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Ge277-2010Stress in the crust
Implications for fault mechanics and earthquake physics
• Motivation
• Basics of Rock Mechanics
• Observational constraints on the state of stress in the crust
Landers (1992, Mw=7,3)
Hernandez et al., J. Geophys. Res., 1999
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Hernandez et al., J. Geophys. Res., 1999
Observed and predicted waveforms
Strong motion data
Hernandez et al., J. Geophys. Res., 1999
(Bouchon et al., 1997)
(Bouchon et al., 1997)
(Bouchon et al., 1997)
Heterogeneity on fault
1 0
De
pth
[km
] 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0
S tr ik e [k m ]
- 1 2 -8 -4 0 4 8 1 2s tre s s [M P a ]
00
Initial stress in simulation by Peyrat et al. (2000)
(Aochi et al, 2003; Peyrat et al, 2000, 2004, Aagard and Heaton, 2008)
For dynamic models of the rupture to match observation we need:- heterogeneous prestress distribution -or heterogeneities of fault-constitutive parameters
• Dynamic models of the seismic cycle on faults must account for the complexity of seismic ruptures which requires some mechanism to maintain stress heterogeneities.
Dynamic modeling
(Kaneko et et al, in press)
(Kaneko et et al, in press)
• The complexity (sustained heterogeneities of stress distribution) could be all due to the earthquake process itself, or to inerseismic processes.
Basics of Rock Mechanics
n
n
1
2
The stress (red vector) acting on a plane at M is the force exterted by one side over the other side divided by plane area…
The state of stress at a point can be characterizes from the stress tensor defined as …
i, j 11 12 13
21 22 23
31 32 33
The stress tensor
Stress acting on a plane at point M…Let n be the unit vector defining an oriented surface with elementary area da at point M. (n points from side A to side B)
Let dT be the force exerted on the plane by the medium on side B. It can be decomposed into a normal and shear component parallel to the surface. The stress vector is:
2 2
n n n
n n n
n
Normal stress
Shear stress
,( ) i j j
dTn da n n
da
Side B
Side A
Principal stresses
11 1
, 22 2
33 3
0 0 0 0
0 0 0 0
0 0 0 0i j
1 2 3
Engineering sign convention tension is positive,Geology sign convention compression is positive…
Plane perpendicular toprincipal direction has no shear stress…
Because the matrix is symmetric, there is coordinate frame such that….
The deviatoric stress tensor…
Stress tensor = mean stress + deviatoric stress tensor
i, j m 0 0
0 m 0
0 0 m
1 m 0 0
0 2 m 0
0 0 3 m
i, j m i, j
11 22 33 1 2 3
3 3m
Sum of forces in 1- and 2-directions…
2-D stress on all possible internal planes…
The Mohr diagram
Sum of forces in 1- and 2-directions…
2-D stress on all possible internal planes…
Rearrange equations yet again…
Get more useful relationship betweenprincipal stresses andstress on any plane….
Rearrange equations…
The Mohr diagram
Representation of the stress state in 3-D using the Mohr cirles.
n
The state of stress of a plane with any orientation plots in this domain
This circle represent the state of stress on planes parallel to
This circle represent the state of stress on planes parallel to
This circle represent the state of stress on planes parallel to
Elastic deformationBonds are elastically deforming
Deformation is recoverable
Cataclastic flowBonds are reorganizing
Deformation is permanent
Failure:Bonds are broken
Deformation is permanent
The proportionality constant is the modulus of elasticity (in units if stress)
Brittle regime (‘low’ temperature and pressure)
Fracture strength of rocks
The Mohr-Coulomb envelope
Failure of rocks does not depend on the intermediate principal stress.
