1. Preconditioning discrete geostatistical models in flow inverse problems
Addy Satija and Jef Caers
Department of Energy Resources Engineering
Stanford University

2. Traditional PPM We need multiple matched models
Start with randomly chosen prior model every time
Prior m1
Posterior m2

3. Preconditioning Idea Start with better initial model Use scoping runs
Precondition the prior
Without narrowing posterior uncertainty
Use scoping runs
Run forward model on few prior models
Learn data-model relationship
Use learning to precondition

4. Illustration Case
Prior Models
Training Image
Data
100 scoping runs

5. Preconditioning 𝜆 1 𝒊 1 𝒊 2 𝒊 3 𝜆 2 𝜆 3 Use a probability map
0.5
Linear combination coefficients need to be calculated using distance based methods
𝜆 3
𝒊 3
Linear combination of scoping models
Coefficients?

6. Metric Space
Distance: Difference between water cut responses
g( )
g( )
Forward Model
Scoping Model

7. Feature Space Reduced non-linearity ki kdata
We do kernel transformation

8. Calculate coefficients
Solve minimize 𝑲𝜆− 𝑘 𝑑𝑎𝑡𝑎 subject to Σ𝜆=1 𝜆≥0 where 𝜆 = coefficients 𝑘 𝑑𝑎𝑡𝑎 = coordinates of data in feature space 𝑲 = [ 𝑘 1 𝑘 2 … 𝑘 100 ] matrix 𝑘 𝑖 = scoping model response 𝑖 in feature space

9. Gradual Deformation to solve for coefficients 𝝀
Three distinct solutions for 𝝀
Model responses with decreasing coefficients 

10. Multiple Solutions
Different sets of coefficient give different probability maps
Multiple Probability Maps
Preconditioned Distribution
Posterior Distribution
MPS
PPM

11. Uncertainty Comparison with Rejection Sampler
Ensemble average of posterior models
No drastic loss of uncertainty 1
0.5
We will test uncertainty using a prediction well later.
Preconditioned PPM
Rejection Sampler

12. Speed comparison with PPM
3X speed-up is observed
Flow simulations needed for 25 posterior models
Traditional PPM
Preconditioned PPM
Scoping Runs
100
Iterations
634 96
Total
196
Initial Model box plot

13. Reason for Speedup Significant information from preconditioning
Spatial probability maps of channel in initial model 1 1
0.5
0.5
Preconditioned PPM
Traditional PPM

14. Is speedup always obtained?
Three cases
Later water cut  More speedup
Case
Speedup 1 3X 2
2.3X 3
1.06X

15. Late water breakthrough case
Increasing coefficient value 
Dense neighbourhood
Similar models are more informative
Useful info in preconditioning map

16. Early water breakthrough case
Preconditioned PPM is only as good as Traditional PPM
Increasing coefficient value 
Sparse neighbourhood
Faraway models get high weights
Preconditioning not useful 

17. Real Case Uncertainty quantification at early stage
Derived from WCA reservoir (seen yesterday morning)
3D models of size 78x59x116
Match water cut for 2000 days
Injector Well
Producer Well

18. Real Case: Preconditioning maps1
0.5

19. 2X Speed-up on regional PPM

20. Uncertainty
Check using prediction well

21. Conclusions
Preconditioning speeds up PPM without drastic loss of uncertainty
Learn from scoping runs
Use learning to select “better” realization to initialize PPM
Preconditioning can also be extended to other methods such as
Metropolis etc.
Ensemble Kalman Filter
Multi-start optimization (see later)

22. Reason for comparable uncertainty
Ensemble average of all probability maps is very close to the prior probability map
No “net loss” of uncertainty
Ensemble average of prior models used for Scoping Runs
Ensemble average of all preconditioning probability maps

23. Real Case Speed-up