Effects of permeability and porosity evolution on simulated earthquakes
Introduction
Recent laboratory experiments conducted at high slip velocities (about 1 m/s, thus comparable with cosesimic rates) and at moderate normal stresses (1–25 MPa, e.g., Sone and Shimamoto, 2009, Han et al., 2010, Di Toro et al., 2011) reveal a dramatic fault weakening, basically consisting in a severe reduction of the coefficient of friction (μ) with respect to the so-called Byerlee’s value (μ ∼ 0.6; Byerlee, 1978). Several efforts have been spent in order to reproduce, at least qualitatively, this dramatic fault weakening by numerical models (i.e., in simulated earthquake ruptures), which has been proved to be more significant with respect to that predicted by classical, or canonical, formulations of governing models, such as the slip weakening law (Ida, 1972) and rate- and state-dependent friction laws (e.g., Ruina, 1983). As also pointed out by Noda et al. (2009) (see also Lachenbruch, 1980), the two physical phenomena most intensively studied are the thermally activated pore fluid pressurization and the flash heating of micro-asperity contacts. Experimental evidence of thermal pressurization (actually, thermochemical pressurization) has been reported by Ferri et al. (2010) and by De Paola et al. (2011). We also mention here that the contribution of thermal pressurization to the bulk weakening, at least in these experiments performed in non-cohesive rocks at very low normal stresses (<2 MPa), is limited to 10–20% of the total weakening (De Paola et al., 2011). The rest could be related to the so-called powder lubrication process (see Han et al., 2010, Reches and Lockner, 2010, Di Toro et al., 2011), which is not completely understood and which therefore cannot be incorporated in our modeling.
Both these two mechanisms, the thermal pressurization and the flash heating, have been adopted in the theoretical modeling of earthquake faulting (Bizzarri and Cocco, 2006a, Bizzarri and Cocco, 2006b, Rice, 2006, Beeler et al., 2008, Bizzarri, 2009a, Noda et al., 2009 among others) and from these studies it emerged that the two keys parameters controlling the time evolution of the fault traction are i) the spatial extension of the slipping zone (where cosesimic slip localizes; Sibson, 2003), and ii) the hydraulic diffusivity ω (defined in next equation (4)).
Within the coseismic time scale (during which the stress is released and the seismic waves are excited by the earthquake source) the temporal evolution of the porosity tends to counterbalance the effect of the thermal pressurization and it is able to change the evolution of the traction within the cohesive zone (Bizzarri and Cocco, 2006b). Under some special assumptions (namely, when the slipping zone equals —where ω is the hydraulic diffusivity defined in next equation (4) and Ta.f. is the period of the spring-slider system—and when the parameter ɛSR in next equation (6) is large), these effects can eventually become even more relevant during the whole seismic cycle, as recently illustrated by Mitsui and Cocco (2010).
In this paper we will consider the effects on the traction evolution of the time variations of the hydraulic diffusivity. In turn, the latter are caused by temporal changes of porosity and permeability, from which ω depends (see equation (4)). In particular, we will analyze the effects of the changes of ω with time on the evolution of the system in terms of the sliding velocity—which determines the recurrence time (Tcycle), defined as the time interval separating two subsequent instabilities—, of the traction evolution and of the thermal history of the fault.
Section snippets
Solution of the elasto-dynamic equation
In the present study we will consider the spring-slider (or mass-spring) analog fault model, where a block of mass m (per unit surface) is subject to an external load (expressed by the temporally constant loading rate ) and slides on a planar slipping zone of thickness 2w against a frictional resistance τ. The second law of dynamics (i.e., the equation of motion) for such a system is that of a harmonic oscillator:in which the overdots indicate the time derivatives, k
Porosity evolution
The porosity of a porous material (rock or sediment) is the dimensionless ratio between the current fraction of voids (pore volume; Vvoids) with respect to the total volume (Vtot) of the material: . By definition, Φ (sometime indicated with the symbol n) falls in [0,1]; Φ < 0.01 for solid granite and Φ > 0.5 for clay (e.g., Paterson and Wong, 2005). Fault zone porosity is expected to change during a cosesimic process due to the formation of the new cracks, changes to ineffective
The Rice’s model
The permeability physically represents a measure of the ability of a porous rock or an unconsolidated material to transmit fluids. It is often expressed through the hydraulic conductivity (κ) via , where ρfluid is the cubic mass density of the fluid and g is the acceleration of gravity. In the special case of a single-phase porous material the permeability is an intensive property, i.e., it is a function of the material structure only and, as such, it is scale invariant (it
Discussion and conclusions
In this paper we have considered the spring-slider dashpot model to numerically model the whole cycle of a fluid-saturated seismogenic fault, obeying a rate- and state-dependent friction. In particular, we assumed that the fault is described by the Linker–Dieterich’s constitutive model (equation (2)), in which the pore fluid pressure evolution is given by the analytical solution of Bizzarri and Cocco, 2006a, Bizzarri and Cocco, 2006b; equation (3)).
Rock–fluid interaction is an important
Acknowledgments
Part of this work originated from the insightful discussions I had the pleasure to have during the workshop “Physico-chemical processes in seismic faults” held in Padova, November 2010, founded by Progetti di Eccellenza, Fondazione Cassa di Risparmio di Padova e Rovigo (CARIPARO). L. Fucci is kindly acknowledged for assisting in the preparation of some figures. Finally, I thank the Associate Editor, G. Di Toro, and two anonymous referees for their useful comments.
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2012, Earth-Science ReviewsCitation Excerpt :Unfortunately, though permeability measurements were performed in different rock types (e.g., Zhang et al., 1999; Wibberley and Shimamoto, 2003), because of technical difficulties, this fundamental parameter is difficult to estimate at seismic deformation conditions (i.e., slip rate of 1 m/s). As pointed out by Bizzarri (2012b), the time variations of hydraulic diffusivity only due to porosity changes (through Eq. (12)) do not markedly affect the earthquake recurrence (cycle time), the traction evolution and the thermal history of the fault, regardless the values of the two free parameters εSR and LSR. On the contrary, the time variations of the permeability alone, in agreement to the Rice (1992)'s model (Eq. (13)) cause large increases in ω that tend to anticipate an instability and therefore tend to reduce the seismic cycle, as it emerges from Fig. 8.
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