Creep events at the brittle ductile transition

Authors


Abstract

[1] We present an analytic formulation to model creep events at the transition between brittle behavior in the crust and viscous behavior in ductile shear zones. We assume that creep events at the brittle ductile transition (BDT) are triggered by slip on optimally oriented fractures or network of fractures filled with weak ductile material. These events are expressed as transient flow in ductile shear zones likely aided by the release of crustal fluids. We show that the creep in the shear zone can be modeled as the motion of a forced damped oscillator composed of a brittle viscoelastic crust, a ductile shear zone and a creeping zone of fractures at the BDT. The time scale of the events varies between seconds to thousands of years depending on the viscous, elastic and brittle-plastic properties of the fractured BDT, the shear zone and the crust. The nature of the events depends on the aspect ratio, γ of the shear zone thickness, Hw to the length of the fractured zone, w. We find that thick shear zones with small fractures at the BDT are stiff and generate creep oscillations. Thin shear zones with well-connected fractures over a large width have very small stiffness and are well lubricated. They generate slow creep events or steady creep event. The former are similar to transient slip events and the latter to creep at the far field tectonic rates. The viscosity of the shear zone, ηw enhances lubrication if it is small and stiffness if it is large.

1. Introduction

[2] Recent work shows that localization at the brittle ductile transition (BDT) arises naturally as a consequence of brittle failure in polymineralic rocks where both brittle and viscous behavior (semibrittle) occur in at least two different mineral phases (i.e., quartz and feldspar between 300°C and 450°C) [e.g., Mancktelow and Pennacchioni, 2005; Pennacchioni, 2005; Pennacchioni and Manktelow, 2007; Fusseis and Handy, 2008; Brander et al., 2011]. Assuming that seismogenic fault zones may be loaded by localized ductile shear at the BDT [e.g., Gilbert et al., 1994; Lister and Davis, 1989; Scholz, 2002], variations in creep rates may be expressed as strain rate variations in the seismogenic fault rooted in the ductile shear zone. Such strain rate variations may partly explain the increasing body of evidence that slip rates vary in space and time [e.g., Wallace, 1987; Friedrich et al., 2003]. Here we present a model of nonsteady strain accumulation in the form of creep events within localized shear zones at the base of the seismogenic lithosphere. Practically, we model the initiation of a network of shear fractures as a stress perturbation at the transition from brittle to ductile behavior. We study the different creep behaviors predicted by the model for physical properties suitable for the Earth's crust. We model large variations in creep rate at the BDT on both short time scales (seconds to years) and long (years to thousands of years) time scales. Finally, we apply this model to well-documented strain rate variations occurring over long time scales along the Wasatch fault and adjacent fault zones which appear to have experienced rapid increases in Holocene slip rate relative to late Pleistocene rates [Friedrich et al., 2003].

1.1. Observational Background

[3] We seek to model regions of the crust where temperatures and pressures are such that mineral assemblages undergo a transition between brittle and viscous behavior. For example, for quartzofeldspathic crust, we would expect the viscous behavior to be exhibited by quartz, which behaves ductilely at ∼200°C, whereas feldspar should remain brittle until ∼450°C. The resulting media in this transition zone has the attributes of a brittle material forming the load bearing framework and a viscous material allowing for creep [e.g., Handy, 1990; Handy et al., 1999; Handy et al., 2007].

[4] Field observations have shown that the formation of fractures in the load-bearing framework of such media is a likely precursor to the development of localized ductile shear and flow [Simpson, 1985, 1986; Segall and Simpson, 1986; Mancktelow and Pennacchioni, 2005; Pennacchioni, 2005; Wightman et al., 2006; Pennacchioni and Manktelow, 2007; Fusseis and Handy, 2008; Brander et al., 2011]. The main assumption is that the presence of compositional heterogeneities and brittle discontinuities (i.e., fractures filled with veins) leads to localization in ductile shear. Often, the evolution of the ductile shear zone is accompanied by the formation of a network of (dilatant) fluid-filled fractures at near-lithostatic fluid pressure (Figure 1a). Fractures and fluid infiltration in fractures creates veins filled with relatively weak ductile minerals by chemical alteration and other fluid assisted mechanisms. These processes decrease the stiffness of the shear zone and increase the amount of viscous material over time, decreasing the ability of the strong phase to bear the load [e.g., Segall and Simpson, 1986; Stel, 1986; Tourigny and Tremblay, 1997; Christiansen and Pollard, 1997; Fusseis and Handy, 2008; Brander et al., 2011]. Under strain a subset of fractures links through new fractures or microcracks (Figure 1b). Eventually, this generates a reorientation of the stresses so that shear stresses are resolved on the fractures and veins. As a consequence the strain localizes as creep within the weak material filling the fractures. The progressive increase in surface area generated by new fractures intensifies the efficiency of fluid rock interaction and facilitates mass transfer and viscous flow in the ductile shear zone (Figure 1c). Multiple fracture events lead to a reduction of grain size [Rutter and Brodie, 1988; Stewart et al., 2000] (Figure 1d). One can envision a shear zone that grows through time and becomes progressively more ductile as fractures increase the ratio of surface area to volume and the ratio of viscous to brittle material (Figure 1). In the long term and at large spatial scales, multiple discrete viscous shear zones are thought to organize into a system of ductile shear bands that in turn forms a larger zone of localized shear [e.g., Segall and Simpson, 1986; Lister and Davis, 1989; Christiansen and Pollard, 1997].

Figure 1.

