11

Optimization of Optimization of Oil Field OperationsOil Field Operations

Louis J. Durlofsky

Department of Energy Resources EngineeringStanford University

2

Collaborators

Jerome Onwunalu(now at BP)Jincong He Jon Saetrom (NTNU)

3Smart Field Modeling

Reservoir Data

Update Model

Optimize Well Settings

Set Well Controls

Field Development Optimization

Optimization highly intensive computationally

44

Outline

Field development (well placement) optimization Particle swarm optimization (PSO) algorithm Well pattern optimization

Production optimization Trajectory piecewise linearization (TPWL) for

surrogate modeling Generalized pattern search method with TPWL

Conclusions

5Optimization of Well Type and Placement

6

Solution Representation for Field Development Optimization

2N optimization variables Representation can be generalized to handle deviated,

horizontal, or multilateral wells:

Concatenation of well variables; (, ) are spatial locations:},,,,,,{ 2211 NN K=x

well 1 well 2 well N

},),,(,),,{( 11 Kttthhh =x

7Particle Swarm Optimization (PSO) Developed originally by Kennedy & Eberhardt (1995) Models social behavior in animals and entails a cooperative

search strategy (population-based like Genetic Algorithm) Successfully applied for subsurface flow optimization

(groundwater remediation) by Mattot et al. (2006)

http://inlinethumb61.webshots.com

8

PSO Solution Iteration

xi solution, vi particle velocity, k iteration, t = 1

Particle velocity has 3 contributions:

9Particle Swarm Optimization (2D Search Space)

10

PSO Neighborhood Topologies

11

Genetic Algorithm (GA) Operations

}Population and selection:

Crossover:

Mutation:

12

PSO versus GA for Well Placement

In our tests, PSO generally outperformed GA 2 dual-lateral producers

Average PSO NPV (from 5 runs) 19% higher than GA 4 deviated producers

Average PSO NPV (from 5 runs) 7% higher than GA

13

Optimization Example: PSO versus GA

Find well location and type (20 wells) to maximize net present value (NPV)

2D model, 100 x 100 blocks, oil-water simulation Swarm (population): 50; iterations (generations): 100 Perform 4 runs for each algorithm 60 optimization variables

},,,,,,,,,{ 222111 NNN iii K=x

14

Optimization Results: PSO and GA

- - PSO GA

15

Well Locations and Types: PSO and GA

16

Solution Representation for Multiple Wells

Number of optimization variables increases with well count high computational expense

Well count N must be specified (this should also be an optimization variable)

May be difficult to enforce distance constraints

Concatenation of well variables:

},,,,,,{ 2211 NN K=xwell 1 well 2 well N

17

Optimization with Well Patterns(Well Pattern Description)

18

Repeated Five-Spot Pattern

19

Optimize Parameters Associated with Pattern

Basic parameters: (, , a, b)

((((, )

a

b

20

Allow for Rotation

21

Shearing

22

Then Replicate and Evaluate

23

Pattern Operators for WPD

Tin

Tout MWW =

=

cossinsincos

rotateM

Scale Rotate Shear

Can be expressed using transformation matrix M:

24

Switch Operator and Extension to Other Patterns

Switch from inverted to

regular pattern

Operators also defined for other pattern types:

25

Illustration of WPD with Two Operators

26

Solution Representation in Well Pattern Description (WPD)

Fixed number of optimization parameters

Number of wells determined as part of optimization

Distance constraints easily satisfied

Can be used with a variety of optimization algorithms

Optimized solution is always a repeated pattern

27

Well-by-Well Perturbation (WWP)

Same number of variables as concatenation approach but much smaller search space, and N is specified

28

Example 1: Problem Set Up

2D model, 100 x 100 grid blocks Oil-water system, 10 years of production Injector BHP: 6000 psi, Producer BHP: 1000 psi Maximize NPV; run optimization multiple times

permeability field

29

Algorithm Performance Pattern Optimization(one pattern operator)

Best NPV using standard well patterns: $2151 MM

30

Example 1: Optimization Results

injector

31

Comparison of Concatenation and WPD+WWP(5 runs, 4 operators, 8000 total simulations)

WPD+WWP outperformed concatenation for all 5 runs

Concatenation (average, # of wells specified)

WWP (average)

WPD (best)

Number of simulations

32

Optimized Well Locations

32

Concatenation WPD+WWPinjector

33

Example 2: Problem Set Up

2D model, 80 x 132 grid blocks Oil-water-gas system, 5 years of production Injector BHP: 2900 psi, Producer BHP: 1200 psi Use 40 PSO particles, perform 5 runs using 3DSL

log permeability field

34

Example 2: Optimization Results

35

Example 2: Optimization Results

WPD (pattern)

