1. The tube wave reflection from borehole fracture
Author: Andrey Ponomarenko,
laboratory for elastic media dynamics
Faculty of Physics
Saint-Petersburg State University
JASS 2008

2. The effective width of the fracture is important to be estimated. Why?
How can we obtain knowledge about it???
from the reflection coefficient of the tube wave!

3. the analytical model considering
calculation of the coefficients
comparing results with FD modeling

4. 1) The model of the tube and fracture
Fracture is infinite in the horizontal plane and perpendicular to the borehole
Fracture
z- coordinate is z=0

5. 2) Symmetrical guided wave in the fracture
can be obtained from the dispersion equation for the slab of fluid bounded on each side by a semi-infinite elastic media

6. Fracture is infinite in the horizontal plane and perpendicular to the borehole
z- coordinate is z=0

7. Continuity of pressure in the “cylinder”:
3) equations:
Continuity of pressure in the “cylinder”:
Continuity of liquid flow through the “cylinder”:
Euler equation for non-viscous liquid:

8. 4) Equations for calculations and obtained results

9. Finite-difference model
The model have cylindrical symmetry
The value of the grid steps is less than the smallest ratio
medium
Elastic
(lay1)
Fluid
Longitudinal velocity (m/s)
5000
1500
Shear velocity (m/s)
3000 -
Density (kg/m³)
2500
1000

10. Finite-difference seismogram

11. Comparison of the finite-difference modeling results and analytic approach (task I)
Excellent agreement between analytic approach and
finite-difference modeling
black – FD, red - analytic

12. Longitudinal velocity (m/s)
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
There is discrepancy in results when thickness of fracture increases

13. Longitudinal velocity (m/s)
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!

14. 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.

15. References
S.Kostek, D.Johnson, K.Winkler, B.Hornby. “The interaction of tube wave with borehole fracture”. Geophys.vol.63,#3,
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.

16. Thank you for attention!