Fracture strength of rocks
Fracture strengths of dry sedimentary rocks…
So n
This straight-line is the Coulomb fracture criterion. Its slope demonstrates the property of most rocks to increase in strength as confining pressure (is increased
Laboratory studies reveal = 0.6 to 0.85 for the majority of rocks
For , failure is by
tensile fracture perpendicular
to the Coulomb
fracture criteria does not apply
This can change the conditions along a fault from stable to unstable (= such that displacement occurs)
The presence of fluids also induces chemical reactions that may contribute to weaken the fault
Pore fluid pressure
Water in the pores of rocks produces a pore pressure Pf which acts outward, whereas confining pressure acts inward. Hence, fluid pressures can support part of the load across a fault. The pore pressure (eff) thus moves the center of the Mohr circle toward the origin without changing the radius (shear stress keeps the same)
= 0.85 N
= 0.50+ 0.6 N
for N > 200 MPa(Surface roughness becomes less important)
‘base’ coefficient of friction
‘base’ coefficient of friction
The frictional behavior of rocks can be described by an empirical relation (Byerlee’s law) which, with the exception of some clays, is independent (to first order) of rock type, sliding velocity, surface
roughness and temperature (up to 400°C).
Frictional strength is related to normal stress
(Byerlee, 1978)
Dc
3) Submitted to a sudden change in sliding velocity, friction evolves to its new steady-state value over a characteristic slip distance Dc
1’) Slide-hold-slide experiments at constant sliding velocity= 3 m/s; hold times in seconds.Effect of healing on s is visible
1) Static friction increases logarithmically with hold time, due to healing. Hence, static friction depends on the fault history, and rocks strengthen with time
2) Dynamic friction decreases with sliding velocity. Hence, most rocks weaken with sliding
Static and Dynamic Fricion…
Granite & gouge
(Marone, 1998)
Rock & gouge
Time strengthening
Slip weakening
Two competing effects!
Dc for real EQs is estimated to 1-100 cm, about 5 orders of magnitude more than the values derived from Laboratory experiments
• Static friction measured in lab is generally of the order of 0.6-0.8 (static friction of friction at very slow sliding rate)
• Dynamic friction (at seismic sliding rates of m/s) can be way lower (<0.1) due to various weakening mechanism.
(see Marone 1998, and papers by Toshi Saimamoto’s group on dynamic weakening)
Observational constraints on the state of stress in the crust
(Brudy et al, 1997)
Solid and open squares : Stress magnitudes derived from hydraulic fracturing tests. Between 3 km and 6.8 km depth the results of the combined analysis of breakouts and drilling-induced fractures are presented for each depth for the least (open triangles), intermediate (crosses) and greatest possible Sh magnitude (open diamonds). Below 7 km the SH magnitude could only be estimated from drilling-induced fractures (solid triangles). At 7 km and 7.7 km only the estimation for the least possible Sh magnitude is presented.
The stress profile demonstrates that below 1 km depth to at least 7.7 km depth, a strike-slip stress regime is prevailing at the KTB site and the differential stress is increasing with depth
Stress magnitudes derived from the KTB drill hole (Germany)
Stress magnitudes derived from the KTB drill hole
(Brudy et al, 1997 )
Mohr circles at five different depths compared to the failure lines for a coefficient of friction of 0.6 and 0.8.
The Mohr circles are drawn for the following combinations of Sh and SH magnitudes: least Sh value with respective least and greatest SH value, intermediate Sh value with respective least and greatest SH value, and greatest Sh value with respective SH value. (e) Mohr circle for the stress estimation at 7.7 km depth. The circles are drawn for lower and upper bound estimates of the Sh magnitude and the respective lower and upper bounds of the SH magnitude. The effective normal stress is the normal stress minus the hydrostatic pore pressure at the receptive depth.
The Mohr circles reach or overcome the failure lines for optimally oriented faults. This means that in the entire investigated depth section, the hypotheses of a frictional equilibrium on preexisting optimally oriented faults with a coefficient between 0.6 and 0.8 is correct.
(Townend and Zoback, 2000 )
Dependence of differential stress on effective mean stress at six locations where deep stress measurements have been made. Dashed lines illustrate relationships predicted using Coulomb frictional-failure theory for various coefficients of friction
(Kohlstedt et al., 1995)
Even in stable tectonic area the state of stress is controlled by frictional strength (as predicted from experimental friction laws) of preexisting faults rather than by the strength of intact rocks.