Formation of a shear zone at the BDT that consists of both brittle (gray) and weak ductile (blue) minerals. A network of fractures filled with the ductile mineral progressively grows over time. uo and u2 are the velocity boundary conditions applied on the top and bottom of the upper plate and shear zone, respectively. (a) A network of fractures at the BDT likely associated with preexisting brittle heterogeneities is subject to shear. (b) Fractures grow and connect with one another. (c) As the network grows the ratio of weak to strong material also grows until the ductile minerals dominate the effective viscosity of the semibrittle shear zone. (d) The ratio of ductile to brittle material continues to grow as long as fluids are available.

[5] Field examples of fractures or veins that are occupied by weaker minerals and eventually form an interconnected weak viscous phase such as calcite, quartzite, serpentinized peridotite are observed [e.g., Vissers et al., 1991; Herwegh and Kunze, 2002; Nüchter and Stöckhert, 2008; Fusseis and Handy, 2008]. For example, in the greenschist metamorphic facies typical of the middle crust, the alteration of feldspar to mica-rich aggregates along shear fractures can provide the softening necessary for the initiation of ductile shearing [Tourigny and Tremblay, 1997; Fusseis and Handy, 2008]. The formation of quartz-filled veins along fractures [Wightman et al., 2006; Nüchter and Stöckhert, 2008] may also provide weakening necessary for a shear zone to creep at the BDT. Dehydration reaction in a subducting slab also provide the fluids necessary to form fractures filled with weak serpentine mineral that eventually form mylonites anastomosing around strong olivine peridotite [e.g., Vissers et al, 1991; Evans, 2004].

[6] Herein, we hypothesize that strain at the BDT accumulates over multiple creep events over a fracture or network of fractures that are oriented in the direction of the shear stress in the shear zone. We do not model the period preceding creep when the fractures and shear stresses are not aligned. However, we model single creep events in the shear zone that occur when the fractures are optimally oriented and can accumulate slip. We present an analytic formulation that allows us to explore the effect of the physical properties of the crust, the shear zone and the BDT in between the crust and ductile shear zone, on creep at the BDT.

2. Shear Zone Model Setup, Formulation, and Model Parameter Space

2.1. Assumptions

2.1.1. Semibrittle Approximation

[7] In the shear zone, we envision that the polymineralic semibrittle crust near and below the BDT deforms via: (a) crystal-plastic deformation of individual phases whose viscosity is approximated by phenomenological creep laws (i.e., quartz) in the ductile shear zone below the BDT, (b) creep on fractures or a network of fractures in a stronger phase (i.e., feldspar) at the BDT. The fractures creep on the weak ductile material filling them. We consider the shear zone to behave effectively as a semibrittle material.

2.1.2. Approximation of Creep on the Fracture Network as a Stress Drop

[8] Here we approximate crystal plasticity with a Newtonian viscosity and slip on fractures or network of fractures is approximated by displacement on a circular crack representative of the stiffness of the fractured or damaged zone [Chinnery, 1969]. Following geological observation we assume the fractures are filled with weak minerals (i.e., calcite, quartz, or serpentine) deposited when dilatant fractures are formed. Strain in the heterogeneous media and the formation of new fractures locally reorient stresses imposing shear stress on some of the preexisting fractures filled with weak mineral (Figure 2a). As a result the set of adequately oriented weak fractures can accumulate strain (Figure 2b). The shear strain localizes on the weak fracture filling material and generates creep. The stress begins to drop concurrently with the initiation of creep, S in the ductile fractures (Figure 2c). Before the stress is optimally oriented the shear stress will likely increase as a function of strain. When the stress is released, the semibrittle media undergoes a transition from a jammed body to a flowing semibrittle medium. During that transition, the network of fractures and veins also serves to decrease the size of the grains or clasts.

Figure 2.

Evolution of fracture orientation, stress, and creep during a creep event. The stresses are oriented such that they impose a shear stress along previously formed fractures and cause shear strain to localize on the fracture filling material. (a) Preexisting fractures are present as brittle discontinuities. Initially, the orientation of the shear stress and the fractures are misaligned (the relative orientation is >0). At close to lithostatic pore pressure under strain the preexisting fractures connect and fill with weak aqueous fluid. Eventually the stresses reorient and a subset of fracture are aligned with shear stress (the relative orientation is 0). (b) The shear zone is jammed and strain accumulates as the stresses reorient. When the shear stress is orientated into the direction of the fractures, creep occurs and the stress drops. (c) Shear across the shear zone generates creep in the direction of shear when the fracture network is connected. Before that a small amount of creep may or may not accumulate. (d) Propagation of the fracture network in a shear zone as a function of strain. The shear zone may propagate episodically into the heterogeneous wall rock in slow transient events.

[9] We chose to express the stress drop, inline image at a pregiven fractured or damaged zone. The fracture length, deff can be a microfracture on the microscale (thin section), a fracture on the mesoscale (outcrop) and a network of interconnected fractures at the macroscale (lithospheric). We assume that the stress drop is concurrent with the creep, S. We follow Chinnery [1969] and Dieterich [1986] and approximate the stress drop as one caused by a circular dislocation (crack) of radius w representative of the damaged zone. In Chinnery [1969], the stress drop is given for a material with a Poisson's ratio of v = 0.25 and maximum displacement across the crack. The maximum strain across the shear zone is equal to the total creep S divided by the dislocation radius, w [Chinnery, 1969]. Taking the radius of the dislocation as w = deff/2, we obtain the stress drop as:

display math(1)

where S is the creep across. A schematic diagram of the fracture process that leads to the stress drop is given in Figure 2. The shear modulus, μ is characteristic of the fractured or damaged zone as a whole. The fractured medium is weak with a low shear modulus μ that accounts for the presence of aqueous fluids or weak minerals in the fractures [Berryman, 2007]. Throughout the paper we fix the shear modulus at the BDT at μ = 3.109 Pa. This value of the modulus μ corresponds to an intermediate shear wave velocity in fault zones believed to contain a large fraction of hydrous melt or aqueous fluid [e.g., Song et al., 2009; Van Avendonk et al., 2010].

