ReducedOrder Modeling Applied for 3D Reservoir Flow Simulation

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Reduced-Order Modeling Applied for 3-D Reservoir
Flow Simulation

• Marco Cardoso
• Lou Durlofsky
• Pallav Sarma (Chevron ETC)

Smart Fields Consortium January 16, 2008
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? Reservoir management often requires many
reservoir flow simulations ? Simulation time
impacted by - Number of grid blocks
- Complexity of the reservoir model
? Reduced-order models represent
reservoir response using fewer degrees of freedom
Motivation
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Previous Work
? Sirovich (1987) introduced basic approach for
turbulence modeling? POD - Proper Orthogonal
Decomposition for production optimization
(Jansen et al., 2006)? CVT Centroidal
Voronoi Tessellation (Burkardt et al.,
2006)? MPE Missing Point Estimation
(Astrid, 2004)
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Outline
? Description of basic procedure ? Application of
ROM to 3-D problem ? Clustering procedure to
reduce number of snapshots ? Missing Point
Estimation to improve computational efficiency
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? Basic idea - Project n-dimensional state
space onto an ?-dimensional subspace (? ltlt n)
POD Proper Orthogonal Decomposition
? Approach - Run full simulation - Record k
snapshots - Compute reduced basis
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Snapshots (Two-phase flow)
Snapshot 1 nc gridblocks
Snapshot 2
Snapshot k
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POD Development (1)
Mean
Data matrix
(k x k matrix)
Covariance matrix
Eigenvalue problem
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POD Development (2)
j-th POD basis vector
Basis matrix
Energy captured
Transformation
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Reduced Order Model Representation
(?)
(?2nc)
(2nc)
Solve for ? unknowns instead of 2nc
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Reduced Jacobian Matrix
? J is a 2nc 2nc block hepta-diagonal in
3D ? Jreduced is an ? ? full matrix
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Implementation of ROM in GPRS

• ? Preprocessing runs to generate data for F?
• Matlab code
• ? POD option implemented in GPRS
• ? MPE option also implemented
• ? All results here generated using GPRS

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Reservoir Stanford-VI
? Synthetic 3-D reservoir model ? Fluvial channel
system ? dx dy 80 ft, dz 15 ft ?
751008 ? 60,000 cells ? ?oil ?water
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Reservoir Layers
P5
P5
P5
P4
P4
P4
P3
P3
P3
P2
P2
P2
P1
P1
P1
I1
I2
I1
I2
I3
I4
I3
I4
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Flow Scenario Selection
? Idea of ROM procedure is to use reduced basis
for predictions ? Selection of scenarios is a
key issue ? Heuristic approach vary pressures
consistent with expected ranges for the
operating conditions of interest ? Following
examples suggest robustness can be achieved
using this approach
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Base Case (Producers)
Injection pressure 6500 psia (cte.)
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Eigenvalues (Base)
Energy ignored 110-10 ?p 23 basis functions
for pressure
Energy ignored 110-7 ?s 35 basis functions
for saturation
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Prediction Schedule I
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Oil Rate GPRS
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Oil Rate POD
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Simulation Time
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Clustering Snapshots
? Large number of snapshots leads to large
eigenvalue problem ? Clustering procedure
treats each snapshot as an object in space and
finds centroids that minimize distance to each
observation ? Useful for eigenproblem and for
reducing number of basis vectors
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Application
Snapshot set
Clusters (Centroids)
? Apply POD to centroids
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Eigenvalues (50 Clusters)
Previous ?p 23 and ?s 35
14 basis functions for pressure
25 basis functions for saturation
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Oil Rate Clusters POD
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Simulation Time (Reduction)
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MPE Missing Point Estimation
? Construction of FT J F is time consuming
? MPE estimates POD coefficients from a
selected number of points in the spatial domain,
reducing cost of FT J F ? Intent is for zMPE
zPOD (reduced vector)
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MPE Condition Number
eMPEk (fMPEk )T (fMPEk ) I 2
f1
f2
f?
fMPE1
1
1
1
M
2
2
2
M 2 S S Mij 2
k 1,,nc
i
j
Reordered index eMPE1 eMPE2 eMPE2nc
K 5 32,000 grid blocks selected
2nc
2nc
2nc
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MPE Grid Block Selection
Layer 1
Layer 2
Layer 3
Layer 4
Layer 5
Layer 6
Layer 7
Layer 8
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Oil Rate Clusters POD MPE
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Simulation Time (Reduction)
More iterations to converge
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Prediction Schedule II
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Oil Rate GPRS, Clusters, POD, MPE
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Simulation Time
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Prediction Schedule III
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Oil Rate GPRS, Clusters, POD, MPE
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Simulation Time
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Prediction Schedule IV
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Oil Rate GPRS, Clusters, POD, MPE
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Simulation Time (Reduction)
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Observations on Timings
? CPR highly specialized solver, very well
developed for reservoir flow simulation ? Curren
t adjoint implementation in GPRS not compatible
with CPR, uses ILU(0) instead ? Comparison of
ROM to GPRS with ILU(0) may be appropriate in
this case
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Schedule IV ILU(0) x CPR x ROM
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Conclusions
? POD provides mechanism for using reduced
basis for simulation ? Clustering provides
efficient means for using many snapshots in
formation of basis ? MPE reduces computational
load of ROM ? ROM implemented in GPRS and applied
to a 3-D reservoir model with good results ? ROM
3x faster than GPRS with CPR and gt 100x
faster with ILU(0)
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Future Work
? Explore alternative approaches for
MPE procedure ? Improve algorithm for cases with
strong gravity and highly variable injection
pressure ? Further testing
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