressure, so that only fault-normal fluid flow may occur in spirit with the 1D spring-slidersystem. The fault zone is considered a “reservoir” with effective fault zone storage S f w . The fault zone isin hydraulic communication with the intact host rock, so that the pore pressure is governed by a diffusionequation: ∂p∂t = κηS ∂ p∂y , (5)where κ is host rock permeability, S is the storage capacity of the host rock, and η is the fluid viscosity.The coordinate y is the fault-normal spatial dimension, with y = 0 being the upstream end of the sample,and y = L the fault position (i.e., the half-length of the sample) (Figure 7a). The experimental boundaryconditions impose that p ( y = 0 , t ) = p . Note that the system is symmetric here so we only consider onehalf of the sample. A mass balance leads to the following boundary condition at y = L : ∂p∂t + 2 κS f wη ∂p∂y = − ˙∆ φS f ( y = L ) , (6)where ˙∆ φ is the rate of inelastic porosity change inside the fault zone – describing dilation as a functionof slip. Recall that from the experimental data, we expressed dilation as a linear increase in fault zonewidth with slip dw/dδ . We then obtain the rate of porosity change as ˙∆ φ = ( dw/dδ ) /w × v . For undrainedconditions, equation (6) reverts back to equation (1) for the undrained isothermal pore pressure drop. Furtherdetails to account for triaxial conditions, normalisation of variables, and numerical implementation to solvethe problem are presented in the next paragraph. Spring-slider model: Application to triaxial experiments
The spring-slider model as defined byequations (3) and (4) are for a constant normal stress and a spring load parallel to the sliding direction.Here, the experiments were conducted under triaxial conditions, so that the normal and shear stress are botha function of the differential stress Q and confining pressure P c : τ = ( Q/
2) sin(2 ψ ) , (7) σ n = P c + ( Q/ (cid:0) − cos(2 ψ ) (cid:1) , (8)where ψ is the angle between loading direction and the fault zone. From this, we obtain τ = τ (1 − f ( δ/δ c )) + µ ( P c − p ) / (cid:18) − µ − cos(2 ψ )sin(2 ψ ) (cid:19) , (9)with τ = τ p / (cid:18) − µ − cos(2 ψ )sin(2 ψ ) (cid:19) . (10)We understand τ to be the shear stress drop measured in triaxial failure experiments (Figure S2b). Theloading piston and intact parts of the rock sample act as the spring load. Differential stress decreases fromelastic relaxation of the loading piston by ∆ Q = − k (cid:48) (cid:15) , with k (cid:48) being the machine stiffness and (cid:15) the axialelastic lengthening of the piston. As Q decreases, the rock sample increases in length by elastic rebound as (cid:15)k (cid:48) /E × L , where L is the half length of the sample and E is the Young’s modulus of the sample. Becausethe movement of the top of the piston was arrested during shear failure, (cid:15) and the relaxation of the rocksample are both accommodated by slip along the fault δ so that (cid:15) = δ cos( ψ ) / (1 + 2 k (cid:48) E/L ). From this andequation 7, it follows that the fault parallel stiffness k is k = k (cid:48) sin(2 ψ ) cos( ψ )2(1 + 2 k (cid:48) L/E ) . (11)22 pring-slider model: Numerical solution for partially drained case We nondimensionalise ourgoverning equations prior to numerical implementation for partially drained conditions. We use the followingscales: t ← t/t diff , (12) y ← y/L, (13) p ← p/ ∆ p undrained , (14) τ ← τ /τ (15) δ ← δ/δ c , (16) v ← v/ ( δ c /t diff ) , (17)where t diff = L ηSκ (18)is the diffusion timescale across the sample, the undrained isothermal pore pressure drop ∆ p undrained is givenby equation (1) (Figure S2b) (with dw/dδ normalised w.r.t. δ c already), and τ is given by equation (10).The static equilibrium (equation (3)) is thus rewritten as K ( v ∞ t − δ ) − (1 − f ( δ )) − τ D ( λ − p ) − Xv = 0 , (19)where K = kδ c /τ , (20) τ D = µ ∆ p undrained /τ p , (21) λ = P c / ∆ p undrained , (22) X = χδ c κ/ ( L τ ηS ) . (23)Note that τ D is to be understood as the cohesion loss over the undrained pore pressure change times thefriction coefficient – i.e., the undrained dilatancy-induced increase in shear resistance (Figure S2b). Thegoverning equation for pore pressure becomes ∂p∂t = ∂ p∂y , (24)with boundary conditions p (0) = p / ∆ p undrained , (25) ∂p∂t + (cid:96) ∂p∂y = − v ( y = L ) , (26)where (cid:96) = 2 SLS f w . (27)The equation