Modeling SH Wave Scattering from Fractures in 2D Heterogeneous Medium

1. Modeling SH Wave Scattering from Fractures in 2D Heterogeneous Medium Earth Resources Laboratory

2. Passive monitoring hydraulic fractures

3. Active detecting hydraulic fracture Single hole image SVP image

4. Mathematical model of a 2D Fracture Linear slip condition for a 2D fracture: Traction is continuous Displacement is discontinuous Relation between displacement discontinuity and traction Schoenberg, M.; 1980, Elastic wave behavior across linear slip interfaces, J. Acoust. Soc. Am., 68, 1516-1521

5. Comparison between BEM and An Analytical Solution for A Traction-free Crack Sanchez-Sesma, F, et al, “Scattering and diffraction of SH waves by a finite crack: an analytical solution”, GJI, 2001

6. Comparison of the Displacement Discontinuity

7. Comparison of the Displacement Discontinuity

8. Comparing the Scattered field between BEM and FDM FD method is based on Coates, R., and M. Schoenberg (1995), Finite-difference modeling of faults and fractures, Geophysics, 60, 1514.

9. Waveform Comparisons between BEM and FD Incident arrivals

13. Illustration of the hybrid Concept Monopole Dipole values along boundary

14. Coupling from BE to FD Monopole Dipole values along boundary : monopole and dipole sources implemented by FD

15. Example of Single Scattering

16. Comparison between BEM and the Hybrid Method

17. Multiple Scatterings Fractures surrounded in one sub-domain Fractures surrounded in multi sub-domains

18. Example of Multiple Scattering

19. Comparing with Pure BE

20. Fractures in Layered Model

21. Incident Field and Scattered Field

22. Fractures in Marmousi Model

23. Incident Field and Scattered Field

24. Conclusions • Applied BEM with a linear slip boundary condition to model SH wave scattering from 2D fractures and verified our method by comparing it with the analytical model. • Proposed a hybrid method in the frequency domain to model the SH wave scattering from fractures embedded in a 2D heterogeneous medium by coupling BEM and FDM.

25. Two Boundaries

26. Wave Propagation by FD 2-D Wave Propagation in Frequency-domain Discretization with 4th Order Spatial Operator

27. Scattering Formulations Where For SH scattering Plug in Given u2(ξ), integration solved by BE

28. Coupling from FD to BE Huygen’s Priciple, secondary sources x0 ΓFD Dipole Monopole FD ΓFD where X BE & Anal. • Implications: • Using FD to propagate from X0 to Γ to obtain and • Using and to propagate from Γ to X onto the fracture • Calculating scattering from fracture with BE

29. Coupling from BE to FD Monopole Dipole Propagate outward through FD G ΓBEM u2sca Outward radiation boundary

30. Fracture Interactions Secondary incident wavefield on A B A where

31. Simulating A Traction-free Fracture with Large Compliances

32. Comparing the Scattered field between BEM and FDM FD method is based on Coates, R., and M. Schoenberg (1995), Finite-difference modeling of faults and fractures, Geophysics, 60, 1514.

33. Waveform Comparisons between BEM and FD Incident arrivals

34. Fractures in Marmousi Model

35. Incident Field and Scattered Field

36. Why Use Boundary Element Method in A Homogeneous Media? • Advantage: • Accuracy • Computational cost • Complex geometries • Intrinsically parallelizable • Drawback: • Requiring analytical expression of Green’s function. • Solution: Hybrid method.

37. Why Using Boundary Element Method in A Homogeneous Media? • Advantage: • Accuracy • Computational cost • Complex geometries • intrinsically parallelable • Disadvantage: • Need analytical expression of Green’s function. • Solution: Hybrid method.