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: max

ζ,un [J = N−1

• n=0

Ln(xn+1, ζ, un)], subject to: ζd ≤ ζ ≤ ζu, ud ≤ un ≤ uu, x0 = x0, gn(xn+1, xn, ζ, un) = 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: max

ζ,un [J = N−1

• n=0

Ln(xn+1, ζ, un)], subject to: ζd ≤ ζ ≤ ζu, ud ≤ un ≤ uu, x0 = x0, gn(xn+1, xn, ζ, un) = 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 Cwd : R∗

i,j ≥ dmin

Well length Cwl : Li = ζh

i − ζt i2, li min ≤ Li ≤ li max

Reservoir bound Crb : ζh

i ∈ Rh i ,

ζt

i ∈ Rt i

Well orientation Cwo : θi,j = arccos

• (ζh

i −ζt i)·(ζh j −ζt j)

ζh

i −ζt i2ζh j −ζt j2

• ≤ θmax

4 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

Problem Formulation

Well Placement Constraints

Well distance Cwd : R∗

i,j ≥ dmin

Well length Cwl : Li = ζh

i − ζt i2, li min ≤ Li ≤ li max

Reservoir bound Crb : ζh

i ∈ Rh i ,

ζt

i ∈ Rt i

Well orientation Cwo : θi,j = arccos

• (ζh

i −ζt i)·(ζh j −ζt j)

ζh

i −ζt i2ζh j −ζt j2

• ≤ θmax

4 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

Problem Formulation

Well Placement Constraints

Well distance Cwd : R∗

i,j ≥ dmin

Well length Cwl : Li = ζh

i − ζt i2, li min ≤ Li ≤ li max

Reservoir bound Crb : ζh

i ∈ Rh i ,

ζt

i ∈ Rt i

Well orientation Cwo : θi,j = arccos

• (ζh

i −ζt i)·(ζh j −ζt j)

ζh

i −ζt i2ζh j −ζt j2

• ≤ θmax

4 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

Problem Formulation

Well Placement Constraints

Well distance Cwd : R∗

i,j ≥ dmin

Well length Cwl : Li = ζh

i − ζt i2, li min ≤ Li ≤ li max

Reservoir bound Crb : ζh

i ∈ Rh i ,

ζt

i ∈ Rt i

Well orientation Cwo : θi,j = arccos

• (ζh

i −ζt i)·(ζh j −ζt j)

ζh

i −ζt i2ζh j −ζt j2

• ≤ θmax

4 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

Problem Formulation

General Form of Well Placement Problem

min −NPV, subject to: Ci(ζ) ≥ 0, i ∈ {wd, wl, rb, wo}, ud ≤ un ≤ uu, x0 = x0, gn(xn+1, xn, ζ, un) = 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)

νi(k + 1) =νi(k) + c1ρ1(k)(pl,i(k) − xi(k)) + c2ρ2(k)(pg,i(k) − xi(k)), xi(k + 1) =xi(k) + νi(k + 1).

7 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

PSO algorithm

Inertia Weight

ˆ νi(k + 1) =w(k)νi(k) + c1ρ1(k)(pl,i(k) − xi(k)), + c2ρ2(k)(pg,i(k) − xi(k)), νj

i (k + 1) =sign(ˆ

νj

i (k + 1)) min{|ˆ

νj

i (k + 1)|, νj max},

xi(k + 1) =xi(k) + νi(k + 1), νj

max =λ(uj − lj),

w(k) = w0 − k K (w0 − w1).

8 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

PSO algorithm

Method 1: Penalty function

Merit function φ1(ζ, µ) = −(NPV)sc + µ

• i

max{0, −(Ci)sc}, 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

• 1

1

• 1

ro S

Φ(y)=?

y

10 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

PSO algorithm

Introducing Decoder for Placing one Horizontal Well

Constraints: Both toe and heel should stay in the circle (feasible region), Variables: Cartesian coordinate for both heel (xh, yh) and toe (xt, yt)

−100 −50 50 100 −100 −50 50 100

11 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

PSO algorithm

Introducing Decoder for Placing one Horizontal Well

Step 1: Define reference r0 =

• 35

35 −35 −35

• Step 2: The input of

decoder should stay in the cube [−1, 1]4 y =

• 0.4

0.6 −0.3 0.5

• −100

−50 50 100 −100 −50 50 100

12 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

PSO algorithm

Introducing Decoder for Placing one Horizontal Well

Step 3: Calculate y/ymax =

1 0.6

• 0.4

0.6 −0.3 0.5

• Step 4: Map g(y) to s

s = g(y/ymax) =

• 66.7

100 −50 83.3

• g(y) = (y − (u−l)

