Matthew A. d'Alessio, Ann E. Blythe, and Roland Bürgmann
Model Overview (above)
|More About Fission Track Modeling|
1. Earthquake Energy Balance (below)
For our model of heat generation, we essentially follow the method outlined by Lachenbruch (1986). We start by computing the amount of heat generated by the earthquake, essentially invoking the formula for work from freshman physics (Work = Force * Distance).
, where Q is the amount of heat generated per unit area of the fault, tau is the average shear stress, D is the total displacement during the earthquake, and e is an efficiency term. Tau is a force per unit area and D is a distance, so we get a work per unit area. However, not all of the work done during the earthquake is converted into heat. The illustration below shows the distribution of work into frictional heat generation, seismic wave radiation, and grain size reduction. While there is some debate over the magnitude of e, laboratory experiments (Lockner & Okubo, 1983) and calculations from seismic waveform spectra of real earthquakes (McGarr, 1999) both indicate that greater than 90% of earthquake work is converted into heat. We use 0.9 for e in our calculations. Also, Q, as expressed above, is the total heat. We assume that heat is distributed equally between the two sides of the fault, so Q for a one-dimensional heat conduction equation like we use below is one half the total heat.
Computing Shear Stress
2. Heat Transport
The next assumption we make is that all heat transport is by conduction on the time scales of interest. While pore pressure gradients may be very high and permeability may increase during rupture (making fluid flow fairly rapid), there simply is not enough water present to carry away an appreciable amount of the heat. Even if fluids are moving around, during faulting, one likely scenario is that flow will be along the fault towards the surface. In that case, hot fluids from below will replace the heat transported away by pore fluids that are expelled during rupture. Both Lachenbruch (1980) and Mase and Smith (1986) who address the role of pore fluids in dynamically weakening faults ignore advective heat transport in their calculations. Further, Evans and Chester (1995) show that our specific sample locality has relatively little evidence for fluids or high fluid pressure. In other words, because of the short duration of high temperature pulses from individual earthquakes (minutes to hours), we feel that our estimates are insensitive to the heat transport by fluids. This stands in stark contrast to surface heat flow.
Solutions to the one-dimensional conduction equation for an instantaneous plane source of heat are common in some of the famous references such as Carslaw and Jaeger (1959) and Ingersoll et al. (1954):
When this equation is plotted for typical conditions for an earthquake at the depth of interest for this study, the time-temperature history produces an extremely localized temperature pulse. Note that only areas within about 10 cm of the fault surface are ever exposed to temperature increases greater than 100 degrees C, and that these temperatures only persist for less than an hour.
Parameters: Depth, 2.3 km; Slip, 4m; Apparent coefficient of friction, 0.35
|More About the Sample Locality|
Read more about the project
d'Alessio, M.A., Blythe, A.E., and Bürgmann R., 2003, No frictional heat along the San Gabriel fault, California: Evidence from fission-track thermochronology: Geology, v. 31, n. 6, p. 541-544.
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