WWP after WPD

WPD+WWP performance

36

Example 2: Well Locations

injector

3737

Production Optimization Problem

Seek to minimize:

u controls, Qj cumulative production/injectionro , cw oil revenue, water costs

)()()()(NPV)( uuuuu wiwiwpwpoo QcQcQrJ ++==

subject to bound & linear/nonlinear constraints

38

Penalty function method: )()(min uuu

hJ +

h constraint violation, penalty parameter

39

Oil-Water Flow Equations

( ) 0 =+

jjj qp

t

Sk

Mass balance equations for j = oil, water

Sj - phase saturation (volume fraction), p - pressurej (Sj ) - phase mobility, k - permeability tensor, qj - source

Discretize: x - states (p, Sw), u - controls (pwell), O(105-106) grid blocks

( ) ( ) ( ) ( ) 0,,,, 111111 =++= ++++++ nnnnnnnn uxQxFxxAuxxg, gJ = xgJ = Newtons method:

4040

Use states and Jacobians generated and saved during training run(s) to represent new solutions

Trajectory Piecewise Linearization (TPWL)

Run training simulations (g(x,u) = 0) Record states and Jacobian matrices (xi, gi/xi) Represent new solutions (xn+1) as expansions around

saved states (xi+1) Map into l-dim reduced space z using POD (xz)

Approach

Basic idea

References: Rewienski & White (2003), Vasilyev et al. (2003), Qu & Chapman (2006), Cardoso & Durlofsky (2010), He et al. (2010)

4141

Linearization around Saved States

x1

x22D state space

i = 1 i = 2

i = 3i = 4

i = 5

i = 6i =7

i = 8

u0

Save xi and gi/xi (u0)

u1

Represent solutions for u1 using xi and gi/xi

4242

TPWL for Reservoir Flow Equations

01111 =++= ++++ nnnn QFAgDiscretized flow equations:

Linearized representation for new state xn+1:

( ) ( ) ( )11111

111

111 ++

+

++++

+

+++

+

+

+ ini

iin

i

iin

i

iin uu

u

gxx

x

gxx

x

ggg

x: states (p, Sw) u: controls (BHPs)

4343

Expansion around Saved States

Linearized representation:

( ) ( ) ( )

+

=

++

+

+++++ 11

1

11111 in

i

iin

i

iini uu

u

Qxx

x

AxxJ

POD (SVD) applied to snapshot matrix: x z

TPWL representation (reduced space, multiply by T ):

( ) ( ) ( )

+

=++

+

+++++ 11

1

111111 in

r

i

iin

r

i

iir

in uuu

Qzz

x

AJzz

JJ 11 ++ = iTir (llll llll) llll ~ O(102 103)

44

Test Case Portion of SPE 10 Model

606030 = 108,000 cells (216,000 unknowns) w = 60 lb/ft3, o = 45 lb/ft3 High resolution for all 72 well blocks llll = 304 (basis optimization applied); 448 unknowns

45

Training and Test Runs

Training input

Target input

= 1 = 0

Test runs: (1 ) Training Targetu u u = +

46

Production Rates for = 0.3

P1 P2

P3 P4

47

Production Rates for = 0.5

GPRS/CPR TPWLRun Time ~1 hr ~2 sec

P1 P2

P3 P4

48

TPWL as a Proxy for Optimization

(Kolda et al., 2003)

Apply TPWL for direct search methods Perform an initial training simulation Retrain TPWL after specified number of iterations,

distance from last training, etc.Generalized Pattern Search

(GPS)

TrainingRetrain

49

Production Optimization: Case 1

Optimization set up Optimize NPV using generalized pattern search (GPS) Oil: $80/bbl, prod. water: $-36/bbl, inj. water: $-18/bbl

Geological model: portion of Stanford VI model 30x40x4 = 4800 grid blocks 4 producers and 2 injectors 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

50

Optimization Result: NPV Evolution

51

Method NPV (initial)$106NPV (final)

$106# of full

simulations

Full-order GPS 49.9 170.1 2500TPWL-guided

GPS 49.9 169.0 15

TPWL model construction ~ 2time for training run

Optimization Result: NPV Summary

52

Optimization Results: Final BHP Schedules

53

Production Optimization: Case 2

Optimization set up Oil: $80/bbl, prod. water: $-10/bbl, inj. water: $-5/bbl GPS with incremental penalty

Geological model: larger portion of Stanford VI 20,400 grid blocks, 4 producers and 2 injectors Simulation time: 1800 days (200 day intervals) Prod: (BHP)min = 1,000 psia; (BHP)max = 3,000 psia Inj: (BHP)min = 5,500 psia; (BHP)max = 7,500 psia Nonlinear constraints: water fractions < 50%

54

Optimization ResultsW

ater

Cu

t Vio

latio

n

34%

55

Optimization Results: Injector BHP Schedules

Method NPV (initial)$106NPV (final)

$106# of full

simulationsTPWL-guided

GPS 729 975 ~12

5656

Summary and Future Work

Applied particle swarm optimization (PSO) for determining placement of new wells

Devised new treatments for optimizing multiwell (field) development problems

Demonstrated use of TPWL (trajectory piecewise linearization) procedure for fast reservoir simulation

Incorporated TPWL into generalized pattern search optimization of oil production

Future work: meta-optimization techniques for use with PSO; enhance TPWL and clarify criteria for retraining; combine field development & production optimization