parameterization and order reduction of geological models …...2012/12/08 · sgems realizations...
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Parameterization and Order Reduction of Geological Models
for History Matching
Hai Xuan Vo
CME-334
Outline
� Research motivation
� Karhunen-Loève (K-L) Expansion or Principal Component
Analysis (PCA)
� Limitations of existing methods
� Optimization-based PCA (O-PCA)
� Application of O-PCA to history matching using AD-GPRS
� Conclusions and future work
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History Matching Problem
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���� �����������������
� � ��, ��, … . , ��#$
� Idea is to find a geological model �such that prediction matches production and honors prior information about
geological model (regularization)
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Challenges in History Matching
� Real models can have millions of gridblocks
� Updating permeabilities for all blocks independently is
expensive and may not maintain geology
� Useful to have algorithms that represent reservoir model as
� % and maintain geology
� Now write history matching problem as:
���% � � � ��� ���� �%� ����
���
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� % implicitly honors regularization
Types of Geological Systems
� Goal is to determine parameterization � % forsuch systems
(Caers, 2011)
Delta
s
Meandering rivers
Tidal and bars
Carbonate reefs
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Previous Work
� KPCA: Sarma et al. (2008), Ma and Zabaras (2011): perform
PCA in high-dimensional feature space
� KPCA can generate more “channelized” realizations but
� Pre-image problem is strongly nonlinear, nonconvex
� Resulting histograms may not honor geology
� PCA: Oliver (1996), Sarma et al. (2006)
� Linear representation: � � )% � �*
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Constructing PCA Representation (1)
� Run Gaussian or training-image based geostatistical
algorithms (SGeMS, GSLIB) to create a set of N realizations
SGeMS realizations
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Training image
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Constructing PCA Representation (2)
� Construct:
� Conceptually, define covariance matrix as:
� Eigen-decomposition of + to give: (in
practice use SVD of X)
� K-L (PCA) representation
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, � ��, ��, … . , ��
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Application of PCA: Gaussian Fields (1)
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SGeMS realizations ���0 � )� % � �*PCA realizations:
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Application of PCA: Gaussian Fields (2)
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SGeMS realization ���0 � )� % � �*PCA realization:
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Application of PCA: non-Gaussian Fields
� Direct use of PCA leads to inconsistency for non-Gaussian
models
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SGeMS realization
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PCA realization
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Modified PCA Procedure
� PCA is simple and linear but gives Gaussian-looking
models and histograms
� Goal: modify PCA procedure to better represent complex
geology
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Optimization-based PCA (O-PCA)
� Standard PCA: ���0 � -�.��/�% � �*
� Apply regularization + bound constraints :
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�� Formulate PCA as an optimization problem:
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� Separable quadratic optimization problem (convex, unique,
analytical solution for ���0 and )!�/%
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O-PCA Characteristics
� Define . Now minimize:
= � � � � �� � 9�$�� ��}}}} ;;;;��1��1��1��1 9�����$zzzz 2{2{2{2{� 9/�}� � DE7FG27G
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� Separable quadratic optimization problem (convex, unique,
analytical solution for ���0 and )!
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�/%
O-PCA Realization
value
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SGeMS
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O-PCA for a non-Stationary Model
Training image(Honarkhah and Caers, 2011)
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SGEMS realizations
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O-PCA Results for a non-Stationary Model (1)
� 40,000 gridblock SGeMS realizations
� Represented using only 70 components of vector %
One realization Another realization
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O-PCA Results for a non-Stationary Model (2)
SGeMS
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Honoring Hard Data with O-PCA
� Hard data is honored by O-PCA
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Hard Data
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O-PCA
O-PCA Analytical Gradient
� Need for history matching
� Construct using:
� O-PCA gives analytically:
�/%
I��I%H
������ ��� I��I�H
J� � ��� � K� �9�� J��� � �����9 ��
�/%
J�, K� = Lagrange multipliers
�% � �
���
�% from O-PCA
from simulator
� � � ∗ �MN � ∗ �
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Differentiability of O-PCA Representation
%� � %� � ∆%
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Current realization
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Application of O-PCA to a HM Problem� Black-oil formulation with AD-GPRS
� Nx = 60, NY = 60; l = 30 components
� 4 water injectors at 1,000 m3/d
� 12 producers, BHP control at 150 bar
� ksand = 2,000 md, kmud = 20 md
� Using injection pressures and
production rates for history matching
� SGeMS realizations conditioned to hard
data at wells
� Using off-the-shelf LBFGS optimizer
(not tuned)
SGeMS realization
Hard data
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Hard data
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Producer
Injector
Results of O-PCA History Matching
Producer
Injector
INJ-1PROD-1 PROD-3
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INJ-2
PROD-1 PROD-3
PROD-4 PROD-6
PROD-7 PROD-2
PROD-5
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History matched
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True model
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Iteration
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unction
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unction
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Results of O-PCA History MatchingInjection BHP’s
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HM
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Initial Guess
History Matched
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Results of O-PCA History MatchingWater Rates
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Water Rate, PROD-12
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ate
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ate
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Results of O-PCA History MatchingOil Rates
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PROD-11 PROD-2
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Initial Guess
History Matched
Initial
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Results of O-PCA History MatchingField Rates
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Water Oil
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ate
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ate
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il R
ate
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HM Results for Another Initial Guess
Producer
Injector
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PROD-4
PROD-7
PROD-5
INJ-1
INJ-2
PROD-1 PROD-3
PROD-4 PROD-6PROD-5
INJ-2
PROD-1 PROD-3
PROD-4 PROD-6
PROD-7 PROD-2
PROD-5
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INJ-1
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PROD-1
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True model
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ate
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ate
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HM
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Field Oil Production Rate
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Initial Guess
History Matched
Initial
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HM
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Fie
ld O
il R
ate
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3/d
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History matched
Conclusions
� Introduced optimization-based PCA (O-PCA) to address
limitations of standard PCA
� O-PCA representation honors hard data and is
continuously differentiable
� O-PCA provides realistic realizations and can be solved
analytically
� Results from history matching show O-PCA capabilities
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