1. The TBIE method and its applications To borehole acoustics
in rocks with parallel fractures
or tilted anisotropy
Pei-cheng Xu
Datatrends Research Corp.
April 14, 2009

2. TBIE
Transformed
Boundary
Integral
Equations

3. Model I - Borehole in rocks with parallel fractures
fluid
Receiver
Source
fractures
rock

4. Model II - Borehole in rocks with tilted anisotropy
Axis of borehole
Symmetry axis
of anisotropy of
surrounding medium
Receiver
fluid
Source
rock

5. Objectives Develop an analytical formulation to predict the
full acoustic waves in a fluid-filled borehole
surrounded by rocks with parallel fractures or
tilted anisotropy.
Implement robust numerical solution for this
formulation.
Study the effect of the fractures or tilted anisotropy
on the borehole acoustic waves.

6. Borehole in exploration geophysics
Technical background
Borehole in exploration geophysics
Borehole acoustics is used in exploration
geophysics to estimate petrophysics parameters
of rocks in the scale of a foot.
Anisotropy of rock properties can be a result of
vertical fractures (HTI) or laminated thin
bedding (VTI) or both.
Deviated boreholes are often drilled from offshore
platforms and some in-land sites.

7. Horizontal, deviated and curved boreholes in oil and gas exploration
fractures

8. acoustic tool Technical background monopole dipole
Source-receiver offsets（m)
（1）（2） （3）（4） （5） （6）
（7）（8）（9）（10）（11）（12)
Receivers
borehole
Dipole source frequency = 3000 Hz
Source time function：Ricker wavelet
fluid
monopole
dipole
Source

9. Types of borehole waves
Technical background
Types of borehole waves
GENERATION OF BOREHOLE SONIC WAVES z r q
W– water wave
P– head P wave
S – head S wave
receiver
G – guided waves
rock
borehole W
zr-zs G S P
water
source 2a

10. Types of borehole waves
Technical background
Types of borehole waves
TYPICAL SEQUENCE OF
BOREHOLE SONIC WAVES G P S W
Pressure
Time (ms)
P – Head P wave
S – Head S wave
W– Water wave
G – Guided waves (pseudo-Rayleigh, Stoneley, flexural)

11. Example of borehole full waveform due to a monopole
VP=3.305 m/ms
VS=1.969 m/ms
Pseudo-Rayleigh
Stoneley
phase
group P S
Water

13. Special case: vertical borehole in VTI rock
When the axis of borehole and axis of anisotropy
symmetry coincide, and there no fractures, classic analytical solution is available in the form of wavenumber integrals.
The wavenumber integrals have irregularly oscillatory integrands and infinite integration domains. They must be evaluated numerically.
We have developed the Modified Clenshaw-Curtis (MCC) integration method to evaluate wavenumber integrals accurately and efficiently.

14. Wavenumber integrals
Irregularly oscillatory
Regularly oscillatory

15. The MCC integration method
F(kr, z) is fitted by Chebyshev polynomials in each interval. An infinite interval is transformed to finite through change of variable.
Then the integration is carried out exactly or asymptotically with desired accuracy in each interval.
When subdividing the interval or doubling the order of polynomials, no previous sampling is wasted.
The fitting is independent of x (2D case) or r (3D case). This method is most efficient when involving a large number of different x or r.

16. Existing approaches to the boundary value problems of Models I and II
The Finite Difference method (Leslie and Randall, 1991; Sinha et al. , 2006).
The Variational method (Ellefsen et al., 1991)
The Perturbation method (Sinha et al. ,1994).
The conventional Boundary Integral Equations (BIE) method (Bouchon, 1993)

17. The conventional BIE method
The original 3D problem becomes a 2D problem on the cylindrical surface.
The coefficients in the boundary integrals involve fundamental solutions in the full spaces of the solid and fluid.
The fundamental solutions (Green’s functions and associated stresses) in the solid are wavenumber integrals (I3D) when the rock is layered or anisotropic.
Has difficulty handling the infinity in z.

18. Boundary conditions

19. The conventional BIE for borehole acoustics
fundamental
solutions
unknowns

20. Integral transform of BIE: from z to kz
original
unknowns
original
known
coefficients

21. Transformed BIE in Cartesian coordinates

22. Transformed BIE in cylindrical coordinates

23. Angular phase transform
Transformed BIE in matrix form

24. Summary of the TBIE approach
Set up conventional BIE: reducing the domain of unknowns from 3D full space to the cylindrical surface.
From BIE to TBIE: replacing z by kz; reducing cylindrical surface to a line circle.
Replace Cartesian (x,y,z) by cylindrical (r,q,z).
From TBIE to linear system of equations.
Solve TBIE for unknown nodal displacements and pressure on the line circle.
Obtain displacements and pressure at any field location from the displacements and pressure on the line circle through direct evaluation of boundary integrals.
Take inverse integral transform of the above result: from kz back to z.

