11
Optimization of Optimization of
Oil Field OperationsOil Field Operations
Louis J. Durlofsky
Department of Energy Resources Engineering
Stanford University
2
Collaborators
Jerome Onwunalu
(now at BP)
Jincong He
Jon Saetrom (NTNU)
3
Smart 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
5
Optimization of Well Type and Placement
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Solution Representation for Field Development Optimization
• 2N optimization variables
• Representation can be generalized to handle deviated, horizontal, or multilateral wells:
Concatenation of well variables; (ξ,η ) are spatial locations:
},,,,,,{2211 NN
ηξηξηξ K=x
well 1 well 2 well N
},),,(,),,{(11K
ttthhhζηξζηξ=x
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Particle 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
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PSO Solution Iteration
xi – solution, vi – particle velocity, k – iteration, ∆t = 1
• Particle velocity has 3 contributions:
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Genetic Algorithm (GA) Operations
}Population and selection:
Crossover:
Mutation:
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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 ηξηξηξ K=x
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Optimization Results: PSO and GA
- · - PSO –– GA
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Well Locations and Types: PSO and GA
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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
ηξηξηξ K=x
well 1 well 2 well N
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Optimize Parameters Associated with Pattern
• Basic parameters: (ξ, η, a, b)
(ξ(ξ(ξ(ξ, ηηηη)
a
b
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Allow for Rotation
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Pattern Operators for WPD
T
in
T
outMWW =
−=
θθ
θθ
cossin
sincosrotate
M
Scale Rotate Shear
• Can be expressed using transformation matrix M:
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‘Switch’ Operator and Extension to Other Patterns
Switch from inverted to
regular pattern
• Operators also defined for other pattern types:
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Illustration of WPD with Two Operators
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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
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Well-by-Well Perturbation (WWP)
• Same number of variables as concatenation approach but much smaller search space, and N is specified
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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
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Algorithm Performance – Pattern Optimization(one pattern operator)
• Best NPV using standard well patterns: $2151 MM
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Example 1: Optimization Results
injector
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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
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Optimized Well Locations
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Concatenation WPD+WWPinjector
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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
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Example 2: Optimization Results
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Example 2: Optimization Results
WPD (pattern)
WWP after WPD
WPD+WWP performance
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Example 2: Well Locations
injector
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Production Optimization Problem
• Seek to minimize:
u – controls, Qj – cumulative production/injection
ro , cw – oil revenue, water costs
)()()()(NPV)( uuuuu wiwiwpwpoo QcQcQrJ ++−=−=
subject to bound & linear/nonlinear constraints
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• Penalty function method: )()(min uuu
hJ ρ+
h – constraint violation, ρρρρ – penalty parameter
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Oil-Water Flow Equations
( ) 0
=+∇⋅∇−∂
∂jj
jqp
t
Skλφ
• Mass balance equations for j = oil, water
Sj - phase saturation (volume fraction), p - pressure
λλλλj (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
, gδJ −= xgJ ∂∂=• Newton’s method:
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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 (x≈≈≈≈ΦΦΦΦz)
Approach
Basic idea
References: Rewienski & White (2003), Vasilyev et al. (2003),
Qu & Chapman (2006), Cardoso & Durlofsky (2010), He et al. (2010)
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Linearization around Saved States
x1
x2
2D state spacei = 1 i = 2
i = 3i = 4
i = 5
i = 6
i =7
i = 8
u0
• Save xi and ∂gi/∂xi (u0)
u1
• Represent solutions for u1 using xi and ∂gi/∂xi
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TPWL for Reservoir Flow Equations
01111 =++= ++++ nnnn
QFAg
Discretized flow equations:
Linearized representation for new state xn+1:
( ) ( ) ( )11
1
11
11
1
1
11 ++
+
++++
+
+++ −
∂
∂+−
∂
∂+−
∂
∂+≅ in
i
iin
i
iin
i
iin uu
u
gxx
x
gxx
x
ggg
x: states (p, Sw) u: controls (BHPs)
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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 ≈≈≈≈ ΦΦΦΦz
• TPWL representation (reduced space, multiply by ΦΦΦΦT ):
( ) ( ) ( )
−
∂
∂+−
∂
∂−= ++
+
++−+++ 11
1
111111 in
r
i
iin
r
i
ii
r
inuu
u
Qzz
x
AJzz
ΦJΦJ 11 ++ = iTi
r(llll ×××× llll) llll ~ O(102 –103)
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Test Case – Portion of SPE 10 Model
• 60××××60××××30 = 108,000 cells (216,000 unknowns)
• ρw = 60 lb/ft3, ρo = 45 lb/ft3
• High resolution for all 72 well blocks
• llll = 304 (basis optimization applied); 448 unknowns
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Training and Test Runs
Training input
Target input
α = 1α = 0
• Test runs: (1 )Training Target
u u uα α= − +
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Production Rates for αααα = 0.3
P1 P2
P3 P4
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Production Rates for αααα = 0.5
GPRS/CPR TPWL
Run Time ~1 hr ~2 sec
P1 P2
P3 P4
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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
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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
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Optimization Result: NPV Evolution
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MethodNPV (initial)
$106
NPV (final)
$106
# of full
simulations
Full-order GPS 49.9 170.1 2500
TPWL-guided GPS 49.9 169.0 15
TPWL model construction ~ 2×time for training run
Optimization Result: NPV Summary
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Optimization Results: Final BHP Schedules
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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%
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Optimization ResultsW
ate
r C
ut
Vio
lati
on
34%
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Optimization Results: Injector BHP Schedules
MethodNPV (initial)
$106
NPV (final)
$106
# of full
simulations
TPWL-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