1. Jincong He, Louis Durlofsky, Pallav Sarma (Chevron ETC)
Efficient Production Optimization and History Matching using Reduced Order Modeling
Jincong He, Louis Durlofsky, Pallav Sarma (Chevron ETC)
SUPRI-HW/Smart Fields Annual Meeting
November 15-16, 2010

2. Reservoir Simulation Applications
Field development & operations
Production optimization
History matching
Uncertainty quantification
Sensitivity studies

3. Outline
Reduced order modeling and trajectory piecewise linearization (TPWL)
TPWL for production optimization
TPWL for history matching (first-stage assessment)
Summary and future work

4. Governing Flow Equations
Oil-water flow equations
Residual equation after discretization
x: States (p, Sw); u: Parameters (BHP, k, T)
Solve at each iteration, nonlinear with O(105~106) unknowns

5. Trajectory Piecewise Linearization (TPWL)
2D state space
i = 3
First order accuracy
i = 2
i = 4
i = 5 u1
i = 1 u0
i = 8
i = 6
i =7 Sw
u0 –Training Simulation u1 –Test Simulation
(Cardoso, 2009)

6. Linearized Model Discretized equations (u: parameters)
Linearization around saved point (xi+1, xi, ui+1)
Full-order linearized equation

7. Linear Reduction of State Space
2nb unknowns for a two-phase problem
Project 2nb unknowns to  unknowns
( from POD in this work)
nb ~  ~ 100

8. Proper Orthogonal Decomposition (POD)
Snapshot 1
nc gridblocks
Snapshot 2
Snapshot k
Optimal in terms of reconstruction error
(Cardoso, 2009)

9. TPWL Formulation Order reduction Recursive formula (highly efficient!)
where

10. Observations Based on physics and makes use of gradient info
Exact solution at the training point, first order accuracy around the training point
Inline runtime only takes 0.5s~1s, not sensitive to the dimension of the problem
Can be used in production optimization and history matching

11. TPWL for Production Optimization
Replace general parameter u with PBH
Suitable for use with gradient-based and direct search methods such as Generalized Pattern Search (GPS)
PBH z x Q

12. TPWL as a Proxy for Optimization
Apply TPWL for direct search methods
Perform a training simulation to start
Retrain TPWL when far from the training
Generalized Pattern Search
Retrain
Training
(Kolda et al., 2003)

13. Optimization Example Optimization set up Optimize NPV using GPS
Oil: $80/bbl, prod. water: $-36/bbl, inj. water: $-18/bbl
Geological model: portion of Stanford VI model
30x40x4 = 4800 grid blocks
Simulation time: 1800 days (200 day intervals)
9 control variables for each producer (36 in total)
(BHP)min = 1,000 psia; (BHP)max = 3,000 psia

14. Optimization Example 1

15. Optimization Result: NPV Summary
Method
NPV (initial)
$106
NPV (final)
# of Full
Simulations
Full order GPS
49.9
170.1
2500
TPWL guided GPS
169.0 15
TPWL overhead ~ 5 Full Simulations

16. TPWL for History Matching
Method 1: Use transmissibility T as parameters
Method 2: Use log transmissibility ln(T) as parameters
ln(T) can be reduced by PCA,
Ideal for ensemble methods with multiple trainings T z x Q

17. TPWL for History Matching
Method 1: Use transmissibility T as parameters
Method 2: Use log transmissibility ln(T) as parameters
ln(T) can be reduced by PCA,
Ideal for ensemble methods with multiple trainings T z x Q

18. Example 1: 30x30x10 Synthetic Model
Training: <k > = 320 md, σ(k ) = 80 md
Target: <k > = 480 md, σ(k ) = 120 md
Linearization with T is used P3 P2 P1 I1 I2 P4 P1 I2 P4 P3 P2 I1
 = 0
 = 1

19. Oil Production Rates for α = 0.5

20. Water Production Rates for α = 0.5

21. Water Injection Rates for α = 0.5

22. Ensemble Kalman Filter (EnKF)
State vector contains model parameters, dynamic variables and production data
P(y) and P(dobs|y) assumed to be Gaussian
Maximum likelihood estimate of ya given prior yp and dobs
Kalman gain is given by

23. EnKF Introduction

24. EnKF Introduction
Forecast Step: yp

25. EnKF Introduction
Assimilation Step: yp
Assimilation Step ya

26. EnKF Introduction
Forecast Step

27. EnKF Introduction
Assimilation Step

28. EnKF Limitations
Kalman gain from small ensemble (<100) can be corrupted, resulting in collapse in ensemble variability and implausible updates
Option 1: Large ensemble (costly)
Option 2: Localization (violates the geological constraints)
Option 3: Use TPWL to provide a large ensemble for EnKF
(from Chen 2010)

29. TPWL with EnKF Demonstration
Forecast Step

30. TPWL with EnKF Demonstration
Forecast Step

31. TPWL with EnKF Demonstration
Assimilation Step

32. TPWL with EnKF Demonstration
Forecast Step
Assimilation Step

33. TPWL with EnKF Demonstration
Forecast Step

34. Algorithm Flow Chart EnKF+TPWL Basic EnKF Run NHF simulations
Run N simulations
Build TPWL proxy
Run NTPWL simulations
Update states
Update states
More data?
More data?

35. Numerical Example 2-D Gaussian field (45x45x1)
<ln(k )>=5, σ(ln(k ))=1
3960 T’s are reduced into 300 principal components
Update 300 variables to match Qo, Qw, Qinj every 50 days
4050 state variables are reduced to 500 variables
Case 1. Ensemble consists of 200 high fidelity (HF) models
Case 2. Ensemble consists of 50 HF models
Case 3. Ensemble consists of 50 HF TPWL models

36. True Solution

37. Initial Ensemble

38. HM and Prediction: Oil Production Rates
Initial
HF200
HF50
HF50+TPWL150

39. HM and Prediction: Water Production Rates
Initial
HF200
HF50
HF50+TPWL150

40. HM and Prediction: Water Injection Rates
Initial
HF200
HF50
HF50+TPWL150

41. Final Ensemble

42. Final Ensemble

43. Example realizations

44. Conclusions
TPWL method provides a reduced-order, linearized proxy for reservoir simulation
Implemented with GPS for production optimization problem, gave around 100x overall speedup
Applied for history matching problem with EnKF, preliminary results are promising

45. Future Work Continue to improve the accuracy and stability of TPWL
Further develop TPWL for use in history matching
Apply TPWL to real reservoir models
Consider use of TPWL for optimization under uncertainty

46. Acknowledgement
Jon Sætrom

47. The End
Thank You!