Comparison between Different heat source functions in thermal conduction problems

I.N.G.V.Published3.1. Fisica dei terremotiope

Toward the Formulation of a Realistic Fault Governing Law in Dynamic Models of Earthquake Ruptures

Dynamic earthquake models can help us in the ambitious understanding, from a deterministic point of view, of how a rupture starts to develop and propagates on a fault, how the excited seismic waves travel in the Earth crust and how the high frequency radiation can damage a site on the ground. Since analytical solutions of the fully dynamic, spontaneous rupture problem do not exist (even in homogeneous conditions), realistic and accurate numerical experiments are the only available tool in studying earthquake sources basing on Newtonian Mechanics. Moreover, they are a credible way of generating physics– based ground motions. In turn, this requires the introduction of a fault governing law, which prevents the solutions to be singular and the crack tip and the energy flux to be unbounded near the rupture front. Contrary to other ambits of Physics, Seismology presently lacks knowledge of the exact physical law which governs natural faults and this is one of the grand challenges for modern seismologists. While for elastic solids it exists an equation of motion which relates particle motion to stresses and forces through the material properties (the scale–free Navier–Cauchy’s equation), for a region undergoing inelastic, brittle deformations this equation is presently missed and scientists have yet to fully decipher the fundamental mechanisms of friction. The traction evolution occurring during an earthquake rupture depends on several mechanisms, potentially concurrent and competing one with each other. Recent laboratory data and field observations revealed the presence, and sometime the coexistence, of thermally–activated processes (such as thermal pressurization of pore fluids, flash heating of asperity contacts, thermally–induced chemical reactions, melting of rocks and gouge debris), porosity and permeability evolution, elasto–dynamic lubrication, etc. In this chapter we will analyze, in an unifying and comprehensive sketch, all possible chemico–physical mechanisms that can affect the fault weakening and we will explicitly indicate how they can be incorporated in a realistic governing model. We will also show through numerical simulations that simplified constitutive models that neglect these phenomena appear to be inadequate to describe the details of the stress release and the consequent high frequency seismic wave radiation. In fact, quantitative estimates show that in most cases the incorporation of such nonlinear phenomena has significant effects, often dramatic, on the dynamic rupture propagation, that finally lead to different damages on the free surface. Given the uncertainties in the relative weight of the various competing processes, the range of variability of the value of some parameters, and the difference in their characteristic time and length scales, we can conclude that the formulation of a realistic governing law still requires multidisciplinary efforts from theoretical models, laboratory experiments and field observations

Rake rotation introduces ambiguity in the formulation of slip-dependent constitutive models: slip modulus or slip path?

The linear slip–weakening (SW) law, predicting that the traction decreases for increasing fault slip, is one of the most widely adopted governing models to describe the traction evolution and the stress release processes occurring during coseismic slip failures. We will show that, contrary to other constitutive models, the SW law inherently poses the problem of considering the Euclidean norm of the slip vector or its cumulative value along its path. In other words, it has the intrinsic problem of its analytical formulation, which does not have a solution a priori. By considering a fully dynamic, spontaneous, 3–D rupture problem, with rake rotation allowed, in this paper we explore whether these two formulations can lead to different results. We prove that, for homogeneous configurations, the two formulations give the same results, with a normalized difference less than 1%, which is comparable to the numerical error due to grid dispersion. In particular, we show that the total slip, the resulting seismic moment, the fracture energy density, the slip–weakening curve and the energy flux at the rupture front are practically identical in the two formulations. These findings contribute to reconcile the results presented in previous papers, where the two formulations have been differently employed. However, this coincidence is not the rule. Indeed, by considering models with a highly heterogeneous initial shear stress distribution, where the rake variation is significant, we have also demonstrated that the overall rupture history is quite different by assuming the two formulations, as well as the fault striations, the traction evolution and the scalar seismic moment. In this case the choice of the analytical formulation of the governing law does really matter

On the point-source approximation of earthquake dynamics

The focus on the present study is on the point-source approximation of a seismic source. First, we compare the synthetic motions on the free surface resulting from different analytical evolutions of the seismic source (the Gabor signal (G), the Bouchon ramp (B), the Cotton and Campillo ramp (CC), the Yoffe function (Y) and the Liu and Archuleta function (LA)). Our numerical experiments indicate that the CC and the Y functions produce synthetics with larger oscillations and correspondingly they have a higher frequency content. Moreover, the CC and the Y functions tend to produce higher peaks in the ground velocity (roughly of a factor of two). We have also found that the falloff at high frequencies is quite different: it roughly follows ~−2 in the case of G and LA functions, it decays more faster than ~−2 for the B function, while it is slow than ~−1 for both the CC and the Y solutions. Then we perform a comparison of seismic waves resulting from 3-D extended ruptures (both supershear and subshear) obeying to different governing laws against those from a single pointsource having the same features. It is shown that the point-source models tend to overestimate the ground motions and that they completely miss the Mach fronts emerging from the supershear transition process. When we compare the extended fault solutions against a multiple point-sources model the agreement becomes more significant, although relevant discrepancies still persist. Our results confirm that, and more importantly quantify how, the point-source approximation is unable to adequately describe the radiation emitted during a real world earthquake, even in the most idealized case of planar fault with homogeneous properties and embedded in a homogeneous, perfectly elastic medium

Determination of the temperature field due to frictional heating on a sliding interface

