Nicholas M. Beeler

Frictional behavior of large displacement experimental faults

The coefficient of friction and velocity dependence of friction of initially bare surfaces and 1-mm-thick simulated fault gouges ( 400 mm at 25°C and 25 MPa normal stress. Steady state negative friction velocity dependence and a steady state fault zone microstructure are achieved after ∼18 mm displacement, and an approximately constant strength is reached after a few tens of millimeters of sliding on initially bare surfaces. Simulated fault gouges show a large but systematic variation of friction, velocity dependence of friction, dilatancy, and degree of localization with displacement. At short displacement (<10 mm), simulated gouge is strong, velocity strengthening and changes in sliding velocity are accompanied by relatively large changes in dilatancy rate. With continued displacement, simulated gouges become progressively weaker and less velocity strengthening, the velocity dependence of dilatancy rate decreases, and deformation becomes localized into a narrow basal shear which at its most localized is observed to be velocity weakening. With subsequent displacement, the fault restrengthens, returns to velocity strengthening, or to velocity neutral, the velocity dependence of dilatancy rate becomes larger, and deformation becomes distributed. Correlation of friction, velocity dependence of friction and of dilatancy rate, and degree of localization at all displacements in simulated gouge suggest that all quantities are interrelated. The observations do not distinguish the independent variables but suggest that the degree of localization is controlled by the fault strength, not by the friction velocity dependence. The friction velocity dependence and velocity dependence of dilatancy rate can be used as qualitative measures of the degree of localization in simulated gouge, in agreement with previous studies. Theory equating the friction velocity dependence of simulated gouge to the sum of the friction velocity dependence of bare surfaces and the velocity dependence of dilatancy rate of simulated gouge fails to quantitatively account for the experimental observations.

Earthquake triggering by transient and static deformations

Observational evidence for both static and transient near-field and far-field triggered seismicity are explained in terms of a frictional instability model, based on a single degree of freedom spring-slider system and rate- and state-dependent frictional constitutive equations. In this study a triggered earthquake is one whose failure time has been advanced by Δt (clock advance) due to a stress perturbation. Triggering stress perturbations considered include square-wave transients and step functions, analogous to seismic waves and coseismic static stress changes, respectively. Perturbations are superimposed on a constant background stressing rate which represents the tectonic stressing rate. The normal stress is assumed to be constant. Approximate, closed-form solutions of the rate-and-state equations are derived for these triggering and background loads, building on the work of Dieterich [1992, 1994]. These solutions can be used to simulate the effects of static and transient stresses as a function of amplitude, onset time t0, and in the case of square waves, duration. The accuracies of the approximate closed-form solutions are also evaluated with respect to the full numerical solution and t0. The approximate solutions underpredict the full solutions, although the difference decreases as t0 approaches the end of the earthquake cycle. The relationship between Δt and t0 differs for transient and static loads: a static stress step imposed late in the cycle causes less clock advance than an equal step imposed earlier, whereas a later applied transient causes greater clock advance than an equal one imposed earlier. For equal Δt, transient amplitudes must be greater than static loads by factors of several tens to hundreds depending on t0. We show that the rate-and-state model requires that the total slip at failure is a constant, regardless of the loading history. Thus a static load applied early in the cycle, or a transient applied at any time, reduces the stress at the initiation of failure, whereas static loads that are applied sufficiently late raise it. Rate-and-state friction predictions differ markedly from those based on Coulomb failure stress changes (ΔCFS) in which Δt equals the amplitude of the static stress change divided by the background stressing rate. The ΔCFS model assumes a stress failure threshold, while the rate-and-state equations require a slip failure threshold. The complete rate-and-state equations predict larger Δt than the ΔCFS model does for static stress steps at small t0, and smaller Δt than the ΔCFS model for stress steps at large t0. The ΔCFS model predicts nonzero Δt only for transient loads that raise the stress to failure stress levels during the transient. In contrast, the rate-and-state model predicts nonzero Δt for smaller loads, and triggered failure may occur well after the transient is finished. We consider heuristically the effects of triggering on a population of faults, as these effects might be evident in seismicity data. Triggering is manifest as an initial increase in seismicity rate that may be followed by a quiescence or by a return to the background rate. Available seismicity data are insufficient to discriminate whether triggered earthquakes are “new” or clock advanced. However, if triggering indeed results from advancing the failure time of inevitable earthquakes, then our modeling suggests that a quiescence always follows transient triggering and that the duration of increased seismicity also cannot exceed the duration of a triggering transient load. Quiescence follows static triggering only if the population of available faults is finite.