2 ) + u+l 2

−100 −50 50 100 −100 −50 50 100

13 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

PSO algorithm

Introducing Decoder for Placing one Horizontal Well

Step 5: Define line segment between s and r0: L(r0, s) = r0 + t(s − r0) Step 6: Find t0 where L intersects the boundary of circle: t0 = 0.72

−100 −50 50 100 −100 −50 50 100

14 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

PSO algorithm

Introducing Decoder for Placing one Horizontal Well

Step 5: Define line segment between s and r0: L(r0, s) = r0 + t(s − r0) Step 6: Find t0 where L intersects the boundary of circle: t0 = 0.72

−100 −50 50 100 −100 −50 50 100

14 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

PSO algorithm

Introducing Decoder for Placing one Horizontal Well

Step 7: Calculate φ(y): φ(y) = r0 +ymaxt0(s−r0)

−100 −50 50 100 −100 −50 50 100

15 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

PSO algorithm

Introducing Decoder for Placing one Horizontal Well

g(y) g(y/ymax) r0 + ymaxt0(s − r0)

−100 −50 50 100 −100 −50 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: [t1, t2], · · · [t2k−1, t2k] Define new map: γ : (0, 1] → ∪k

i=1(t2i−1, t2i]

17 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

PSO algorithm

Non-Convex Feasible Space

γ : (0, 1] → ∪k

i=1(t2i−1, t2i]

18 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

PSO algorithm

General Form of Decoder

φ(y) =

• ro + to · (g(y/ymax − ro))

if y = 0 ro if y = 0 ymax =

n

max

i=1 | yi |,

t0 = γ(| ymax |).

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,

• riginates from the reference point, intersect the feasible

search space just at one point.

ro

20 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

Simulation Results

Case Study I

I1 I2 I3 I4 Permeability

200 400 600 800 1000 1200 1400 200 400 600 800 1000 1200 1400 mD 0.01 0.1 1 10 100 1000

200 400 600 800 1000 1200 100 200 300 400 500 600 NPV ($MM) Number of simulation Decoder Penalty (tune I) Penalty (tune II)

Algorithm Best Mean Relative standard (×108) (×108 ) 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,

• ne realization,

fixed production settings, 40 × 64 × 14 = 35, 840 grid cells.

1000 2000 3000 4000 5000 6000 7000 8000 9000 1000 2000 3000 4000 5000 6000 7000 8000 9000

OP−1 OP−2 OP−3 OP−4 OP−5 WI−1 WI−2 WI−3

22 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

Simulation Results

Case Study II: Regions Setting for Decoder

Initial search regions

1000 2000 3000 4000 5000 6000 7000 8000 9000 1000 2000 3000 4000 5000 6000 7000 8000 9000

OP−1 OP−2 OP−3 OP−4 OP−5 WI−1 WI−2 WI−3

Improved search regions

1000 2000 3000 4000 5000 6000 7000 8000 9000 1000 2000 3000 4000 5000 6000 7000 8000 9000

OP−1 OP−2 OP−3 OP−4 OP−5 WI−1 WI−2 WI−3

23 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

Simulation Results

Case Study II: results

np = 49, ng = 50

500 1000 1500 2000 2500 9.2 9.4 9.6 9.8 10.0 10.2 10.4 10.6 10.8 NPV ($b) Number of simulation 24 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

Simulation Results

Conclusion and Future Work

Conclusion:

Improve the decision-making support by introducing realistic well placement constraints, Couple decoder with the PSO algorithm, Compare to the penalty method, the decoder can be used efficiently.

Future work:

Applying this methodology to more complex cases, Geological uncertainty, Variable production strategy.

25 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

Simulation Results

Thank You!

26 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO

Simulation Results

Koziel, S., Michalewicz, Z., Mar. 1999. Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evol. Comput. 7 (1), 19–44. URL http://dx.doi.org/10.1162/evco.1999.7.1.19 Onwunalu, J., Durlofsky, L., 2010. Application of a particle swarm optimization algorithm for determining optimum well location and type. Comput. Geosci. 14 (1), 183–198.

26 Mansoureh Jesmani, Norway Well Placement Optimization Using PSO