Bayesian AVO Inversion and Application to a Case Study
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Transcript of Bayesian AVO Inversion and Application to a Case Study
Bayesian AVO Inversion and Application to a Case Study
Pål Dahle*, Ragnar Hauge, and Odd KolbjørnsenNorwegian Computing Center
Nam H. PhamStatoil
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
Reservoir
Pro
bab
ilit
y
Geology Seismic
Combined
Combining models3)
Summary4)
Earth model2)
Geophysical model1) Bayesian inversion
Rapid spatially coupled AVO inversion
Case study5)
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
d(x,t,) = w t cpp(x,t,) + (x,t,)*
Convolutional model:
Reflectivity
cpp(x,t,) = aVp() lnVp(x,t) + aVs
() lnVs(x,t) + a() ln(x,t)
Weak contrast approximation (continuous version):
t
t
t
d(x,t,) = w t cpp(x,t,) + (x,t,)*
Convolutional model:
Matrix formulation: d = Gm +
m(x,t) = [ lnVp(x,t), lnVs(x,t) , ln(x,t) ]
Assuming Normal Distributions
m(x,t) = [ lnVp(x,t), lnVs(x,t) , ln(x,t) ]
d~ N( md, d)
m ~ N( m, m) ~ N(0, e)
Matrix formulation: d = Gm +
Earth Model
m(x,t) = mBG(x,t) +mH(x,t)
Isotropic, inhomogeneous earth:
Vp
m = Cov mH (x1,t1), mH (x2,t2)
Vs
m ~ N(mBG, m)
lnVs
ln
7.70
7.80
7.75
7.0 7.2 7.4
m : Inter-parameter Dependence
Cov mH (x1,t1), mH (x2,t2) = 0( t1 - t2 ) ( x1 - x2 )
lnVp
7.70
7.80
7.75
ln
7.8 7.9 8.0
lnVs
lnVp
7.8 7.9 8.0
7.0
7.2
7.4
7.6
m : Vertical Dependence
2100
2200
2300-20 0 20
0
1
Vp
2000 2500 3000
Cov mH (x1,t1), mH (x2,t2) = 0( t1 - t2 ) ( x1 - x2 )
m: Lateral Dependence
1250
1350
1500 1600 1700
1300
1250
1350
Vp
-400
40 -400
40
1
0
Cov mH (x1,t1), mH (x2,t2) = 0( t1 - t2 ) ( x1 - x2 )
Combining the Models
d~ N( md, d)
m ~ N( m, m) ~ N(0, e)
m d ~ N( mm|d , m|d)
The Posterior Distribution
mm|d = mBG+mG*(GmG* + e )-1(d - GmBG)
m|d = m - mG*(GmG* + e )-1G m
m,d m dtoo much time ....
Solving in Frequency Space
m,d m d
m,d
m d
3D FFT 3D inverse FFT
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
well
AI Prediction Conditioned to Wells
Well
AI Conditional Simulation 1
Well
AI Conditional Simulation 2
Well
Case Study Conclusions
• Good match for AI used for modelling of– Facies– Porosity