• It has long been argued that friction on active faults must actually be low, less than about 0.1 :– The absence of heat flow anomaly associated
with the SAF (Brune, 1969; Lachenbruch and Sass, 1973) suggests an ambient shear stress<15 Mpa (lithostatic gradient is 27 Mpa/km). Similarly the thermal structure of the Himalaya requires a friction less than 0.1 (Hermann et al, JGR, in Press).
NB: Lithostatic gradient is about 27 MPa/km.
• It has long been argued that friction on active faults must be low, less than about 0.1 :– Thrust Sheet mechanics: the thickness-length
aspect ration of thrust sheet requires a very low basal friction (for internal deviatoric stress not to exceed crustal rock strength) (Hubert and Ruby, 1959). Analysis based on the critical taper theory generally yield friction less than 0.1 or even lower on decollement (Davis et al, 1983).
• It has long been argued that friction on active faults must be low, less than about 0.1 :– The maximum horizontal stress near the San
Andreas Fault is nearly orhogonal to the fault strike (Zoback et al, 1987).
(Hardebeck and Hauksson, 2001)
Coseismic stress change during the Landers earthquake has induced a
rotation of 1 by 15°. This implies that
the ratio of the coseismic shear stress change on the fault, to the preexisting deviatoric stress amplitude, is of the order of
Δ= 0.65
Given that Δis estimated to 8 MPa, we infer a
=12 Mpa
This value is lower by a factor 10 than that predicted from Byerlee’s Law for hydrostatic pore pressure. The fault seems to be ‘weak possibly because of high pore pressure
(Hardebeck and Hauksson, 2001)
Role of Fluids in the brittle crust
(a) An example of temperature profile and (b) the de-trended temperature data utilizing a moving window as described in the text. In (a) T is temperature and in (b) is the difference between the measured temperature and the average temperature to enhance thermal anomalies. We identified thermal anomalies (denoted by arrows in Fig. (b)), which exceeded a cutoff value of ºC.
(Ito and Zoback , 2000)
Shear stress versus effective normal stress, normalized by the vertical stress Sv at each fracture depth, for (a) hydraulically conductive and (b) non-conductive fractures in the KTB main hole for the depth range of 3 -7 km. The open square in (a) represents the shear and normal stresses for a major Mesozoic age thrust fault.
(Ito and Zoback , 2000)
Shear stress versus effective normal stress, normalized by the vertical stress Sv at each fracture depth, for hydraulically conductive fractures found in the depth ranges of (a) 3 – 4 km, (b) 4 – 5 km, (c) 5 – 6 km and (d) 6 – 7 km. The stresses are represented by using open circles with three different sizes depending on the amplitude of the thermal anomaly associated with each fracture.
(Ito and Zoback , 2000)
Shear and effective normal stresses on fractures identified using borehole imaging techniques in the Cajon Pass (red diamonds and dots), Long Valley (yellow triangles and dots), Nevada Test Site (green circles and dots), and KTB (blue squares and dots) boreholes. The larger, filled symbols represent hydraulically conductive fractures and faults, and the dots represent non-conductive fractures. The inset figure illustrates the range in shear to normal stress ration for all four datasets combined. The number of data in each dataset is normalized so that each dataset has equal weight. Original data
from Barton et al. (1995) and Ito and Zoback (2000).
(Zoback and Townend, 2000)
These measurements show that :– the crust is at the critical stress level for
frictional sliding of faults with μ 0.6– hydraulically conductive fractures are those
optimally oriented for frictional sliding– pore pressure is quasi hydrostatic down to a
depth of 10km, probably due to percolation along critically stressed fractures,
– This maintain high effective stresses in the upper crust which can then sustain most of the plate driving tectonic forces.
Stress orientation in S. California
(Hardebeck and Hauksson, 1999)
