Group B Heat Flow Across the San Andreas Fault. Chris Trautner Bill Savran

Transcription

1 Group B Heat Flow Across the San Andreas Fault Chris Trautner Bill Savran

2 Outline Background informa?on on the problem: Hea?ng due to fric?onal sliding Shear stress at depth Average stress drop along fault Solu?on of heat equa?on in 2D Fourier series Method of images Compare with Lachenbruch and Sass 1980 Consider hydrothermal hea?ng to explain heat flow anomaly

3 Earthquake Energy Release Energy released during an earthquake should be related to: Size of fault Amount of stress on the fault Displacement of fault E=AτΔ Majority of energy is converted into heat Assuming this heat can be averaged over many cycles, can express heat flow as a func?on of plate velocity E=τV

4 Fault Heat Produc?on Plot of fault heat produc?on q(z)=τv f=0.60 v=5cm/yr Depth from Surface (km) Fault Heat Production (mw/m^2) Seismogenic Depth q(12km)=290 mw/m Heat Generation (mw/m^2)

5 Shear Stress on Fault Typical stress drop during a major earthquake is about 3-5MPa (30-50 bar) About two orders of magnitude lower than shear stresses at the bo_om of the seismogenic zone Fault is not restored to a stress- free state even a`er very large events Fault is under significant shear stress at all?mes

6 Shear Stress on Fault Seismogenic zone: surface to 12 km depth Shear stress is a large por?on of the overburden stress τ(z)=fρ c gz f=0.60 ρ c =2600kg- m - 3 g=9.81m- s - 2 Depth From Surface (km) Shear Stress on Fault Shear Stress (MPa) Seismogenic Depth τ(12km)=183 MPa Shear Stress (MPa)

7 Deriva?on of Heat equa?on Star?ng with the first law of thermodynamics: du = δq +δw Consider the control element: (Conserva?on of Energy) z q z + z q y q y q y + y q x + x q z y x

8 Deriva?on of the Heat equa?on Rewrite using the heat flux through the control element: ρc p (ΔxΔyΔz) dt dt = q x q x +Δx + q y q y +Δy + q z q z+δz Subs?tute in Fourier s Law: + Q(x, y,z,t) ρc p (ΔxΔyΔz) dt dt, + - k(δxδz) T. y, + - k(δxδy) T. z, T = - k(δyδz). x & T ( k(δxδz) y + ' y & T k(δxδy) z + ' ( z & T k(δyδz) x + ' ( x T k(δxδz) y Δy )/ + 0 * 1 T k(δxδy) z Δz )/ * T k(δyδz) x Δx )/ * + 0 1

9 Deriva?on of the Heat equa?on Simplify: ρc p dt dt = x ( k x T x ) y ( k y T y ) z ( k z T z ) + Q(x,z) Assume steady- state and isotropic condi?ons. Q(x,z) 1 k = 2 T x + 2 T 2 z 2 With infinitely long line- source: δ(x)δ(z + a) 1 k = 2 T x + 2 T 2 z 2

10 Equa?on: Solving the Heat equa?on δ(x)δ(z + a) 1 k = 2 T x + 2 T 2 z 2 Boundary Condi?ons: T(x,0) = 0 T(x, ) = 0 T(x, ) = 0

11 Solving the heat equa?on Take Fourier Transform of both sides: I 2 $ % δ(x)δ(z + a) 1 & k ' ( ) = $ 2 T I2 % x + 2 T & 2 z 2 ' ( ) T(k) = e i2πk z a (2π) 2 (k x 2 + k z 2 ) Now we need to perform inverse transforms to get back to spa?al coordinates:

12 Solving the heat equa?on Take inverse transform with respect to k z : T(k x,z) = 1 e i2πkz (z+a ) (2π) 2 (k 2 x + k 2 z ) dk z = e 2π k x (z+a ) (2π) k x Take inverse transform with respect to k x : e 2π k x (z+a) T(x,z) = 2π k x = 1 & ' 2π ln x 2 + (z + a) 2 ( [ ] 1 2 e i2πk xx dk x ) * +

13 Solving the heat equa?on Use method of images to sa?sfy boundary condi?on: T(x,z) = 1 q source 2π k $ % ln x 2 + (z a) 2 & 1 1 [ ] 2 ln[ x 2 + (z + a) 2 ] 2 ' ( ) This solu?on will provide the scalar temperature field from the line- source. We can use this to determine heat- flux at the surface.

