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