Transcript No Slide Title

Geol 4068
Class presentation to accompany reading
of
Chapter 2- Waves in Fluids
Elements of 3D Seismology,
2nd Edition
by Christopher Liner
September 6, 2005
Juan M. Lorenzo
Outline
•What is a Fluid?
•Elastic Moduli
•Acoustic Solution to the Wave
equation
•Density
•Velocity
Outline
•What is a Fluid?
•Elastic Moduli
•Acoustic Solution to the Wave
equation
•Density
•Velocity
TYPES OF SEISMIC MATERIALS
Fluid
(no shear
strength)
Porous solid
(Solid+fluid BIOT
THEORY)
Solid (shear
strength and
high and
finite
resistance to
compressibility
Outline
•What is a Fluid?
•Elastic Moduli
•Acoustic Solution to the Wave
equation
•Density
•Velocity
Key Physical Parameters of the
Acoustic Wave Equation
VP  
(Pa)
Po
  P
V
V
Bulk Modulus or modulus of
incompressibility
Vo
Key Physical Parameters of the
Acoustic Wave Equation
VP  
Pf
  P
V
V
(Pa)
Bulk Modulus or modulus of
incompressibility
Vf
P  Pf  Po
V Vf Vo Vref
.m  N
Pa  kg
s2m2 m2
Example: Ocean Water
VP 1500m / s
   VP
density = 1,035 kg.m^3
2
Incompressibility modulus is
of order 10^9 Pa
Outline
•What is a Fluid?
•Elastic Moduli
•Acoustic Solution to the Wave
equation
•Density
•Velocity
Different Approaches
•Hooke’s Law: stress (Pa)= Y (Pa). strain
+ Newton’s Law (F=ma)
•Lagrangian Mechanics (Energy
Relationships)
Outline
•What is a Fluid?
•Elastic Moduli
•Acoustic Solution to the Wave
equation
•Density
•Velocity
Density is given minor importance with
respect to velocity. Usually velocity and
denisty increase together.
Gardner’s Rule (Gardner et a., 1974)
density = a .V^0.25, where a=0.31
Salt: 4500-5000
m/s; 2.0-2.1 g/cc
Outline
•What is a Fluid?
•Elastic Moduli
•Acoustic Solution to the Wave
equation
•Density
•Velocity
j
Vaverage ( j) 
i Viti
Vrms( j) 
j
i ti










1/2
j
Vi2t
1
j
ti
1










Vbackus(Vi, i ) Vbackus (Vi ) Vaverage(Vi ) Vrms
Vbackus  j  













1/2
j


2




i




 Viti  
 2 2 
 Vi ti i  

 
 Vit

1
j
j
1
1
 iViti 
j
Vaverage ( j) 
Vrms( j) 










i Viti
i ti
1/2
j
Vi2t
1
j
Mean velocity;
traditional
j
ti
1










Very important for basic seismic
processing. Can be obtained
directly from seismic field data
or GPR field data. Errors ~10%
V=330 m/s, rho =0
i=1
z=100000m
s = 200m; V=1000
m/s, rho =1.6
V=1500 m/s, z=
500m rho =1.8
i=2
i=3=j
layer
V
1
2
3
sums
rho
330
1000
1500
0.013
1.6
1.8
z(m)
time thickness(s)
V*t
V*V*t
rho*V*t
z*z
z/(VVvrho)
100000 303.0303
100000 33000000
1300
1E+10 70.63643
200
0.2
200
200000
320
40000 0.000125
500 0.333333
500
750000
900
250000 0.000123
TIMES
Vt
VVt
rhoVt
ZZ
z/(Vvrho)
100700 303.5636
100700 33950000
2520
1E+10 70.63668
VrmsVrms
Vrms
111838.2
334.4221
Vavg
331.7262
VbacVbac
Vbac
56180
237.0232
Excel macro
j
Vaverage ( j) 
i Viti
Vrms( j) 
j
i ti










1/2
j

2 
i 






Vi2t
1
j
ti
1
Vbackus(Vi, i ) Vbackus (Vi ) Vaverage(Vi ) Vrms
Vbackus  j  













1/2
j


2




i




 Viti  
 2 2 
 Vi ti i  

 
 Vit

1
j
j
1
1
 iViti 