1. Heat Flow Across The San Andreas Fault
Brendan Crowell
Scott DeWolf
SIO 234
November 9th, 2007

2. Problem Outline
Calculate average shear stress due to lithostatic load of fault
Use a linear heat source to calculate heat flow anomaly
Comparison of surface heat flow calculations to in situ measurements
Possibility for hydrothermal circulation?

3. Shear Stress from Lithostatic Load
The average shear stress for the seismic zone is found by:
For D=12 km, ρc=2600 kg/m3 and f=0.6, one obtains 92 MPa for the average shear stress

4. Shear Stress from Lithostatic Load
An upper limit on the stress drop along the San Andreas was placed at 25 to 30 MPa for slip rates of 5 cm/yr [Brune et al., 1969]
This discrepancy in the shear stress has been attributed to thermal effects along the fault

5. Heat Anomaly from an Infinite Linear Heat Source
The differential equation:
The boundary conditions:
Requires adding
mirror source!

6. Heat Anomaly from an Infinite Linear Heat Source
Find analytic solution by Fourier:
Use derivative property:

7. Heat Anomaly from an Infinite Linear Heat Source
Apply Cauchy Residue Theorem:
Eyes on the prize - surface heat flow:

8. Heat Anomaly from an Infinite Linear Heat Source
Rewrite obscure identity:
Simplify to final form:

9. Surface Heat Flow
To obtain the surface heat flow, the following equation needs to be integrated over the entire depth of the fault zone
The internal heat generation, q, can be generalized as 2vτ

10. Surface Heat Flow The solution to the integral is
If we look only at the fault (x=0), we obtain heat flows of Q=300*f mW/m2

11. Surface Heat Flow

12. Surface Heat Flow
Lachenbruch and Sass [1980] measured heat flows between 40 and 100 mW/m2, which gives f values of 0.13 and 0.32 respectively.
This means more heat needs to be removed by other means, such as hydrothermal circulation

13. Hydrothermal Circulation
If hydrothermal circulation exists to a depth d removing all of the excess heat, we obtain the following for the surface heat flow
To obtain a f=0.6 for the fault, the depth of hydrothermal circulation would need to be km.

14. Hydrothermal Circulation
In order for hydrothermal circulation to be viable, advection must dominate diffusion over the scale of d
The advective time is ~d/u and the diffusive time is ~d2/κ, so the ratio of advective to diffusive times is ~ κ/ud
The Darcian velocity must be greater than m/s for hydrothermal heat removal to be viable

15. Hydrothermal Circulation
Darcy’s Law states
In order for u to be greater than m/s, the permeability k needs to be greater than m2
k for crust at surface is ~10-15 m2 and at 10 km depth ~10-18 m2. Upon inspection, advection cannot dominate diffusion, and hydrothermal circulation cannot remove all the excess heat

16. What is happening?
Hydrothermal removal of heat could exist if the pressure head was large enough from rugged topography [Saffer et al., 2003]
Frictional coefficient may be much less than 0.6 (Evidence of talc in boreholes?)