2.2. General Model Setup

[10] We seek to describe the displacements that occur at the BDT as a function of time during the formation and slip of optimally oriented fractures. The traction at the BDT provides the force necessary to initiate creep, S on the damaged zone at the BDT. To simplify the development of our model we will assume that there is no strain variation in the x direction. This assumption limits our analysis to tractions at the interface in the model.

[11] The system is a two-layer model in which a viscoelastic plate (upper crust) is coupled to a viscous layer (shear zone) of thickness, Hw (Figure 3a). The upper plate has a thickness, Hb. Hw can vary from a few millimeters to kilometers. Let us consider the velocity at the top of the upper plate is uo (Figure 3a) and the bottom of the shear zone is moving at a constant velocity, u2 assuming that flow may be occurring below the shear zone in the ductile lower crust or lower plate. The velocity u2 may be constant or equal to zero. We model the upper plate as a Maxwell viscoelastic solid of shear modulus, G and of viscosity, ηL approaching values where the material would flow extremely slowly similarly to a brittle material. The shear zone below behaves as a viscous material with viscosity, ηw. The in-plane geometry of the lithosphere and the reference coordinate axes x and z are sketched in Figure 3; the y axis is normal to the plane of the figure. Deformation in the model is plane strain since the dimensions of the lithosphere along the y axes, perpendicular to x-z plane shown in Figure 3, is considered very large compared to the thickness of the layers measured in the z direction. We further assume that deformation is independent of distance along the y axis. Under these conditions, only the components of traction σzx and strain inline image along the longitudinal (x axis) direction are important.

Figure 3.

(a) Shear zone model dragged by an upper viscoelastic plate. (b) Mechanical equivalent of the semibrittle granular shear zone model.

2.3. Model Equations

2.3.1. Analytic Formulation

[12] The model setup is that of a two layer Couette model with negligible inertia. Initially, we use stress continuity at the interface between the layers to calculate equilibrium for the creep, S at the BDT. We then perturb this equilibrium by a stress drop due to creep on shear fractures at the BDT. Note that in this paper since ux is the only nonzero component of velocity we skip the subscript x, that is, u = ux. We assume that flow in both upper plate and shear zone conserve momentum and follow Couette flow. By doing so, we assume that the velocity varies linearly with depth. Ideally, the top surface of the upper layer is a free surface and has zero shear stress. In that case a boundary layer should form and the velocity should vary nonlinearly with depth. However, we approximate the velocity variation as linear with depth for a first-order estimate. In the upper plate (up) the velocity is expressed as:

display math(2)

where u1 is the horizontal velocity component at the bottom surface of the upper plate as shown in Figure 3a. In the shear zone (w), we have:

display math(3)

where u2 is the constant horizontal velocity component at the bottom of the shear zone. While writing equations (2) and (3) we assumed no slip conditions at the BDT interface, that is, the velocity at the bottom surface of the upper plate is equal to the velocity at the upper surface of the shear zone and so both are denoted by u1. The resulting shear strain rates in the upper plate and the shear zone are given by,

display math(4)
display math(5)

[13] The upper plate behaves as a Maxwell viscoelastic body and the corresponding constitutive relationship is given by,

display math(6)

where σ denotes traction. The effective traction in the viscous shear zone is:

display math(7)

[14] Since the strain is assumed to be constant in the x direction, consequently there is no variation of shear stresses in the x direction. Under plane strain conditions, the force equilibrium leads to the shear traction continuity at the BDT interface between the upper plate and the shear zone, that is,

display math(8)

[15] Taking the derivative of equation (8) with time and using the constitutive relationships (equations (6) and (7)) for each layer, we get:

display math(9)
display math(10)
display math(10a)
display math(10b)

[16] Assuming that the creep rate in the semibrittle shear zone at the BDT is inline image, inline image and inline image where C is a constant, we obtain:

display math(11)

[17] Using equation (8) we can write,

display math(12)

[18] Substituting the above equation in equation (11) and upon rearranging the terms we get,

display math(13)

[19] Since u2 is a constant and it only results in a shift of the driving stress, we chose u2 = 0. Upon rearrangement, we find that the traction continuity at the interface can be written as:

display math(14)

[20] Equation (14) represents creep at the interface between upperplate to viscous shear zone. We now perturb that equilibrium with a stress drop that is proportional to the creep in a damaged zone with optimally oriented fractures. The traction continuity at the interface incorporating the stress drop is given by:

display math(15)

[21] Introducing the stress drop term from equation (1) into the above equation and following the procedure from equations (2) to (14), we can derive the perturbed traction continuity equation as:

display math(16)

[22] The above second-order differential equation representing the creep, S, at the interface as an oscillatory behavior provides a hypothetical mechanism for episodically initiating slow localized creep events such as that described in the introduction. We simulate the release of stresses (i.e., unloading) at the BDT by forcing the system to unload elastic stress by creeping over an amount given by the steady state solution of equation (16). The oscillator is represented in Figure 3b with inline image representing a fictitious equivalent mass, m per unit length perpendicular to the x-z plane, which contributes to the inertial force of the oscillator system with dimensions of [M/L]. The viscous resistance (damping) of the system, inline image has the dimensions of inline image. The effective stiffness of the oscillator system inline image has units of inline image and depends exclusively on the perturbation term. Equation (15) may be rewritten in standard form of a forced damped oscillator:

display math(17)

where inline image, inline image, and inline image. The decay constant, D and natural frequency, ωo of the oscillator depend on the effective elastic and viscous structure of the lithosphere. In addition the frequency is determined by the shear modulus, the shear zone thickness and the radius of the characteristic shear fracture. The right hand side term in equation (17) has the dimensions of force per unit length and represents an external force F applied to the oscillator system due to the velocity uo.

