Jincong He, Louis Durlofsky, Pallav Sarma (Chevron ETC)

1. Jincong He, Louis Durlofsky, Pallav Sarma (Chevron ETC) Efficient Production Optimization and History Matching using Reduced Order Modeling 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) P 2D state space i = 3 First order accuracy i = 2 i = 4 u1 i = 5 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. ( from POD in this work) nb~ 106 ~ 100 Linear Reduction of State Space • 2nb unknowns for a two-phase problem • Project 2nb unknowns to  unknowns

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

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

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) Q x z PBH

12. Retrain Training 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 (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 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 Q x z T

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 Q x z T

18. P1 P2 P1 P2 I2 I1 I2 I1 P4 P4 P3 P3 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  = 0  = 1

19. P1 P2 P3 P4 Oil Production Rates for α = 0.5

20. P1 P2 P3 P4 Water Production Rates for α = 0.5

21. I1 I2 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 ypand 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 Build TPWL proxy Run N simulations 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 + 150 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!