Post on 22-Feb-2016
description
Karen L. Ricciardi
The effects of uncertainty on aground water management
problem involving saltwater intrusion
Department of MathematicsUniversity of Massachusetts in Boston
Boston, MA, USAKaren.Ricciardi@umb.edu
Ann MulliganWoods Hole Oceanographic Institution
Marine Policy CenterWoods Hole, MA, USAamulligan@whoi.edu
Objective
De
Determine a groundwater supply plan that:* meets the demands of the community * minimizes the risk of salt water intrusion
Difficulties:* hydrologic parameters are uncertain* salt water interface responds nonlinearly to
pumping changes computational effort
Lower Capeof
Massachusetts
(2004, Masterson)
1
23
45
6
Truro, MAPumping in 2004:1. Knowles(2): 757 m3/d2. S. Hollow(3/8): 2,158 m3/d3. N. Truro Air Force Base 4:
312 m3/d4. N. Truro Air Force Base 5:
312 m3/d5. N. Unionfield: off6. CCC-5: off
Discharge in Provincetown.Total Supply Needs: 3,540 m3/d
Salt water Interface ModelingUsing the Ghyben-Herzberg approximation (1:40), the SWinterface is determined using the following iterative method.
Pumping design Modflow
Heads at cells
Update transmissivity
SW interface
Convergence criteria met?
Yes No
How does uncertainty affect the management of
coastal water supply?
Uncertainty in Layers 1 and 2
Modeler’s uncertainty
Spatial variability (1992, Hess et al.; SGSIM)
Wells OFF (Modeler’s uncertainty)
1
2 345
60 1.5 3.0 4.5 6.0 0 1.5 3.0 4.5 6.0
0 1.5 3.0 4.5 6.00 1.5 3.0 4.5 6.0
0 1.5 3.0 4.5 6.0 0 1.5 3.0 4.5 6.0
0
0
00
0
0
Wells ON (Modeler’s uncertainty)
1
2 345
6
95% dryout
0 1.5 3.0 4.5 6.00
0 1.5 3.0 4.5 6.00
0 1.5 3.0 4.5 6.00
0 1.5 3.0 4.5 6.00
0 1.5 3.0 4.5 6.00
0 1.5 3.0 4.5 6.00
Head(m) with “known” K
Modeler’s UncertaintyPumping(m3/d)
Mean Head (m)
S.D. Pumping(m3/d)
Mean Head (m)
S.D.
Well 1 0.86 0 0.84 0.20 757 0.82[1] 0.02Well 2 2.37 0 2.29 0.46 2158 0.99 0.10Well 3 1.63 0 1.58 0.33 312 1.27 0.25Well 4 1.29 0 1.25 0.29 312 0.98 0.22Well 5 2.62 0 2.54 0.51 0 2.30 0.45Well 6 2.77 0 2.68 0.49 0 2.63 0.47
[1] Only feasible in 5% of the scenarios
Management Model
ii headminmax
dm
i
i
dm
m3
3
0pumping86.0head540,3supply
constraints
objective
Results
Pumping(m3/d)
Headm
Well 1 Off 0.86
Well 2 39 2.27
Well 3 116 1.47
Well 4 356 1.02
Well 5 56 2.43
Well 6 2,973 0.86
K1 = 656 m/d; K2 = 246 m/d757 OFF
2,158 39312 116
312 356
OFF 56
OFF 2,973
Modeler’s Uncertaintymean= 0std= 0
mean= 43std= 100
mean= 36std= 38
mean= 518std= 120
mean= 3std= 9
mean= 2939std= 204
Pumping (m3/d)
Freq
uenc
y
Modeler’s Uncertainty
Modeler’s Uncertainty
Modeler’s UncertaintyCorrelation Coefficients
well 2 well 3 well 4 well 5 well 6
well 1 n/a n/a n/a n/a n/a
well 2 1.00 0.60 0.25 -0.13 -0.68
well 3 1.00 0.52 0.20 -0.79
well 4 1.00 0.21 -0.86
well 5 1.00 -0.17
1
2 345
6
well 2 well 3 well 4 well 5 well 6
well 1 n/a n/a n/a n/a n/a
well 2 0 5.3 E -4 2.0 E -1 4.9 E -1 4.4 E-5
well 3 0 3.7 E -3 2.9 E -1 2.6 E -7
well 4 0 2.8 E -1 2.6 E-9
well 5 0 3.8 E -1
Modeler’s UncertaintyP-values (< 0.05 is significant)
1
2 345
6
Modeler’s Uncertainty
MEAN VALUES OFMULTIPLE SOLUTIONS
Well 1: offWell 2: 43 m3/dWell 3: 36 m3/dWell 4: 518 m3/dWell 5: 3 m3/dWell 6: 2,939 m3/d
NO UNCERTAINTY
Well 1: offWell 2: 39 m3/dWell 3: 116 m3/dWell 4: 356 m3/dWell 5: 56 m3/dWell 6: 2,973 m3/d
Risk (50% reliable)
well 1 well 2 well 3 well 4 well 5 well 6
0 389 65 587 0 2499
0 301 91 587 0 2561
0 230 91 587 0 2632
0 177 91 587 0 2685
0 133 91 587 0 2729
0 71 91 604 0 2774
0 0 73 666 35 2765
Uncertainty in the hydraulic conductivity should be considered when developing a management program where salt water intrusion may be an issue.