2.3.2. Analytical Solutions

[23] The solutions of equation (17) are composed of a steady state solution and a transient solution. An oscillator can have three different behaviors depending on the roots of a quadratic equation, which in turn depend on D and ωo. The steady state solution is:

display math(18)

[24] If inline image the oscillator is under-damped and the solution becomes:

display math(19)

[25] Where S(0) and inline image are the initial creep and velocity of the shear across the semibrittle zone and inline image.

[26] If inline image the oscillator is critically damped and the solution is:

display math(20)

[27] If inline image the oscillator is overdamped and the solution is:

display math(21)

[28] With inline image, inline image, and inline image.

[29] The characteristic duration of the events before the oscillator returns to equilibrium are given by

[30] For the under-damped and critically damped cases:

display math(22)

[31] For the overdamped case:

display math(23)

2.4. Model Parameter Space and Creep Behaviors

[32] The equivalent mass per unit length, inline image is not a real mass but rather a measure of the rate of response of the oscillator system to a change in motion. Herein, we consider a fixed value of ηL (maximum stickiness of the brittle upper plate at the BDT) and of elastic shear modulus G of the upper plate; therefore m varies only as a function of the viscosity of the shear zone ηw. Thus, a low viscosity of shear zone (i.e., small ηw) decreases m leading to fast response and large ηw increases m leading to slow response. The parameter ηw is defined by crystalline creep flow laws dependent on temperature, strain rate, grain size [e.g., Handy, 1990; Hirth and Tullis, 1994; Kohlstedt et al., 1995; Handy et al., 1999, 2007]. It also depends on the recovery behavior of dislocations by processes like dynamic recrystallization that control, for example, the recrystallized steady state grains size depending on the deformation conditions [e.g., Braun et al., 1999; Montesi and Hirth, 2003; Handy et al., 2007]. The viscoelastic resistance of the oscillator system inline image is a measure of damping by the viscous resistance at the BDT caused by both upper plate and shear zone and is dependent on the ratio of the thickness of the two layers.

[33] Two characteristic time scales can be extracted from this formulation. The time scale Tu (equation (22)) is the decay time related to the resistance of the shear zone and of the upper plate to creep. The natural period of the oscillator is given by inline image. The natural period is dependent on Ke that is itself dependent on the aspect ratio inline image of the shear zone thickness to the radius of the damaged zone, w. A thin and wide layer with a small value for γ has a small effective stiffness, consequently, a long time period; whereas, a thick shear zone has a small effective stiffness, consequently, a small period.

[34] We distinguish several creep behaviors depending on the properties of the shear zone (Figure 4):

  1. If no creep occurs on the fractures at the BDT the shear zone is jammed. The displacement is constant in the shear zone (indicated by the blue region in Figure 4a).
  2. If the stiffness inline image is negligible, the shear zone creeps at the tectonic rates (Figure 4b). The semibrittle shear zone eventually reaches a quasi steady state with a slip rate inline image equal to uo. Steady creep is more likely to occur if γ is very small because the stiffness is proportional to γ. The third term of the left-hand side in equation (16) can be neglected and the solution is approximately

    display math(24)

    [35] and

    display math(25)

    When ηL is several orders of magnitude larger than ηw (i.e., weak shear zone) the term inline image the expression in equation (25) can be simplified to inline image. If no initial displacement has occurred before creep the shear zone initially creeps at a rate uo. If the decay constant is very small and the decay time, Tu is very large (>105−106 years), the creep rate will be nearly equal to the tectonic rate uo over very long time scales. The top of the shear zone appears to creep at uo (Figure 4b) and the shear zone appears to creep at the imposed rate (tectonic) uo. Eventually the creep should reach a steady state, Ss.

  3. If inline image and the stiffness Ke is not negligible the shear zone is overdamped and the shear zone creeps gradually with no oscillations. A transition from steady creep to slow transient creep occurs (Figure 4c). The time scales Tc and Tu compete to control the creep rate and creep amount in the shear zone. After creeping by Ss creep rate decreases to zero and the BDT is jammed. In the case that inline image, the shear zone may strain at a constant rate.

    [37] When the stiffness Ke is larger the shear zone is critically damped. The shear zone returns to equilibrium in the minimum possible amount of time. In that case the natural period inline image of the oscillator is close to the relaxation time generated by the viscous properties of the media. The analytic solution for the critically damped case is:

    display math(26)

    A close look at the analytical solution shows that the creep rate is initially very fast and then reduces to zero after an amount of time proportional to the natural period inline image.If the creep events repeat the shear zone will appear to have a jammed-slow creep event behavior. The flow behavior is dominated by the contacts at the fractures. As in the previous case the BDT creep rates returns to zero after an amount of creep proportional to Ss and inline image.

  4. When inline image, the shear zone is very stiff and oscillation period become smaller than the decay time. Physically, it means that unloading from the fractures occurs faster than the viscosity can dissipate the resulting oscillatory motion. The creep events generate motion that may be larger or smaller than the steady state creep rate in the shear zone. This results in what appears to be creep oscillations (Figure 4d). After some creep the shear zone at the BDT is jammed.
Figure 4.

Possible creep behaviors in the semibrittle shear zone displayed with velocity profiles. (a) The shear zone is jammed and the upper and lower plates are coupled. (b) Steady creep event for which the shear zone creeps over long time scales at the upper plate velocity before returning to the jammed state. (c) Slow creep event for the shear zone creeps at a rate faster than the upper plate rate for a given duration before returning to jammed state. (d) Creep oscillations for which the shear zone creeps faster and reverses sense of motion compare to the upper plate rate before returning to jammed state.

