Transcript Numerical modeling: Tube-wave reflections in cased borehole Alexandrov Dmitriy,

Numerical modeling:
Tube-wave reflections in
cased borehole
Alexandrov Dmitriy,
Saint-Petersburg State University
Outline
Modeling approaches
Model 2
Model 3
Model 1
1D effective wavenumber approach
Conclusions
Limitations
Outline







Modeling approaches:


1D effective wavenumber approach
finite-difference

wave field in isotropic homogeneous fluid
Wave field in cased borehole

wave field in isotropic homogeneous elastic media


Idealized disk-shaped perforation
Idealized zero-length disk-shaped perforation
Reflection from geological interfaces behind casing;
Reflection from corroded section of the casing;
Response of perforation in cased borehole:
1D approach limitations;
Conclusions.
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Introduction
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Model 1
Modeling approaches
Model 2
Model 3
1D effective wavenumber approach
Conclusions
Limitations
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
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
1D effective wavenumber approach

Helmholtz equations:
 2 ( z )
2

k
1  ( z)  0
2
z
 2 ( z )
2

k
2 ( z)  0
2
z
z  0, z  L
0 zL
P   f  2
U
Solution form:
1 =eik z  R1eik z ,
1
1
2 =T2eik z  R2eik z ,
2
2
3 =T3eik z
3
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
d ( z )
dz
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
1D effective wavenumber approach
Boundary conditions:

continuity of pressure:
P1 (0)  P2 (0), P2 ( L)  P3 ( L)

continuity of fluid flow:
 V NdS  0
S
2i (k22 s22  k12 s12 ) sin(k2 L)
R1 
(k2 s2  k1s1 ) 2 e ik2 L  (k2 s2  k1s1 ) 2 eik2 L
4k2 s2 k1 s1 e ik1L
T3 
,
2  ik2 L
2 ik2 L
(k2 s2  k1s1 ) e
 (k2 s2  k1s1 ) e
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
2
s1,2   R1,2
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
1D effective wavenumber approach

Multilayered model
 2i ( z )
2

k
i i ( z )  0
2
z
Boundary conditions:

continuity of pressure:
Pi ( zi )  Pi 1 ( zi )

continuity of fluid flow:
 V NdS  0
i =Ai e
iki z
 Bi e
 iki z
AN
B1
R , T 
A1
A1
 Bi 
 Bi 1 
   Gi 

 Ai 
 Ai 1 
 BN 
 BN 
 B1 
 B2 

G

...

G
G
...
G

G


 

1
1 2
N 1 
T 
A
A
A
A
 1
 2
 N
 N
S
BN  0 
R
(GT )12
1
, T
(GT ) 22
(GT ) 22
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Wave field in isotropic homogeneous fluid
x3
Motion equation:
 ui tik
f 2 
t
xk
2
x2
tik   f  ik divu   p ik
x1
2 u
 f 2   f grad div u
t
n
T
Ti  tik nk
1 2
p( x, y, z, t )  2  t p( x, y, z, t )   (t ) ( x, y, z )
vf
p  div u ,
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
1 f

2
vf  f
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Wave field in isotropic homogeneous fluid
P (r , k ,  ) 
2
v
2
f
P(r , k ,  )   (r )
i (2)
P(r , k ,  )  C f J 0 (i f r )  H 0 (i f r ),
4
U r 
   Cf
U z 
k2
1
f  2  2
 vf
(2)

J
(

i

r
)



H
 f 1

1
f
f
1 ( i f r )



(2)
2 

 kJ 0 (i f r )  4  f   kH 0 (i f r ) 
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Wave field in isotropic homogeneous elastic media
Motion equation:
 2ui tik
f 2 
t
xk
tik   f  ik divu  2 ik ,
1  ui uk 
 ik  


2  xk xi 
 u
 2  (  2 )grad div u   rot rot u
t
2
  p H1(2) (i p r ) 
 kH1(2) (i s r ) 
U r 
  Cs 

   C p 
(2)
(2)
U z 
  s H 0 (i s r ) 
  kH 0 (i p r ) 
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
1D effective wavenumber approach
Conclusions
Limitations
Modeling approaches
Results
Wavefield in cased borehole
Boundary conditions
 Continuity of displacement:
U r ( R  0)  U r ( R  0)
U ( R  a  0)  U ( R  a  0)
 Continuity of stress vector:
trr
R 0
 trr
R 0
trr
Ra 0
 trr
R  a 0
trz
R 0
 trz
R 0
trz
R a 0
 trz
R  a 0
M
D,
det M
C  C f , C pc  , C pc  , Csc , Csc , C pe  , Cse 
MC  D

C
Dispersion equation:
det M  0  k  k ( )
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Reflection from geological interfaces behind casing
Reflection coefficient for tube wave
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Reflection from corroded section of the casing
Reflection of tube wave from three
different types of corroded section.
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Idealized perforation in cased borehole
Considered models:

Finite-length perforation
(10 cm)

Zero-length perforation
(break in casing)
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Idealized perforation in cased borehole
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 borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Limitations
Low frequency approximation
for tube-wave slowness
(White J.E. 1984):
1 
1
cT    

B M 
ur
p

0
R 2M
ur
p

2 R 2M
deviation 
p 
u
max  r 

 2 R 2M 
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Limitations
Relative error defined as:
fmax
|R

2
 R
( f )  RFD ( f ) | df
fmin
fmax
1D
fmin
1D
( f )  RFD ( f ) df
Considered model:
R
h
Relative error of 1D approach
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Reflection coefficients
Finite-difference code
1D approach
h
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
R
 0.5
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Reflection coefficients
Finite-difference code
1D approach
h
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
R
2
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Reflection coefficients
Finite-difference code
1D approach
h
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
R
4
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Conclusions
 Validated 1D approach for
 multi-layered media (cased boreholes)
 inhomogeneous borehole casing
 idealized perforations in cased
borehole
 Defined the limitations for 1D
approach
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Modeling approaches
Results
Wavefield in cased borehole
1D effective wavenumber approach
Conclusions
Limitations
Thank you for attention!
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Model 1
Modeling approaches
Model 2
Model 3
1D effective wavenumber approach
Conclusions
Limitations
References









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 Stoneleywave 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.
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.
Outline
Model 1
Modeling approaches
Model 2
Model 3
1D effective wavenumber approach
Conclusions
Limitations
Formation parameters
Longitudinal
velocity (m/s)
Shear velocity
(m/s)
Density
(kg/m3)
Elastic halfspaces
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
Tube-wave reflections in cased borehole
AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.