25. The triple-fold infinite integration

26. Model I geometry in the kz domain
water q
water x d
fracture
reduced borehole
rock

27. Symmetry axis of the fractured rock
Borehole in HTI formation
Symmetry axis
of the borehole z x
Symmetry axis of the fractured rock

28. Effect of a fracture on borehole waves
Borehole and fracture form a composite waveguide.
Fracture causes wave anisotropy.
Distinguish a fracture from anisotropy: dual flexural waves and leaky fracture mode.
Dual flexural waves - channel flexural wave followed by borehole flexural wave in the waveform.
Leaky fracture mode - sharp dip in the spectrum
Effects of fracture aperture, orientation and distance are as expected.

29. Effects of a fracture 0.5 cm 0.5 cm Uniform Fractured Fractured

30. Effects of a fracture 0.5 cm 0.3 m 0.5 cm 0.3 m Uniform Fractured

31. Effects of a fracture
1 cm
Uniform
Fractured
Fractured
Uniform
1 cm

32. Effects of a fracture Flexural wave azimuthally anisotropic z=4.35 m
d = 0 m
h=0.5 cm d
d = 0.2 m
d = 0.3 m
d = 0.5 m
d = 2 m
5 %
10 %
azimuthally
anisotropic
15 %
20 %

33. Effects of a fracture Flexural wave azimuthally anisotropic ISO
z=4.35 m
d = 0
h=0.5 cm
d = 0.2 m d
d = 0.3 m
d = 0.5 m
d = 2 m
5 %
10 %
azimuthally
anisotropic
15 %
20 %

34. Model II geometry in the kz domain
Top View (against z-axis) x’ q x
water
rock
reduced borehole x’ z’ z
Side View (along y-axis)
rock
formation x
reduced borehole

35. Transformation between coordinate systems: the borehole and the rockx3 x2
x’3
x’2
Rock x1
x’1

36. Borehole in rocks with tilted anisotropy
Symmetry axis
of the borehole z z
Symmetry axis
of the rock x x

37. Oblique body waves in TI media
Technical background
Oblique body waves in TI media

38. Study of the effect of tilted anisotropy
On borehole waves
Amplitude spectrum: magnitude and shape change gradually with increased tilted angle.
Waveforms: arrivals of events shift gradually with increased tilted angle.
Azimuthal anisotropy reaches maximum at f=90o and reduces to none at f=0o.

39. Dipole spectra at different tilted angles (q=0o-0o)
Effects of tilted anisotropy
Dipole spectra at different tilted angles (q=0o-0o)
f=0o
f=10o
f=80o
f=90o

40. Dipole spectra at different tilted angles (q=90o-90o)
Effects of tilted anisotropy
Dipole spectra at different tilted angles (q=90o-90o)
f=0o
f=10o
f=80o
f=90o

41. Dipole spectra at different tilted angles
Effects of tilted anisotropy
Dipole spectra at different tilted angles
(q=90o-90o vs 0o-0o)
f=0o
f=10o
f=80o
f=90o

42. Dipole amplitude spectra at different tilted angle
Effects of tilted anisotropy
Dipole amplitude spectra at different tilted angle
q=0o-0o

43. at the fast and slow principal azimuths
Effects of tilted anisotropy
Dipole waveforms
at the fast and slow principal azimuths
f=90o

44. Dipole waveforms at different tilted angle
Effects of tilted anisotropy
Dipole waveforms at different tilted angle
q=0o-0o

45. Conclusions
The Integral transform successfully overcomes the numerical difficulty of other methods in dealing with the infinitely long borehole.
The MCC method is ideal for handling the three-fold infinite,
irregularly oscillatory integrals involved in the TBIE approach.
The TBIE method enables us to study the effects of a
vertical fracture on the borehole waves, which no other researchers have been able to do.
The TBIE method enables us to produce synthetic borehole waves in tilted anisotropic rocks more accurately and efficiently than other methods.

46. References
P.-C. Xu and J. O. Parra, Effects of single vertical fluid-filled fractures on full waveform dipole sonic logs, Geophysics, 68(2), (2003).
P.-C. Xu and J. O. Parra, Synthetic multipole sonic logs and normal modes for a deviated borehole in anisotropic formations, Expanded Abstracts, SEG 77th Annual Meeting, San Antonio, Texas, Sept (2007).
K. J. Ellefsen, C. H. Cheng, and M. N. Toksoz, Effects of anisotropy upon the normal modes in a borehole, J. Acoust. Soc. Am., 89(6), (1991).
B. H. Sinha, E. Simsek, and Q.-H. Liu, Elastic-wave propagation in deviated wells in anisotropic formations, Geophysics, 71(6), D191-D202 (2006).
H. D. Leslie, and C. J. Randall, Multipole sources in boreholes penetrating anisotropic formations: Numerical and experimental results, J. Acoust. Soc. Am., 91(1), (1992).
M. Bouchon, A numerical simulation of the acoustic and elastic wavefields radiated by a source on a fluid-filled borehole embedded in a layered medium, Geophysics, 58(4), (1993).
P.-C. Xu and A. K. Mal, An adaptive integration scheme for irregularly oscillatory functions, Wave Motion, 7, (1985).
P.-C. Xu and A. K. Mal, Calculation of the inplane Green's functions for layered solids, Bull. Seism. Soc. Am., 77(4), (1987).
J. O. Parra, V. R. Sturdivant and P.-C. Xu, Interwell seismic transmission and reflection through a dipping low-velocity layer, J. Acoust. Soc. Am., 93(4), (1993).