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