Bayesian AVO Inversion and Application to a Case Study

1. Bayesian AVO Inversion and Application to a Case Study Pål Dahle*, Ragnar Hauge, and Odd Kolbjørnsen Norwegian Computing Center Nam H. Pham Statoil

2. Contents Objective • Constrain high resolution 3D reservoirs by seismic AVO data Method • Bayesian inversion, merging of geophysical and geological models Contribution • Fast algorithm • Spatial coupling • Uncertainty assessment Vp Vs 

3. Outline 1) Geophysical model Bayesian inversion 2) Earth model 3) Combining models Combined 4) Summary Geology 5) Case study Rapid spatially coupled AVO inversion Probability Seismic Reservoir

4. Convolutional model: d(x,t,) = wtcpp(x,t,) + (x,t,) *  d(x,t,) AVO-trace, surface point x, “offset” w (t)Seismic wavelet, angle dependentcpp(x,t,) Seismic reflectivity(x,t,) Error term  w (t) cpp(x,t,) d(x,t,)  Geophysical Model

5. Weak contrast approximation (continuous version): Convolutional model: m(x,t) = [ lnVp(x,t), lnVs(x,t), ln(x,t) ] d(x,t,) = wtcpp(x,t,) + (x,t,) * cpp(x,t,) = aVp() lnVp(x,t)+ aVs() lnVs(x,t) + a() ln(x,t)     t t t Reflectivity Matrix formulation:d = Gm + 

6. m(x,t) = [ lnVp(x,t), lnVs(x,t), ln(x,t) ] m ~ N( m,m)  ~ N(0,e) d~ N( md,d) Assuming Normal Distributions Matrix formulation:d = Gm + 

7.  m= CovmH (x1,t1), mH (x2,t2) Earth Model Vp Isotropic, inhomogeneous earth: m(x,t) = mBG(x,t)+mH(x,t) Vs m ~ N(mBG,m)

8. CovmH (x1,t1), mH (x2,t2) =0(t1 - t2 ) (x1 - x2 ) ln lnVs ln 7.6 7.80 7.80 7.4 7.75 7.75 7.2 7.70 7.70 7.0 7.8 7.9 8.0 7.8 7.9 8.0 7.2 7.4 7.0 lnVp lnVp lnVs m: Inter-parameter Dependence

9. CovmH (x1,t1), mH (x2,t2) =0(t1 - t2 ) (x1 - x2 ) m: Vertical Dependence Vp 2100 1 2200 0 -20 0 20 2300 2000 2500 3000

10. CovmH (x1,t1), mH (x2,t2) =0(t1 - t2 ) (x1 - x2 ) 1350 1350 1 1300 1250 1250 0 40 -40 0 0 -40 40 1500 1600 1700 m: Lateral Dependence Vp

11. m ~ N( m,m)  ~ N(0,e) d~ N( md,d) m d ~ N( mm|d , m|d) Combining the Models

12. mm|d = mBG+mG*(GmG* + e)-1(d - GmBG) m|d = m - mG*(GmG* + e)-1G m m,d m d The Posterior Distribution too much time ....

13. m,d m d 3D FFT 3D inverse FFT m,d m d Solving in Frequency Space

14. Summary • Bayesian inversion • Convolutional model,weak contrast • Spatial dependencies of earth parameters • Fast inversion • 100 million grid cells ~ 1 hour • More than inversion • Consistent merging of well logs • High resolution reservoirs

15. Smørbukk Case Study

16. The Smørbukk Case • 32 mill grid cells • 3 angles • 2.5 h

17. Frequency Split • Background freq < 6Hz • Inversion 6Hz ≤ freq ≤ 40Hz • Simulation freq > 40Hz

18. Background Modelling

19. Background Model Vp6 Vs6 RHOB6

20. Inversion Input Data • Background model: Vp, Vs, and Rho • Well data: TWT, DT, DTS, and Rho • Seismic Data • Wavelets

21. Predicted AI From Inversion

22. AI Prediction in Wells Well 1Well 2Well 3

23. SI Prediction in Wells Well 1Well 2Well 3

24. Density Prediction in Wells Well 1Well 2Well 3

25. AI Cross Sections: Horisontal

26. AI Background

27. AI Prediction

28. AI Prediction Kriged to Wells

29. AI Conditional Simulation1

30. AI Conditional Simulation 2

31. AI Cross Sections: Vertical

32. AI Background Well

33. AI Prediction Well

34. AI Prediction Conditioned to Wells Well well

35. AI Conditional Simulation 1 Well

36. AI Conditional Simulation 2 Well

37. Case Study Conclusions • Good match for AI used for modelling of • Facies • Porosity