Estimation of Porosity and Permeability

1
Estimation of Porosity and Permeability from
4D-Seismic and Production Data Using Principal
Component Analysis
Smart Fields Seminar Stanford University
July 31, 2008
2
Outline

• The Norne Field
• The history matching problem
• Integrating production and seismic data
• The optimization problem
• Principal Component Analysis
• Results
• Conclusions

3
Norne field

• Discovered at 1991
• Size 9 3 Km with 110 m oil and 75 m gas
  column
• Seismic Survey
• Base 1991 (Conventional)
• 1th July 2001 Q-Marine survey
• 2th August 2003 Q-Marine survey
• 3th August 2004 Q-Marine survey
• 4th August 2006 Q-Marine survey
• 5th July 2008 Q-Marine survey

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Norne Simulation Model
E
D
G
C

• Model is redesigned based on 2004 geo model
• 46 x 122 x 22, DX DY 80-100 m
• 46 development wells which only 22 are available
• 15 producer
• 8 injector

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Survey Difference 2003 - 2001
well CHT2
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• Semi Synthetic Model

injector
producer
Porosity
Permeability
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Outline

• The Norne Field
• The history matching problem
• Integrating production and seismic data
• The optimization problem
• Principal Component Analysis
• Results
• Conclusions

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Inversion process
Initial Guess (d)
Response of the system M(?)
Forward Modeling
OPTIMIZATION min M(?)- M(?)
Predicted parameters (K,F)
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Outline

• The Norne Field
• The history matching problem
• Integrating production and seismic data
• The optimization problem
• Principal Component Analysis
• Results
• Conclusions

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• Time-Lapse Seismic Data

MONITOR
BASE
MONITOR-BASE
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• Adding 4D seismic data

• Am f(X,?V,??)
• (?V,??) f(T,Sf,P,C)
• X Acquisition effects
• T Temperature
• Sf Fluid Saturation
• P Pressure/Stress
• C Compaction
• Assumptions
• No temperature variation
• No Compaction

Am
Time(sec)

• Variations in S and P affect 4D Am
• Variations in K and ? affect S and P
• Variations in K and ? affect 4D Am

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Adding 4D seismic data
Real 4D Seismic
Processed 4D Seismic Real Production Data Sim.
Production Data Sim Seismic Data
Processing
Match
Petro Elastic model
Reservoir Properties
Flow Simulation
?S, ?P
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Forward Models Used

• Production
• Fluid Flow Simulation (Eclipse 100)
• Seismic
• Petro Elastic Model
• Gassmanns equation (Saturation)
• Hertz Mindlin Model (Pressure)
• Forward seismic Modeling (Matrix propagation
  techniques)

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Outline

• The Norne Field
• The history matching problem
• Integrating production and seismic data
• The optimization problem
• Principal Component Analysis
• Results
• Conclusions

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Objective Function
Observed Data
Production
Seismic
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Bound Constraints
Geological bound Constraints
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Bound Constraints
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Four Optimization Strategies
v
v
v
v
v
v
v
v
v
v
production data is added after 15 iterations
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All Strategies Reduce Cost Function
SZG
P
ALT
SZGP
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Production Matching
WOPR
WBHPI
WBHPP
WWPR
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4D Seismic Matching
AVO Gradients Mismatch
Zero Offset Amplitude Mismatch
121
140
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20
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Estimated Porosity/Permeability
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Estimated Porosity/Permeability
K
F
Real
Real
Estimated
Estimated
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Estimated Porosity/Permeability
permeability error
porosity error
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Outline

• The Norne Field
• The history matching problem
• Integrating production and seismic data
• The optimization problem
• Principal Component Analysis
• Results
• Conclusions

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Motivation for PCA
CPU-TIMENumber of forward Simulation x Number of
Iterations

• reduce CPU time
• have a geologically acceptable estimate

expected
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Principal Component Analysis

• Orthogonal linear transformation
• Other names
• Karhunen-Loeve Transform (KLT)
• Proper Orthogonal Decomposition (POD)
• Hotelling Transform
• Involves eigenvalue decomposition / singular
  value decomposition of a covariance matrix
• Application
• reduces dimension in multidimensional data sets
• Introduces naturally geologic constraints

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Principal Component Analysis

• Interpretation as data compression

with
orthonormal
and
so that
is maximized
from D. Echeverria Ciaurri et al. 2008
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Porosity Realizations

• Matrix Approach when size of the problem is huge
• Turning Ban has artifacts and conditioning to
  local data is difficult
• Fractals conditioning to local data is difficult
• Annealing recommended for permeability
• Sequential Gaussian Simulation

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Available Statistical Data

• Log porosity of the wells
• Permeability-porosity relation
• Porosity distribution variogram

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Sequential Gaussian Simulation

• All conditional distribution is Gaussian and the
  mean and variance is given by kriging.
• Procedure
• Transform data to normal scores
• Establish grid network and coordinate system
• Compute the variogram corresponding to available
  well data
• Simulate realization by ordinary kriging which is
  conditioned to
• variogram
• local well data
• Back transform all values

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Realizations
Porosity 1
Porosity 2
Porosity 3
Permeability 1
Permeability 2
Permeability 3
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Effect of PCA (Porosity)
Original
200 components
Normalized variance
100 components
50 components
Number of components
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Effect of PCA (Permeability)
Original
200 components
Normalized variance
50 components
100 components
Number of components
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Optimization strategies using PCA

• PCA-STG1

PCA-STG2
i
Optimization loop
Optimization loop
Iteration loop
Iteration loop
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Outline

• The Norne Field
• The history matching problem
• Integrating production and seismic data
• The optimization problem
• Principal Component Analysis
• Results
• Conclusions

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Cost Function
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Estimated Porosity/Permeability
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Estimated Porosity
real porosity
estimated porosity NOT-PCA
estimated porosity PCA-STG1
estimated porosity PCA-STG2
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Estimated Permeability
estimated permeability NOT-PCA
real permeability
estimated permeability PCA-STG1
estimated permeability PCA-STG2
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Outline

• The Norne Field
• The history matching problem
• Integrating production and seismic data
• The optimization problem
• Principal Component Analysis
• Results
• Conclusions

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Conclusions

• Adding 4D seismic to production data yields a
  better history match
• If geologic constraints are not considered, the
  matched solutions might not be geologically
  realistic
• If numerical gradients are used in the history
  matching, the computational load can be
  prohibitive for practical applications

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Conclusions

• By Principal Component Analysis (PCA) we can
  speed up the gradient-based optimization
  considerably and at the same time take into
  account geologic constraints
• The good results obtained with this PCA-based
  technique in a semi synthetic case from the Norne
  field encourage to apply to the history matching
  of the complete field

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Future Work

• Apply PCA-based optimization to the complete
  field
• Use distributed computing in gradient
  approximation
• Test alternative optimization algorithms
• Study efficient methods (adjoints) for gradient
  computation
• Extension of PCA to Kernel PCA

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Acknowledgements
Stanford
NTNU
NFR

• STATOIL for permission to use the reservoir model
• Schlumberger-GeoQuest for the use of the Eclipse
  simulator.
• Alexey Stovas (NTNU) for the seismic forward
  modeling
• Jan Ivar Jensen (NTNU) For assistance

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Thank You!
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Estimation of Porosity and Permeability from
4D-Seismic and Production Data Using Principal
Component Analysis
Smart Fields Seminar Stanford University
July 31, 2008