In the recent years we assisted to an increasing number of studies devoted to the quantification of the effects of temperature developed as a consequence of frictional heat on a sliding interface. The temperature field generated on the fault surface is responsible of a large number of physical and chemical dissipative process, summarized in Bizzarri (2010a). Among these we mention here the flash heating of micro–asperity contacts, basically consisting in a different behavior of fault friction at high fault slip velocities [e.g., Bizzarri, 2009a; Noda et al., 2009], the melting of rocks and gouge particles [Nielsen et al., 2008; Bizzarri, 2010b], the thermally–induced pressurization of fluids in saturated fault structures [Andrews, 2002; Bizzarri and Cocco, 2006; Rice, 2006]. A key issue of all these studies is the proper calculation of the temperature distribution on the fault surface and its temporal evolution. In this study we compare two different analytical solutions proposed in the literature with the special aim to clarify their prominent features, the numerical advantages and the different physical implications of each of them. In particular, we will compare the temporal evolution of the obtained temperature in the case of spontaneously spreading, fully dynamic rupture on a fault of finite width and we will show how the solutions can be reconciled

Earthquake dynamics and fault interactions

The goal of this Ph.D. Thesis is to show and discuss implications of the main aspects of different governing laws in the description of all phases of seismogenic processes: nucleation, dynamic propagation and energy release, healing, arrest and mutual interactions between faults. This has been done by implementing or developing numerical codes and algorithms able to solve, in various dimensionalities, the fully dynamic, spontaneous problem. The next stage of the research project in which this work is inserted is to try to infer some constitutive details from real events, in order to discriminate between different models

Determination of the temperature field due to frictional heating on a sliding interface

In the recent years we assisted to an increasing number of studies devoted to the quantification of the effects of temperature developed as a consequence of frictional heat on a sliding interface. The temperature field generated on the fault surface is responsible of a large number of physical and chemical dissipative process, summarized in Bizzarri (2010a). Among these we mention here the flash heating of micro–asperity contacts, basically consisting in a different behavior of fault friction at high fault slip velocities [e.g., Bizzarri, 2009a; Noda et al., 2009], the melting of rocks and gouge particles [Nielsen et al., 2008; Bizzarri, 2010b], the thermally–induced pressurization of fluids in saturated fault structures [Andrews, 2002; Bizzarri and Cocco, 2006; Rice, 2006]. A key issue of all these studies is the proper calculation of the temperature distribution on the fault surface and its temporal evolution. In this study we compare two different analytical solutions proposed in the literature with the special aim to clarify their prominent features, the numerical advantages and the different physical implications of each of them. In particular, we will compare the temporal evolution of the obtained temperature in the case of spontaneously spreading, fully dynamic rupture on a fault of finite width and we will show how the solutions can be reconciled

On the slip-weakening behavior of rate- and state-dependent constitutive laws

We study the dynamic traction behavior within the cohesive zone during the propagation of earthquake ruptures adopting rate and state dependent constitutive relations. The resulting slip weakening curve displays an equivalent slip weakening distance (D0_eq), which is different from the parameter L controlling the state variable evolution. The adopted constitutive parameters (a, b, L) control the slip weakening behavior and the absorbed fracture energy. The dimension of the nucleation patch scales with L and not with D0_eq. We propose a scaling relation between these two lengthscale parameters which prescribes that D0_eq/L ~ 15

The role of radiation damping in the modeling of repeated earthquake events

We have investigated the role of the radiation damping term (RDT) on repeated earthquake ruptures by modeling the faulting process through a single one-dimensional analog fault system governed by different constitutive laws. The RDT expresses the energy lost by the seismic waves. The RDT is inherently accounted for in more elaborated, fully dynamic models of extended fault, whereas it is neglected in one-dimensional fault models. In this study, we adopt various formulations of the laboratoryderived rate-dependent and state-dependent friction constitutive laws: the Dieterich-Ruina law, the Ruina-Dieterich law and the Chester and Higgs law. Our numerical results clearly indicate that the RDT significantly affects the system dynamics. More specifically, the more the RDT is effective, the more frequent the slip failures are (with a cycle-time reduction of ca. 30%). We also show that inclusion of the RDT tends to promote smaller but more frequent earthquake instabilities, irrespective of the choice of the governing law. Our data shed light on the limitations implied by the conventional formulation of the equation of motion for the spring system, in which the energy radiation is ignored

A thermal pressurization model for the spontaneous dynamic rupture propagation on a three-dimensional faul. 1. Methodological approach

We investigate the role of frictional heating and thermal pressurization on earthquake ruptures by modeling the spontaneous propagation of a three-dimensional (3-D) crack on a planar fault governed by assigned constitutive laws and allowing the evolution of effective normal stress. We use both slip-weakening and rate- and state-dependent constitutive laws; in this latter case we employ the Linker and Dieterich evolution law for the state variable, and we couple the temporal variations of friction coefficient with those of effective normal stress. In the companion paper we investigate the effects of thermal pressurization on the dynamic traction evolution. We solve the 1-D heat conduction equation coupled with Darcy’s law for fluid flow in porous media. We obtain a relation that couples pore fluid pressure to the temperature evolution on the fault plane. We analytically solve the thermal pressurization problem by considering an appropriate heat source for a fault of finite thickness. Our modeling results show that thermal pressurization reduces the temperature increase caused by frictional heating. However, the effect of the slipping zone thickness on temperature changes is stronger than that of thermal pressurization, at least for a constant porosity model. Pore pressure and effective normal stress evolution affect the dynamic propagation of the earthquake rupture producing a shorter breakdown time and larger breakdown stress drop and rupture velocity. The evolution of the state variable in the framework of rate- and state-dependent friction laws is very different when thermal pressurization is active. In this case the evolution of the friction coefficient differs substantially from that inferred from a slip-weakening law. This implies that the traction evolution and the dynamic parameters are strongly affected by thermal pressurization