Alexandrov Dmitriy , Saint-Petersburg State University
description
Transcript of 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
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
Introduction
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
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 approach1D effective wavenumber approachModeling approachesModeling approaches
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
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:
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
Modeling approachesModeling approaches 1D effective wavenumber approach1D effective wavenumber approach
1D 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
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
Modeling approachesModeling approaches 1D effective wavenumber approach1D effective wavenumber approach
1D effective wavenumber approach
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
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
Modeling approachesModeling approaches 1D effective wavenumber approach1D effective wavenumber approach
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��������������
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
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
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
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
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
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
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
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
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
Reflection from geological interfaces behind casing
Reflection coefficient for tube wave
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
Reflection from corroded section of the casing
Reflection of tube wave from three different types of corroded section.
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
Idealized perforation in cased borehole
Considered models:
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
Finite-length perforation (10 cm)
Zero-length perforation (break in casing)
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
Reflection of the tube wave from perforation with 10 cm length .
Reflection of the tube wave from zero-length perforation.
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
Idealized perforation in cased borehole
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
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
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
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:
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
Limitations
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
0.5hR Finite-difference code
1D approach
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
Reflection coefficients
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
2hR
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
Finite-difference code1D approach
Reflection coefficients
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
4hR
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
Finite-difference code1D approach
Reflection coefficients
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
Validated 1D approach for multi-layered media (cased boreholes) inhomogeneous borehole casing idealized perforations in cased
borehole Defined the limitations for 1D
approach
Conclusions
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
Thank you for attention!
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
OutlineOutline
LimitationsLimitationsWavefield in cased boreholeWavefield in cased borehole ResultsResults ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
References
OutlineOutline
LimitationsLimitationsModelModel 1 1 ModelModel 2 2 ModelModel 3 3 ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .
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
LimitationsLimitationsModelModel 1 1 ModelModel 2 2 ModelModel 3 3 ConclusionsConclusions
1D effective wavenumber approach1D effective wavenumber approachModeling approachesModeling approaches
Tube-wave reflections in cased boreholeTube-wave reflections in cased boreholeAlexandrovDmitriyAlexandrovDmitriy, , StPSUStPSU, , Saint-PetersburgSaint-Petersburg, , RussiaRussia. .