wave modeling, tomography, geostatistics and edge detection youli quan
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Wave Modeling, Tomography, Geostatistics and Edge
Detection
Youli Quan
Modeling Waves in a Borehole
First Topic
Modeling Waves in a Borehole
• Borehole models
• Mathematical description
• Examples
• Conclusions
Borehole models
• Mathematical description
• Examples
• Conclusions
Modeling Waves in a Borehole
BOREHOLE RELATED SEISMIC MEASUREMENTS
o o
Vertical seismic profiling
o
Cross-boreholeprofiling
Singleboreholeprofiling
o
Soniclogging
Fluid-filledborehole
xSource
Fluid-filledborehole
BOREHOLE MODELS
Radially layered model
z
r
z
r
Complex radially symmetric model
Formation
Cement
Fluid
Steel
• Borehole models
Mathematical description
• Examples
• Conclusions
Modeling Waves in a Borehole
u(r, z, t) (e )
˜ (r, k, ) cp
( j)ei( j)(r( j) r) ˜ H o
(2)(( j)r) cp
( j)ei( j) (r r( j) ) ˜ H o
(1)(( j)r)
˜ (r, k, ) cs
( j)ei
( j)(r( j) r ) ˜ H 1(2)(
( j)r) cs
(j )ei
( j) (r r( j) ) ˜ H o(1)(
(j )r)
(j ) (
( j)
)2 k2 (j ) (
( j)
)2 k 2
In the radially symmetric medium, the displacement
where
The general solutions in the jth layer are
Modified R/T matrices
J J+1
c( j1) = T+
( j)c( j) + R-+
( j)c( j1)
R-+
(j)c( j1)
c( j1)
T-
( j)c( j1)
T+
( j)c( j)
c( j)
R+-
(j)c( j)
c( j) = R+-
(j)c( j) + T -
( j)c( j1)
J+1J
Generalized R/T matrices
J+2J+1Jc
( j)
c( j1) = ˆ T +
( j)c( j)
c( j) = ˆ R +-
( j)c( j)
A recursive relation
with the initial condition
ˆ R (j ) R
( j) T( j) ˆ R
( j1) ˆ T ( j)
ˆ T ( j) [I R
( j) ˆ R ( j1)]T
( j)
ˆ R (N1) 0
j = N, N-1, .., 1
• Borehole models
• Mathematical description
Examples
• Conclusions
Modeling Waves in a Borehole
0
10
20
30
40
50
0 0.1 0.2 0.3 0.4
Qp
Radius (m)
Qs
1000
2000
3000
4000
5000
6000
0 0.1 0.2 0.3 0.4
Vp
Radius (m)
Vs
V e l o c i t i e s ( m / s e c )
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4
Density
Radius (m)
A simple fluid-filled open borehole
Time (ms)
0 1 3 422.44
5.44
Sou
rce-
rece
iver
off
set
(m)
Seismograms in this simple borehole
1000
2000
3000
4000
5000
6000
0 0.1 0.2 0.3 0.4
Vel
ocit
ies
(m/s
ec)
Radius (m)
Vp
Vs
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4
Density
Radius (m)
0
10
20
30
40
50
0 0.1 0.2 0.3 0.4
Qp
Radius (m)
Qs
Time (ms)0 1 2 3 4
Sou
rce-
rece
iver
off
set
(m)
2.44
5.44
Seismograms in this damaged borehole
A damaged fluid-filled open borehole
Seismograms in this flushed borehole
Time (ms)0 1 2 3 4
Sou
rce-
rece
iver
off
set
(m)
2.44
5.44
A flushed fluid-filled open borehole
1000
2000
3000
4000
5000
6000
0 0.1 0.2 0.3 0.4
Vel
ocit
ies
(m/s
ec)
Radius (m)
Vp
Vs
0
10
20
30
40
50
0 0.1 0.2 0.3 0.4Radius (m)
Qp
Qs
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4Radius (m)
Density
o Source
Receivers
100 m
150 m
10 m Formation I Formation II
Borehole
xxxxxxxxxx
REFLECTION DUE TO AN OUTER-CYLINDRICAL FORMATION
Model
Vp=3 km/sVs=1.8 km/s
Vp=1.9 km/sVs=1.4 km/s
Seismograms (fo = 800 Hz)
SP-SS-P
P-P-P-P
S-SP
Tube wave
P-P
10
150
10 50 100 150Time (ms)
source - receiver offset (m)
A SYNTHETIC CROSSWELL SURVEY
ooooooooooooooo
30 m
ReceiversSources
Cased borehole
Vp=5 km/sVs=2.9 km/s
Vp=5.8 km/s
Vs=3.3 km/s100 m
Formation
(a) Model. There is a fault in the formation (b) A common receiver gather
Tube Wave
S-wave
P-wave
xxxxxxxxxxxxxxx
30 m
10 20 30 400Time (ms)
0
S-Roffset
(m)
100
• Borehole models
• Mathematical description
• Examples
Conclusions
Modeling Waves in a Borehole
• A new wave modeling method based on the generalized R/T coefficients is developed for complex borehole simulations.
• This method is efficient, robust, and accurate. It has been applied to sonic logging, crosswell profiling, and single borehole profiling.
Attenuation Tomography
Second Topic
Acoustic Sources
Acoustic Receivers
Lower Absorption
Higher Absorption
Higher frequencyWaveform
Lower frequencyWaveform
Waveform and Attenuation
Measure Attenuation from Waveform
Medium ResponseH(f)
Incident WaveS(f)
Transmitted WaveR(f)=S(f)H(f)
H ( f ) exp [ f odl ]
xxxxxxxxxxxx
oooooooooooooo
Sources Receivers
Vf=5 kft/sQf=20
70 ft
100 ftVp=11.8 kft/sVs=6.9 kft/sQp=30
Vp=12 kft/sVs=7 kft/sQp=60
25 65
(a) Original model (b) Reconstruction
SYNTHETIC EXAMPLE ON ATTENUATION TOMOGRAPHYCrosswell geometry, RT method for modeling
Geostatistics
Third Topic
Geostatistics
• Introduction
• Variogram
• Kriging
Introduction
• Variogram
• Kriging
Geostatistics
• Geostatistics is the study of phenomena that fluctuate in space.
• It offers tools aimed at understanding and modeling spatial variability.
• These tools include histogram, covariance, variogram, kriging, simulation, and etc.
• Introduction
Variogram
• Kriging
Geostatistics
hh
ji
ij
vvhN
h 2)()(2
1)(
x
x
x xx
x
x
x
xx
xx
x
x
xx
x
x
xx
xx
x
x
vivjhij
Experimental Models:
Theoretical Models:
otherwise
if ])(5.0)(5.1[ )(
3
A
a h ahahAh )]
3exp(1[ )(
a
hAh
Exponential Model Spherical Model
• Introduction
• Variogram
Kriging
Geostatistics
x
x
x xx
x
x
xx
x x
x
x
x
x
xx
x
x
x
o
Kriging is a linear estimator with following features:
vi
)ˆ( i
iivwv
(a) Weighting factors are solved based on the selected variogram.
(b) It has minimum variance of the estimation errors.
(c) The estimation is unbiased.
(d) Estimated values has the same statistical properties as given
data