PSO Algorithm for Optimum Well Placement subject to Realistic Field Development Constraints Mansoureh Jesmani, NTNU, Mathias C. Bellout, NTNU, Remus Hanea, Statoil, and Bjarne Foss, NTNU June 10, 2015 1 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
Problem Formulation Well Placement Problem Common formulation of well placement problem: N − 1 � L n ( x n +1 , ζ , u n )] , max ζ , u n [ J = n =0 subject to: ζ d ≤ ζ ≤ ζ u , u d ≤ u n ≤ u u , x 0 = x 0 , g n ( x n +1 , x n , ζ , u n ) = 0 , n = 0 , 1 , · · · , N − 1 . 2 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
Problem Formulation Well Placement Problem Common formulation of well placement problem: N − 1 � L n ( x n +1 , ζ , u n )] , max ζ , u n [ J = n =0 subject to: ζ d ≤ ζ ≤ ζ u , u d ≤ u n ≤ u u , x 0 = x 0 , g n ( x n +1 , x n , ζ , u n ) = 0 , n = 0 , 1 , · · · , N − 1 . 2 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
Problem Formulation Motivation Problem: Engineering experiences are not included. Valuable solution depends on Identification of limitations, Translation of them into constraints. The success of the optimization effort relies on Efficient search algorithm, Constraint-handling techniques. 3 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
Problem Formulation Well Placement Constraints Well distance C wd : R ∗ i,j ≥ d min Well length C wl : L i = � ζ h i − ζ t i � 2 , l i min ≤ L i ≤ l i max Reservoir bound C rb : ζ h ζ t i ∈ R h i ∈ R t i , i Well orientation � ( ζ h i − ζ t i ) · ( ζ h j − ζ t � j ) C wo : θ i,j = arccos � ≤ θ max � � � ζ h i − ζ t i � 2 � ζ h j − ζ t j � 2 � 4 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
Problem Formulation Well Placement Constraints Well distance C wd : R ∗ i,j ≥ d min Well length C wl : L i = � ζ h i − ζ t i � 2 , l i min ≤ L i ≤ l i max Reservoir bound C rb : ζ h ζ t i ∈ R h i ∈ R t i , i Well orientation � ( ζ h i − ζ t i ) · ( ζ h j − ζ t � j ) C wo : θ i,j = arccos � ≤ θ max � � � ζ h i − ζ t i � 2 � ζ h j − ζ t j � 2 � 4 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
Problem Formulation Well Placement Constraints Well distance C wd : R ∗ i,j ≥ d min Well length C wl : L i = � ζ h i − ζ t i � 2 , l i min ≤ L i ≤ l i max Reservoir bound C rb : ζ h ζ t i ∈ R h i ∈ R t i , i Well orientation � ( ζ h i − ζ t i ) · ( ζ h j − ζ t � j ) C wo : θ i,j = arccos � ≤ θ max � � � ζ h i − ζ t i � 2 � ζ h j − ζ t j � 2 � 4 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
Problem Formulation Well Placement Constraints Well distance C wd : R ∗ i,j ≥ d min Well length C wl : L i = � ζ h i − ζ t i � 2 , l i min ≤ L i ≤ l i max Reservoir bound C rb : ζ h ζ t i ∈ R h i ∈ R t i , i Well orientation � ( ζ h i − ζ t i ) · ( ζ h j − ζ t � j ) C wo : θ i,j = arccos � ≤ θ max � � � ζ h i − ζ t i � 2 � ζ h j − ζ t j � 2 � 4 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
Problem Formulation General Form of Well Placement Problem min − NPV , subject to: C i ( ζ ) ≥ 0 , i ∈ { wd, wl, rb, wo } , u d ≤ u n ≤ u u , x 0 = x 0 , g n ( x n +1 , x n , ζ , u n ) = 0 , n = 0 , 1 , · · · , N − 1 . 5 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Particle Swarm Optimization (PSO) PSO provides comparable or better results than binary GA (Onwunalu and Durlofsky, 2010). 6 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Particle Swarm Optimization (PSO) ν i ( k + 1) = ν i ( k ) + c 1 ρ 1 ( k )( p l,i ( k ) − x i ( k )) + c 2 ρ 2 ( k )( p g,i ( k ) − x i ( k )) , x i ( k + 1) = x i ( k ) + ν i ( k + 1) . 7 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Inertia Weight ν i ( k + 1) = w ( k ) ν i ( k ) + c 1 ρ 1 ( k )( p l,i ( k ) − x i ( k )) , ˆ + c 2 ρ 2 ( k )( p g,i ( k ) − x i ( k )) , ν j ν j ν j i ( k + 1) | , ν j i ( k + 1) = sign (ˆ i ( k + 1)) min {| ˆ max } , x i ( k + 1) = x i ( k ) + ν i ( k + 1) , w ( k ) = w 0 − k max = λ ( u j − l j ) , ν j K ( w 0 − w 1 ) . 8 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Method 1: Penalty function Merit function � φ 1 ( ζ , µ ) = − ( NPV ) sc + µ max { 0 , − ( C i ) sc } , i Penalty parameter ( µ ) grows with iteration number. 