PSO Algorithm for Optimum Well Placement subject to Realistic Field - PowerPoint PPT Presentation

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