The TBIE method and its applications To borehole acoustics in rocks with parallel fractures

$: The TBIE method and its applications To borehole acoustics in rocks with parallel fractures$
The TBIE method and its applicationsTo borehole acoustics

in rocks with parallel fractures or tilted anisotropy

Pei-cheng XuDatatrends Research Corp.

April 14, 2009

Transformed Boundary Integral

Equations

TBIE

x1

x2

x3

Source

Receiver

fractures

fluid

Model I - Borehole in rocks with parallel fractures

rock

x 1

x 2

x 3

x3’

x1’

x2’

Symmetry axisof anisotropy of

surrounding medium

Axis of borehole

Source

Receiver

fluid

Model II - Borehole in rocks with tilted anisotropy

rock

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.

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.

Horizontal, deviated andcurved boreholes in

oil and gas exploration

fractures

Source-receiver offsets （ m)

（ 1 ）（ 2 ）（ 3 ）（ 4 ）（ 5 ）（ 6 ）2.70 2.85 3.00 3.15 3.30 3.45 （ 7 ）（ 8 ）（ 9 ）（ 10 ）（ 11 ）（ 12)3.60 3.75 3.90 4.05 4.20 4.35

Dipole source frequency = 3000 Hz

Source time function ： Ricker wavelet

Source

Receivers

Technical background acoustic tool

boreholefluid

monopole dipole

source

receiver

z r

borehole

water

rock

W S PG

W– water wave

P– head P waveS – head S wave

G – guided waves

GENERATION OF BOREHOLE SONIC WAVES

2a

zrzs

Technical background Types of borehole waves

Time (ms)Pres

sure

P S W

G

TYPICAL SEQUENCE OF BOREHOLE SONIC WAVES

W– Water wave

P – Head P wave

S – Head S wave

G – Guided waves (pseudo-Rayleigh, Stoneley, flexural)

Technical background Types of borehole waves

0 0.5 1 1.5 2 2.5 3 3.5

Time (ms)

Elastic case A freq=10 kHz Q=50 scale=0.001

Stoneley

P S

Example of borehole full waveform due to a monopole

Pseudo-Rayleigh

Water

VP=3.305 m/msVS=1.969 m/ms

phase group

Dispersion: Water-Filled Borehole in Uniform Isotropic Medium

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

0 5 10 15 20

Frequency (KHz)

Pha

se V

elo

city

(km

/s) M-1

M-2

M-3

D-1

D-2

D-3

Q-1

Q-2

Host: Vp=5.87 Vs=2.92 den=2.7 a=0.20 m

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.

Wavenumber integrals

rxxD dkxikzkFzxI )exp(),(),(2

rrmrm

mD dkfrkJzkFzrI )()(),(),,(0

2

03

Irregularly oscillatory Regularly oscillatory

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.

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)

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.

Boundary conditions

),,(),,( zauzau rf

r

),,(),,( zazap rr

0),,( zarz 0)( z,a,rz

)()](),()(),([)( dSnuxTxGxCu jlm

ljlj

S

lmm )()()](

),()(),([)( 0

2 xpdSpn

xQnuxQxpC mmf

S

f

On S: Field point of

Glm(,x)

z r

x

S: boreholesurface

On or off S: Load point of

Glm(,x)

The conventional BIE for borehole acoustics

fundamentalsolutions

unknowns

s

Integral transform of BIE: from z to kz

)(),,(),(

)(),,(),(

)(),,(),(

)(),,(),(

000)(

0

0002)(

0

000)(

0

000)(

0

0

0

0

0

zzdzzn

Qe

n

q

zzdzzQeq

zzdnzzTet

zzdnzzGeg

zzik

fzzik

jm

ljzzik

ml

llmzzik

m

z

z

z

z

originalunknowns

originalknown

coefficients

dke

kr,θ,P

kr,θ,U

kr,θ,U

kr,θ,U

=

zr,θp

zr,θu

zr,θu

zr,θu

zzik

z

z

z

z

z

3

2

1

3

2

1

,

,

,

,

dUtPgCU lmlmm )](),()(),([)( 00

2

0

0

)()](),()()(),([)( 000

2

0

00

PdPn

qnUqPC llf

Transformed BIE in Cartesian coordinates

)()](),()()(),([)(

)](),()(),()(),()(),([)(

)](),()(),()(),()(),([)(

)](),()(),()(),()(),([)(

0000

2

0

0

0000

2

0

0

0000

2

0

0

0000

2

0

0

PdPqn

nUqPC

dUtUtUtPgCU

dUtUtUtPgCU

dUtUtUtPgCU

iib

zzzzrzrzz

zzrr

zrzrrrrrr

Transformed BIE in cylindrical coordinates

)](cos[)(

)](cos[)(

)](sin[)(

)](cos[)(

)(

)(

)(

)(

MP

MU

MU

MU

P

U

U

U

A

Az

A

Ar

z

r

02

1 0

PP

U

n

qICq

gtcI

f

Transformed BIE in matrix form

Angular phase transform

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,,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.

The triple-fold infinite integration

rrmrm

mD dkfrkJzkFzrI )()(),(),,(0

2

03

)(),,(),( 000)(

00 zzdnzzGeg llm

zzikm

z

dkekr,θ,U =zr,θu zzik

zz

11 ,

fracture

reduced borehole

rock

d

2a

h

x

y

water

water

Model I geometry in the kz domain

x

z

Symmetry axis of the fractured rock

Symmetry axisof the borehole

Borehole in HTI formation

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 modesharp dip in the spectrum

• Effects of fracture aperture, orientation and distance are as expected.

