alexandrov dmitriy , saint-petersburg state university

Alexandrov Dmitriy, Saint-Petersburg State University

Numerical modeling:Numerical modeling:Tube-wave reflections in Tube-wave reflections in

cased boreholecased borehole

Outline

Modeling approaches: 1D effective wavenumber approach finite-difference

Wave field in cased borehole wave field in isotropic homogeneous fluid wave field in isotropic homogeneous elastic media

Reflection from geological interfaces behind casing; Reflection from corroded section of the casing; Response of perforation in cased borehole:

Idealized disk-shaped perforation Idealized zero-length disk-shaped perforation

1D approach limitations; Conclusions.

Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .

OutlineOutlineLimitationsLimitationsModelModel 1 1 ModelModel 2 2 ModelModel 3 3 ConclusionsConclusions

1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches


Introduction

OutlineOutline

LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions


Modeling approaches

Finite-difference (FD) code flexible little analytical insight

1D effective wavenumber approach Attractive for analysis Approximate Validity for cased borehole is

unknown Validate 1D approach using FD code

OutlineOutline

LimitationsLimitationsModelModel 1 1 ModelModel 2 2 ModelModel 3 3 ConclusionsConclusions



1D effective wavenumber approach

2

212

2222

( )( ) 0 0,

( )( ) 0 0

zk z z z L

z

zk z z L

z

Helmholtz equations:

2 ( ) f

d zP U

dz

1 1

32 2

1 1

2 2 2 3 3

= ,

= , =

ik z ik z

ik zik z ik z

e R e

T e R e T e

Solution form:


OutlineOutline


Modeling approachesModeling approaches 1D effective wavenumber approach1D effective wavenumber approach


Boundary conditions:Boundary conditions:

continuity of pressure: continuity of fluid flow: 0

S

V NdS ��

1 2 2 3(0) (0), ( ) ( )P P P L P L

2 2

1

2 2

2 2 2 22 2 1 1 2

1 2 22 2 1 1 2 2 1 1

22 2 1 13 1,2 1,22 2

2 2 1 1 2 2 1 1

2 ( )sin( )

( ) ( )

4,

( ) ( )

ik L ik L

ik L

ik L ik L

i k s k s k LR

k s k s e k s k s e

k s k s eT s R

k s k s e k s k s e


OutlineOutline




Multilayered model

Boundary conditions: continuity of pressure:

continuity of fluid flow:

= i iik z ik zi i iAe B e

22

2

( )( ) 0i

i i

zk z

z

1

1 1

, NABR T

A A

0S

V NdS ��

1( ) ( )i i i iP z P z

1

1

i ii

i i

B BG

A A

1 21 1 2 1

1 2

... ... N NN T

N N

B BB BG GG G G

A AA A

12

22 22

( ) 10 ,

( ) ( )T

NT T

GB R T

G G


OutlineOutline



2

2i ik

fk

u t

t x

Motion equation:

divik f ik ikt u p

1x2x

3x

i ik kT t n

n

T��


Wave field in isotropic homogeneous fluid

2

2grad div f f

uu

t

22

1( , , , ) ( , , , ) ( ) ( , , )t

f

p x y z t p x y z t t x y zv

2

1div , f

f f

p uv

OutlineOutline




2(2)

0 0 2 2

1( , , ) ( ) ( ),

4f f f ff

i kP r k C J i r H i r

v

2

2( , , ) ( , , ) ( )

f

P r k P r k rv

(2)1 1

(2)20 0

( ) ( )1( ) ( )4

f f f frf

f fz f

J i r H i rUC

kJ i r kH i rU

OutlineOutline



Wave field in isotropic homogeneous fluid

Wave field in isotropic homogeneous elastic media

2

2( 2 )grad div rot rot

uu u

t

2

2i ik

fk

u t

t x

Motion equation:1

div 2 , 2

i kik f ik ik ik

k i

u ut u

x x

(2) (2)1 1(2) (2)0 0

( ) ( )

( ) ( )p pr s

p spz s s

H i rU kH i rC C

kH i rU H i r


OutlineOutline



Boundary conditions

0 0 0 0

0 0 0 0

Continuity of displacement:

