Optimal Design of Groundwater Remediation Systems with Sampling Methods C. T. Kelley Department of Mathematics Center for Research in Scientific Computation North Carolina State University Raleigh, North Carolina, USA Chinese University of Hong Kong Hong Kong, January 25, 2006 Supported by NSF, ARO, DOEd. C. T. Kelley – p.1
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Optimal Design of GroundwaterRemediation Systems with Sampling
MethodsC. T. Kelley
Department of Mathematics
Center for Research in Scientific Computation
North Carolina State University
Raleigh, North Carolina, USA
Chinese University of Hong Kong
Hong Kong, January 25, 2006
Supported by NSF, ARO, DOEd.
C. T. Kelley – p.1
Outline
• Collaborators
• General formulation and example problem• Formulation• Optimization Landscapes• Results
• Implicit Filtering
• Community problems
• Conclusions
C. T. Kelley – p.2
Collaborators
• IFFCO developers from NCSU Math:Tony Choi, Owen Eslinger, Paul Gilmore,Alton Patrick, Vincent Bannister
• NCSU Math: Corey Winton, Dan Finkel, Jörg Gablonsky,Katie Fowler , Chris Kees, Jill Reese, Todd Coffey
• Other Places:
• Boeing: Andrew Booker, John Dennis
• UNC: Casey Miller, Matt Farthing, Glenn Williams
• ERDC: Stacy Howington
• Univ. Trier: Astrid Battermann
• Mich Tech: Alex Mayer
C. T. Kelley – p.3
What’s the problem.
• Control flow of contaminants in groundwater.• Keep plume on site.• Keep concentrations at acceptable levels.• Minimize cost, mass of contaminant,
contaminant concentration . . .
• Control flow and pressure.• Municipal water supplies.• Agriculture.
Number and location of wells, pumping rates.Pumping rates and well locations go in the source term forflow
∫
ΩS (t)dΩ =
n
∑i=1
Qi
and for concentration∫
ΩS
C(t)dΩ =n
∑i=1
C(xi)Qi.
Examples:
• Sum of δ functions at well locations.
• Well model with well diameter, well type, ...
C. T. Kelley – p.11
Example: Hydraulic Capture
Minimize total cost:
f T (Q) =n
∑i=1
c0db0i + ∑
Qi<−10−6
c1|Qim|b1(zgs−hmin)b2
︸ ︷︷ ︸
f c
+
∫ t f
0
(
∑i,Qi<−10−6
c2Qi(hi− zgs)+ ∑i,Qi>10−6
c3Qi
)
dt
︸ ︷︷ ︸
f o
,
to keep a contaminant inside a “capture zone”.Ω = [0,1000]× [0,1000]
C. T. Kelley – p.12
Notation
• (xi,yi) are well locations.
• Qi is pumping rate(> 0 for injection, < 0 for extraction.
• di is depth of well i
• hi is head at well i (MODFLOW)
• zgs is elevation of ground surface
• Qm is design pumping rate.
• hmin is minimum allowable pumping rate.
C. T. Kelley – p.13
Boundary conditions: Unconfined aquifer
∂h∂x
∣∣∣∣x=0
=∂h∂y
∣∣∣∣y=0
=∂h∂ z
∣∣∣∣z=0
= 0, t > 0
K∂h∂ z
(x,y,z = h, t > 0) =−1.903×10−8 (m/s).
h(1000,y,z, t > 0) = 20−0.001y(m),h(x,1000,z, t > 0) = 20−0.001x(m),h(x,y,z,0) = hs.
C. T. Kelley – p.14
Constraints I
Simple bounds:
Qemax ≤ Qi ≤ Qimax, i = 1, ...,n
Limits on the pumps.Simple linear inequality:
∑i
Qi ≥ QmaxT ,
limit on total net extraction rate.
C. T. Kelley – p.15
Constraints II
Keep wells away from Dirichlet boundary
0≤ xi,yi ≤ 800.
Bounds on h
hmin ≤ hi ≤ hmax, i = 1, ...,n
No dry holes.Velocity Highly nonlinear function of well locations.50×50×10 grid.
C. T. Kelley – p.16
Formulation Decisions I
• Contain plume: constrain velocity at zone boundary.Test velocity at five downstream locations.Approximate velocity with difference of h.Five new constraints.Need only flow code. Better simulations in progress.
• Implicit filtering deals with bounds naturally.
• Treat constraints as yes/no for sampling method• Stratify by cost.• Avoid simulator if infeasible wrt cheap (linear)
constraints.
• Well is de-installed if pumping rate is suff small.
C. T. Kelley – p.17
Formulation Decisions II
• Discontinuous objective.• 50×50×10 grid. Wells must be on grid nodes.
Move to nearest.• Remove well from array (di = 0) if pumping rate is
too small.
• Treat head constraint and linear constraintsas hidden or yes-no.
• Initial iterate: two extraction, two injection
C. T. Kelley – p.18
Initial iterate
No wells
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
24
23
22.5
22
21.5
21
20.5
20
19.5
Four well configuration
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
20
20.5
21
22
22.5
22
23
21
20.5
22
22.5
C. T. Kelley – p.19
Initial/Final Plumes
Initial Plume Plume after 5 years
C. T. Kelley – p.20
Results
Optimal configuration
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
22
20.5
19.5
20 19.5
20
Cost of Optimization
0 50 100 150 200 250 300 350 4002
3
4
5
6
7
8
9x 10
4
Calls to MODFLOW
Fun
ctio
n V
alue
C. T. Kelley – p.21
Landscapes
Vary (x1,y1) near initial iterate
400450
500550
600650
700
100
150
200
250
300
350
4007.88
7.9
7.92
7.94
7.96
7.98
8
8.02
x 104
xy
Fun
ctio
n V
alue
Vary pumping rate initial iterate
−1.5−1
−0.50
0.51
1.5
x 10−3
−1.5
−1
−0.5
0
0.5
1
1.5
x 10−33
4
5
6
7
8
x 104
well 4
Well 1F
unct
ion
Val
ue
C. T. Kelley – p.22
Other Approaches
100
101
102
103
2
3
4
5
6
7
8
9
10
11
12x 10
4
Calls to MODFLOW
Fun
ctio
n V
alue
HC
IFFCO DIRECT−L Nomad2N Nomad2N+1GA DE APPS
C. T. Kelley – p.23
Wait a minute!
• Optimal point has one well, we start with four.Was this fair?
• How does performance depend on initial iterate?
• Do some methods benefit from special choices?
• How can you construct a “rich” set of initial iterates fortesting?
• We’re trying:• Use DIRECT to find feasible points.• Use statistics to identify clusters.• Sample wisely within the clusters.
C. T. Kelley – p.24
Optimization strategy
minx∈D
f (x)
• Conventional gradient-based methods can fail if f is• multi-modal,• non-convex,• discontinuous,• non-deterministic, or if
• D is not determined by smooth inequalities.
Sampling methods attempt to address these problems.
C. T. Kelley – p.25
Stencil-based sampling methods
• Begin with a base point x.
• Examine points on a stencil;reject or adjust points not in D .