1. (Introduction to) Earthquake Energy Balance
Mechanical energy, surface energy and the Griffith criteria
Seismic energy and seismic efficiency
The heat flow paradox
Apparent stress drop

2. Earthquake energy balance: related questions
Are faults weaker or stronger than the surrounding crust?
Do earthquakes release most, or just a small fraction of the strain energy that is stored in the crust?
time
stress
time
stress
recurrence
time
stress
drop

3. Earthquake energy balance: A crack within an elastic medium
We first consider the energy balance of a crack embedded within an elastic medium.
Why cracks?
Because “there’s a crack in everything” (Anthem, Leonard Cohen).
At any given moment the rupture may be envisioned as a shear crack with well defined crack tips, beyond which the slip velocity is equal to zero.

4. Earthquake energy balance: Griffith criteria
The static frictionless case:
UM is the mechanical energy.
UMpotential is the potential energy of the external load applied on the system boundary.
UMinternal is the internal elastic strain energy stored in the medium
US is the surface energy.
crack extends if:
crack at equilibrium if
crack heals if:

5. Earthquake energy balance: dynamic shear crack
Dynamic shear crack with non-zero friction:
Here, in addition to UMinternal, UMpotential and US:
UK is the kinetic energy.
UF is the work done against friction.
During an earthquake, the partition of energy (after less before) is as follows:
where ES is the radiated seismic energy.

6. Earthquake energy balance: dynamic shear crack
Since earthquake duration is so small compared to the inter seismic interval, the motion of the plate boundaries far from the fault is negligible, and UMpotential=0. Thus, the expression for the radiated energy simplifies to:
Question: what are the signs of UMinternal, US and UF?
Let us now write expressions for UMinternal, US and UF .

7. Earthquake energy balance: elastic strain energy
To get a physical sense of what UMinternal is, it is useful to consider the spring-slider analog.
The reduction in the elastic strain energy stored in the spring during a slip episode is just the area under the force versus slip curve.
For the spring-slider system, UMinternal is equal to:

8. Earthquake energy balance: elastic strain energy
Similarly, for a crack embedded within an elastic medium, UMinternal is equal to:
where 1 and 2 are initial and final stresses, respectively, and the minus sign indicates a decrease in elastic strain energy.

9. Earthquake energy balance: frictional dissipation and surface energy
The frictional dissipation:
where F is the friction, V is sliding speed, and t is time. Frictional work is converted mainly to heat.
The surface energy:
where  is the energy per unit area required to break the atomic bonds, and A is the rapture dimensions. Experimental studies show that  is very small, and thus surface energy is very small compared to the radiated energy (but not everyone agrees with this argument).

10. Earthquake energy balance: the simplest model
Consider the simplest model, in which the friction drops instantaneously from 1 to 2.
In such case: F=2, and we get:

11. Earthquake energy balance: seismic efficiency
We define seismic efficiency, , as the ratio between the seismic energy and the negative of the elastic strain energy change, often referred to as the faulting energy.
which leads to:
with  being the static stress drop. While the stress drop may be determined from seismic data, absolute stresses may not.

12. Earthquake energy balance: seismic efficiency
The static stress drop is equal to:
where G is the shear modulus, C is a geometrical constant, and (the tilde) L is the rupture characteristic length.
The characteristic rupture length scale is different for small and large earthquakes.
For small earthquakes, and Combining this with the expression for seismic moment we get:
Both M and r may be inferred from seismic data.