3. Application to the Brittle Ductile Transition

3.1. Possible Range of Parameters and Creep Behaviors for the Lithosphere

[39] The parameters m, R, Ke, Tc, and Tu previously defined are dependent on μ, ηw, ηL, γ, G, and Hb. G, μ, ηw, ηL can be defined for the Earth's Lithosphere for some specific minerals, temperatures and strain rate conditions. The geometrical aspect ratio γ is the ratio of the shear zone thickness to radius of the fracture and has a strong influence on the effective stiffness, Ke of the shear zone. Hw can vary from millimeters to several kilometers and w can be microscale to kilometer scale as well. Within these ranges, we find 10−7 < γ < 105 defining creep zones at the interface with very small to almost infinite stiffness. The former case can be viewed as a highly lubricated interface with large connected fractures in a damaged zone and the latter as a very strong BDT with only microfractures. In turn we can define a range of values for m, R, Ke, Tc, and Tu admissible for the BDT in the Earth's lithosphere.

[40] We fix the viscosity of the upper plate to ηL = 1023 Pa s, high enough that the plate is behaving elastically on the time scale of a hypothetical event of ∼50,000 of thousands of years. A viscosity of 1021 Pa s and 1022 Pa s would correspond to relaxation times of ∼500 and 5000 years, respectively, and such viscosities may be adequate for shorter events in a weaker crust. The parameter ηw is the viscosity of the weak phase occupying the shear zone and likely filling the creeping shear fractures. We consider quartzite, calcite and serpentine for modeling the weak phase viscosity (Table 1). In Figure 5, we explore a special case behavior for the shear zone for values of γ = 10 and G and Hb that are sometimes observed for the lithosphere (see Table 1, Figure 5).

Table 1. Viscosities, Damping, and Natural Periods for Potential Mineral Phases in the Semibrittle Domain for γ = 1000 (High-Stiffness Shear Zone)a
Fixed Parameters: w = 1 m, G = 3.1010 Pa, Hb = 15 km, Hw = 1 km, μ = 3.109 PaViscosity, ηw log10[Pa.s], inline image = 10−15−10−10 s (∼Maximum, Minimum/Maximum, Minimum)Decay Time, Tu, log10[s] (∼Maximum, Minimum/Maximum, Minimum)Natural Period, Tc, log10[s]/2(π) (∼Maximum, Minimum/Maximum, Minimum)
  1. a

    The first column lists the mineral assemblages, mechanisms (dislocation or diffusion creep), the references, and pressure and temperature conditions used to calculate the viscosities. The second column lists the viscosities for the minimum and maximum temperature listed column 1 and a minimum and maximum strain rate of 10−15 s−1 (0.1 mm yr−1 over 3 km thick shear zone) and 10−10 s−1 (10 m yr−1 over 3 km thick shear zone), respectively. The third and fourth columns list the damping and contact time scales for the same set of maximum and minimum values of temperature and strain rates (column 2). The time scales are given in seconds. Time scales greater or equal to 3 × 1013 s corresponds to millions of years. Those less than 3 × 1013−3 × 1010 s corresponds to thousands of years, less than 3 × 1010−3 × 107 s are hundreds to years, less than 3 × 107−3 × 105 s are months to days. Below the time scales are in hours to seconds. For each column the first set of values correspond to the minimum temperature and the min strain rate, minimum temperature and the maximum strain rate. The second set of values correspond to the maximum temperature and the minimum strain rate, maximum temperature and the maximum strain rate.

Quartzite (300°C–800°C)
Dislocation creep   
Shelton and Tullis [1981]21.4, 12.2/18.8, 9.812.0, 3/9.5, 0.510.2, 5.6/8.9, 4.4
Kronenberg and Tullis [1984]22.2, 14.7/18.8, 11.412.4, 5.4/9.5, 210.6, 6.9/8.9, 5.2
Gleason and Tullis [1995]20.7, 14.7/17.1, 1111.5, 5.5/7.8, 29.9, 6.9/8.1, 5
Gleason and Tullis [1995] Melt (2%)17.5, 13.5/13.5, 9.88, 4.3/4.3, 0.58.3, 6.3/6.3, 4.4
Calcite (200°C–700°C)
Dislocation creep: Carrara Marble   
Schmid et al. [1980]27, 17/23, 12.512.5, 7.7/12.5, 3.513, 8/11, 5.8
Schmid et al. [1980]37, 18.5/33, 1512.5, 9.3/12.5, 5.518, 8.8/16, 7
Diffusion creep: Synth. Calcite   
Walker et al. [1990]35, 14.3/33, 12.312.5, 5/12.5, 317, 6.7/16, 5.6
Serpentine (200°C–500°C)   
Dislocation creep   
Hilairet et al. [2007], 2 GPa18, 17.5/14.5, 13.78.7, 8.1/5, 4.58.5, 8.3/6.8, 6.4
Figure 5.

(a) Fracture ratio as a function of viscosity scaled by event duration and phase field for the flow behavior of the shear zone, (b) Creep duration as a function of accumulated deformation. (c) Creep duration as a function of moment per unit rupture area. (d) Creep rate as a function of creep deficit.

[41] Using crystalline creep laws from the literature (Table 1) we find ηw varies between 108 and 1037 Pa s with values greater than 1023 Pa.s, corresponding to flow on Maxwell time scales greater than that of the strong upper plate. When varying ηw in this range, we find that the viscous resistance R varies between 1022 and 1024 Pa.s, a viscosity close or higher to that of the upper plate. Correspondingly, m varies between 1020 and 1036 kg m−1. Because R varies little relative to m, we find that m controls the behavior of the shear zone through the natural frequency of the oscillator.