14 Temperature Distribu?on Using solu?on for a line source, can solve for temperature distribu?on due to fault heat produc?on First approxima?on: discre?ze into a few line sources Depth from Surface (km) Fault Heat Production (mw/m^2) Seismogenic Depth i q source = d 1 τ(z)dz d 0 Temperature is then the sum of temperature fields due to each line source: Heat Generation (mw/m^2) T tot ( x, z) n := T x, z, a i i = 1 ( )

15 Temperature Distribu?on Temperature distribu?on from discre?zed sources: Sum of infinitesimally small line sources can be described by an integral Heat flow Depth a D=12km T gen ( a, x, z) := 1 q( a) ln ( x) 2 ( z + a) ln ( x) 2 + ( z a) 2 2π k D T tot ( x, z) := T gen ( a, x, z) da

16 Temperature Distribu?on Integral can be evaluated analy?cally or numerically: Highest temperatures are a few km above depth of the seismogenic zone Temperature map meets required boundary condi?ons:

17 Surface Heat Flow Surface heat flow can be found using Fourier s law: Heat Flow Anomaly (mw/m^2) Distance From Fault (km) z- direc?on z- direc?on (surface, f=0.6)

18 Surface Heat Flow - Measurements Measurements of heat flow above San Andreas have been compiled by Lachenbruch and Sass (1980) Measurements are from geographically large area Study from 1973 found San Andreas is within a ~100km- wide band of high heat flow, but there is no thermal anomaly along main heat trace Data have significant sca_er: μ=69.8 mw/m 2 σ=19.8 mw/m 2 Heat flow values are inconsistent with reasonable value of 0.6 assumed for f

19 Surface Heat Flow - Measurements Coefficient of fric?on of 0.4 is consistent with the mean of the data, 0.3 approaches 84 th percen?le of the data Heat Flow Anomaly (mw/m^2) f=0.1 f=0.2 f=0.3 f=0.4 f=0.5 f=0.6 Measurements Distance From Fault (km)

20 Surface Heat Flow - Discussion Turco_e et al (1980), using a more sophis?cated model with bri_le and plas?c zones determined that a coefficient of fric?on of 0.05 would be consistent with no measurable heat flow anomaly Coefficient of fric?on would have to be applicable to depths of 25 to 30 km Turco_e et al propose three possible explana?ons for this Hydrosta?c pressure is nearly equal to the lithosta?c pressure (might give a coefficient of fric?on as low as 0.05, but this situa?on is deemed unlikely) Proposed model is inaccurate perhaps strain in fault system occurs over a broad region? Measurements have failed to iden?fy the heat flow anomaly (groundwater effects) also dismissed as unlikely

21 Surface Heat Flow - Discussion Can determine whether hydrothermal circula?on is even a plausible theory to explain lack of thermal anomaly by removing heat sources to depth d: Heat flow removed by circula?on Depth Heat flow d T gen ( a, x, z) := D=12km 1 q( a) ln ( x) 2 + ( z + a) 2 2π k T tot ( x, z) 1 D := T gen ( a, x, z) 0 2 ln ( x) 2 + ( z a) 2 da 1 2

22 Surface Heat Flow - Discussion Temperature and ver?cal heat flow calcula?ons indicate match to measured values is possible with a depth of heat flow dele?on to about 3.5 km with f=0.6: 150 Surface heat flow (f=0.6, d=3.5km) Measurements Heat flow (mw/m^2) Distance from fault (km)

23 Conclusions Solu?on to infinite plane source of heat with func?onal dependence on z is possible using solu?on to line source Based on reasonable assump?ons regarding coefficient of fric?on, shear stress on the fault, and plate veloci?es, San Andreas should produce a measureable heat anomaly. However, no such anomaly has been measured. Possible explana?ons include: High hydrosta?c stresses which reduce the coefficient of fric?on Hydrothermal effects which dilute or spread the heat anomaly Model errors Based on the simple calcula?ons in this presenta?on, low coefficient of fric?on and hydrothermal circula?on appear to be plausible explana?ons for the lack of a measured heat anomaly