13. Earthquake energy balance: seismic efficiency
Stress drops vary between 0.1 and 10 MPa over a range of seismic moments between 1018 and 1027 dyn cm.
Figure from: Schlische et al., 1996

14. Earthquake energy balance: seismic efficiency
constraints on absolute stresses: In a hydrostatic state of stress, the friction stress increases with depth according to:
where  is the coefficient of friction, g is the acceleration of gravity, and c and w are the densities of crustal rocks and water, respectively.
Laboratory experiments show:
Byerlee, 1978

15. Earthquake energy balance: seismic efficiency
Using:
, the coefficient of friction = 0.6
c, rock density = 2600 Kg m-3
w, water density = 1000 Kg m-3
g, the acceleration of gravity = 9.8 m s-2
D, the depth of the seismogenic zone, say 12x103 m
We get an average friction of:
and the inferred seismic efficiency is:

16. Earthquake energy balance: seismic efficiency
So, the radiated energy makes only a small fraction of the energy that is available for faulting.
Based on this conclusion a strong heat-flow anomaly is expected at the surface right above seismic faults.

17. Earthquake energy balance: the heat flow paradox
At least in the case of the San-Andreas fault in California, the expected heat anomaly is not observed.
A section perpendicular to the SAF plane:
Figure from: Scholz, 1990
The disagreement between the expected and observed heat-flow profiles is often referred to as the HEAT FLOW PARADOX.

18. Earthquake energy balance: the heat flow paradox
A section parallel to the SAF plane:
Figure from: Scholz, 1990

19. Earthquake energy balance: the heat flow paradox
A possible way out of the heat-flow paradox is to question the validity of Byerlee law for geologic faults.
What can get wrong with Byerlee law?
It turned out that at any given moment during rupture, the slip occurs over a small portion of the total rupture area.
The 1992 Landers earthquake:
Wald and Heaton, 1994

20. Earthquake energy balance: the heat flow paradox
To many seismologists, this “pulse-like” slip reminds a moving carpet wrinkle.
The reason the carpet wrinkle can slide under very low shear stress is because the normal stress is locally (i.e., under the wrinkle) much smaller than the average normal stress.

21. Earthquake energy balance
The assumptions underlying the ''simple model'' are:
Instantaneous drop from static to kinetic friction, and constant friction during slip.
Uniform distribution of slip and stresses.
Zero overshoot.
Constant sliding velocity.
No off fault deformation.
The first point means that continuity is violated...

22. Earthquake energy balance
Other conceptual models:
Static-kinetic friction
slip weakening
quasi-static
The (simple) static-kinetic model.
The slip-weakening model. Significant amount of energy is dissipated in the process of fracturing the contact surface. In the literature this energy is interchangeably referred to as the break-down energy, fracture energy or surface energy.
A silent (or slow) earthquake - no energy is radiated.

23. Earthquake energy balance
In reality, things are probably more complex than that.
We now know that the distribution of slip and stresses is highly heterogeneous, and that the source time function is quite complex.

24. Earthquake energy balance: radiated energy versus seismic moment and the apparent stress drop
Radiated energy and seismic moment of a large number of earthquakes have been independently estimated. It is interesting to examine the radiated energy and seismic moment ratio.
M=4.3
M=7.3
Remarkably, the ratio of radiated energy to seismic moment is fairly constant over a wide range of earthquake magnitudes.
Figure from: Kanamori, Annu. Rev. Earth Planet. Sci., 1994

25. Earthquake energy balance: radiated energy versus seismic moment and the apparent stress drop
What is the physical interpretation of the ratio ES to M0? Recall that the seismic moment is:
and the radiated energy for constant friction (i.e., F = 2):
Thus, ES/M0 multiplied by the shear modulus, G, is simply:
This is often referred to as the 'apparent stress drop'.

26. Earthquake energy balance: radiated energy versus seismic moment and the apparent stress drop
Thanks to the improvement of seismic data, it is now possible to directly measure the radiated energy of small earthquakes.
In recent years, the constancy of the apparent stress drop is a matter of debate.
Figure from: Figure from Kanamori and Brodsky, Rep. Prog. Phys., 2004

27. Further reading
Scholz, C. H., The mechanics of earthquakes and faulting, New-York: Cambridge Univ. Press., 439 p., 1990.
Kanamori, H., Mechanics of Earthquakes: Ann. Rev. Earth and Planetary Sciences, v. 22, p , 1994.