( 0) ( 0)

( 0) ( 0)

Continuity of stress vector:

r r

rr rr rr rrR R R a R a

rz rz rz rzR R R a R a

U R U R

U R a U R a

t t t t

t t t t

��

,

det

, , , , , ,c c c c e ef p p s s p s

MMC D C D

M

C C C C C C C C

��

��

Dispersion equation:

det 0 ( )M k k


OutlineOutline



Reflection from geological interfaces behind casing

Reflection coefficient for tube wave


OutlineOutline



Reflection from corroded section of the casing

Reflection of tube wave from three different types of corroded section.


OutlineOutline



Idealized perforation in cased borehole

Considered models:


Finite-length perforation (10 cm)

Zero-length perforation (break in casing)

OutlineOutline



Reflection of the tube wave from perforation with 10 cm length .

Reflection of the tube wave from zero-length perforation.


Idealized perforation in cased borehole

OutlineOutline



LimitationsLow frequency approximation

for tube-wave slowness (White J.E. 1984):

2 2

max2 2

r

r

u pR Mdeviationu pR M

1 1Tc B M

02

ru p

R M


OutlineOutline



Relative error defined as:

Relative error of 1D approach

h

R

1

1

| ( ) ( ) |2

( ) ( )

fmax

D FDfmin

fmax

D FDfmin

R f R f df

R f R f df

Considered model:


Limitations

OutlineOutline



0.5hR Finite-difference code

1D approach


Reflection coefficients

OutlineOutline



2hR


Finite-difference code1D approach


OutlineOutline



4hR


Finite-difference code1D approach


OutlineOutline



Validated 1D approach for multi-layered media (cased boreholes) inhomogeneous borehole casing idealized perforations in cased

borehole Defined the limitations for 1D

approach

Conclusions


OutlineOutline



Thank you for attention!


OutlineOutline



References

OutlineOutline




References Bakulin, A., Gurevich, B., Ciz, R., and Ziatdinov S., 2005, Tube-wave reflection from a

porous permeable layer with an idealized perforation: 75th Annual Meeting, Society of Exploration Geophysicists, Expanded Abstract, 332-335.

Krauklis, P. V., and A. P. Krauklis, 2005, Tube Wave Reflection and Transmission on the Fracture: 67th Meeting, EAGE, Expanded Abstracts, P217.

Medlin, W.L., Schmitt, D.P., 1994, Fracture diagnostics with tube-wave reflections logs: Journal of Petroleum Technology, March, 239-248.

Paige, R.W., L.R. Murray, and J.D.M. Roberts, 1995, Field applications of hydraulic impedance testing for fracture measurements: SPE Production and Facilities, February, 7-12.

Tang, X. M., and C. H. Cheng, 1993, Borehole Stoneley waves propagation across permeable structures: Geophysical Prospecting, 41, 165-187.

Tezuka, K., C.H. Cheng, and X.M. Tang, 1997, Modeling of low-frequency Stoneley-wave propagation in an irregular borehole: Geophysics, 62, 1047-1058.

White, J. E., 1983, Underground sound, Elsevier. Winkler, K. W., H. Liu, and D.L. Johnson, 1989, Permeability and borehole Stoneley

waves: Comparison between experiment and theory: Geophysics, 54, 66–75.

Formation parametersLongitudinal velocity (m/s)

Shear velocity (m/s)

Density (kg/m3)

Elastic half-spaces

3500 2500 3400

Fluid 1500 - 1000

Casing 1 (steel) 6000 3000 7000

Casing 2 (plastic)

2840 1480 1200

Layer 1 3100 1800 2600

Layer 2 3700 2400 3000

Corroded section 1

1200 600 1400

Corroded section 2

3000 1500 3500

Corroded section 3

4200 2100 4900

OutlineOutline




alexandrov dmitriy , saint-petersburg state university

Documents

cased boreholewave field

length diskshaped perforation

fd codeoutlinelimitationsmodel

casingresponse of perforation

continuity of pressure

continuity of fluid

geological interfaces

corroded section