Examining the solutions for the scenarios representing the uncertainty allows one to ascertain information about the correlation between wells.
Examining modeler’s uncertainty using a multi-scenario approach provides a means by which it is possible to determine reliable management designs.
There are multiple designs that provide reliable solutions to the management problem.
Conclusions
Equivalent solutions of the fixed supply problem
Maximum supply problem.
Spatial variability affects.
Boundary affects: Single boundary problem.
Well locations and numbers as a decision variable.
Current Work
Thank you.
Truro, MA• 4 layers• 39x85 nodes/layer (2157 active)• Constant head at the oceans• Streams are modeled as drains
with conductance = 149 m2/d, head = 0.6 m
• MODFLOW 2000: water table on; convertible boundaries
• Steady state• Recharge 0.0015 ft/d
Truro, Massachusetts
Layer Elevation (m) Hydraulic conductivity (m/d)
1 7.6 to -1.5 with topographic relief 61.0
2 -1.5 to -24.4 22.9
3 -24.4 to -61.0 15.2
4 -61.0 to -152.4 0.3
Truro, MA
Head Results for Layer 1• Mean head values used• Fixed pumping• SS not reached, no
convergence of the iterative method
26 m
18 m
9 m
0 m
Heads at wells when wells are OFF (100 spatially variable fields)
1
2 345
6
Well 11
2 345
6
Well 2
Well 3 Well 4
Well 5 Well 6
Wells ON (Spatially variable fields)
1
2 345
6
Head(m) with “known” K
Spatial Variability
Pumping(m3/d)
Mean Head (m)
S.D. Pumping(m3/d)
Mean Head (m)
S.D.
Well 1 0.86 0 0.89 0.04 757 0.42[2] 0.06Well 2 2.37 0 2.40 0.02 2158 1.04 0.13Well 3 1.63 0 1.65 0.04 312 1.34 0.05Well 4 1.29 0 1.31 0.05 312 1.02 0.05Well 5 2.62 0 2.65 0.02 0 2.42 0.06Well 6 2.77 0 2.80 0.02 0 2.76 0.02
[2] Only feasible in 21% of the scenarios.
Management Model
One perfectly homogeneous K field for each layer.
well 1: offwell 2: variable qwell 3: offwell 4: offwell 5: variable qwell 6: variable q
total supply = 3,540 m3/dmax head = 0.91 m (not 0.86 m as
in the model)well 2: 1,076 m3/dwell 6: 2,464 m3/d
Management Model
Objective function is:• Piecewise linear• Not differentiable• Minimum head varies
over different wells in the feasible region
well 6
well 4
wel
l 2
Management Model
Constraints are:• Linear• Variable well
dependencepumping
well 2 intrusion
well 6intrusion
well 1intrusion
ConstraintsToo much pumping:
If q1+q2+…+q5 > 3,540,then set q6 = q1+q2+…+q5+3,540.
This will cause the drawdown to be much larger than it would be naturally.
Salt water intrusion:If head at welli < 0.86 mthen penalize the objective function by 1-eqi/1,000.
This will increase the value of the objective function an amount related to the pumping at the well where there is a violation.
Pattern Search Solver
1997, Torczon
2003, Kolda and Torczon
2004, Gray and Kolda (APPSPACK)
Spatial Variability
mean= 0std= 0
mean= 57std= 188
mean= 179std= 130
mean= 2813std= 419
mean= 105std= 165
mean= 385std= 139
Pumping (m3/d)
Freq
uenc
y
Spatial Variability
Spatial VariabilityCorrelation Coefficients
well 2 well 3 well 4 well 5 well 6
well 1 n/a n/a n/a n/a n/a
well 2 1.00 0.25 -0.12 0.31 -0.56
well 3 1.00 0.06 0.70 -0.79
well 4 1.00 -0.03 -0.38
well 5 1.00 -0.78
well 2 well 3 well 4 well 5 well 6
well 1 n/a n/a n/a n/a n/a
well 2 0 1.9 E -1 5.2 E -1 1.0 E -1 1.7 E-3
well 3 0 7.8 E -1 2.7 E -5 3.7 E -7
well 4 0 8.9 E -1 4.0 E -2
well 5 0 5.0 E -7
Spatial VariabilityP-values (< 5.0 E -2 is significant)
1
2 345
6