Structured multiscale history matching workflow, parameterization, streamlines, GTTI

1. Multiscale Parameterization and
History Matching in Structured and
Unstructured Grids:
Theory and Field Application
E. W. Bhark, A. Rey, A. Datta-Gupta and B. Jafarpour

2. Motivation
• Develop structured history matching workflow
• Coarse (regional) scale
 Novel grid-connectivity-based
parameterization
• Flexible, efficient application for
large models, complex geology
 Calibrate multiscale heterogeneity
 Avoid traditional regional multipliers
• Local (grid cell) scale
 Established streamline-based method
• Vasco et al. (1998); Datta-Gupta and King (2007)
 Refine prior preferential flow paths
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3. Outline of presentation
• Parameterization in history matching
 Methods of linear transformation
 Grid-connectivity-based parameterization
• Structured history matching workflow
• Field application
 Offshore reservoir model (Rey et al. [2009], SPE124950)
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4. Why re-parameterization?
• Reduce redundant model information
 Preserve important heterogeneity
~5,000 Unknowns 100 Unknowns 50 25
Ex., high-resolution
(3D) abs. permeability
• Improves:
 Solution non-uniqueness and stability, computational efficiency
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5. Parameterization by linear transform
= v1 + v2 + v3 + … + v50 + … + vN
N-parameter
high-resolution
model
 u= v Φ M N u  v for M << N
 u1 
 1 
u 
 2   v1 
• Required basis properties
 2     v2 
       Compression power: most
        
        energy in fewest coefficients vi
      

 M 
   v M 
   Amenable to efficient
u 
 N application for very large grids
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6. Highlights of new basis
 u1 
1 u 
   2 v
 2    v
    
Grid-connectivity-based transform basis = 

 

    
     
    

 M 
   v M

(1) Model (or prior) independent u 
 N
 Can benefit from prior model information
(2) Applicable to any grid geometry (e.g., CPG, irregular unstructured,
NNCs, faults)
(3) Efficient construction for very large grids
(4) Strong, generic compression performance
(5) Geologic spatial continuity
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7. Basis development
Concept: Develop as generalization of discrete Fourier basis
KEY: Perform Fourier transform of function u by (scalar) projection
on eigenvectors of grid Laplacian (2nd difference matrix)
• Interior rows
 Second difference
 Periodic operator (circulant matrix)
• Exterior rows
 Boundary conditions control
eigenvector behavior
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8. Basis development
CPG Unstructured Grid Laplacian
5
10
15
20
25
30
35
40
45
50
5 10 15 20 25 30 35 40 45 50
2-point connectivity (1/2/3-D)
• Decompose L to construct basis functions (rows of )
 Always symmetric, sparse
 Efficient (partial) decomposition by restarted Lanczos method
 Orthogonal basis functions; Φu  v  u  Φ1 v  ΦT v
• In general (non-periodic) case
 Eigen(Lanczos)vectors  vibrational modes of the model grid
 Eigenvalues represent modal frequencies
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9. Basis functions: Examples
Corner-point Grid
(Brugge) • Modal shape  modal frequency
• Constant basis
 Zero frequency
• Discontinuities honored
Basis vec. 1 Basis vec. 2 Basis vec. 3 Basis vec. 4 Basis vec. 5 Basis vec. 9
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10. Basis functions: Examples
Unstructured grid Basis function 1 Basis function 3 Basis function 5 Basis function 8 Basis function 10
Unstructured grid
(local refinement)
Channel structure
Multiple subdomains
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11. Structured multiscale workflow
(1) START: Prior model (2) Regional update (3) Local update
Prior spatial hydraulic Parameterize
property model Streamline-,
multiplier field
sensitivity-based
inversion (GTTI)
Update in transform
domain
Multiscale iterate
Gradient-based
iterate
Back-transform
Unit-multiplier field at multiplier field to
grid cell resolution spatial domain
Calibrated Model
FINISH
Flow and transport
Add higher-
simulation
frequency modes to
basis
NO Data misfit
tolerance?
YES
Additional YES
spatial
detail?
NO
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12. Field application: Offshore reservoir
Reservoir
• > 300,000 cells
• Mature waterflood
• 8 years of production history
• 4 producers and 4 water injectors
• Complex depositional sequence of
turbidite sand bodies / facies
• Rey et al. (2009), SPE124950
Parameter
• Permeability
Data
• Water cut
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13. Conceptual heterogeneity model
Prior model facies (5)
Prior geo-model P2
I2
P1
P3 I1
Initial Kx: I3
Average of measurements
at wells per facies (5) Facies ID
P4
I4
Next objective:
Use parameterization to assist
in heterogeneity identification
and updating
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14. Workflow: Prior model & multiplier field
F2
Prior geo-model
Multiplier field
F6
F5
F3
F1
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15. Facies basis functions
Facies 5:
Prior geo-model • Multiplier field is linear
combination of basis functions
Multiplier field
Basis functions F5 multiplier field:
u=
1 3 6 8 15
v1 …+ v3 …+ v6 …+ v8 …+ v15
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16. Adaptive multiscale inversion
Prior geo-model • Sequentially refine within-facies heterogeneity
 From coarse to finer scales
 Adaptive inclusion of basis functions
Multiplier field
1 5 10
Basis functions
Multiscale inversion
• End refinement when production data become
insensitive to addition of basis functions
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17. Multiscale update
Number of leading basis
Kx: Adaptive multiscale
functions per facies
10
10
10
1
5
36
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18. Comparison with previous calibration
This study Rey et al. (2009)
Tx multiplier Facies zonation Tx multiplier
Adaptive multiscale Manual zonation
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19. Data misfit: WCT
Initial and multiscale
P2 P3
P4 P1
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20. Streamline-based inversion
High-resolution
Prior geo-model permeability model
• Refine at grid-cell scale
• Streamline paths determined by
Multiplier field
heterogeneity, well pattern
Basis functions
Multiscale inversion
Streamline-based
inversion
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21. Streamline-based update
Final Kx match Kx change
Kx (md) Kx (md)
• Local updates
• Minimal updates along prior preferential flow paths
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22. Final Data misfit: WCT
Multiscale and streamline
P2 P3
P4 P1
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23. Comparison of data misfit: WCT
Multiscale/SL and Business Unit
P2 P3
P4 P1
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24. Comparison with previous calibration
P3
This study • Regional
I3 I4
SOURCE parameterization
I2
more consistent with
model constraints
I3
I4
TMX: Rey et al. (2009) Figure 26: Rey et al. (2009)
TMX
mult.
High perm
(> upper limit
near P3)
Potential
channel
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25. Summary
• Multiscale approach to history matching
 Builds on well-established ‘structured’ workflow
 Regional heterogeneity
 Generalized grid-connectivity-based parameterization
 Efficient, flexible application to any reservoir model geometry
 Refine local heterogeneity
 Prior preferential flow paths captured by streamlines
• Field application
 Demonstrates practical feasibility
 Improvement upon heterogeneity characterization using
standard zonation approaches
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