9 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Method 2: Decoder A homomorphous mapping between an n -dimensional cube and a feasible search space (Koziel and Michalewicz, 1999). 1 r o Φ(y)=? -1 1 0 y -1 S 10 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Introducing Decoder for Placing one Horizontal Well 100 Constraints: Both toe and 50 heel should stay in the circle (feasible region), 0 Variables: Cartesian coordinate for both heel − 50 ( x h , y h ) and toe ( x t , y t ) − 100 − 100 − 50 0 50 100 11 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Introducing Decoder for Placing one Horizontal Well 100 Step 1: Define reference r 0 = 50 � � 35 35 − 35 − 35 Step 2: The input of 0 decoder should stay in the cube [ − 1 , 1] 4 − 50 y = − 100 � � 0 . 4 0 . 6 − 0 . 3 0 . 5 − 100 − 50 0 50 100 12 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Introducing Decoder for Placing one Horizontal Well 100 Step 3: Calculate y/y max = 50 1 � � 0 . 4 0 . 6 − 0 . 3 0 . 5 0 . 6 0 Step 4: Map g ( y ) to s s = g ( y/y max ) = − 50 � � 66 . 7 100 − 50 83 . 3 g ( y ) = ( y − ( u − l ) 2 ) + u + l − 100 2 − 100 − 50 0 50 100 13 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Introducing Decoder for Placing one Horizontal Well 100 Step 5: Define line 50 segment between s and r 0 : 0 L ( r 0 , s ) = r 0 + t ( s − r 0 ) − 50 Step 6: Find t 0 where L intersects the boundary of − 100 circle: t 0 = 0 . 72 − 100 − 50 0 50 100 14 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Introducing Decoder for Placing one Horizontal Well 100 Step 5: Define line segment between s and 50 r 0 : 0 L ( r 0 , s ) = r 0 + t ( s − r 0 ) Step 6: Find t 0 where L − 50 intersects the boundary of circle: t 0 = 0 . 72 − 100 − 100 − 50 0 50 100 14 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Introducing Decoder for Placing one Horizontal Well 100 50 Step 7: Calculate φ ( y ) : 0 φ ( y ) = r 0 + y max t 0 ( s − r 0 ) − 50 − 100 − 100 − 50 0 50 100 15 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Introducing Decoder for Placing one Horizontal Well 100 50 g ( y ) 0 g ( y/y max ) r 0 + y max t 0 ( s − r 0 ) − 50 − 100 − 100 − 50 0 50 100 16 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Additional Constraints and Non-Convex Feasible Set Non-convex feasible set if: Non-convex feasible region, Include other constraints. In the case of non-convex feasible set: All steps are same, Several feasible interval: [ t 1 , t 2 ] , · · · [ t 2 k − 1 , t 2 k ] Define new map: γ : (0 , 1] → ∪ k i =1 ( t 2 i − 1 , t 2 i ] 17 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Non-Convex Feasible Space γ : (0 , 1] → ∪ k i =1 ( t 2 i − 1 , t 2 i ] 18 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm General Form of Decoder � r o + t o · ( g ( y /y max − r o )) if y � = 0 φ ( y ) = if y = 0 r o n y max = max i =1 | y i | , t 0 = γ ( | y max | ) . 19 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
PSO algorithm Decoder There is no need for any additional parameters, Always return a feasible solution, The map has locality feature, if any line segment, originates from the reference point, intersect the feasible search space just at one point. r o 20 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
Simulation Results Case Study I Permeability 1000 1400 I 3 I 4 600 1200 100 500 1000 400 Decoder NPV ($MM) Penalty (tune I) 10 800 Penalty (tune II) 300 600 200 1 100 400 0.1 0 200 200 400 600 800 1000 1200 I 1 I 2 Number of simulation 0 0.01 0 200 400 600 800 1000 1200 1400 mD Algorithm Best Mean Relative standard ( × 10 8 ) ( × 10 8 ) deviation ( % ) Decoder 5 . 28 5 . 19 2 . 8 Penalty(tune I) 5 . 26 5 . 17 2 . 7 Penalty(tune II) 5 . 24 4 . 86 6 . 8 21 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
Simulation Results Case Study II: Regions Setting for Decoder 5 producers and 3 injectors, one realization, fixed production settings, 40 × 64 × 14 = 35 , 840 grid cells. 9000 8000 OP − 3 WI − 1 7000 OP − 5 6000 OP − 4 5000 OP − 1 4000 OP − 2 3000 WI − 3 WI − 2 2000 1000 1000 2000 3000 4000 5000 6000 7000 8000 9000 22 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO
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