Transcript Rapid spatially coupled AVO inversion

Bayesian AVO Inversion and
Application to a Case Study
Pål Dahle*, Ragnar Hauge, and Odd Kolbjørnsen
Norwegian Computing Center
Nam H. Pham
Statoil
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

Outline
Rapid spatially coupled AVO inversion
Bayesian inversion
2) Earth model
3) Combining models
Probability
1) Geophysical model
Combined
4) Summary
5) Case study
Geology
Seismic
Reservoir
Geophysical Model
Convolutional model:
d(x,t,) = w t cpp(x,t,) + (x,t,)
*
d(x,t,)
w(t)
cpp(x,t,)
(x,t,)
AVO-trace, surface point x, “offset” 
Seismic wavelet, angle dependent
Seismic reflectivity
Error term
d(x,t,)
w(t)
cpp(x,t,)
Reflectivity
Convolutional model:
d(x,t,) = w t cpp(x,t,) + (x,t,)
*
Weak contrast approximation (continuous version):
cpp(x,t,) = aVp()  lnVp(x,t) + aVs()  lnVs(x,t) + a()  ln(x,t)
t
t
t
Matrix formulation: d = Gm + 
m(x,t) = [ lnVp(x,t), lnVs(x,t) , ln(x,t) ]
Assuming Normal Distributions
m ~ N( m, m)
 ~ N(0, e)
d ~ N( md, d)
Matrix formulation: d = Gm + 
m(x,t) = [ lnVp(x,t), lnVs(x,t) , ln(x,t) ]
Earth Model
Vp
Isotropic, inhomogeneous earth:
m(x,t) = mBG(x,t) +mH(x,t)
Vs
m ~ N(mBG, m)
m = Cov

mH (x1,t1), mH (x2,t2)
m : Inter-parameter Dependence
Cov
mH (x1,t1), mH (x2,t2) =
lnVs
0( t
ln
1 - t2
) ( x1 - x2 )
ln
7.6
7.80
7.80
7.4
7.75
7.75
7.2
7.0
7.70
7.70
7.8 7.9 8.0
lnVp
7.8 7.9 8.0
lnVp
7.0 7.2
7.4
lnVs
m : Vertical Dependence
Cov
mH (x1,t1), mH (x2,t2) =
0( t
1 - t2
) ( x1 - x2 )
Vp
2100
1
2200
0
-20
0
20
2300
2000 2500 3000
m: Lateral Dependence
Cov
mH (x1,t1), mH (x2,t2) =
0( t
1 - t2
) ( x1 - x2 )
Vp
1350
1350
1
1300
0
-40
0
0
40
-40
40
1250
1250
1500
1600
1700
Combining the Models
m ~ N( m, m)
 ~ N(0, e)
d ~ N( md, d)
m d ~ N( mm|d , m|d)
The Posterior Distribution
mm|d = mBG+mG*(GmG* + e )-1(d - GmBG)
m|d = m - mG*(GmG* + e )-1G m
m,d
m d
too much time ....
Solving in Frequency Space
m,d
3D FFT
m d
3D inverse FFT

m,d
 
m d
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
Smørbukk Case Study
The Smørbukk Case
• 32 mill grid cells
• 3 angles
• 2.5 h
Frequency Split
• Background
freq < 6Hz
• Inversion
6Hz ≤ freq ≤ 40Hz
• Simulation
freq > 40Hz
Background Modelling
Background Model
Vp6
Vs6
RHOB6
Inversion Input Data
• Background model:
Vp, Vs, and Rho
• Well data:
TWT, DT, DTS, and Rho
• Seismic Data
• Wavelets
Predicted AI From Inversion
AI Prediction in Wells
Well 1
Well 2
Well 3
SI Prediction in Wells
Well 1
Well 2
Well 3
Density Prediction in Wells
Well 1
Well 2
Well 3
AI Cross Sections:
Horisontal
AI Background
AI Prediction
AI Prediction Kriged to Wells
AI Conditional Simulation 1
AI Conditional Simulation 2
AI Cross Sections:
Vertical
AI Background
Well
AI Prediction
Well
AI Prediction Conditioned to Wells
Well
well
AI Conditional Simulation 1
Well
AI Conditional Simulation 2
Well
Case Study Conclusions
• Good match for AI used for modelling of
– Facies
– Porosity