we are solving numerically is simply: dvdt = [ K ( v ∞ − v ) − f (cid:48) [ δ ] v + τ D ˙ p ] /χ, (28) dδdt = v, (29)where we compute the pressure rate in the fault by solving the full diffusion problem (Equation 24) usingfinite differences in space and the method of lines, so we can use very efficient stiff ODE solvers for the task.23elabelling the space coordinate y = 1 − y , so that the fault is at position y = 0 and sample edge is at y = 1,the diffusion problem reads ∂p∂t = ∂ p∂y , (30) p ( y = 1 , t ) = 1 , (31) ∂p∂t (cid:12)(cid:12)(cid:12) y =0 ,t − (cid:96) ∂p∂y (cid:12)(cid:12)(cid:12) y =0 ,t = − v. (32)We discretise space into N + 1 nodes at positions y , . . . , y N uniformly spaced with spacing ∆ y , and usecentered finite differences to evaluate spatial derivatives: dp n dt = p n +1 − p n + p n − ∆ y , (33)valid for n = 0 , . . . , N −
1, and boundary conditions read dp N dt = 0 (34)and dp dt − (cid:96) p − p − y = h (cid:48) [ δ ] v/δ , (35)where a ghost node at n = − n = 0, we find the time derivativeof pore pressure in the fault as dp dt = (cid:20) p − p ∆ y + h (cid:48) [ δ ] v/δ (cid:96) (cid:21) / (∆ y/ /(cid:96) ) . (36)Equations (28), (29), (33) for n = 1 , . . . , N −
1, (34) and (36) form a system of N + 3 ODEs in timefor unknowns { v, δ, p , . . . , p N } . We solve this system using a 5th order, A-L stable, stiffly-accurate, explicitsingly diagonal implicit Runge-Kutta method with splitting (see Kennedy and Carpenter (2003), methodimplemented as
KenCarp5() in the
DifferentialEquations.jl package, see
Rackauckas and Nie (2017)).All physical parameters, with the exception of the storage S f w and characteristic cohesion-weakeningdistance δ c , have been measured or imposed during the shear failure experiments: Intact hydraulic parameters κ and S were measured prior to deformation (Figure S1), the shear stress drop and coefficient of frictionwere obtained from mechanical data, and P c and p were imposed (Table 1).We assume that dilation is linear with slip, and we set the dilation rate to zero for fault slip beyondthe cohesion weakening distance δ c , thereby assuming that dilation is largest during the formation of thefault and neglecting dilation during fluid recharge driven slip. This is supported by the order of magnitudedifference between dilation rates measured for failure and for slip (Figure 3). To ensure that the total increasein pore volume after shear failure in the simulations is equal to the measured increase in pore volume, wenormalised the dilation rate dw (i.e., increase in fault zone width) by the input value used for δ c . Hence,the dilation rates for shear failure given in Table 1 are a lower bound. Note that more elaborate functionsto describe dilation with slip have been proposed (e.g., Rudnicki and Chen , 1988;
Segall and Rice , 1995),based on dilation rates measured in gouge. However, for shear failure of intact rock, dilation was onlymeasured afterwards, so that a linear increase during failure and subsequent slip is the most straigthforwardapproximation.We use the spring-slider model to simulate the slip rate and pore pressure during shear failure for a rangeof values for δ c and S f w . We then calculate the misfit between the observed maximum slip rate and minimumpore pressure during shear failure (Table 1) and the simulated maximum slip rate and minimum pore pressure,using a least absolute criterion and assuming a Laplacian probability density function ( Tarantola , 2005). Weuse uncertainties of 0.1 mm s − and 5 MPa for the slip rate and pore pressure, respectively. From theresulting probability density, we pick the best fitting pair of values for S f w and δ c (Figure 7d, Table 1). Forthe simulation results of the other stable ruptures, see Figure S4.24 ore pressure recovery versus slip for triaxial experiments: For the relation between pore pressurerecovery and slip after the largest drop in pore pressure (Figure 6) we consider the fault strength is givenas τ = µ ( σ n − p ), and is in balance with the imposed load − kδ . Using the triaxial relations for shear andnormal stress and machine stiffness (equations (7) and (11)), we obtain: p = δk (cid:18) µ − (1 − cos(2 ψ ))sin(2 ψ ) (cid:19) . (37)With E ≈
40 GPa, measured during unloading of the sample at the end of the experiment, L = 5 cm, ψ = 30 ◦ , k (cid:48) = 382 GPa m − , and µ = 0 .
