Transcript Prezentace aplikace PowerPoint

Tensile earthquakes:
mathematical description and
graphical representation
Václav Vavryčuk
Institute of Geophysics, Prague
Tensile earthquakes: definition and observations
Definition:
• slip does not lie in the fault
• fault is opening or closing during a rupture process
Observations:
• geothermal and volcanic areas (injection of fluids/magma)
• hydro-fracturing in oil/gas fields
• collapses in mines
• complex fault geometry – tensile wing cracks
• tensile vibrations of a fault
Tensile faulting due to fluid injection
DC + CLVD + ISO


[u]


fluid pressure
fault
• high pore pressure can cause opening faults during the rupture
process (CLVD and ISO are then positive).
Tensile wing cracks
DC + CLVD + ISO
shear faulting
on a main fault
tensile
wing crack
tensile
wing crack
• wing cracks are induced by shear faulting
• predominant frequency of shear and tensile motions can be
significantly different
Tensile vibrations of a fault
DC + CLVD + ISO
shear faulting
on a main fault
tensile
vibrations
• vibrations can be induced by small scale irregularities on a fault
• predominant frequency of shear and tensile motions can be
significantly different
Mathematical model
Source tensor & moment tensor
Area with a continuous displacement
Stress tensor
 ij  cijkl uk ,l
Strain tensor
1
ekl   uk ,l  ul , k 
2
Hooke’s law
 ij  cijkl ekl
Area with a discontinuous displacement
Moment tensor
M ij  cijkl S nk ul
Source tensor
S
Dkl  nl uk  nk ul 
2
Hooke’s law
M ij  cijkl Dkl
Shear earthquakes in isotropy
Source tensor (potency)
n
0 0 1 
1
Dt   S u 0 0 0
2
1 0 0
S
u
u  ut 
Moment tensor
double-couple
0 0 1 
M t    S u 0 0 0
1 0 0
u  ut 
double-couple
u – slip vector
S – fault area
 – Lamé coefficient
n – fault normal
cijkl – elastic parameters
fault
Tensile earthquakes in isotropy
Source tensor (potency)
n
0
Dt   S  0
u1
0 u1 
0 0 
0 u3 
u  ut 
u
S

fault
non-double-couple
Moment tensor
0
 u3
Mt   S  0  u3

  u1
0
 u1


0

  2  u3 
u  ut 
non-double-couple
u – slip vector
S – fault area
λ,  – Lamé coefficients
n – fault normal
cijkl – elastic parameters
Eigenvectors of the source and moment tensors
Shear earthquakes
Shear source
P-axis
T-axis
n
45 
45 

[u]
P/T axes form angle
of 45 ° with the fault
normal and slip
P/T axes bisect the
angle between the
fault normal and slip
Eigenvectors of the source and moment tensors
Tensile earthquakes
Shear-tensile source
P-axis
T-axis
n
 /2
 /2


[u]
P/T axes form a
general angle with
the fault normal and
slip
P/T axes bisect the
angle between the
fault normal and slip
Radiation pattern
Shear and tensile radiation patterns I
Shear earthquakes
+
Tensile earthquakes
-
+
fault
-
+
fault
+
pure extensive source
Shear and tensile radiation patterns II
Shear earthquakes
Tensile earthquakes
α = 0°
x
α = 5°
x
x
z
α = 10°
z
z
Shear and tensile radiation patterns III
Tensile earthquakes
α = 10°
x
α = 30°
x
z
α = 90°
x
z
no nodal lines
z
no nodal lines
Graphical representation
of tensile sources
Focal spheres and nodal lines
Shear earthquakes
Tensile earthquakes
α = 0°
α = 20°
fault
T
P
fault
T
slip plane
Φ = 45°, δ = 50°, λ = -45°
P
slip plane
Nodal lines and source lines
Source lines
Nodal lines
α = 20°
α = 20°
fault
T
P
fault
T
slip plane
Φ = 45°, δ = 50°, λ = -45°
P
slip plane
Nodal lines and source lines
Source lines
Nodal lines
α = 30°
α = 30°
fault
T
P
fault
T
slip plane
Φ = 45°, δ = 50°, λ = -45°
P
slip plane
Decomposition of
source and moment tensors
DC and non-DC components
Source and moment tensors:
• tensile earthquakes generate DC, CLVD and ISO
• CLVD and ISO have the same sign
• CLVD and ISO are positive (negative) for extensive
(compressive) sources
• CLVD and ISO are linearly dependent
Moment tensors:
• direction of the CLVD-ISO line depends on the P to S
velocity ratio of the focal zone
ISO versus CLVD
Decomposition of source and moment tensors for tensile sources with
slope ranging from -90° to 90°
Source tensors
Moment tensors
40
2.0
1.7
ISO [%]
20
1.5
0
-20
-40
-80
-40
0
40
80
CLVD [%]
Direction of the ISO-CLVD line is
independent of the vP / vS ratio
Direction of the ISO-CLVD line is
dependent on the vP / vS ratio
Errors of ISO and CLVD
Decomposition of noisy source and moment tensors for 1000
realizations of random noise
Source tensors
Moment tensors
CLVD-ISO errors
CLVD-ISO errors
50
1
0.8
0.8
0.6
20
0.4
40
ISO [%]
ISO [%]
30
1
0.6
30
0.4
10
0.2
0
0
20
40
CLVD [%]
60
0
0.2
20
0
20
40
CLVD [%]
CLVD is about 2-3 times less accurate than ISO!
60
0
Conclusions
• tensile faulting is a frequently observed mechanism
• source tensors and moment tensors are non-DC
• nodal lines of the DC part have no physical meaning
• new lines called ‘source lines’ are introduced for
representing the tensile source on the focal sphere
• CLVD is 2-3 times more sensitive to errors than ISO
• the P to S velocity ratio can be retrieved from the ISOCLVD function