[42] From Table 1, the range of viscosities obtained for a range of temperature and minerals provides us with a range of decay time, Tu and natural period, Tc. We calculate the maximum and minimum time, Tu using the maximum and minimum value of temperature and strain rate, respectively.

[43] Generally, diffusion creep or the presence of melts leads to lower time scales since ηw is smallest (Table 1). Quartzite, serpentine, and calcite can have decay time and natural periods (equation (22) and natural period of the oscillator) of the order of 100s of thousands of years to minutes at both high and low strain rates (Table 1). The smallest Tu durations are reached in cases for which ηw is smallest, especially in the cases of quartzite with 2% melt and for serpentine. For γ = 1000, at small strain rates Tu > Tc and at high strain rates Tu < Tc.The former case generates creep oscillations in the shear zone, the latter case, slow or steady creep events. The serpentine flow law at 2 GPa (pressure condition for amphibole to eclogite) has little temperature dependence and the time scales are of the order of years to hours depending on the strain rate.

3.2. Lubrication Effect, the Aspect Ratio γ

[44] We now analyze the behavior of our model for a wide range of values of the aspect ratio γ. Localization and creep on optimally oriented fractures at all scales may occur at a very wide range of temperatures (viscosity) and effective stresses. However, the aspect ratio, γ, exerts a strong control on the effective stiffness and creep behavior in the shear zone. In Figure 5 we plot the behavior of the shear zone for γ varying between 10−7 and 105 (very soft to very stiff BDT, a range that also represents macro to microfractures) and a wide range of ηw (obtained from Table 1) divided by the period or characteristic duration of an event (equations (22) and (23)). The velocity of the top surface of the upper plate uo is taken as 1 cm yr−1, a value characteristic of tectonic rates. The damage zone radius w varies between 10−6 and 104 m values that correspond to a microfractured BDT zone with little damage all the way to a very damaged BDT zone connected by multiple optimally oriented fractures. The thickness of the shear zone Hw is calculated from the values of γ and w and is constrained to vary between 1 mm and 10 km.

[45] Figure 5a shows that creep can oscillate (blue crosses), be slow and transient (red crosses) or be almost constant (black crosses). We find that a stiff BDT zone with γ > 0 is more likely to oscillate than a soft one. However, when ηw is small and the stiffness Ke large, the BDT zone can be overdamped and respond with a slow creep event. Steady creep behavior (black crosses) occurs when the natural period is many orders of magnitude larger than the decay time (i.e., when the BDT is very soft). The results can be understood in the following way: (1) Small w is representative of a BDT zone that is slightly damaged and stiff. As a result the shear zone is not lubricated. (2) Large w corresponds to a damaged BDT with connected and optimally oriented fractures (in the sense of shear). As a result the shear zone is well lubricated. (3) In addition, a very thin and viscous weak shear zone contributes to render the interface softer by possibly connecting the fractures through the weak viscous matrix of shear zone. When the viscosity increases or the shear zone thickens, the strength of the matrix increases and oscillations occur on a time scale smaller than the decay time.

[46] Using the same range of γ and ηw, we plot the characteristic durations (decay time) for steady creep, overdamped (equation (22)), and under-damped (equation (23)) creep as a function of Ss (Figure 5b). Steady creep duration is taken as a maximum value of 1014 s (Millions of years). The characteristic duration of the creep events varies between seconds to thousands of years. The oscillator defines a domain in which oscillations occur and a domain where slow creep events with no oscillations dominate. The boundary between these two domains defines the critical behavior of the oscillator. The upper transition between oscillations and slow transient creep occurs when the viscosity of the shear zone becomes similar to that of the upper plate. The lower transition occurs for an optimal γ that is smaller for longer events. This suggests that for a given shear zone thickness there is an optimal aspect ratio that leads to the fastest damping of the creep events. Slow critical creep is following a linear relationship between characteristic durations and creep. The same relationship appears in Figure 5c where we plot the moment per unit area of creep as a function of characteristic duration of creep. The moment per unit area of rupture (equivalent earthquake moment) is written as:

display math(27)

[47] The moment can be expressed as a function of the characteristic duration for a critical event, knowing that the fracture length for critical event can be calculated from inline image and is expressed as:

display math(28)

[48] For critical events the duration of the slow creep episode is given by equation (22). For the case of critical creep we may combine equations (22) and (28) with equation (18) to obtain:

display math(29)

[49] As shown Figure 5, this relationship shows that moment is linearly proportional to the duration of the event.

[50] The case of critical damping is of particular importance since in that state the system returns to equilibrium in a minimum amount of time. The viscous flow in the shear zone optimally dissipates the load generated by slip at the BDT. The durations vary from seconds to thousands of years with the largest events being systematically the longest. We also show that γ has a strong control on the nature of the creep (Figure 5a).

[51] Creep events can have durations spanning minutes to tens of thousands of years. One of the main assumptions of our model is the presence of aqueous fluids to sustain creep across the fracture at the BDT. It shows that to sustain an event over hours to years or even thousands of years a continuous flow of fluids passing through the BDT must be maintained. Therefore, the duration is also likely dependent on the availability of fluids in the lithosphere.

