Application of a particle swarm optimization algorithm for determining optimum well location and type
Jerome Onwunalu
Louis J. Durlofsky
Smart Fields Meeting
April 8, 2009
Outline
Introduction Optimization of well placement Algorithms for well placement optimization Genetic algorithm (GA) Particle swarm optimization (PSO) Examples Conclusions
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Introduction
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Introduction
Oil eld development Optimize placement of wells Optimize well type Incorporate eld constraints Account for geological uncertainty Expensive simulation runs
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Well placement optimization problem
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Well placement optimization problem
Features Multidimensional Multimodal - many local optima Constrained Mixed variables Lack of analytical derivatives Discontinuous objective functions
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Well placement optimization algorithms
Adjoint-based algorithms Stochastic approximation (SA) algorithms
Simultaneous perturbation SA algorithm (SPSA) Finite difference SA algorithm (FDSA)
Genetic algorithm (GA) Particle swarm optimization (PSO) algorithm
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Well placement optimization algorithms
Adjoint-based algorithms Stochastic approximation (SA) algorithms
Simultaneous perturbation SA algorithm (SPSA) Finite difference SA algorithm (FDSA)
Genetic algorithm (GA) Particle swarm optimization (PSO) algorithm
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Genetic algorithm (GA)
Overview Based on Darwinian evolutionary theory Population-based Members of the population are called individuals Determine the tness (quality) of each individual using an objective function Rank population according to tness Produce new population using best individuals GA operators Selection, Crossover, Mutation
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GA owchart
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Particle swarm optimization (PSO) algorithm
Overview Developed by Kennedy and Eberhardt (1995) Based on the social interaction exhibited by animals Particle refers to a potential solution Collection of particles is called swarm Population-based Update particles using a cognitive and social model Improve performance using different particle neighborhood topologies
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Examples of PSO neighborhood topologies
6 7 8 1 2
(a) Star
5 4 3
7 8 1
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3 2 1
(c) Cluster Smart Fields
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2
(b) Ring
PSO variants and update equation
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PSO variants and update equation
PSO variants Global best PSO (gBest) - one neighborhood Local best PSO (lBest)- several neighborhoods
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PSO variants and update equation
PSO variants Global best PSO (gBest) - one neighborhood Local best PSO (lBest)- several neighborhoods Solution update equation xi (k + 1) = xi (k) + vi (k + 1)
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PSO variants and update equation
PSO variants Global best PSO (gBest) - one neighborhood Local best PSO (lBest)- several neighborhoods Solution update equation xi (k + 1) = xi (k) + vi (k + 1) vi (k + 1) is the velocity of particle i
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PSO solution update in 2D
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PSO solution update in 2D
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PSO solution update in 2D
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PSO solution update in 2D
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PSO solution update in 2D
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PSO solution update in 2D
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Velocity update equation
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Velocity update equation
Three velocity components
1 2 3
Inertia Cognitive Social
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Velocity update equation
Three velocity components
1 2 3
Inertia Cognitive Social v(k + 1) = 1 v(k) + 2 r1 vc (k) + 3 r2 vs (k)
inertia cognitive social
weights 1 , 2 , 2 r1 , r2 U (0, 1)
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Velocity update equation
Three velocity components
1 2 3
Inertia Cognitive Social v(k + 1) = 1 v(k) + 2 r1 vc (k) + 3 r2 vs (k)
inertia cognitive social
weights 1 , 2 , 2 r1 , r2 U (0, 1)
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Velocity update equation
Three velocity components
1 2 3
Inertia Cognitive Social v(k + 1) = 1 v(k) + 2 r1 vc (k) + 3 r2 vs (k)
inertia cognitive social
weights 1 , 2 , 2 r1 , r2 U (0, 1) Velocity pushes the search towards better regions
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Standard PSO
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Standard PSO
Features Update global best particle after each function evaluation
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Standard PSO
Features Update global best particle after each function evaluation Random topology - changes with iteration
6 7 8 1
5 4 3
Example of random topology. Links change with iteration
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Standard PSO
Features Update global best particle after each function evaluation Random topology - changes with iteration Reordering of particles 6 7 8 1 3 2
5 4
Example of random topology. Links change with iteration
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PSO owchart
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Examples
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Example 1: Single vertical producer
Problem Maximize NPV by optimizing a single producer over 10 realizations of reservoir model Optimization 2 optimization variable Reservoir model dimension: 40 40 1 Duration of simulation: 2000 days Compare GA and PSO Sensitivity on population size and number of iterations
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Example 1: Permeability models
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Example 1: Objective function surface
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Example 1: Results
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Example 2: 20 vertical wells
Problem Maximize NPV by optimizing the type and location of 20 vertical wells Optimization 3 unknown variables per well Reservoir model dimension: 100 100 1 Duration of simulation: 2000 days Compare GA and PSO Population size - 50, iterations - 100 Perform 4 runs for each algorithm
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Example 2: Results
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Example 2: Well locations
(d) PSO
(e) GA
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Example 3: 4 deviated wells
Problem Maximize NPV by optimizing the location of 4 deviated producers Optimization 6 unknown variables per well Reservoir model dimension: 63637 Duration of simulation: 2000 days Compare GA and PSO Population size - 50, iterations - 100 Perform 5 runs for each algorithm
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Example 3: Permeability eld
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Example 3: Results
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Example 3: Well locations
(f) GA - 3D view
(g) PSO - 3D view
(h) GA - top view
(i) PSO - top view
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Example 4: Choice of topology
Problem Maximize NPV by optimizing the location of 10 vertical wells Optimization 10 wells: 8 producers, 2 injectors Reservoir model dimension: 5050 Duration of simulation: 2000 days Compare PSO topology: Star, Ring, Cluster, Random Population size - 20, iterations - 50 Perform 8 runs for each topology type
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Example 4: PSO topologies
(j) Star
(k) Ring
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Example 4: Results - Avg. Obj. func
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Example 4: Results - Avg. diversity
Diversity - degree of dispersion of the particles in each iteration
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Conclusions
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Conclusions
Evaluation of optimization algorithms For the problems considered, PSO performs better than GA Superior performance of PSO may be due to:
technique of updating the global best position increased communication between particles
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Conclusions
Evaluation of optimization algorithms For the problems considered, PSO performs better than GA Superior performance of PSO may be due to:
technique of updating the global best position increased communication between particles
Investigated PSO topologies on a well placement problem
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Questions/comments
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Particle diversity
Diversity degree of dispersion of the particles
D(S(t)) =
1 Ns
Ns
(xi,j (k) xj (k))2
i=1 j=1
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