On rate-state and Coulomb failure models

We examine the predictions of Coulomb failure stress and rate-state frictional models. We study the change in failure time (clock advance) Δt due to stress step perturbations (i.e., coseismic static stress increases) added to “background” stressing at a constant rate (i.e., tectonic loading) at time t0. The predictability of Δt implies a predictable change in seismicity rate r(t)/r0, testable using earthquake catalogs, where r0 is the constant rate resulting from tectonic stressing. Models of r(t)/r0, consistent with general properties of aftershock sequences, must predict an Omori law seismicity decay rate, a sequence duration that is less than a few percent of the mainshock cycle time and a return directly to the background rate. A Coulomb model requires that a fault remains locked during loading, that failure occur instantaneously, and that Δt is independent of t0. These characteristics imply an instantaneous infinite seismicity rate increase of zero duration. Numerical calculations of r(t)/r0 for different state evolution laws show that aftershocks occur on faults extremely close to failure at the mainshock origin time, that these faults must be “Coulomb-like,” and that the slip evolution law can be precluded. Real aftershock population characteristics also may constrain rate-state constitutive parameters; a may be lower than laboratory values, the stiffness may be high, and/or normal stress may be lower than lithostatic. We also compare Coulomb and rate-state models theoretically. Rate-state model fault behavior becomes more Coulomb-like as constitutive parameter a decreases relative to parameter b. This is because the slip initially decelerates, representing an initial healing of fault contacts. The deceleration is more pronounced for smaller a, more closely simulating a locked fault. Even when the rate-state Δt has Coulomb characteristics, its magnitude may differ by some constant dependent on b. In this case, a rate-state model behaves like a modified Coulomb failure model in which the failure stress threshold is lowered due to weakening, increasing the clock advance. The deviation from a non-Coulomb response also depends on the loading rate, elastic stiffness, initial conditions, and assumptions about how state evolves.

The roles of time and displacement in velocity‐dependent volumetric strain of fault zones

The relationship between measured friction μA and volumetric strain during frictional sliding was determined using a rate and state variable dependent friction constitutive equation, a common work balance relating friction and volume change, and two types of experimental faults: initially bare surfaces of Westerly granite and rock surfaces separated by a 1 mm layer of <90 μm Westerly granite gouge. The constitutive equation is the sum of a constant term representing the nominal resistance to sliding and two smaller terms: a rate dependent term representing the shear viscosity of the fault surface (direct effect), and a term which represents variations in the area of contact (evolution effect). The work balance relationship requires that μA differs from the frictional resistance that leads to shear heating by the derivative of fault normal displacement with respect shear displacement, dδn/dδs. An implication of this relationship is that the rate dependence of dδn/dδs contributes to the rate dependence of μA. Experiments show changes in sliding velocity lead to changes in both fault strength and volume. Analysis of data with the rate and state equations combined with the work balance relationship preclude the conventional interpretation of the direct effect in the rate and state variable constitutive equations. Consideration of a model bare surface fault consisting of an undeformable indentor sliding on a deformable surface reveals a serious flaw in the work balance relationship if volume change is time-dependent. For the model, at zero slip rate indentation creep under the normal load leads to time-dependent strengthening of the fault surface but, according to the work balance relationship, no work is done because compaction or dilatancy can only be induced by shearing. Additional tests on initially bare surfaces and gouges show that fault normal strain in experiments is time-dependent, consistent with the model. This time-dependent fault normal strain, which is not accounted for in the work balance relationship, explains the inconsistency between the constitutive equations and the work balance. For initially bare surface faults, all rate dependence of volume change is due to time dependence. Similar results are found for gouge. We conclude that μA reflects the frictional resistance that results in shear heating, and no correction needs to be made for the volume changes. The result that time-dependent volume changes do not contribute to μA is a general result and extends beyond these experiments, the simple indentor model and particular constitutive equations used to illustrate the principle.