4. Timescales of Slow Events

4.1. Effect of “Stick” Along the Semibrittle Shear Zone

[52] It is likely that the shear zone at the BDT creeps after a period during which it is jammed or deforming at a rate slower than the tectonic rate. During that “stick” period strain is accumulated that needs to be released by creep. We can safely assume that an initial negative deficit amount of creep xo accumulates during the corresponding jamming period at the BDT. The deficit creep is then released by shear fracture events as per equation (17). In that case the total amount of creep experienced by the shear zone before it jams again is Ss minus the amount of deficit creep accumulated elastically inline image and the average creep rate is proportional to frequency of the oscillator, ωo as inline image.The natural frequency ωo is a function of the viscous, elastic and plastic (fracture) properties at the BDT and whether the oscillator is under-damped, overdamped, or critically damped. Depending on the frequency of the oscillator and the deficit creep the average creep rate can be much larger than the rate imposed by the boundary velocity, uo (Figure 5d). For the case of the critical oscillator (red cross, Figure 5d) and a creep deficit varying between 10−6 and 10,000 m the average creep rate is of the order of hundreds of meters per year. In the steady creep case the creep rate is nearly equal to uo (black cross). When the shear zone is under-damped the average creep rate varies between imperceptible to tens of meters per year as well as when the viscosity of the shear zone is nearing that of the upper plate (Figure 5d).

[53] In Figure 6, we plot the creep and creep rate for the case of a semibrittle shear zone at the BDT of 1 km thickness that has experienced 3 m of deficit creep at 15 km depth. The granular viscosity is ηw = 1016 Pa s which corresponds to a weak wet quartizite dominated shear zone at the BDT for a temperature of 300°C [Gleason and Tullis, 1995]. We chose three values of γ equal to 0.001 (Hw = 5 km, w = 50 km), 100 (Hw = 1 km, w = 10 m), and 10,000 (Hw = 1m, w = 10−4 m). The first case corresponds to a highly damaged, connected and soft BDT and the last case corresponds to a very stiff, unconnected BDT. Other parameters are the same as above.

Figure 6.

(a–c) Schematic representation of the characteristic dimensions of a shear zone, the creep history (in black) and the creep rate history (in red) of creep (a-), critically damped (b-), then under-damped (c-) cases for a specific shear zone in which the ratio of fracture radius versus thickness of shear zone is varied

[54] The steady creep case (Figure 6a) may correspond tectonically to a continuous detachment or décollement structure at the BDT. The amount of creep and the near 1 cm yr−1 creep rate is consistent with the boundary condition of 1 cm yr−1 applied on the crust. The critical creep case could occur on a network of mesoscale (cm to m) fractures connected through the BDT (Figure 6b). This type of structure is well described in ductile shear zone that experienced large amounts of strain and at the BDT [e.g., Mitra, 1978, 1979, 1984; Gapais et al., 1987; Carreras et al., 2010]. Initially the creep rate increases to 6 cm yr−1 and over 250 years decreases back to zero (jammed state). The amount of creep accumulated is equal to the deficit creep xo in addition to Ss (Figure 6b). Finally, for the under-damped case, the fracture radius is of the order of a fraction of a millimeter. If such a network of fractures is activated, the BDT is still very stiff and the natural period is much smaller than the decay time. The creep oscillates over 20 years but would eventually be damped after a period proportional to the decay time (a few 100s of years) (Figure 6c). The creep rate varies between 1000 and −1000 cm yr−1 at the BDT. The creep and creep rate have a period of about 2 years and the shear zone creep at the BDT oscillates back and forth by up to 6 m. This last event does not sound realistic tectonically since the observations of such large tectonic rates and inversions in creep rates are so far unknown. It may be however that such oscillations may occur on a smaller time scale (weeks or days).

5. Example of Slow Creep Event: Wasatch Fault System

[55] The set of possible creep behaviors (Figures 5 and 6) and the time scales predicted from weak phase rheologies for the lithosphere are compatible with the observations of secular rate variations on fault zones varying from years to thousands of years [e.g., Wallace, 1987; Friedrich et al., 2003]. The decay time scales are of the order of minutes to thousands of years which spans a larger range and may be observable at subduction zone. However the model developed here is so far not intended to replicate such observations. Secular variations rates are low and in the range of tectonic rates. Table 1 suggests that if creep events occur at the BDT they are occurring on very long time scale of hundreds of years to thousands of years. The likely creep mechanism is dislocation creep.

[56] One likely case where fluid-filled fracture network forming in an extensional ductile detachment likely generates a time variable strain release over 10 kyr is the Wasatch fault system [Friedrich et al., 2003]. During an observed Holocene transient event, strain rate on the Wasatch fault system is in excess of the regional tectonic rate. Earthquakes appear to cluster with an intracluster rate two to four times greater than the Pleistocene rate of about 1 mm yr−1. The observed faster strain release could be caused by a creep transient in a ductile shear zone at the depth of 10–15 km. The presence of fluid at the brittle to ductile transition in detachment systems in the Basin and Range Province (Western US) is very likely since the upper crust experience large amount of brittle faulting opening downward paths for the percolation of fluid. The high thermal gradient in the area is also conducive to sustained hydrothermal circulation. One of the best examples of such an area is the Northern Snake Range that shows evidence of hydrothermal circulation between the upper crust and the mylonitic detachment for 4 million years (27–23 Ma) when the detachment was active [Gébelin et al., 2011].

