The tube wave reflection from borehole fracture

Author: Andrey Ponomarenko, laboratory for elastic media dynamics

Faculty of Physics Saint-Petersburg State University

JASS 2008

• How can we obtain knowledge about it???

The effective width of the fracture is important to be estimated. Why?

• from the reflection coefficient of the tube wave!

• the analytical model considering

• calculation of the coefficients

• comparing results with FD modeling

2

2

car

cacr ,

1) The model of the tube and fracture

• Fracture is infinite in the horizontal plane and perpendicular to the borehole

GWr

TWT

rlayer

zikII

zikzikI

Vk

Vk

rkHQP

eTP

eRePT

TT

,

)(

)(

)(

)2(0

deftf ti

)(21)(

• Fracture z- coordinate is z=0

21

22

1

ss

f

fTW VVV

2) Symmetrical guided wave in the fracture

2

2

2

1tP

VP

GW

GWV can be obtained from the dispersion equation for the slab of fluid bounded on each side by a semi-infinite elastic media

dpeP ti

21

02

2

pV

pGW

2

2

2

2

22

2 11zrrrr rrr

12

2

rV

CHp

pV

prrr

p

GW

GW

)2(0

2

2

2

2

01

rVi

r GWCe

tir

Vi

ti eCedepP GW

~

21

2

2

car

cacr ,

• Fracture is infinite in the horizontal plane and perpendicular to the borehole

GWr

TWT

rlayer

zikII

zikzikI

Vk

Vk

rkHQP

eTP

eRePT

TT

,

)(

)(

)(

)2(0

deftf ti

)(21)(

• Fracture z- coordinate is z=0

21

22

1

ss

f

fTW VVV

U

dUVdiv

0,, dSnVdSnVs

in

s

ex

• Continuity of pressure in the “cylinder”:

akHQTR r)2(

01

0Vdiv

• Euler equation for non-viscous liquid:

iPgradVPgrad

dtVd

deVtV ti

)(21)(

• Continuity of liquid flow through the “cylinder”:

21

2 ,02

VVV

VhaVVa

I

layerIII

3) equations:

4) Equations for calculations and obtained results

akHQTR r)2(

01

akaH

akHihkk

kT

akaHakHihkk

akaHakHihk

R

r

rrT

T

r

rrT

r

rr

)2(0

)2(1

)2(0

)2(1

)2(0

)2(1

akHakaHakHihkk

kQ

rr

rrT

T

)2(0)2(

0

)2(1

022 layerIII VhaVVa

iPgradV

rkHQP

eTP

eReP

rlayer

zikII

zikzikI

T

TT

)2(0)(

)(

)(

21 VVVI

Finite-difference model

• The model have cylindrical symmetry

• The value of the grid steps is less than the smallest ratio 14min

medium Elastic(lay1)

Fluid

Longitudinal velocity (m/s)

5000 1500

Shear velocity (m/s)

3000 -

Density (kg/m³)

2500 1000

34;;2

2

222

2

2

psp

i

k

k

iikik

k

iki

VKVV

urotrotudivgradtu

xu

xuudiv

xtu

Finite-difference seismogram

Comparison of the finite-difference modeling results and analytic approach (task I)

black – FD, red - analytic

TWT

r

rrT

T

GWr

r

rrT

r

rr

Vk

akaHakHihkk

kT

Vk

akaHakHihkk

akaHakHihk

R

,

,

)2(0

)2(1

)2(0

)2(1

)2(0

)2(1

Excellent agreement between analytic approach and

finite-difference modeling

Comparison of the finite-difference modeling results and analytical approach (task II)

• Another tube radius (0.031 m), other media parameters, different fractures

medium Elastic(lay1)

Fluid

Longitudinal velocity

(m/s)

4200 1500

Shear velocity

(m/s)

2500 -

Density (kg/m³)

2700 1000

• The discrepancy is increasing but not too much!

Conclusions• The analytical formula of the wave coefficients were derived

• We can obtain better physical insight into the interaction of the tube wave with the fracture

• The analytic approach were made which showed excellent agreement with finite-difference modeling both for the absolute values of reflection coefficient and for the seismogram

• It is clear that we can use analytical formula for the cases with wide range of models

• It is possible to use obtained formula for the estimating of the well-fracture's system parameters and, consequently, the well productivity. And it requests too less time than the time of finite-difference code's similar calculations.

References• S.Kostek, D.Johnson, K.Winkler, B.Hornby. “The interaction of tube wave with borehole

fracture”. Geophys.vol.63,#3, 809-815

• B.Plyushchnkov, V. Turchaninov. “Finite-difference code for acoustic logging modeling. Operation instructions”. KIAM RAS. Moscow, 2003

• S.Ziatdinov, A.Bakulin, B.Kashtan. “Tube waves from a horizontal fluid-filled fracture of a finite radius”. SEG New Orleans 2006 Annual Meeting abstracts.

Thank you for attention!