Inferring fault rheology from low‐frequency earthquakes on the San Andreas

Families of recurring low-frequency earthquakes (LFEs) within nonvolcanic tremor (NVT) on the San Andreas fault in central California show strong sensitivity to shear stress induced by the daily tidal cycle. LFEs occur at all levels of the tidal shear stress and are in phase with the very small, ~400 Pa, stress amplitude. To quantitatively explain the correlation, we use a model from the existing literature that assumes the LFE sources are small, persistent regions that repeatedly fail during shear of a much larger scale, otherwise aseismically creeping fault zone. The LFE source patches see tectonic loading, creep of the surrounding fault which may be modulated by the tidal stress, and direct tidal loading. If the patches are small relative to the surrounding creeping fault then the stressing is dominated by fault creep, and if patch failure occurs at a threshold stress, then the resulting seismicity rate is proportional to the fault creep rate or fault zone strain rate. Using the seismicity rate as a proxy for strain rate and the tidal shear stress, we fit the data with possible fault rheologies that produce creep in laboratory experiments at temperatures of 400 to 600°C appropriate for the LFE source depth. The rheological properties of rock-forming minerals for dislocation creep and dislocation glide are not consistent with the observed fault creep because strong correlation between small stress perturbations and strain rate requires perturbation on the order of the ambient stress. The observed tidal modulation restricts ambient stress to be at most a few kilopascal, much lower than rock strength. A purely rate dependent friction is consistent with the observations only if the product of the friction rate dependence and effective normal stress is ~ 0.5 kPa. Extrapolating the friction rate strengthening dependence of phyllosilicates (talc) to depth would require the effective normal stress to be ~50 kPa, implying pore pressure is lithostatic. If the LFE source is on the order of tens of meters, as required by the model, rate-weakening friction rate dependence (e.g., olivine) at 400 to 600°C requires that the minimum effective pressure at the LFE source is ~ 2.5 MPa.

The role of fluid pressure on frictional behavior at the base of the seismogenic zone

To characterize stress and deformation style at the base of the seismogenic zone, we investigate how the mechanical properties of fluid-rock systems respond to variations in temperature and strain rate. The role of fluids on the processes responsible for the brittle-ductile transition in quartz-rich rocks has not been explored at experimental conditions where the kinetic competition between microcracking and viscous flow is similar to that expected in the Earth. Our initial analysis of this competition suggests that the effective stress law for sliding friction should not work as efficiently near the brittle-ductile transition as it does at shallow conditions.

32 - Rock Failure and Earthquakes

This chapter summarizes experimental observations and related theoretical developments of faulted and intact rock properties related to earthquake nucleation, failure, and dynamic slip. It primarily considers earthquakes occurring in the brittle crust. Intermediate and deep-focus earthquakes having unique mechanical considerations are discussed. Further, the chapter focuses on repeatable laboratory observations and their direct implications for fault strength and stability. Natural faults in the Earths crust are zones of localized shear that are inherently weaker than the surrounding intact rock. Slip can occur slowly as aseismic creep or rapidly as earthquakes with long interevent times during which little or no slip occurs. Crustal faults exhibit complex and irregular geometry, typically fractal in nature that includes irregular, interlocked surfaces, offset segments, bends, and junctions. Experimental and theoretical rock mechanics studies greatly influence the understanding of earthquakes and related crustal processes. Many key theoretical advances have either originated or been tested and refined through observations of brittle deformation under controlled laboratory conditions. Various parameters that affect strength of both fault surfaces and intact rock include mineralogy, porosity, cementation, packing geometry of gouge, and surface roughness.

