bhark, e.w., structured history matching workflow using parameterization and streamline methods
DESCRIPTION
Structured multiscale history matching workflow, parameterization, streamlines, GTTITRANSCRIPT
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
• 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
Motivation
2
• 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)
Outline of presentation
3
• Reduce redundant model information
Preserve important heterogeneity
• Improves:
Solution non-uniqueness and stability, computational efficiency
Why re-parameterization?
~5,000 Unknowns 100 Unknowns 50 25
Ex., high-resolution
(3D) abs. permeability
4
Parameterization by linear transform
5
v1 + v2 + v3 + … + v50 + … + = vN
M
N
M v
v
v
u
u
u
2
12
1
2
1
u = v
• Required basis properties
Compression power: most
energy in fewest coefficients vi
Amenable to efficient
application for very large grids
for M << NvuΦ NM
N-parameter
high-resolution
model
Grid-connectivity-based transform basis
(1) Model (or prior) independent
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
6
Highlights of new basis
M
N
M v
v
v
u
u
u
2
12
1
2
1
=
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)
Basis development
• Interior rows Second difference
Periodic operator (circulant matrix)
• Exterior rows Boundary conditions control
eigenvector behavior
7
• Decompose L to construct basis functions (rows of )
Always symmetric, sparse
Efficient (partial) decomposition by restarted Lanczos method
Orthogonal basis functions;
• In general (non-periodic) case
Eigen(Lanczos)vectors vibrational modes of the model grid
Eigenvalues represent modal frequencies
Basis development
vΦvΦuvΦuT 1
5 10 15 20 25 30 35 40 45 50
5
10
15
20
25
30
35
40
45
50
Grid LaplacianCPG Unstructured
2-point connectivity (1/2/3-D)
8
• 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
Corner-point Grid
(Brugge)
Basis functions: Examples
Basis vec. 9
9
Unstructured grid
Unstructured grid(local refinement)
Basis function 1 Basis function 3 Basis function 5 Basis function 8 Basis function 10
Basis functions: Examples
Multiple subdomains
10
Channel structure
Parameterize
multiplier field
Additional
spatial
detail?
NO
Add higher-
frequency modes to
basis
YES
Calibrated Model
(1) START: Prior model
Prior spatial hydraulic
property model
Update in transform
domain
Back-transform
multiplier field to
spatial domain
Flow and transport
simulation
Mu
ltis
cale
ite
rate
Unit-multiplier field at
grid cell resolutionG
rad
ien
t-b
ased
itera
te
Streamline-,
sensitivity-based
inversion (GTTI)
Structured multiscale workflow
Data misfit
tolerance?
NO
YES
(2) Regional update (3) Local update
FINISH
11
Field application: Offshore reservoir
12
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
Prior model facies (5)
Conceptual heterogeneity model
Next objective:
Use parameterization to assist
in heterogeneity identification
and updating
P2I2
P3
I3
P1I1
I4
P4
Initial Kx:
Average of measurements
at wells per facies (5)
Prior geo-model
Facies ID
13
Prior geo-model
Multiplier field
Workflow: Prior model & multiplier field
F6F5
F3
F1
F2
14
Prior geo-model
Multiplier field
Basis functions
Facies 5:
• Multiplier field is linear
combination of basis functions
1 3 6 8 15
v1 …+ v3 …+ v6 …+ v8 …+ v15
F5 multiplier field:
u =
Facies basis functions
15
Adaptive multiscale inversion
Prior geo-model
Multiplier field
Basis functions
Multiscale inversion
• Sequentially refine within-facies heterogeneity
From coarse to finer scales
Adaptive inclusion of basis functions
• End refinement when production data become
insensitive to addition of basis functions
1 5 10
16
Multiscale update
Kx: Adaptive multiscaleNumber of leading basis
functions per facies
10
10
5
1
10
36
17
Adaptive multiscale
Comparison with previous calibration
Manual zonation
Rey et al. (2009)This study
Facies zonation Tx multiplier
18
Tx multiplier
Data misfit: WCTInitial and multiscale
P2
P4
P3
P1
19
Prior geo-model
Multiplier field
Basis functions
Multiscale inversion
Streamline-based inversion
High-resolution permeability model
Streamline-based inversion
• Refine at grid-cell scale
• Streamline paths determined by
heterogeneity, well pattern
20
Streamline-based update
Kx changeFinal Kx match
• Local updates
• Minimal updates along prior preferential flow paths
21
Kx (md)Kx (md)
Final Data misfit: WCTMultiscale and streamline
P2
P4
P3
P1
22
Comparison of data misfit: WCTMultiscale/SL and Business Unit
23
P2
P4
P3
P1
24
Comparison with previous calibration
This study
I4
I3
I2
• Regional
parameterization
more consistent with
model constraints
High perm
(> upper limit
near P3)
Potential
channel
Figure 26: Rey et al. (2009)
SOURCE
TMX: Rey et al. (2009)
TMX mult.
P3
I3 I4
• 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
25
Summary