Multi-Objective Optimisation Techniques in Reservoir Simulation. Mike Christie Heriot-Watt University

Transcription

1 Multi-Objective Optimisation Techniques in Reservoir Simulation Mike Christie Heriot-Watt University

2 Outline Introduction Stochastic Optimisation Model Calibration Forecasting Reservoir Optimisation Summary

3 Outline Introduction Stochastic Optimisation Model Calibration Forecasting Reservoir Optimisation Summary

4 Mathematics of Flow in Porous Media Conservation of Mass Conservation of Momentum replaced by Darcy s law v k x p Conservation of Energy most processes isothermal Equation of State

5 Equations governing flow Parabolic equation for pressure p kro( S) krw( S) c. k( ) p t x o w Hyperbolic equation for saturation oxiso g yisg x. oxi vo g yiv g 0 t

6 Data Collection x? k x?

7 Model Calibration: Teal South simulated observed time (days) simulated observed time (days) simulated observed time (days)

8 Range of Possible Values for Unknown Parameters log(kh 1 ) log(kh 2 ) boundary porosity log(kv/kh 1 ) log(kv/kh 1 ) log(c r ) aquifer strength

9 Outline Introduction Stochastic Optimisation Model Calibration Forecasting Reservoir Optimisation Summary

10 Particle Swam Optimization (PSO) A swarm intellegence algorithm (Kennedy & Eberhart,1995). Particles are points in parameter space. Particles move based on their own experience and that of the swarm. PSO equations v wv c r p x c r g x k 1 k best k k k i i 1 1 i i 2 2 best i x x v k 1 k k 1 i i i r 1, r 2 are random vectors w is the inertial weight c 1, c 2 are the cognition and social acceleration components

11 Obj 2 Dominance and Pareto Optimality Solution x 1 dominates solution x 2, if : 1. x 1 is no worse than x 2 in all objectives, and 2. x 1 is strictly better than x 2 in at least one objective Pareto front Obj 1 Image of Pareto optimal set in objective space

12 MO Particle Swam Optimization (PSO) MOPSO equations v wv c r p x c r g x k 1 k best k k k i i 1 1 i i 2 2 best i x x v k 1 k k 1 i i i r 1, r 2 are random vectors w is the inertial weight c 1, c 2 are the cognition and social acceleration components Pbest and Gbest now sampled from Pareto archive

13 Why Use Multi-Objective? Sum of objectives limited exploration of Pareto front Example minimising two objectives Objective sum Multi-Objective Figure from Hajizadeh, PhD Thesis (Chapter 6, Figure12), Heriot-Watt, 2011

14 Outline Introduction Stochastic Optimisation Model Calibration Forecasting Reservoir Optimisation Summary

15 Model Calibration Called History Matching in oil business Synthetic example IC Fault Model Real example Zagadka Field

16 IC Fault Model Synthetic 2D Model 2 Wells: 1 Inj, 1 Prod 6 Layers 1,3,5 Poor Sand (blue) 2,4,6 Good Sand (red) 1 Fault * 3 Uncertain Inputs: 1. k high = [100,200] md 2. k low = [0,50] md 3. throw = [0,60] ft * Data from Z. Tavassoli, Jonathan N. Carter, and Peter R. King, Imperial College, London

17 IC Fault Model Observed k high = md k low = 1.3 md throw = 10.4 ft Misfit Definition: Truth Profile (Observed) M 1 n 0.03 obs p t sim obs p : oil/water rates Simulator controlled to match BHP at the wells

18 IC Fault Model Misfit Surface Database (DB) 159,661 uniformly generated models throw k low misfit k high truth low high

19 Convergence Speed 20 Runs SOPSO Iteration 28 MOPSO Iteration 12

20 Zagadka Field Waterflood/aquifer support 95 wells in 10 groups 15 years history Compartmentalized

21 Misfit Model 4 Convergence (PSO) Iteration

22 Model 4 Field Level History Match Match with PSO (single objective) 250 simulations (overnight, 12 core workstation) Field oil rate Field water rate

23 Match on Group Rates Single Objective

24 Minimum Misfit Multi-Objective vs Single Objective PSO MOPSO Iteration

25 Multi-Objective Trade Off MO balances changes to fit one quantity vs another

26 MO Not Always Faster than SO Study on EoN field

27 Histogram of 10 Final Misfits

28 Outline Introduction Stochastic Optimisation Model Calibration Forecasting Reservoir Optimisation Summary

29 Forecasting with MO Generic Bayesian integral J g( m) P( m) dm eg mean oil rate Q ( m) P( m) dm MC approximation Resample to calculate P(m k )dv k mk P mk m k 1 k k 1 o N N 1 g 1 J g m P m dv N h N k k k

30 Calculating Weights for Sample Gibbs sampler Assume misfit constant in Voronoi cell

31 Single & Multi Objective HM for PUNQ-S3 Based on real field An industry standard benchmark Multiple wells and production variables Algorithm used: PSO

32 FWPT (sm3) FWPT (sm3) FOPT (sm3) FOPT (sm3) SO vs. MO Forecasting Time (days) Time (days) Time (days) Time (days)

33 Outline Introduction Stochastic Optimisation Model Calibration Forecasting Reservoir Optimisation Summary

34 Field Development Optimisation Based on Scapa field Original HM done as part of MSc project HM to field rates (from DECC website) Wells are different from real field 4 injectors, 4 producers

35 Field Level History Match Match to first 10 years of history Remaining data used to check forecast

36 Optimisation Multi-Objective Maximise cumulative oil (FOPT) Minimise maximum water rate (FWPR) Optimisation variables Well locations (16 variables i, j for each well) Injection well rates

37 Field Development Optimisation (Scapa)

38 Optimise Field Development Plan 77 models 10% Original MSc development plan (4 injectors, 4 producers) Current Scapa production 55%

39 Scapa: Optimise Mid-Life Infill Wells Trade-off : 1.5 bbls oil in year 10 for 1 bbl oil in year 2

40 Optimisation Under Uncertainty PUNQ-S3 Multiple HM models Optimise infill wells

41 Mean Cumulative Oil Optimisation Under Uncertainty Uncertainty in Cumulative Oil

42 Summary History Matching Multi-objective may increase convergence speed, but not always Forecasting Multi-objective provides greater coverage of pareto front; can lead to better forecasts Optimisation Explore the full range of trade-offs Can include uncertainty

43 Acknowledgements EoN for funding the SO vs MO comparison BG, EoN, JOGMEC, RFD for funding the uncertainty group at HW Schlumberger for licences of ECLIPSE RFD for licences of tnavigator