Geodynamics
Brittle deformation and faulting
Lecture 11.6 - Predicting fault orientations
Lecturer: David Whipp
[email protected]
Geodynamics
www.helsinki.fi/yliopisto
1
Goals of this lecture
•
Introduce Anderson’s theory and how it can be applied to
predict fault orientations
2
Faulting
Beartooth Plateau, Wyoming, USA
3
Faulting
Beartooth Plateau, Wyoming, USA
4
Anderson’s theory
•
Anderson formulated the Mohr-Coulomb criterion in terms of
principal stresses and lithostatic pressure (𝜎𝑦𝑦 = 𝜌𝑔𝑦)
•
Anderson assumed 𝜎𝑦𝑦 = 𝜎3 for a reverse fault, 𝜎𝑦𝑦 = 𝜎1 for a
normal fault and 𝜎𝑦𝑦 = 𝜎2 = (𝜎1 + 𝜎3)/2 for a strike-slip fault
c
b
a
!1
!
!3
!2
!
!3
!2
!
Normal fault
Fig. 5.8, Stüwe, 2007
!1
!
!3
!
!1
!
!2
!
Reverse fault
Strike slip fault
5
Anderson’s theory
•
In terms of differential stress, Anderson’s theory
is
𝑐 = 0; 𝑓𝑠 = 0.85
Fig. 5.7, Stüwe, 2007
Reverse:
Normal:
Strike-slip:
d
d
=
=
d
=
1
1
3
3
1
2 (c + fs ( yy
p
=
fs2 + 1
=
3
⇢w gy)
fs
2 (c fs ( yy ⇢w gy)
p
fs2 + 1 + fs
2 (c + fs ( yy ⇢w gy)
p
=
fs2 + 1
6
Predicting fault orientation
•
Now we’ll look at how to apply Anderson’s theory to predict
the dip angle 𝛽 of normal and reverse faults
•
Assume principal stresses
yy = ⇢gy
xx = ⇢gy +
xx
where 𝛥𝜎𝑥𝑥 is the tectonic deviatoric stress, which is positive
for a reverse fault and negative for a normal fault
7
638
Predicting fault orientation
Faulting
Fig. 8.8, Turcotte and Schubert, 2014
•
Figure We
8.8 first
Principal
and
normal
tangential
on to
a dip–
needstresses
to relate
𝜎𝑥𝑥
and 𝜎and
𝜎n and 𝜎stresses
𝑦𝑦 to
s in order
slip fault.
apply Amonton’s law by using the equation for the normal and
shear stresses in a coordinate system rotated by angle 𝜃 with
respect to the principal stresses (see Lecture set 3)
with the vertical
1
1 − pw )
2f
(ρgy
s
+ yy ) + ( xx
2✓
n =∆σ(xxxx
yy ) cos
=
.
(8.32
2
2
± sin 2θ − fs (1 + cos 2θ)
1
( xxmanyyypreexisting
) sin 2✓ joints and faults. W
s =contain
Continental crustal rocks
2
hypothesize that under a tectonic stress these preexisting zones of weaknes
Note that here, 𝜃 is with respect to vertical, 𝜃 = 𝜋/2 -­‐ 𝛽
will be reactivated to form a dip–slip fault at an angle requiring the min
mum value of the tectonic stress. In other words, thrust faulting and norm
8
faulting will occur at angles that minimize |∆σxx |. The angle θ that give
•
638
Predicting fault orientation
Faulting
Fig. 8.8, Turcotte and Schubert, 2014
If we plug in the values for 𝜎 and 𝜎 , we find
•
Figure 8.8 Principal stresses and normal and tangential stresses on a dip–
𝑥𝑥
slip fault.
with the vertical
n
= ⇢gy +
2
xx
(1 + cos 2✓)
sin 2✓
2
2fs (ρgy − pw )
Inserting the∆σ
values
law that(8.32
.
xx = above into the form of Amonton’s
± sin 2θ − fs (1 + cos 2θ)
includes pore fluid pressure, |𝜏| = 𝑓𝑠(𝜎n -­‐ 𝑝𝑤), yields

Continental
crustal
rocks contain many preexisting
joints and faults. W
xx
xx
sin 2✓
= fs ⇢gy
(1 + cos
2✓) of weaknes
w + preexisting
hypothesize±that2 under
a tectonic
stresspthese
zones
2
will be reactivated to form a dip–slip fault at an angle requiring the min
Note that the upper sign is for reverse faults (𝛥𝜎𝑥𝑥 > 0) and
mum value of the tectonic stress. In other words, thrust faulting and norm
9
the lower for normal faults (𝛥𝜎𝑥𝑥 < 0)
faulting will occur at angles that minimize |∆σxx |. The angle θ that give
•
•
s
=
xx
𝑦𝑦
638
Predicting fault orientation
Faulting
Fig. 8.8, Turcotte and Schubert, 2014
The previous expression can be rearranged to solve for 𝛥𝜎 •
Figure 8.8 Principal stresses and normal and tangential stresses on a dip–
𝑥𝑥
2fs (⇢gy pw )
xx =
± sin 2✓ fs (1 + cos 2✓)
If we assume that faulting will occur with the minimum
with the vertical
tectonic stress, then |𝛥𝜎𝑥𝑥2f
| should
bep minimized
(ρgy
−
s
w)
∆σxx =
.
(8.32
± sin 2θ
fs (1 + cos 2θ)
By setting 𝑑𝛥𝜎𝑥𝑥/𝑑𝜃 = 0,
we−find
1 many preexisting1joints and faults. W
Continental
crustal rocks contain
tan 2✓ = ⌥
tan 2 = ±
or
hypothesize that under a tectonic
fs stress these preexisting
fs zones of weaknes
will be reactivated
to form a dip–slip fault at an angle requiring the min
where the upper sign again applies to reverse faults and the
mum value of the tectonic stress. In other words, thrust faulting and norm
10
lower to normal faults
faulting will occur at angles that minimize |∆σxx |. The angle θ that give
slip fault.
•
•
Predicting fault orientation
8.4 Anderson Theory of Faulting
Dip angle
Figure 8.9 Dependence
of the
angle
of dip and
β onSchubert,
the coefficient
Figs. 8.9,
8.10,
Turcotte
2014 of fr
fs for normal and
639 thrust faults.
Tectonic stress
⇢ = 2700 kg m
3
pw = ⇢w gy
⇢w = 1000 kg m
3
y = 5 km
•
Figure 8.9 Dependence of the angle of dip β on the coefficient
of friction
Figure
8.10 Dependence of the deviatoric stress on the coeffi
fs for normal and thrust faults.
of static friction for thrust and normal faults with pw =
ρ = 2,700 kg m−3 ,ρw = 1,000 kg m−3 , g = 10 m s−2 , and y = 5 km.
Finally, the two equations from the previous slide can be
combined to yield the tectonic stresses corresponding to angle
𝜃 or 𝛽
±fmagnitude,
pwdoes
) normal faulting. Based on laborat
absolute
s (⇢gy than
surements, a typical
value for the coefficient of friction would be f
xx =
2
1/2
+ 8–7).
fs )From ⌥
fs (8–34) the corresponding angle of
(see (1
Figure
Equation
thrust fault is β = 24.8◦ . At a depth of 5 km the deviatoric stress f
ure 8–10 is ∆σxx = 305 MPa. The angle of dip of a normal fault is β
where the upper sign again corresponds to reverse faults and
the lower to normal faults
11
Let’s see what you’ve learned…
•
If you’re watching this lecture in Moodle, you will now be
automatically directed to the quiz!
12