0

0.2

0.4

0.6

0.8

2 3 4 5 6 7 8frequency (kHz)

Uniform

Fractured

0.5 cm

0

0.2

0.4

0.6

0.8


Uniform

Fractured

0.5 cm

Effects of a fracture

0

0.2

0.4

0.6

0.8


0

0.2

0.4

0.6

0.8


0.5 cm

0.5 cm

0.3 m

0.3 m

Uniform

Uniform

Fractured

Fractured


0

0.2

0.4

0.6

0.8


0

0.2

0.4

0.6

0.8


1 cm

1 cm

Uniform

Fractured

Fractured

Uniform


h=0.5 cmd

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Time (ms)

d = 0.2 m

d = 0.3 m

d = 0.5 m

d = 2 m

5 %

10 %

15 %

20 %

d = 0 m

ISOz=4.35 m

Flexural wave

azimuthallyanisotropic


h=0.5 cm

d

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6Time (ms)

d = 0

d = 0.2 m

d = 0.3 m

d = 0.5 m

d = 2 m

5 %

10 %

15 %

20 %

ISOz=4.35 m

Flexural wave

azimuthallyanisotropic


reduced borehole

formation

2a

x

y

water

Model II geometry in the kz domain

x

z

reduced borehole

z’ x’

Side View (along y-axis)

Top View (against z-axis)

rock

x’

rock

'jiji xx

''''''

''''

nmlknjlimijk

nlknilik

TT

GG

Transformation between coordinate systems: the borehole and the rock

x1

x2

x3

x’1

x’2

x’3

Borehole

Rock

x

zSymmetry axisof the borehole

Borehole in rocks with tilted anisotropy

Symmetry axis

of the rock

Technical backgroundOblique body waves in TI media

VTI model - Limestone

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Deviation Angle (deg.)

Ve

loc

ity

(m

/ms

)

V_qP

V_qSV

V_SH

(C33/r)1/2

(C11/r)1/2

(C44/r)1/2 (C44/r)1/2

(C66/r)1/2

VTI model - Cotton Valley Shale

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Deviation Angle (deg.)

Ve

loc

ity

(m

/ms

)

V_qP

V_qSV

V_SH

(C33/r)1/2

(C11/r)1/2

(C44/r)1/2

(C66/r)1/2

(C44/r)1/2

Study of the effect of tilted anisotropyOn 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 =90o and reduces to none at =0o.

HardShale ( =0o) =0o/0o

0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7 8 9 10

Freq. (kHz)

Am

plit

ud

e

1 2

3 4

5 6

7 8

9 10

11 12


0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7 8 9 10

Freq. (kHz)

Am

plit

ud

e

1 2

3 4

5 6

7 8

9 10

11 12

Effects of tilted anisotropy

=0o =10o


0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7 8 9 10

Freq. (kHz)

Am

plit

ud

e

1 2

3 4

5 6

7 8

9 10

11 12


0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7 8 9 10

Freq. (kHz)

Am

plit

ud

e1 2

3 4

5 6

7 8

9 10

11 12

=80o =90o

Dipole spectra at different tilted angles (=0o-0o)


0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7 8 9 10

Freq. (kHz)

Am

plit

ud

e

1 2

3 4

5 6

7 8

9 10

11 12


0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7 8 9 10

Freq. (kHz)

Am

plit

ud

e1 2

3 4

5 6

7 8

9 10

11 12


=80o =90o

=0o =10o

Dipole spectra at different tilted angles (=90o-90o)


0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7 8 9 10

Freq. (kHz)

Am

plit

ud

e

1 2

3 4

5 6

7 8

9 10

11 12


0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7 8 9 10

Freq. (kHz)

Am

plit

ud

e

1 2

3 4

5 6

7 8

9 10

11 12

HardShale (=10o) 00-00 vs 90-90

-1800

-1500

-1200

-900

-600

-300

0

300

600

900

1200

1500

1800

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Time (msec)

90-90

0-0

HardShale (=0o) 00-00 vs 90-90

-1800

-1500

-1200

-900

-600

-300

0

300

600

900

1200

1500

1800

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Time (msec)

90-90

0-0


=0o =10o

HardShale (=80o) 00-00 vs 90-90

-1800

-1500

-1200

-900

-600

-300

0

300

600

900

1200

1500

1800

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Time (msec)

90-90

0-0

HardShale (=90o) 00-00 vs 90-90

-1800

-1500

-1200

-900

-600

-300

0

300

600

900

1200

1500

1800

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Time (msec)

90-90

0-0

=80o =90o

Dipole spectra at different tilted angles (=90o-90o vs 0o-0o)

=60o, 90o

0 1 2 3 4 5 6 7 8 9 10

Frequency (kHz)

No

rmal

ized

am

plit

ud

e


Dipole amplitude spectra at different tilted angle

=0o0o

Dipole waveforms at the fast and slow principal azimuths

Receiver 1 @11300 ft

-1000

-750

-500

-250

0

250

500

750

1000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Time (ms)

@fast azimuth

@slow azimuth


=90o

= 60o, 70o, 80o, 90o

1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5

Time (ms)

No

rma

lize

d a

mp

litu

de


Dipole waveforms at different tilted angle

=0o0o

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.

References • P.-C. Xu and J. O. Parra, Effects of single vertical fluid-filled fractures on full waveform

dipole sonic logs, Geophysics, 68(2), 487-496 (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. 23-28 (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), 2597-2616 (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), 12-27 (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), 475-481 (1993).

• P.-C. Xu and A. K. Mal, An adaptive integration scheme for irregularly oscillatory functions, Wave Motion, 7, 235-243 (1985).

• P.-C. Xu and A. K. Mal, Calculation of the inplane Green's functions for layered solids, Bull. Seism. Soc. Am., 77(4), 1823-1837 (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), 1954-1969 (1993).

The TBIE method and its applications To borehole acoustics in rocks with parallel fractures

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