6, we observe that this relation follows the slip versus pore pressuredata from the stable failure experiments, where deviations for samples WG4 and WG5 are explained byvariations in µ , which are near to 0.7 (Figure 6). 25 ac k e t sample prospectivefailureplaneP eff transducer (a) P eff transducer(b) Figure S1: (a) : Schematic of the sample setup, with two effective pressure transducers on the prospectivefailure plane, and two transducers on the hanging block of the sample. (b) : Photo of an effective pressuretransducer. slip s h ea r s t r e ss , po r o s it y c h a ng e δ c τ peak τ r τ p ∆ φ = d w / d δ w δ (a) effective pressure s h ea r s t r e ss τ obsr τ r τ peak fr i c ti o n a l s t r e n g t h fr a c t u r e s t r e n g t h ∆ p undrained τ p τ (b) Figure S2: (a) : Sketch of the cohesion shear stress drop τ p from peak stress τ peak down to residual frictionalstrength τ r (black curve) for a constant normal stress and a linear increase in porosity (gray curve) asfunctions of slip. (b) : The components of the resistance to sliding on the fault depicted in a normal stressversus shear stress sketch. One component is the cohesion shear stress drop τ p for a constant normal stress.The normal stress is not constant for triaxial deformation experiments, so that the observed shear stressdrop τ is larger than τ p . The other component is caused by a localised pore pressure reduction, here weshow an example of how the undrained isothermal pore pressure drop ∆ p undrained increases the resistance tofrictional sliding on the fault. 26
50 100 150 effective pressure [MPa] p e r m ea b ilit y [ m ] (a) effective pressure [MPa] s t o r a g eca p ac it y [ P a − ] (b) Figure S3: Permeability κ (a) and storage capacity S (b) as a function of effective pressure, measuredon undeformed thermally cracked Westerly granite using the transient pore pressure step method. Fittedexponential functions describing effective pressure dependence are shown as grey curves. Dashed verticallines at experimental imposed effective pressures of 40 and 80 MPa. Left-pointing triangle = WG3, diamond= WG4, circle = WG5, square = WG7, inverted triangle = WG8, hexagon = WG10, right-pointing triangle= WG12. See Table 1 for sample reference. 27 s h ea r s t r e ss [ M P a ] (a): stable failure P c = , p = po r e p r e ss u r e [ M P a ] off-faulton-fault s li p [ mm ] (b): dynamic failure P c = , p = s h ea r s t r e ss [ M P a ] (c): stable failure P c = , p = po r e p r e ss u r e [ M P a ] s li p [ mm ] (d): dynamic failure P c = , p = Figure S4: Pore pressure records during (a) dynamic failure (sample WG2, P c = 70 MPa, p = 30 MPa), (b) and (c) stable failure (sample WG5, P c = 100 MPa, p = 60 MPa, and WG12, P c = 110 MPa, p = 70 MPa),and (d) dynamic failure (sample WG7, P c = 160 MPa, p = 80 MPa). See Table 1 for sample reference.28
50 100 P eff d w / d δ [ − ] R u mm e l e t a l . ( ) t h i ss t udy Figure S5: Dilation rate with slip for intact samples versus effective pressure measured in this study(black symbols) and extracted from shear failure data of intact Fichtelgebirge granite from
Rummel et al. (1978). The effective pressure dependency of dw/dδ is estimated by an exponential relation dw/dδ = A × exp( − . P eff ), where the constant A = 0 .
27 for thermally cracked Westerly granite (black curve) and A = 0 . A = 0 .
35 and A = 0 .
16, respectively. Left-pointing triangle = WG3, diamond = WG4,circle = WG5, square = WG7, inverted triangle = WG8, hexagon = WG10, right-pointing triangle = WG12.29 .3 1 10 δ c [ mm ] − − S f w [ P a − m ] (a) sample WG5 t [ s ] v [ mm / s ] ob s e r v e d simulated(b)
100 0 100 200 t [ s ] p [ M P a ] (c) δ c [ mm ] S f w [ P a − m ] (d) sample WG8 t [ s ] v [ mm / s ] ob s e r v e d simulated(e)
200 0 200 t [ s ] p [ M P a ] (f) δ c [ mm ] S f w [ P a − m ] (g) sample WG12 t [ s ] v [ mm / s ] ob s e r v e d simulated(h)
100 0 100 200 t [ s ] p [ M P a ] (i) Figure S6: Spring-slider simulation results for stable shear failure experiment WG5 (top), WG8 (centre), andWG12 (bottom). (a) : Probability density resulting from exploring ( δ c , S f w )-space, computed using a leastabsolute value criterion for the misfit between observed and simulated data. Best fitting simulation indicatedby black marker. (b) : Observed (gray curve) and simulated (black) slip rate over time. (c) : Observed (graycurve) and simulated (black) fault zone pore pressure. Inset: Magnified pore pressure record around thestable failure event. 30 und r a i n e dpo r e p r e ss u r e d r op [ M P a ] s m a ll d il . l a r g e d il . (a) effective pressure [MPa] c r iti ca l s ti ff n e ss [ G P a m − ] und r a i n e d d r a i n e d machinestiffness stabledynamic(b) Figure S7: (a)
Undrained isothermal pore pressure drop ∆ p undrained and (b) critical stiffness k cr versuseffective pressure. Solid black line is based on the average of the estimates for S f w , the large and smalldilation curves (dashed) are based on minimum and maximum estimates for S f w and dw/dδdw/dδ