[57] In Figure 7, we develop a model based on the interpretation and reinterpretation of Consortium for Continental Reflection Profiling seismic profiles in Utah [Allmendinger et al., 1983; McBride et al., 2010] and the interpretation of geodetic measurements from the Basin and Range Geodetic Network continuous Global Positioning System (GPS) network [Niemi et al., 2004] which show that the fault system was very active in the Holocene. In the Salt Lake City part of the Wasatch fault zone GPS data constrain the locking depth of the brittle Sevier Desert detachment and its extension in the middle crust [Velasco et al., 2009] (Figure 7a). The brittle layer thickness between is 15 km. We fix the upper and lower plate shear modulus to 3 × 1010 Pa and the shear zone shear modulus to 3 × 109 Pa. The tectonic rate is fixed at 1 mm yr−1. In order to obtain time scales and amplitude of cumulative displacement similar to the observations [Friedrich et al., 2003] we need to choose a granular viscosity of 1019 Pa s (Figure 7b). The strain accumulated prior to the event or deficit creep is 30 m. It corresponds to a strain accumulation at the BDT at 1 mm yr−1 over 30 kyr. These values correspond to the strain rate and quiescence period observed on the Wasatch fault system. The thickness of the shear zone, Hw, is not known, we therefore vary it between 2 and 6 km to match the potential thickness of ductile shear zone at the BDT and the thickness of a potentially weak middle crust in the Basin and Range Province [e.g., Wernicke, 1990]. We take γ equal to 10 that corresponds to fracture radius w of 200–600 m and Hw = 6 km matches best the observations. The resulting rate of creep on the shear zone varies between 0 and 8 mm yr−1. The fastest rates are obtained for the thinner shear zones. The maximum strain rate as defined by Friedrich et al. [2003] varies between 6 and 9 mm yr−1 and could correspond to a model with a shear zone thickness varying between 5000 and 6000 m. This exercise by no means explored all the available parameter space but is a good indication that our model can potentially explain the changes in tectonic rates and seismic activity observed on the Wasatch front.

Figure 7.

(a) Model geometry and parameters from seismic and geodetic observations. γ is 10 and the shear zone thickness varies between 2 and 6 km. (b) Modeled Holocene creep history over a range of shear zone thicknesses. (c) Creep rate history generated by creep on the shear zone. For the best case, the damaged zone length is 500 m and the shear zone thickness is 5 km. The shear zone viscosity is 1019 Pa s.

6. Discussion and Conclusion

[58] We distinguished four main creep behaviors: (1) Steady creep at a very damaged and weak BDT. In that case inline image and γ is very small and the shear zone lubricated. The shear zone creeps at a rate close to the tectonic rate. (2) Slow creep events occur when inline image, for a stiff or soft BDT. The shear zone is lubricated and experiences extended periods of creep at rates larger than the tectonic rate before returning to a jammed state. (3) Critical slow creep events occur when inline image for a stiff or soft BDT. This corresponds to the state for which the shear zone creeps before returning to steady state (jammed) in the shortest amount of time. (4) Oscillations when inline image and γ is large. The stiffness at the BDT is such that damping cannot diffuse creep efficiently. The shear zone is not lubricated.

[59] The viscosity of the shear zone is dependent on temperature, strain rate, fracture size, Moreover at high strain rates and stresses, viscous shear heating [e.g., Cordonnier et al., 2012] and dynamic processes will change the shear zone viscosity as it creeps. This will need to be taken into account into future formulations of the creep.

[60] To sustain creep in the shear zone, the presence of aqueous fluids at high pore pressure is required to enhance connectivity. Experiments in semibrittle analogues show that the connectivity of a precursory fracture network is dependent on the creation and destruction of fractures (pore space) that are respectively a function of accumulated plastic strain and the precipitation of minerals [e.g., Bauer et al., 2000]. In addition to a thorough analysis of the nonlinear effects of crystal plasticity on the creep of the shear zone, the effect of fluids diffusing through the network of fractures is necessary.

[61] If a system of brittle faults happens to be loaded tectonically by slow creep events in the shear zone, we could define two main tectonic behaviors: (1) Steady creep for which the slip rate on brittle fault is equal to the tectonic rate. When integrated over the Quaternary the strain rate along the San Andreas Fault (SAF), North Anatolian Fault, the Altyn Tagh Fault, or the East California Shear Zone (ESCZ) show little deviation from uniform [e.g., Sieh and Jahns, 1984; Wernicke et al., 2000; Argus and Gordon, 2001; Sella et al., 2002; Bennett et al., 2003; Frankel et al., 2007; Kozaci et al., 2009; Cowgill et al., 2009; Lee et al., 2009; Thatcher, 2009]. These fault systems fit our definition of steady creep. (2) Slow creep events during which brittle faults rooted in such a shear zone may experience an acceleration in slip rate released by earthquakes followed by a period tectonic quiescence during which little or no earthquakes occur. One of the best examples for such a system is the Wasatch fault zone [Friedrich et al., 2003]. However, paleoseismological evidence show that secular variations in strain rate over millenial time scales may occur along the some sections of the SAF (5 kyr) [Weldon et al., 2004], the ECSZ (South of the Garlock fault) and the Los Angeles Basin (LAB) (12 kyr) [Dolan et al., 2007], the Wasatch Front (WF) and the Basin and Range (10–50 kyr) [Wallace, 1987; Friedrich et al., 2003].

[62] An interesting feature of this model is that it follows the scaling relationship between duration and moment of slow slip events described by Ide et al. [2007] for shear zones of kilometer scale in length. This similarity may be coincidental, however the creep events' time scales can be of the order of minutes to years, which would correspond to the durations of the various slow earthquakes observed at subduction zones [Ide et al., 2007]. Such creep events associated with fractures may be possible if γ is optimal or large and the viscosity of the shear zone is of the order of 108−1017 Pa s (Table 1). This could correspond to fractures filled with calcite or quartz for low pressures and temperatures greater than 400°C depending on the creep mechanism and the presence of melt (Table 1). Fluid-filled fractures in serpentinized peridotites can also generate such events but for pressure large enough (>2 GPa) that serpentine can start behaving ductily. However for lower temperatures and pressures it is more likely that the frictional properties of the material dominate the deformation processes.

Acknowledgments

LLL and RB would like to acknowledge the support provided by NSF grant EAR#0510365.

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