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Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of Massachusetts in Boston Boston, MA, USA [email protected] Ann Mulligan Woods Hole Oceanographic Institution Marine Policy Center Woods Hole, MA, USA [email protected]

Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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Page 1: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Karen L. Ricciardi

The effects of uncertainty on aground water management

problem involving saltwater intrusion

Department of MathematicsUniversity of Massachusetts in Boston

Boston, MA, [email protected]

Ann MulliganWoods Hole Oceanographic Institution

Marine Policy CenterWoods Hole, MA, [email protected]

Page 2: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 3: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Lower Capeof

Massachusetts

(2004, Masterson)

Page 4: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

1

2

3

45

6

Truro, MA

Pumping in 2004:

1. Knowles(2): 757 m3/d

2. S. Hollow(3/8): 2,158 m3/d

3. N. Truro Air Force Base 4:

312 m3/d

4. N. Truro Air Force Base 5:

312 m3/d

5. N. Unionfield: off

6. CCC-5: off

Discharge in Provincetown.

Total Supply Needs: 3,540 m3/d

Page 5: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 6: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

How does uncertainty affect the management of

coastal water supply?

Page 7: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Uncertainty in Layers 1 and 2

Modeler’s uncertainty

Spatial variability (1992, Hess et al.; SGSIM)

Page 8: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 9: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 10: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Head(m) with “known” K

Modeler’s Uncertainty

Pumping(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

Page 11: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Management Model

ii headminmax

dm

i

i

dm

m3

3

0pumping

86.0head

540,3supply

constraints

objective

Page 12: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Results

Pumping

(m3/d)

Head

m

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 39

312 116

312 356

OFF 56

OFF 2,973

Page 13: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Modeler’s Uncertainty

mean= 0std= 0

mean= 43std= 100

mean= 36std= 38

mean= 518std= 120

mean= 3std= 9

mean= 2939std= 204

Pumping (m3/d)

Fre

quen

cy

Page 14: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Modeler’s Uncertainty

Page 15: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Modeler’s Uncertainty

Page 16: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 17: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

  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

Page 18: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 19: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 20: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 21: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 22: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Thank you.

Page 23: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 24: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 25: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 26: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 27: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Wells ON (Spatially variable fields)

1

2 345

6

Page 28: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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.

Page 29: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 30: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Management Model

Objective function is:• Piecewise linear• Not differentiable• Minimum head varies

over different wells in the feasible region

wel

l 6

well 4

wel

l 2

Page 31: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Management Model

Constraints are:• Linear• Variable well

dependencepumping

well 2 intrusion

well 6intrusion

well 1intrusion

Page 32: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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.

Page 33: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Pattern Search Solver

1997, Torczon

2003, Kolda and Torczon

2004, Gray and Kolda (APPSPACK)

Page 34: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Spatial Variability

mean= 0std= 0

mean= 57std= 188

mean= 179std= 130

mean= 2813std= 419

mean= 105std= 165

mean= 385std= 139

Pumping (m3/d)

Fre

quen

cy

Page 35: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

Spatial Variability

Page 36: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

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

Page 37: Karen L. Ricciardi The effects of uncertainty on a ground water management problem involving saltwater intrusion Department of Mathematics University of

  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