Laboratory Observations of Fault Strength in Response to Changes in Normal Stress

Changes in fault normal stress can either inhibit or promote rupture propagation, depending on the fault geometry and on how fault shear strength varies in response to the normal stress change. A better understanding of this dependence will lead to improved earthquake simulation techniques, and ultimately, improved earthquake hazard mitigation efforts. We present the results of new laboratory experiments investigating the effects of step changes in fault normal stress on the fault shear strength during sliding, using bare Westerly granite samples, with roughened sliding surfaces, in a double direct shear apparatus. Previous experimental studies examining the shear strength following a step change in the normal stress produce contradictory results: a set of double direct shear experiments indicates that the shear strength of a fault responds immediately, and then is followed by a prolonged slip-dependent response, while a set of shock loading experiments indicates that there is no immediate component, and the response is purely gradual and slip-dependent. In our new, high-resolution experiments, we observe that the acoustic transmissivity and dilatancy of simulated faults in our tests respond immediately to changes in the normal stress, consistent with the interpretations of previous investigations, and verify an immediate increase in the area of contact between the roughened sliding surfaces as normal stress increases. However, the shear strength of the fault does not immediately increase, indicating that the new area of contact between the rough fault surfaces does not appear preloaded with any shear resistance or strength. Additional slip is required for the fault to achieve a new shear strength appropriate for its new loading conditions, consistent with previous observations made during shock loading.

Implications of coulomb plasticity for the velocity dependence of experimental faults

Simulated fault gouges often deform more stably than initially bare surfaces of the same composition. It is important to understand why the sliding stability is enhanced because the presence of gouge on natural faults may have the same effect as seen in experiments, and thus explain the absence of earthquakes at shallow depths. Gouge stabilization in experiments has been attributed to positive contributions to velocity dependence within gouge layers from either dilation (Maroneet al., 1990) or grain fracture (Biegelet al., 1989). In this study we test the hypothesis that some aspects of gouge and initially bare surface velocity dependence are identical by measuring the time-dependent constitutive parameterb. An important result follows however from stress analysis: if both sample configurations are frictional in the Mohr-Coulomb sense, each configuration is required to deform on planes of distinctly different orientation. The measured strength and velocity dependence will reflect this geometric difference. Our observed values ofb for simulated granite and quartz gouge are two to two and a half times smaller thanb for initially bare surfaces. This difference is completely accounted for if gouge is represented as a cohesionless-Coulomb plastic material. The analysis demonstrates the following points: 1) gouge deformation is fully consistent with Coulomb plasticity, 2) observed gouge velocity dependence is a function of observed strength and 3) the constitutive parameterb is the same for both bare surfaces and gouge. Furthermore, the results suggest that there is no time-dependent strengthening associated with stabilizing effects in gouge. These observations provide a framework for understanding how slip on initially bare surfaces and gouge deformation are related.

High‐frequency imaging of elastic contrast and contact area with implications for naturally observed changes in fault properties

During localized slip of a laboratory fault we simultaneously measure the contact area and the dynamic fault normal elastic stiffness. One objective is to determine conditions where stiffness may be used to infer changes in area of contact during sliding on nontransparent fault surfaces. Slip speeds between 0.01 and 10 µm/s and normal stresses between 1 and 2.5 MPa were imposed during velocity step, normal stress step, and slide-hold-slide tests. Stiffness and contact area have a linear interdependence during rate stepping tests and during the hold portion of slide-hold-slide tests. So long as linearity holds, measured fault stiffness can be used on nontransparent materials to infer changes in contact area. However, there are conditions where relations between contact area and stiffness are nonlinear and nonunique. A second objective is to make comparisons between the laboratory- and field-measured changes in fault properties. Time-dependent changes in fault zone normal stiffness made in stress relaxation tests imply postseismic wave speed changes on the order of 0.3% to 0.8% per year in the two or more years following an earthquake; these are smaller than postseismic increases seen within natural damage zones. Based on scaling of the experimental observations, natural postseismic fault normal contraction could be accommodated within a few decimeter wide fault core. Changes in the stiffness of laboratory shear zones exceed 10% per decade and might be detectable in the field postseismically.